Transcriber’s Notes:

  Underscores “_” before and after a word or phrase indicate _italics_
    in the original text.
  Equal signs “=” before and after a word or phrase indicate =bold=
    in the original text.
  Small capitals have been converted to SOLID capitals.
  Illustrations have been moved so they do not break up paragraphs.
  Typographical and punctuation errors have been silently corrected.
  Page 41: “1,000,000 feet or 300,000 kilometres per second” changed
   to “1,000,000,000 feet or 300,000 kilometres per second”




                     THE ATOM AND THE BOHR THEORY
                           OF ITS STRUCTURE

    _Original Title: “Bohr’s Atomteori, almenfatteligt fremstillet”_

                   _Translated from the Danish by $1
            Fellow of the American-Scandinavian Foundation,
                     1923, and RACHEL T. LINDSAY_

                     [Illustration: _Niels Bohr_]




                               THE ATOM
                                  AND
                            THE BOHR THEORY
                           OF ITS STRUCTURE

                      An Elementary Presentation

                                  BY
                             H. A. KRAMERS
        LECTURER AT THE INSTITUTE OF THEORETICAL PHYSICS IN THE
                       UNIVERSITY OF COPENHAGEN
                                  AND
                              HELGE HOLST
        LIBRARIAN AT THE ROYAL TECHNICAL COLLEGE OF COPENHAGEN

                          WITH A FOREWORD BY
                     SIR ERNEST RUTHERFORD, F.R.S.

                            [Illustration]

                         NEW YORK
                      ALFRED A. KNOPF
                           1923




PRINTED IN GREAT BRITAIN BY MORRISON AND GIBB LTD., EDINBURGH




PREFACE


At the close of the nineteenth century and the beginning of the
twentieth, our knowledge of the activities in the interior of matter
experienced a development which surpassed the boldest hopes that
could have been entertained by the chemists and physicists of the
nineteenth century. The smallest particles of chemistry, the atoms of
the elements, which hitherto had been approached merely by inductive
thought, now became tangible realities, so to speak, which could be
counted and whose tracks could be photographed. A series of remarkable
experimental investigations, stimulated largely by the English
physicist, J. J. Thomson, had disclosed the existence of negatively
charged particles, the so-called electrons, ¹/₂₀₀₀ the mass of the
smallest atom of the known elements. A theory of electrons, based
on Maxwell’s classical electrodynamical theory and developed mainly
through the labours of Lorentz in Holland and Larmor in England,
had brought the problem of atomic structure into close connection
with the theory of radiation. The experiments of Rutherford proved,
beyond a doubt, that atoms were composed simply of light, negative
electric particles, and small heavy, positive electric particles.
The new “quantum theory” of Planck was proving itself very powerful
in overcoming grave difficulties in the theory of radiation. The time
thus seemed ripe for a comprehensive investigation of the fundamental
problem of physics—the constitution of matter, and an explanation in
terms of simple general laws of the physical and chemical properties of
the atoms of the elements.

During the first ten years of the new century the problem was attacked
with great zeal by many scientists, and many interesting atomic models
were developed and studied. But most of these had more significance for
chemistry than for physics, and it was not until 1913 that the work of
the Danish physicist, Niels Bohr, paved the way for a really physical
investigation of the problem in a remarkable series of papers on the
spectrum and atomic structure of hydrogen. The ideas of Bohr, founded
as they were on the quantum theory, were startling and revolutionary,
but their immense success in explaining the facts of experience after
a time won for them the wide recognition of the scientific world, and
stimulated work by other investigators along similar lines. The past
decade has witnessed an enormous development at the hands of scientists
in all parts of the world of Bohr’s original conceptions; but through
it all Bohr has remained the leading spirit, and the theory which, at
the present time, gives the most comprehensive view of atomic structure
may, therefore, most properly bear the name of Bohr.

It is the object of this book to give the reader a glimpse of the
fundamental conceptions of this theory, together with some of the most
significant results it has attained. The book is designed to meet
the needs of those who wish to keep abreast of modern developments
in science, but have neither time nor inclination to delve into the
highly mathematical abstract literature in which the developments are
usually concealed. It is with this in mind that the first four chapters
have been devoted to a general survey of those parts of physics and
chemistry which have close connection with atomic theory. No attempt
has been made at a mathematical development, and the physical meaning
of such mathematical formulæ as do occur has been clearly emphasized
in the text. It is hoped, however, that even those readers whose
acquaintance with atomic theory is more than casual, will find the book
a stimulus to further study of the Bohr theory.

Here we wish to record our best thanks to Mr. and Mrs. Lindsay for
the ability and the great care with which they have carried out the
translation from the Danish original.




FOREWORD


During the last decade there has been a great advance in our knowledge
of the structure of the atom and of the relation between the atoms of
the chemical elements. In the later stages, science owes much to the
remarkable achievements of Professor Niels Bohr and his co-workers in
Copenhagen. For the first time, we have been given a consistent theory
to explain the arrangement and motion of the electrons in the outer
atom. The theory of Bohr is not only able to account in considerable
detail for the variation in the properties of the elements exemplified
by the periodic law, but also for the main features of the spectra,
both X-ray and optical, shown by all elements.

This volume, written by Dr. Kramers and Mr. Holst, gives a simple
and interesting account of our knowledge of atomic structure, with
special reference to the work of Professor Bohr. Dr. Kramers is in
an especially fortunate position to give a first-hand account of
this subject, for he has been a valued assistant to Professor Bohr
in developing his theories, and has himself made important original
contributions to our knowledge in this branch of inquiry.

I can confidently recommend this book to English readers as a clearly
written and accurate account of the development of our ideas on atomic
structure. It is written in simple language, and the essential ideas
are explained without mathematical calculations. This book should prove
attractive not only to the general scientific reader, but also to the
student who wishes to gain a broad general idea of this subject before
entering into the details of the mathematical theory.

                                                   E. RUTHERFORD.
    CAVENDISH LABORATORY,
      CAMBRIDGE, _8th October 1923_.




CONTENTS


                                                    PAGE
  PREFACE                                            vii
  FOREWORD                                            xi

  CHAP.
    I. ATOMS AND MOLECULES                             1
   II. LIGHT WAVES AND THE SPECTRUM                   34
  III. IONS AND ELECTRONS                             61
   IV. THE NUCLEAR ATOM                               83
    V. THE BOHR THEORY OF THE HYDROGEN SPECTRUM      105
   VI. VARIOUS APPLICATIONS OF THE BOHR THEORY       153
  VII. ATOMIC STRUCTURE AND THE CHEMICAL PROPERTIES
       OF THE ELEMENTS                               180

  INTERPRETATION OF SYMBOLS AND PHYSICAL CONSTANTS   209

                   COLOURED PLATES
  PLATE
     I. SPECTRUM PLATES ACCORDING TO THE ORIGINAL
        DRAWINGS OF BUNSEN AND KIRCHHOFF          _At end_

    II. PRINCIPAL FEATURES OF ATOMIC STRUCTURE IN
        SOME OF THE ELEMENTS—ATOMIC STRUCTURE
        OF RADIUM                                 _At end_




THE ATOM AND THE BOHR THEORY OF ITS STRUCTURE




CHAPTER I

ATOMS AND MOLECULES


Introduction.

As early as 400 B.C. the Greek philosopher, Democritus, taught
that the world consisted of empty space and an infinite number of small
invisible particles. These particles, differing in form and magnitude,
by their arrangements and movements, by their unions and disunions,
caused the existence of physical bodies with different characteristics,
and also produced the observed variations in these bodies. This
theory, which no doubt antedated Democritus, later became known as the
Atomic Theory, since the particles were called atoms, _i.e._ the
“indivisible.”

But the atomic conception was not the generally accepted one in
antiquity. Aristotle (384-322 B.C.) was not an atomist, and
denied the existence of discontinuous matter; his philosophy had a
tremendous influence upon the ideas of the ancients, and even upon the
beliefs of the Middle Ages. It must be confessed that his conception
of the continuity of matter seemed to agree best with experiment,
because of the apparent homogeneity of physical substances such as
metal, glass, water and air. But even this apparent homogeneity could
not be considered entirely inconsistent with the atomic theory, for,
according to the latter, the atoms were so small as to be invisible.
Moreover, the atomic theory left the way open for a more complete
understanding of the properties of matter. Thus when air was compressed
and thereafter allowed to expand, or when salt was dissolved in water
producing an apparently new homogeneous liquid, salt water, or when
silver was melted by heat, or light changed colour on passing through
wine, it was clear that something had happened in the interior of the
substances in question. But complete homogeneity is synonymous with
inactivity. How is it possible to obtain a definite idea of the inner
activity lying at the bottom of these changes of state, if we do not
think of the phenomenon as an interplay between the different parts of
the apparently homogeneous matter? Thus, in the examples above, the
decrease in the volume of the air might be considered as due to the
particles drawing nearer to each other; the dissolving of salt in water
might be looked upon as the movement of the salt particles in between
the water particles and the combination of the two kinds; the melting
of silver might naturally appear to be due to the loosening of bonds
between the individual silver particles.

The atomic theory had thus a sound physical basis, and proved
particularly attractive to those philosophers who tried to explain
the mysterious activity of matter in terms of exact measurements. The
atomic hypothesis was never completely overthrown, being supported
after the time of Aristotle by Epicurus (_c._ 300 B.C.),
who introduced the term “atom,” and by the Latin poet, Lucretius
(_c._ 75 B.C.) in his _De Rerum Natura_. Even in the Middle Ages it was
supported by men of independent thought, such as Nicholas of Autrucia,
who assumed that all natural activities were due to unions or disunions
of atoms. It is interesting to note that in 1348 he was forced to
retract this heresy. With the impetus given to the new physics by
Galileo (1600) the atomic view gradually spread, sometimes explicitly
stated as atomic theory, sometimes as a background for the ideas of
individual philosophers. Various investigators developed comprehensive
atomic theories in which they attempted to explain nearly everything
from purely arbitrary hypotheses; they occasionally arrived at very
curious and amusing conceptions. For example, about 1650 the Frenchman,
Pierre Gassendi, following some of the ancient atomists, explained the
solidity of bodies by assuming a hook-like form of atom so that the
various atoms in a solid body could be hooked together. He thought of
frost as an element with tetrahedral atoms, that is, atoms with four
plane faces and with four vertices each; the vertices produced the
characteristic pricking sensation in the skin. A much more thorough
treatment of the atomic theory was given by Boscovich (1770). He saw
that it was unnecessary to conceive of the atoms as spheres, cubes, or
other sharply defined physical bodies; he considered them simply as
points in space, mathematical points with the additional property of
being centres of force. He assumed that any two atoms influenced each
other with a force which varied, according to a complicated formula,
with the distance between the centres. But the time was hardly ripe
for such a theory, inspired as it evidently was by Newton’s teachings
about the gravitational forces between the bodies of the universe.
Indeed there were no physical experiments whose results could, with
certainty, be assumed to express the properties of the individual atoms.


The Atomic Theory and Chemistry.

[Illustration: FIG. 1.—The four elements and the four fundamental
characteristics.]

In the meantime atomic investigations of a very different nature had
been influencing the new science of chemistry, in which the atomic
theory was later to prove itself extraordinarily fruitful. It was
particularly unfortunate that in chemistry, concerned as it is with
the inner activities of the elements, Aristotle’s philosophy was
long the prevailing one. He adopted and developed the famous theory
of the four “elements,” namely, the dry and cold earth, the cold
and damp water, the damp and warm air, the warm and dry fire. These
“elements” must not be confused with the chemical elements known at
the present day; they were merely representatives of the different
consistent combinations of the four fundamental qualities, dryness
and wetness, heat and cold. From the symmetry in the system these were
supposed to be the principles by means of which all the properties of
matter could be explained. Neither the four “elements” nor the four
fundamental qualities could be clearly defined; they were vague ideas
to be discussed in long dialectic treatises, but were founded upon no
physical quantities which could be measured.

A system of chemistry which had its theoretical foundations in
the Greek elemental conceptions naturally had to work in the
dark. Undoubtedly this uncertainty contributed to the relatively
insignificant results of all the labour expended in the Middle Ages
on chemical experiments, many of which had to do with the attempt
to transmute the base metals into gold. Naturally there were many
important contributions to chemistry, and the theories were changed
and developed in many ways in the course of time. The alchemists of
the Middle Ages thought that metal consisted only of sulphur and
quicksilver; but the interpretation of this idea was influenced by
the Greek elemental theory which was maintained at the same time;
thus these new metal “elements” were considered by many merely as the
expressions of certain aspects of the metallic characteristics, rather
than as definite substances, identical with the elements bearing these
names. It is, however, necessary to guard against attributing to a
single conception too great influence on the historical development
of the chemical and physical sciences. That the growth of the latter
was hindered for so long a time was due more to the uncritical faith
in authority and to the whole characteristic psychological point of
view which governed Western thought in the centuries preceding the
Renaissance.

Robert Boyle (1627-1691) is one of the men to whom great honour is
due for brushing aside the old ideas about the elements which had
originated in obscure philosophical meditations. To him an element was
simply a substance which by no method could be separated into other
substances, but which could unite with other elements to form chemical
compounds possessing very different characteristics, including that
of being decomposable into their constituent elements. Undoubtedly
Boyle’s clear conception of this matter was connected with his
representation of matter as of an atomic nature. According to the
atomic conception, the chemical processes do not depend upon changes
within the element itself, but rather in the union or disunion of the
constituent atoms. Thus when iron sulphide is produced by heating iron
and sulphur together, according to this conception, the iron atoms
and the sulphur atoms combine in such a way that each iron atom links
itself with a sulphur atom. There is then a definite meaning in the
statement that iron sulphide consists of iron and sulphur, and that
these two substances are both represented in the new substance. There
is also a definite meaning, for instance, in the statement that iron
is an element, namely, that by no known means can the iron be broken
down into different kinds of atoms which can be reunited to produce a
substance different from iron.

The clarity which the atomic interpretation gave to the conception of
chemical elements and compounds was surely most useful to chemical
research in the following years; but before the atomic theory could
play a really great rôle in chemistry, it had to undergo considerable
development. In the time of Boyle, and even later, there was still
uncertainty as to which substances were the elements. Thus, water
was generally considered as an element. According to the so-called
phlogiston theory developed by the German Stahl (1660-1734), a theory
which prevailed in chemistry for many years, the metals were chemical
compounds consisting of a gaseous substance, phlogiston, which was
driven off when the metals were heated in air, and the metallic
oxide which was left behind. It was not until the latter half of the
eighteenth century that the foundation was laid for the new chemical
science by a series of discoveries and researches carried on by the
Swedish scientist Scheele, the Englishmen Priestley and Cavendish,
and particularly by the Frenchman Lavoisier. It was then discovered
that water is a chemical compound of the gaseous elements oxygen
and hydrogen, that air is principally a mixture (not a compound) of
oxygen and nitrogen, that combustion is a chemical process in which
some substance is united with oxygen, that metals are elements, while
metallic oxides, on the other hand, are compounds of metal and oxygen,
etc. Of special significance for the atomic theory was the fact that
Lavoisier made weighing one of the most powerful tools of scientific
chemistry.

Weighing had indeed been used previously in chemical experiments, but
the experimenters had been satisfied with very crude precision, and
the results had little influence on chemical theory. For example, the
phlogiston theory was maintained in spite of the fact that it was well
known that metallic oxide weighed more than the metal from which it
was obtained. Lavoisier now showed, by very careful weighings, that
chemical combinations or decompositions can never change the total
weight of the substances involved; a given quantity of metallic oxide
weighs just as much as the metal and the oxygen taken individually, or
_vice versa_. From the point of view of the atomic theory, this
obviously means that the weight of individual atoms is not changed in
the combinations of atoms which occur in the chemical processes. In
other words, _the weight of an atom is an invariable quantity_.
Here, then, we have the first property of the atom itself to be
established by experiment—a property, indeed, which most atomists had
already tacitly assumed.

Moreover, by the practice of weighing it was determined that _to
every chemical combination there corresponds a definite weight ratio
among the constituent parts_. This also had been previously accepted
by most chemists as highly probable; but it must be admitted that the
law at one time was assailed from several sides.

In comparing the weight ratios in different chemical compounds certain
rules were, in the meantime, obtained. In many ways the most important
of these, the so-called _law of multiple proportions_, was
enunciated in the beginning of the last century by the Englishman,
John Dalton. As an example of this law we may take two compounds of
carbon and hydrogen called methane or marsh gas and ethylene, in which
the quantities of hydrogen compounded with the same quantity of carbon
are as two is to one. Another example may be seen in the compounds of
carbon and oxygen. In the two compounds of carbon and oxygen, carbon
monoxide and carbon dioxide, the weight ratios between the carbon
and oxygen are respectively as three to four and three to eight. A
definite quantity of carbon has thus in carbon dioxide combined with
just twice as much oxygen as in carbon monoxide. No less than five
oxygen compounds with nitrogen are known, where with a given quantity
of nitrogen the oxygen is combined in ratios of one, two, three, four
and five.

These simple number relations can be explained very easily by the
atomic theory, by assuming, first, that all atoms of the same element
have the same weight; and second, that in a chemical combination
between two elements the atoms combine to form an atomic group
characteristic of the compound in question—a _compound atom_, as
Dalton called it, or a _molecule_, as the atomic group is now
called. These molecules consist of comparatively few atoms, as, for
example, one of each kind, or one of one kind and two, three or four
of another, or two of one kind and three or four of another, etc. When
three elements are involved in a chemical compound the molecule must
contain at least three atoms, but there may be four, five, six or
more. The law of multiple proportions thus takes on a more complicated
character, but it remains apparent even in this case.

When Dalton in the beginning of last century formulated the theory of
the formation of chemical compounds from the atoms of the elements, he
at once turned atomic theory into the path of more practical research,
and it was soon evident that chemical research had then obtained a
valuable tool. It may be said that Dalton’s atomic theory is the firm
foundation upon which modern chemistry is built.

[Illustration]

[Illustration]

[Illustration]

[Illustration:

  FIG. 2.—Representation of a part of Dalton’s atomic
         table (of 1808) where the atom of each element
        has its own symbol, and chemical compounds
        are indicated by the union of the atoms of the
        elements into groups by 2, 3, 4 ... (binary,
        ternary, quaternary ... atoms). Below are given
        the designations of the different atoms, and in
        parentheses the atomic weight given by Dalton with
        that of hydrogen as unity and the designations of
        the indicated atomic groups.

    _Atoms of the Elements._—1. Hydrogen (1); 2. Azote (5); 3. Carbon
    (5); 4. Oxygen (7); 5. Phosphorus (9); 6. Sulphur (13); 7. Magnesia
    (20); 8. Lime (23); 9. Soda (28); 10. Potash (42); 11. Strontites
    (46); 12. Barytes (68); 13. Iron (38); 14. Zinc (56); 15. Copper (56);
    16. Lead (95); 17. Silver (100); 18. Platina (100); 19. Gold (140);
    20. Mercury (167).

    _Chemical Compounds._—21. Water; 22. Ammonia; 23. 26. 27. and
    30. Oxygen compounds of Azote; 24. 29. and 33. Hydrogen compounds
    of Carbon; 25. Carbon monoxide; 28. Carbon dioxide; 31.
    Sulphuric acid; 32. Hydro-sulphuric acid.]

While Dalton’s theory could not give information about the absolute
weights in grams of the atoms of various elements, it could say
something about the relative atomic weights, _i.e._, the ratios of
the weights of the different kinds of atoms, although it is true that
these ratios could not always be determined with certainty. If, for
example, the ratio between the oxygen and hydrogen in water is found
to be as eight to one, then the weight ratio between the oxygen atom
and the hydrogen atom will be as eight to one, if the water molecule is
composed of one oxygen atom and one hydrogen atom (as Dalton supposed,
see Fig. 2). But it will be as sixteen to one, if the water molecule is
composed of one oxygen and two hydrogen atoms (as we now know to be the
case). On the other hand, a ratio of seven to one will be compatible
with the experimental ratio of eight to one only if we assume that the
water molecule consists of fifteen atoms, eight of oxygen and seven of
hydrogen, a very improbable hypothesis. In another case let us compare
the quantities of oxygen and of hydrogen which are compounded with the
same quantities of carbon in the two substances, carbon monoxide and
methane respectively. On the assumption that the molecules in question
have a simple structure, we can draw conclusions about the ratio of the
atomic weights of hydrogen and oxygen. Now, if a ratio such as seven
to one or fourteen to one is obtained while the analysis of water
gives eight to one or sixteen to one, then either the structure of the
molecule is more complicated than was assumed, or the analyses must be
improved by more careful experiments. We can thus understand that the
atomic theory can serve as a controlling influence on the analysis of
chemical compounds.

In order to choose between the different possible ratios of atomic
weights, for example, the eight to one or the sixteen to one in the
case of oxygen and hydrogen, Dalton had to make certain arbitrary
assumptions. The first of these is that two elements of which only
one compound is known appear with but one atom each in a molecule.
Partly on account of this assumption and partly on account of the
incompleteness of his analyses, Dalton’s values of the ratios of the
atomic weights of the atoms and his pictures of the structure of
molecules differ from those of the present day, as is obvious from Fig.
2.

A much firmer foundation for the choice made appears later in the
_Avogadro Law_; starting with the fact that different gases show
great similarity in their physical conduct—for instance, all expand by
an increase of ¹⁄₂₇₃ of their volume with an increase in temperature
from 0° C. to 1° C.—the Italian, Avogadro, in 1811, put forward the
hypothesis that equal volumes of all gases at the same temperature and
pressure contain equal numbers of molecules. A few examples suffice to
show the usefulness of this rule.

When one volume of the gas chlorine unites with one volume of hydrogen
there result two volumes of the gas, hydrogen chloride, at the same
temperature and pressure. According to Avogadro’s Law one molecule of
chlorine and one molecule of hydrogen unite to become two molecules of
hydrogen chloride, and since each of these two molecules must contain
at least one atom of hydrogen and one atom of chlorine, it follows
that one molecule of chlorine must contain two atoms of chlorine and
that one molecule of hydrogen must contain two atoms of hydrogen. From
this one can see that even in the elements the atoms are united into
molecules. It is now well established that most elements have diatomic
molecules, though some, including mercury and many other metals, are
monatomic. When oxygen and hydrogen unite to form water, one litre of
oxygen and two litres of hydrogen produce two litres of water vapour at
same temperature and pressure. Accordingly, one molecule of oxygen and
two molecules of hydrogen form two molecules of water. If the oxygen
molecule is diatomic like the hydrogen, then one molecule of water
contains one atom of oxygen and two atoms of hydrogen. Since the weight
ratio between the oxygen and hydrogen in water is eight to one, the
atomic weight of oxygen must be sixteen times that of hydrogen.

Through such considerations, supported by certain other rules, it has
gradually proved possible to obtain reliable figures for the ratios
between the atomic weights of all known elements and the atomic weight
of hydrogen. For convenience it was customary to assign the number 1 to
the latter and to call the ratio between the weight of the atom of a
given element and the weight of the hydrogen atom the atomic weight of
the element in question. Thus the atomic weight of oxygen is 16, that
of carbon 12, because one carbon atom weighs as much as 12 hydrogen
atoms. Nitrogen has the atomic weight 14, sulphur 32, copper 63.6, etc.

A summary of the chemical properties and chemical compounds was greatly
facilitated by the symbolic system initiated by the Swedish chemist,
Berzelius. In this system the initial of the Latin name of the element
(sometimes with one other letter from the Latin name) is made to
indicate the element itself, an atom of the element, and its atomic
weight with respect to hydrogen as unity, while a small subscript to
the initial designates the number of atoms to be used. For example, in
the chemical formula for sulphuric acid, H₂SO₄, the symbolic formula
means that this substance is a chemical compound of hydrogen, sulphur
and oxygen, that the acid molecule consists of two atoms of hydrogen,
one atom of sulphur and four atoms of oxygen, and that the weight
ratios between the three constituent parts is as 2×1 = 2 to 32 to 4×16
= 64, or as 1:16:32. To say that the chemical formula of zinc chloride
is ZnCl₂ means that the zinc chloride molecule consists of one atom of
zinc and two atoms of chlorine. Furthermore the changes which take
place in a chemical process may be indicated in a very simple way. Thus
the decomposition of water into hydrogen and oxygen may be represented
by the so-called chemical “equation” 2H₂O ⇾ 2H₂+O₂, where H₂ and O₂
signify the molecules of hydrogen and oxygen respectively. Conversely,
the combination of hydrogen and oxygen to form water will be given by
the equation 2H₂+O₂ ⇾ 2H₂O.

As a consequence of the development of the atomic theory the atoms of
the elements became, so to speak, the building stones of which the
elements and the chemical compounds are built. It can also be said that
the atoms are the smallest particles which the chemists reckon with in
the chemical processes, but it does not follow from the theory that
these building stones in themselves are indivisible. The theory leaves
the way open to the idea that they are composed of smaller parts. A
belief founded on such an idea was indeed enunciated by the Englishman,
Prout, a short time after Dalton had developed his atomic theory.
Prout assumed that the hydrogen atoms were the fundamental ones, and
that the atoms of the other elements consisted of a smaller or larger
number of the atoms of hydrogen. This might explain the fact that
within the limits of experimental error, many atomic weights seemed
to be integral multiples of that of hydrogen—16 for oxygen, 14 for
nitrogen, and 12 for carbon, etc. This led to the possibility that the
same might hold for all elements, and this hypothesis gave impetus to
very careful determinations of atomic weights. These, however, showed
that the assumption of the integral multiples could not be verified. It
therefore seemed as if Prout’s hypothesis would have to be given up. It
has, however, recently come into its own again, although the situation
is more complicated than Prout had imagined (see p. 97).

Dalton’s atomic theory gave no information about the atoms except that
the atoms of each element had a definite constant weight, and that
they could combine to form molecules in certain simple ratios. What
the forces are which unite them into such combinations, and why they
prefer certain unions to others, were very perplexing problems, which
could only be solved when chemical and physical research had collected
a great mass of information as a surer source of speculation.

From the knowledge of atomic weights it was easy to calculate what
weight ratios might be found to exist in chemical compounds, the
molecules of which consisted of simple atomic combinations. Thus many
compounds which were later produced in the laboratory were first
predicted theoretically, but only a small part of the total number of
possible compounds (corresponding to simple atomic combinations) could
actually be produced. Clearly it was one of the greatest problems of
chemistry to find the laws governing these cases.

It had early been known that the elements seemed to fall into two
groups, characterized by certain fundamental differences, the
metals and the metalloids. In addition, there were recognized two
very important groups of chemical compounds, _i.e._ acids and
bases, possessing the property of neutralizing each other to form a
third group of compounds, the so-called salts. The phenomenon called
electrolysis, in which an electric current separates a dissolved
salt or an acid into two parts which are carried respectively with
and against the direction of the current, indicates strongly that
the forces holding the atoms together in the molecule are of an
electrical nature, _i.e._ of the same nature as those forces
which bring together bodies of opposite electrical charges. One is
led to denote all metals as electropositive and all metalloids as
electronegative, which means that in a compound consisting of a
metal and a metalloid the metal appears with a positive charge and
the metalloid with a negative charge. The chemist Berzelius did a
great deal to develop electrical theories for chemical processes.
Great difficulties, however, were encountered, some proving for the
time being insurmountable. Such a difficulty, for example, is the
circumstance that two atoms of the same kind (like two hydrogen atoms)
can unite into a diatomic molecule, although one might expect them to
be similarly electrified and to repel rather than attract each other.

Another circumstance playing a very important part in determining the
chemical compounds which are possible, is the consideration of what is
called _valence_.

As mentioned above, one atom of oxygen combines with two atoms of
hydrogen to form water, while one atom of chlorine combines with but
one atom of hydrogen to form hydrogen chloride. The oxygen atom thus
seems to be “equivalent” to two hydrogen atoms or two chlorine atoms,
while one chlorine atom is “equivalent” to one hydrogen atom. The atoms
of hydrogen and chlorine are for this reason called monovalent, while
that of oxygen is called divalent. Again an acid is a chemical compound
containing hydrogen, in which the hydrogen can be replaced by a metal
to produce a metallic salt. Thus, when zinc is dissolved by sulphuric
acid to form hydrogen and the salt zinc sulphate, the hydrogen of the
acid is replaced by the zinc and the chemical change may be expressed
by the formula

  Zn+H₂SO₄ ⇾ H₂+ZnSO₄

In this, one atom of zinc changes place with two atoms of hydrogen. The
zinc atom is therefore divalent. This is consistent with the fact that
one zinc atom will combine with one oxygen atom to form zinc oxide. To
take another example, if silver is dissolved in nitric acid, one atom
of silver is exchanged for one atom of hydrogen. Silver, therefore, is
monovalent, and we should expect that one atom of oxygen would unite
with two atoms of silver. Some elements are trivalent, as, for example,
nitrogen, which combines with hydrogen to form ammonia, NH₃; others,
again, are tetravalent, such as carbon, which unites with hydrogen
to form marsh gas CH₄, and with oxygen to form carbon dioxide CO₂. A
valence greater than seven or eight has not been found in any element.

[Illustration: FIG. 3.—Rough illustrations of the valences of
the elements.

A. Hydrogen chloride (HCl); B. Water (H₂O); C. Methane (CH₄); D.
Ethylene (C₂H₄).]

If we consider the matter rather roughly and more or less as Gassendi
did, we can explain the concept of valence by assuming that the atoms
possess hooks; thus hydrogen and chlorine are each furnished with one
hook, oxygen and zinc with two hooks, nitrogen with three hooks, etc.
When a hydrogen atom and a chlorine atom are hooked together, there
are no free hooks left, and consequently the compound is said to be
saturated. When one hydrogen atom is hooked into each of the hooks of
an oxygen or carbon atom the saturation is also complete (see Fig. 3,
A, B, C).

The matter is not so simple as this, however, since the same element
can often appear with different valences. Iron may be divalent,
trivalent or hexavalent in different compounds. In many cases,
however, where an examination of the weight ratios seems to show that
an element has changed its valence, this is not really true. It was
mentioned previously that carbon forms another compound with hydrogen
in addition to CH₄, namely, ethylene, containing half as much hydrogen
in proportion to the same amount of carbon. With the aid of Avogadro’s
Law, it is found that the ethylene molecule is not CH₂ but C₂H₄. Thus
we can say that the two carbon atoms in the molecule are held together
by two pairs of hooks, and consequently the compound can be expressed
by the formula

  H-C-H
    ‖
  H-C-H

where the dashes correspond to hooks (cf. Fig. 3, D). Such a formula is
called a _structural formula_.

Even if we are not allowed to think of the atoms in the molecules
as held together by hooks, it is well to have some sort of concrete
picture of molecular structure. It is possible to represent the
tetravalent carbon atom in the form of a tetrahedron, and to consider
the united atoms or atomic groups as placed at the four vertices. With
such a spatial representation we can get an idea about many chemical
questions which otherwise would be difficult to explain. We know, for
example, that two compound molecules having the same kind and number of
atoms and the same bonds (and hence the same structural formulæ), may
yet be different in that they are images of each other like a pair of
gloves. Substances whose molecules are symmetrical in this way can be
distinguished from each other by their different action on the passage
of light. This molecular chemistry of space, or stereo-chemistry as
it is called, has proved of great importance in explaining difficult
problems in organic chemistry, _i.e._ the chemistry of carbon.
Although there have never been many chemists who really have believed
the carbon atom to be a rigid tetrahedron, we must admit that in this
way it has been possible to get on the track of the secrets of atomic
structure.

In comparing the properties of the elements with their atomic weights,
there has been discovered a peculiar relation which remained for a
long time without explanation, but which later suggested a certain
connection between the inner structure of an atom and its chemical
properties. We refer to the _natural or periodic system of the
elements_ which was enunciated in 1869 by the Russian chemist,
Mendelejeff, and about the same time and independently by the German,
Lothar Meyer. This system will be understood most clearly by examining
the table on p. 23, where the elements with their respective atomic
weights and chemical symbols are arranged in numbered columns so that
the atomic weights increase upon reading the table from left to right
or from top to bottom. It will be seen that in each of the nine columns
there are collected elements with related properties, forming what may
be called chemical families. The table as here given is of a recent
date and differs from the old table of Mendelejeff, both in the greater
number of elements and in the particulars of the arrangement. With each
element there is associated a number which indicates its position in
the series with respect to increasing atomic weight. Thus hydrogen has
the number 1, helium 2, etc., up to uranium, the atom of which is the
heaviest of any known element, and to which the number 92 is given. In
each of the columns the elements fall naturally into two sub-groups,
and this division is indicated in the table by placing the chemical
symbols to the right or left in the column.

On close examination it becomes evident that the regularity in the
system is not entirely simple. First of all some cases will be found
where the atomic weight of one element is greater than that of the
following element. (The cases of argon and potassium on the one hand
and cobalt and nickel on the other are examples.) Such an interchange
is absolutely necessary if the elements which belong to the same
chemical family are to be placed in the same column. As a second
instance of irregularity, attention must be called to Column VIII. in
the table. While in the first score or so of elements it is always
found that two successive elements have different properties and
clearly belong to distinct chemical families, in the so-called iron
group (iron, cobalt and nickel) we meet with a case where successive
elements resemble each other in many respects (for instance, in their
magnetic properties). Since there are two more such “triads” in the
periodic system, however, we cannot properly call this an irregularity.
But in addition to these difficulties there is what we may even
call a kind of inelegance presented by the so-called “rare earths”
group. In this group there follow after lanthanum thirteen elements
whose properties are rather similar, so that it is very difficult to
separate them from each other in the mixtures in which they occur in
the minerals of nature. (In the table these elements are enclosed in a
frame.)

On the other hand, the apparent absence of an element in certain places
in the table (indicated by a dash) cannot by any means be looked upon
as irregular. In Mendelejeff’s first system there were many vacant
spaces. With the help of his table Mendelejeff was, to some extent,
able to predict the properties of the missing elements. An example of
this is the case of the element between gallium and arsenic. This is
called germanium, and was discovered to have precisely the properties
which had been predicted for it—a discovery which was one of the
greatest triumphs in favour of the reality of the periodic system. On
the whole, the elements discovered since the time of Mendelejeff have
found their natural positions in the table. This is seen, for example,
in the case of the so-called “inactive gases” of the atmosphere,
helium, argon, neon, xenon, krypton and niton, which have the common
property of being able to form no chemical combinations whatever.
Their valence is therefore zero, and in the table they are placed by
themselves in a separate column headed with zero.

To explain the mystery of the periodic system, it was necessary to
make clear not only the regularity of it, but also the apparent
irregularities which seemed to be arbitrary individual peculiarities of
certain elements or groups. In the periodic system, chemistry laid down
some rather searching tests for future theories of atomic structure.


THE PERIODIC OR NATURAL SYSTEM OF THE ELEMENTS

  +----------+------------+------------+---------------+------------+--------------+-------------+---------------+-------------------------+
  |   0      |      I.    |      II.   |      III.     |     IV.    |      V.      |      VI.    |      VII.     |          VIII.          |
  +==========+============+============+===============+============+==============+=============+===============+=========================+
  |          | 1 Hydrogen |            |               |            |              |             |               |                         |
  |          | H 1·008    |            |               |            |              |             |               |                         |
  +----------+------------+------------+---------------+------------+--------------+-------------+---------------+-------------------------+
  | 2 Helium | 3 Lithium  | 4 Beryllium|  5 Boron      | 6 Carbon   | 7 Nitrogen   |  8 Oxygen   |  9 Fluorine   |                         |
  |  He 4·00 | Li 6·94    | Be 9·1     |  B 11·0       |   C 12·0   | N 14·0       |  O 16       |  F 19·0       |                         |
  +----------+------------+------------+---------------+------------+--------------+-------------+---------------+------------------------+
  | 10 Neon  | 11 Sodium  |12 Magnesium| 13 Aluminium  | 14 Silicon | 5 Phosphorus | 16 Sulphur  | 17 Chlorine   |                        |
  | Ne 20·2  | Na 23·0    |Mg 24·3     | Al 27·1       |  Si 28·3   | P 31·0       |  S 32·1     | Cl 35·5       |                        |
  +----------+------------+------------+---------------+------------+--------------+-------------+---------------+------------------------+
  | 18 Argon |19 Potassium| 20 Calcium | 21 Scandium   | 22 Titanium| 23 Vanadium  |24 Chromium  | 25 Manganese  | 26 Iron    27 Cobalt   |
  |  A 39·9  | K 39·1     | Ca 40·1    | Sc 44·1       | Ti 48·1    |  V 51·0      | Cr 52·0     | Mn 54·9       | Fe 55·8    Co 59·0     |
  |          |            |            |               |            |              |             |               |       28 Nickel        |
  |          |            |            |               |            |              |             |               |       Ni 58.7          |
  +----------+------------+------------+---------------+------------+--------------+-------------+---------------+------------------------+
  |          |  29 Copper |  30 Zinc   | 31 Gallium    |32 Germanium| 33 Arsenic   | 34 Selenium |  35 Bromine   |                        |
  |          |  Cu 63·6   |   Zn 65·4  |  Ga 69·9      | Ge 72·5    | As 75·0      | Se 79·2     |  Br 79·9      |                        |
  +----------+------------+------------+---------------+------------+--------------+-------------+---------------+------------------------+
  |36 Krypton|37 Rubidium |38 Strontium| 39 Yttrium    |40 Zirconium| 41 Niobium   |42 Molybdenum|  43 --        |44 Ruthenium  45 Rhodium|
  | Kr 82·9  |  Rb 85·4   |Sr 87·6     | Y 88·7        |Zr 90·6     | Nb 93·5      |Mo 96·0      |               | Ru 101·7     Rh 102·9  |
  |          |            |            |               |            |              |             |               |       46 Palladium     |
  |          |            |            |               |            |              |             |               |           Pd 106·7     |
  +----------+------------+------------+---------------+------------+--------------+-------------+---------------+------------------------+
  |          | 47 Silver  | 48 Cadmium | 49 Indium     | 50 Tin     | 51 Antimony  | 52 Tellurium|  53 Jodine    |                        |
  |          |  Ag 107·9  |  Cd 112·4  |  In 114·8     | Sn 118·7   | Sb 120·2     | Te 127·5    |  J 126·9      |                        |
  +----------+------------+------------+--------------=====================================================================================
  | 54 Xenon | 55 Caesium | 56 Barium  | 57 Lanthanum ||58 Cerium        59 Praseodymium   60 Neodymium     61 --           62 Samarium  ||
  |  X 130·2 | Cs 132·8   | Ba 137·3   | La 139·0     || Ce 140·2        Pr 140·6          Nd 144·3                          Sm 150·4    ||
  |          |            |            |              ||       63 Europium        64 Gadolinium     65 Terbium      66 Dysprosium        ||
  |          |            |            |              ||       Eu 152·0           Gd 157·3          Tb 159·2        Dy 162·5             ||
  |          |            |            |              ||67 Holmium     68 Erbium         69 Thulium     70 Ytterbium      71 Cassiopeium ||
  |          |            |            |              ||Ho 163·5       Er 167·7          Tm 168·5       Yb 173·0          Cp 175·0       ||
  +----------+------------+------------+--------------=====================================================================================
  |          |            |            |               | 72 Hafnium| 73 Tantalum   | 74 Tungsten   |  75 --      | 76 Osmium   77 Iridium |
  |          |            |            |               | Hf 179    | Ta 181·5      |  W 184·0      |             | Os 190·9    Ir 192·1   |
  |          |            |            |               |           |               |               |             |        78 Platinum     |
  |          |            |            |               |           |               |               |             |         Pt 195·2       |
  +----------+------------+------------+---------------+-----------+---------------+---------------+-------------+------------------------+
  |          | 79 Gold    | 80 Mercury | 81 Thallium   | 82 Lead   | 83 Bismuth    | 84 Polonium   |  85 --      |                        |
  |          |  Au 197·2  | Hg 200·6   | Tl 204·0      | Pb 207·2  | Bi 209·0      | Po 210·0      |             |                        |
  +----------+------------+------------+---------------+-----------+---------------+---------------+-------------+------------------------+
  | 86 Niton |87 --       | 88 Radium  | 89 Actinium   | 90 Thorium|91 Protactinium| 92 Uranium    |             |                        |
  | Ni 222·0 |            | Ra 226·0   | Ac?           | Th 232·1  | Pa?           |  U 238        |             |                        |
  +----------+------------+------------+---------------+-----------+---------------+---------------+-------------+------------------------+


The Molecular Theory of Physics.

From a consideration of the chemical properties of the elements we
shall now turn to an examination of the physical characteristics,
although in a certain sense chemistry itself is but one special phase
of physics.

If matter is really constructed of independently existing
particles—atoms and molecules—the interplay of the individual parts
must determine not only the chemical activities, but also the other
properties of matter. Since most of these properties are different for
different substances, or in other words are “molecular properties,” it
is reasonable to suppose that in many cases explanations can be more
readily given by considering the molecules as the fundamental parts.
It is natural that the first attempts to develop a molecular theory
concerned gases, for their physical properties are much simpler than
those of liquids or solids. This simplicity is indeed easily understood
on the molecular theory. When a liquid by evaporation is transformed
into a gas, the same weight of the element has a volume several hundred
times greater than before. The molecules, packed together tightly in
the liquid, in the gas are separated from each other and can move
freely without influencing each other appreciably. When two of them
come very close to each other, mutually repulsive forces will arise to
prevent collision. Since it must be assumed that in such a “collision”
the individual molecules do not change, they can then to a certain
extent be considered as elastic bodies, spheres for instance.

From considerations of this nature the kinetic theory of gases
developed. According to this a mass of gas consists of an immense
number of very small molecules. Each molecule travels with great
velocity in a straight line until it meets an obstruction, such as
another molecule or the wall of the containing vessel; after such an
encounter the molecule travels in a second direction until it collides
again, and so on. The pressure of the gas on the wall of the container
is the result of the very many collisions which each little piece
of wall receives in a short interval of time. The magnitude of the
pressure depends upon the number, mass and velocity of the molecules.
The velocity will be different for the individual molecules in a
gas, even if all the molecules are of the same kind, but at a given
temperature an average velocity can be determined and used. If the
temperature is increased, this average molecular velocity will be
increased, and if at the same time the volume is kept constant, the
pressure of the gas on the walls will be increased. If the temperature
and the average velocity remain constant while the volume is halved,
there will be twice as many molecules per cubic centimetre as before.
Therefore, on each square centimetre of the containing wall there will
be twice as many collisions, and consequently the pressure will be
doubled. Boyle’s Law, that the pressure of a gas at a given temperature
is inversely proportional to its volume, is thus an immediate result of
the molecular theory.

The molecular theory also throws new light upon the correspondence
between heat and mechanical work and upon the law of the conservation
of energy, which about the middle of the nineteenth century was
enunciated by the Englishman, Joule, the Germans, Mayer and Helmholtz,
and the Dane, Colding. A brief discussion of heat and energy will be
given here, since some conception of these phenomena is necessary in
understanding what follows.

To lift a stone of 5 pounds through a distance of 10 feet demands an
expenditure of work amounting to 5 × 10 = 50 foot-pounds; but the stone
is now enabled to perform an equally large amount of work in falling
back these 10 feet. The stone, by its height above the earth and by
the attraction of the earth, now has in its elevated position what is
called _“potential” energy_ to the amount of 50 foot-pounds. If
the stone as it falls lifts another weight by some such device as a
block and tackle, the potential energy lost by the falling stone will
be transferred to the lifted one. If the apparatus is frictionless, the
falling stone can lift 5 pounds 10 feet or 10 pounds 5 feet, etc., so
that all the 50 foot-pounds of potential energy will be stored in the
second stone. If instead of being used to lift the second stone, the
original stone is allowed to fall freely or to roll down an inclined
plane without friction, the velocity will increase as the stone falls,
and, as the potential energy is lost, another form of energy, known
as energy of motion or _kinetic energy_, is gained. Conversely,
a body when it loses its velocity can do work, such as stretching a
spring or setting another body in motion. Let us suppose that the stone
is fastened to a cord and is swinging like a pendulum in a vacuum where
there is no resistance to its motion. The pendulum will alternately
sink and rise again to the same height. As the pendulum sinks, the
potential energy will be changed into kinetic energy, but as it rises
again the kinetic will be exchanged for potential. Thus there is no
loss of energy, but merely a continuous exchange between the two forms.

If a moving body meets resistance, or if its free fall is halted by
a fixed body, it might seem as if, at last, the energy were lost.
This, however, is not the case, for another transformation occurs.
Every one knows that heat is developed by friction, and that heat
can produce work, as in a steam-engine. Careful investigations have
shown that a given amount of mechanical work will always produce a
certain definite amount of heat, that is, 400 foot-pounds of work,
if converted into heat, will always produce 1 B.T.U. of heat, which
is the amount necessary to raise the temperature of 1 pound of water
1° F. Conversely, when heat is converted into work, 1 B.T.U. of heat
“vanishes” every time 400 foot-pounds of work are produced. Heat then
is just a special form of energy, and the development of heat by
friction or collision is merely a transformation of energy from one
form to another.

With the assistance of the molecular theory it becomes possible to
interpret as purely mechanical the transformation of mechanical work
into _heat energy_. Let us suppose that a falling body strikes a
piston at the top of a gas-filled cylinder, closed at the bottom. If
the piston is driven down, the gas will be compressed and therefore
heated, for the speed of the molecules will be increased by collisions
with the piston in its downward motion. In this example the kinetic
energy given to the piston by the exterior falling body is used to
increase the kinetic energy of the molecules of the gas. When the
molecules contain more than one atom, attention must also be given to
the rotations of the atoms in a molecule about each other. A part of
any added kinetic energy in the gas will be used to increase the energy
of the atomic rotations.

The next step is to assume that, in solids and liquids, heat is
purely a molecular motion. Here, too, the development of heat after
collision with a moving body should be treated as a transformation
of the kinetic energy of an individual, visible body into an inner
kinetic energy, divided among the innumerable invisible molecules of
the heated solid or liquid. In considering the internal conduct of
gases it is unnecessary (at least in the main) to consider any inner
forces except the repulsions in the collisions of the molecules. In
solids and liquids, however, the attractions of the tightly packed
molecules for each other must not be neglected. Indeed the situation
is too complicated to be explained by any simple molecular theory. Not
all energy transformations can be considered as purely mechanical. For
instance, heat can be produced in a body by rays from the sun or from a
hot fire, and, conversely, a hot body can lose its heat by radiation.
Here, also, we are concerned with transformations of energy; therefore
the law for the conservation of energy still holds, _i.e._
the total amount of energy can neither be increased nor decreased
by transformations from one form to another. For the production of
1 B.T.U. of heat a definite amount of _radiation energy_ is
required; conversely, the same amount of radiation energy is produced
when 1 B.T.U. of heat is transformed into radiation. This change
cannot, however, be explained as the result of mechanical interplay
between bodies in motion.

The mechanical theory of heat is very useful when we restrict ourselves
to the transfer of heat from one body to another, which is in contact
with it. When applied to gases the theory leads directly to Avogadro’s
Law. If two masses of gas have the same temperature, _i.e._,
if no exchange of heat between them takes place even if they are
in contact with each other, then the average value of the kinetic
energy of the molecules must be the same in both gases. If one gas is
hydrogen and the other oxygen, the lighter hydrogen molecules must have
a greater velocity than the heavier oxygen molecules; otherwise they
cannot have the same kinetic energy (the kinetic energy of a body is
one-half the product of the mass and the square of the velocity). Since
the pressure of a gas depends upon the kinetic energy of the molecules
and upon their number per cubic centimetre, at the same temperature
and pressure equal volumes must contain equal numbers of oxygen and of
hydrogen molecules. As Joule showed in 1851, from the mass of a gas
per cubic centimetre and from its pressure per square centimetre, the
average velocity of the molecules can be calculated. For hydrogen at 0°
C. and atmospheric pressure the average velocity is about 5500 feet per
second; for oxygen under the same conditions it is something over 1300
feet per second.

All these results of the atomic and molecular theory, however, gave
no information about the absolute weight of the individual atoms and
molecules, nor about their magnitude nor the number of molecules in a
cubic centimetre at a given temperature and pressure. As long as such
questions were unsolved there was a suggestion of unreality in the
theory. The suspicion was easily aroused that the theory was merely
a convenient scheme for picturing a series of observations, and that
atoms and molecules were merely creations of the imagination. The
theory would seem more plausible if its supporters could say how large
and how heavy the atoms and molecules were. The molecular theory of
gases showed how to solve these problems which chemistry had been
powerless to solve.

Let us assume that the temperature of a mass of gas is 100° C. at a
certain altitude, and 0° C. one metre lower, _i.e._, the molecules
have different average velocities in the two places. The difference
between the velocities will gradually decrease and disappear on account
of molecular collisions. We might expect this “levelling out” process
or equilibration to proceed very rapidly because of the great velocity
of the molecules, but we must consider the fact that the molecules are
not entirely free in their movements. In reality they will travel but
very short distances before meeting other molecules, and consequently
their directions of motion will change. It is easy to understand that
the difference between the velocities of the molecules of the gas will
not disappear so quickly when the molecules move in zigzag lines with
very short straight stretches. The greater velocity in one part of the
gas will then influence the velocity in the other part only through
many intermediate steps. Gases are therefore poor conductors of heat.
When the molecular velocity of a gas and its conductivity of heat are
known, the average length of the small straight pieces of the zigzag
lines can be calculated—in other words, the length of the _mean free
path_. This length is very short; for oxygen at standard temperature
and pressure it is about one ten-thousandth of a millimetre, or 0·1 μ,
where μ is 0·001 millimetre or one _micron_.

In addition to the velocity of the molecules, the length of the mean
free path depends upon the average distance between the centres of two
neighbouring molecules (in other words, upon the number of molecules
per cubic centimetre) and upon their size. There is difficulty in
defining the size of molecules because, as a rule, each contains
at least two atoms; but it is helpful to consider the molecules,
temporarily, as elastic spheres. Even with this assumption we cannot
yet determine their dimensions from the mean free path, since there
are two unknowns, the dimensions of the molecules and their number
per cubic centimetre. Upon these two quantities depends, however,
also the volume which will contain this number of molecules, if they
are packed closely together. If we assume that we meet such a packing
when the substance is condensed in liquid form, this volume can be
calculated from a knowledge of the ratio between the volume in liquid
form and the volume of the same mass in gaseous form (at 0° C. and
atmospheric pressure). Then from this result and the length of the mean
free path the two unknowns can be determined. Although the assumptions
are imperfect, they serve to give an idea about the dimensions of
the molecules; the results found in this way are of the same order
of magnitude as those derived later by more perfect methods of an
electrical nature.

The radius of a molecule, considered as a sphere, is of the order of
magnitude 0·1 μμ, where μμ means 10⁻⁶ millimetre or 0·001 micron. Even
if a molecule is by no means a rigid sphere, the value given shows that
the molecule is almost unbelievably small, or, in other words, that it
can produce appreciable attraction and repulsion in only a very small
region in space.

The number of molecules in a cubic centimetre of gas at 0° C. and
atmospheric pressure has been calculated with fair accuracy as
approximately 27 × 10¹⁸. From this number and from the weight of a
cubic centimetre of a given gas the weight of one molecule can be
found. One hydrogen molecule weighs about 1·65 × 10⁻²⁴ grams, and
one gram of hydrogen contains about 6 × 10²³ atoms and 3 × 10²³
molecules. The weight of the atoms of the other elements can be found
by multiplying the weight of the hydrogen atom by the relative atomic
weight of the element in question—16 for oxygen, 14 for nitrogen, etc.
If the pressure on the gas is reduced as much as possible (to about one
ten-millionth of an atmosphere) there will still be 3 × 10¹² molecules
in a cubic centimetre, and the average distance between molecules will
be about one micron. The mean free path between two collisions will be
considerable, about two metres, for instance, in the case of hydrogen.

The values found for the number, weight and dimensions of molecules
are either so very large or so extremely small that many people,
instead of having more faith in the atomic and molecular theory,
perhaps may be more than ever inclined to suppose the atoms and
molecules to be mere creations of the imagination. In fact, it is only
two or three decades ago that some physicists and chemists—led by the
celebrated German scientist, Wilhelm Ostwald—denied the existence of
atoms and molecules, and even went so far as to try to remove the
atomic theory from science. When these sceptics, in defence of their
views, said that the atoms and molecules were, and for ever would be,
completely inaccessible to observation, it had to be admitted at that
time that they were seemingly sure of their argument, in this one
objection at any rate.

A series of remarkable discoveries at the close of the nineteenth
century so increased our knowledge of the atoms and improved the
methods of studying them that all doubts about their existence had to
be silenced. However incredible it may sound, we are now in a position
to examine many of the activities of a single atom, and even to
count atoms, one by one, and to photograph the path of an individual
atom. All these discoveries depend upon the behaviour of atoms as
electrically charged, moving under the influence of electrical forces.
This subject will be developed in another section after a discussion of
some phenomena of light, an understanding of which is necessary for the
appreciation of the theory of atomic structure proposed by Niels Bohr.

In the molecular theory of gases, where we have to do with neutral
molecules, much progress has in the last years been made by the Dane,
Martin Knudsen, in his experiments at a very low pressure, when the
molecules can travel relatively far without colliding with other
molecules. While his researches give information on many interesting
and important details, his work gives at the same time evidence of a
very direct nature concerning the existence of atoms and molecules.




CHAPTER II

LIGHT WAVES AND THE SPECTRUM


The Wave Theory of Light.

There have been several theories about the nature of light. The
great English physicist, Isaac Newton (1642-1727), who was a pioneer
in the study of light as well as in that of mechanics, favoured
an atomic explanation of light; _i.e._, he thought that it
consisted of particles or light corpuscules which were emitted from
luminous bodies like projectiles from a cannon. In contrast to this
“emission” theory was the wave theory of Newton’s contemporary, the
Dutch scientist, Huygens. According to him, light was a wave motion
passing from luminous bodies into a substance called the ether,
which filled the otherwise empty universe and permeated all bodies,
at least all transparent ones. In the nineteenth century the wave
theory, particularly through the work of the Englishman, Young, and the
Frenchman, Fresnel, came to prevail over the emission theory. Since
the wave theory plays an important part in the following chapters,
a discussion of the general characteristics of all wave motions is
appropriate here. The examples will include water waves on the surface
of a body of water, and sound waves in air.

[Illustration: FIG. 4.—Photograph of the interference between two
similar wave systems.]

[Illustration: FIG. 5.—A section of the same picture enlarged.

(From Grimsehl, _Lehrbuch der Physik_.)]

Let us suppose that we are in a boat which is anchored on a body of
water and let us watch the regular waves which pass us. If there is
neither wind nor current, a light body like a cork, lying on the
surface, rises with the wave crests and sinks with the troughs, going
forward slightly with the former and backward with the latter, but
remaining, on the whole, in the same spot. Since the cork follows the
surrounding water particles, it shows their movements, and we thus see
that the individual particles are in oscillation, or more accurately,
in circulation, one circulation being completed during the time in
which the wave motion advances a _wave-length_, _i.e._, the distance
from one crest to the next. This interval of time is called the _time
of oscillation_, or the _period_. If the number of crests passed in a
given time is counted, the oscillations of the individual particles
in the same time can be determined. The number of oscillations in
the unit of time, which we here may take to be one minute, is called
the _frequency_. If the frequency is forty and the wave-length is
three metres, the wave progresses 3 × 40 = 120 metres in one minute.
The velocity with which the wave motion advances, or in other words
its _velocity of propagation_, is then 120 metres per minute. We thus
have the rule that _velocity of propagation is equal to the product of
frequency and wave-length_ (cf. Fig. 8).

On the surface of a body of water there may exist at the same time
several wave systems; large waves created by winds which have
themselves perhaps died down, small ripples produced by breezes and
running over the larger waves, and waves from ships, etc. The form of
the surface and the changes of form may thus be very complicated; but
the problem is simplified by combining the motions of the individual
wave systems at any given point. If one system at a given time gives
a crest and another at the same instant also gives a crest at the
same point, the two together produce a higher crest. Similarly, the
resultant of two simultaneous troughs is a deeper trough; a crest from
one system and a simultaneous trough from the other partially destroy
or neutralize each other. A very interesting yet simple case of such
“interference” of two wave systems is obtained when the systems have
equal wave-lengths and equal amplitudes. Such an interference can be
produced by throwing two stones, as much alike as possible, into the
water at the same time, at a short distance from each other. When
the two sets of wave rings meet there is created a network of crests
and troughs. Figs. 4 and 5 show photographs of such an interference,
produced by setting in oscillation two spheres which were suspended
over a body of water.

[Illustration: FIG. 6.—Schematic representation of an interference.]

In Fig. 6 there is a schematic representation of an interference of the
same nature. Let us examine the situation at points along the lower
boundary line. At 0, which is equidistant from the two wave centres,
there is evidently a wave crest in each system; therefore there is a
resultant crest of double the amplitude of a single crest if the two
systems have the same amplitude. Half a period later there is a trough
in each system with a resultant trough of twice the amplitude of a
single trough. Thus higher crests and deeper troughs alternate. The
same situation is found at point 2, a wave-length farther from the left
than from the right wave centre; in fact, these results are found at
all points such as 2, 2′, 4 and 4′, where the difference in distance
from the two wave centres is an even number of wave-lengths. At the
point 1, on the other hand, where the difference between the distance
from the centres is one-half a wave-length, a crest from one system
meets a trough from the other, and the resultant is neither crest nor
trough but zero. There is the same result at points 1′, 3, 3′, 5, 5′,
etc., where the difference between the distances from the two wave
centres is an odd number of half wave-lengths. By throwing a stone
into the water in front of a smooth wall an interference is obtained,
similar to the one described above. The waves are reflected from the
wall as if they came from a centre at a point behind the wall and
symmetrically placed with respect to the point where the stone actually
falls.

[Illustration: FIG. 7.—Waves which are reflected by a board and pass
through a hole in it.]

When a wave system meets a wall in which there is a small hole, this
opening acts as a new wave centre, from which, on the other side
of the wall, there spread half-rings of crests and troughs. But if
the waves are small and the opening is large in proportion to the
wave-length, the case is essentially different. Let us suppose that
wave rings originate at every point of the opening. As a result of the
co-operation of all these wave systems the crests and troughs will
advance, just as before, in the original direction of propagation,
_i.e._, along straight lines drawn from the original wave centre
through the opening; lines of radiation, we may call them. It can be
shown, however, that as these lines of radiation deviate more and more
from the normal to the wall, the interference between wave systems
weakens the resultant wave motion. If the deviation from the normal to
the wall is increased, the weakening varies in magnitude, provided that
the waves are sufficiently small; but even if the wave motions at times
may thus “flare up” somewhat, still on the whole they will decrease as
the deviation from the normal to the wall is increased. The smaller the
waves in comparison to the opening, the more marked is the decrease
of the wave motions as the distance from the normal to the wall is
increased, and the more nearly the waves will move on in straight
lines. That light moves in straight lines, so that opaque objects cast
sharp shadows, is therefore consistent with the wave theory, provided
the light waves are very small; though it is reasonable to expect
that on the passage of light through narrow openings there will be
produced an appreciable bending in the direction of the rays. This
supposition agrees entirely with experiment. As early as the middle
of the seventeenth century, the Italian Grimaldi discovered such a
_diffraction_ of light which passes through a narrow opening into
a dark room.

[Illustration: FIG. 8.—Schematic representation of a wave.

A and B denote crests; C denotes a trough.

λ = wave-length. α = amplitude of wave.

If T denotes the time the wave takes to travel from A to B, and ν = 1/T
the frequency, the wave velocity _v_ will be equal to λ/T = λν.

Points P and P′ are points in the same phase.]

In both light and sound the use of such terms as wave and wave motion
is figurative, for crests and troughs are lacking. But this choice of
terms is commendable, because sound and light possess an essential
property similar to one possessed by water waves. What happens when
a tuning-fork emits sound waves into the surrounding air, is that
the air particles are set in oscillation in the direction of the
propagation of sound. All the particles of air have the same period as
the tuning-fork, and the number of oscillations per second determines
the pitch of the note produced; but the air particles at different
distances from the tuning-fork are not all simultaneously in the same
_phase_ or condition of oscillation. If one particle, at a certain
distance from the source of sound and at a given time, is moving most
rapidly away from the source, then at the same time there is another
particle, somewhat farther along the direction of propagation, which is
moving towards the source most rapidly. This alternation of direction
will exist all along the path of the sound. Where the particles
are approaching each other, the air is in a state of condensation,
and where the particles are drawing apart, the air is in a state
of rarefaction. While the individual particles are oscillating in
approximately the same place, the condensations and rarefactions, like
troughs and crests in water, advance with a velocity which is called
the velocity of sound. If we call the distance between two consecutive
points in the same phase a wave-length, and the number of oscillations
in a period of time the frequency, then, as in the case of water waves,
the velocity of propagation will be equal to the product of frequency
and wave-length.

Light, like sound, is a periodic change of the conditions in the
different points of space. These changes which emanate from the
source of light, in the course of one period advance one wave-length,
_i.e._, the distance between two successive points in the same
phase and lying in the direction of propagation. As in the cases of
sound and water waves, the velocity of propagation or the velocity of
light is equal to the product of frequency and wave-length. If this
velocity is indicated by the letter _c_, the frequency by ν and
the wave-length by λ, then

                   _c_        _c_
  _c_ = νλ or ν = ---- or λ = ---- .
                    λ          ν

The velocity of light in free space is a constant, the same for all
wave-lengths. It was first determined by the Danish astronomer Ole
Rømer (1676) by observations of the moons of Jupiter. According to
the measurements of the present day the velocity of light is about
1,000,000,000 feet or 300,000 kilometres per second. In centimetres it is
thus about 3 × 10¹⁰.

Efforts have been made to consider light waves, like sound waves, as
produced by the oscillations of particles, not of the air, but of a
particular substance, the “ether,” filling and permeating everything;
but all attempts to form definite representations of the material
properties of the ether and of the movements of its particles have been
unsuccessful. The _electromagnetic theory of light_, enunciated
about fifty years ago by the Scottish physicist, Maxwell, has
furnished information of an essentially different character concerning
the nature of light waves.

Let us suppose that electricity is oscillating in a conductor
connecting two metal spheres, for instance. The spheres, therefore,
have, alternately, positive and negative charges. Then according to
Maxwell’s theory we shall expect that in the surrounding space there
will spread a kind of _electromagnetic wave_ with a velocity
equal to that of light. Wherever these waves are, there should arise
electric and magnetic forces at right angles to each other and to
the direction of propagation of the waves; the forces should change
direction in rhythm with the movements of electricity in the emitting
conductor. By way of illustration let us assume that we have somewhere
in space an immensely small and light body or particle with an electric
charge. If, in the region in question, an electromagnetic wave motion
takes place, then the charged particle will oscillate as a result of
the periodically changing electrical forces. The particle here plays
the same rôle as the cork on the surface of the water (cf. p. 35);
the charged body thus makes the electrical oscillations in space
apparent just as the cork shows the oscillations of the water. In
addition to the electrical forces there are also magnetic forces in an
electromagnetic wave. We can imagine that they are made apparent by
using a very small steel magnet instead of the charged body. According
to Maxwell’s theory, the magnet exposed to the electromagnetic wave
will perform rapid oscillations. Maxwell came to the conclusion that
light consisted of electromagnetic waves of a similar nature, but much
more delicate than could possibly be produced and made visible directly
by electrical means.

In the latter part of the nineteenth century the German physicist, H.
Hertz, succeeded in producing electromagnetic waves with oscillations
of the order of magnitude of 100,000,000 per second, corresponding to
wave-lengths of the order of magnitude of several metres.

       _c_    3 × 10¹⁰
  (λ = ---- = -------- = 300 cm.).
        ν       10⁸

Moreover, he proved the existence of the oscillating electric forces
by producing electric sparks in a circle of wire held in the path
of the waves. He showed also that these electromagnetic waves were
reflected and interfered with each other according to the same laws
as in the case of light waves. After these discoveries there could be
no reasonable doubt that light waves were actually electromagnetic
waves, but so small that it would be totally impossible to examine the
oscillations directly with the assistance of electric instruments.

But there was in this work of Hertz no solution of the problems about
the nature of ether and the processes underlying the oscillations. More
and more, scientists have been inclined to rest satisfied with the
electromagnetic description of light waves and to give up speculation
on the nature of the ether. Indeed, within the last few years,
specially through the influence of Einstein’s theory of relativity,
many doubts have arisen as to the actual existence of the ether.
The disagreement about its existence is, however, more a matter of
definition than of reality. We can still talk about light as consisting
of ether waves if we abandon the old conception of the ether as a
rigid elastic body with definite material properties, such as specific
gravity, hardness and elasticity.


The Dispersion of Light.

It has been said that the wave-length of light is much shorter than
that of the Hertzian waves. This does not mean that all light waves
have the same wave-length and frequency. The light which comes to us
from the sun is composed of waves of many different wave-lengths and
frequencies, to each of which corresponds a particular colour.

In this respect also light may be compared with sound. In whatever way
a sound is produced, it is in general of a complicated nature, composed
of many distinct notes, each with its characteristic wave-length and
frequency. Naturally the air particles cannot oscillate in several
different ways simultaneously. At a given time, however, we can think
of the condensation and rarefactions of the air or the oscillations
of the particles corresponding to different tones, as compounded with
each other in a way similar to that in which the resultant crests and
troughs are produced on a body of water with several coexistent wave
systems. When we say that the complicated wave-movement emitted from
some sound-producing instrument consists of different tones, this
does not only mean that we may imagine it purely mathematically as
resolved into a series of simpler wave systems. The resolution may
also take place in a more physical way. Let us assume that we have a
collection of strings each of which will produce a note of particular
pitch. Now, if sound waves meet this collection of strings, each
string is set in oscillation by the one wave in the compound sound
wave which corresponds to it. Each string is then said to act as a
_resonator_ for the note in question. The notes which set the
resonator strings in oscillation sound more loudly in the neighbourhood
of the resonators; but, as the wave train continues on its journey the
tones taken out by the strings will become weak in contrast to those
notes which found no corresponding strings. The resonator is said to
_absorb_ the notes with which it is in pitch.

Light which is composed of different colours, _i.e._, of wave
systems with different wave-lengths, can also be resolved or dispersed,
but by a method different from that in the case of sound.

When light passes from one medium to another, as from air to glass
or _vice versa_, it is refracted, _i.e._, the direction of
the light rays is changed; but if the light is composed of different
colours the refraction is accompanied by a “spreading” of the colours
which is called dispersion. If we look through a glass prism so that
the light from the object examined must pass in and out through two
faces of the prism which make not too great an angle with each other,
the light-producing object is not only displaced by the refraction,
but has coloured edges. Newton was the first to explain the relation
of the production of the colours to refraction. He made an experiment
with sunlight, which he sent through a narrow opening into a dark room.
The sunlight was then by a glass prism transformed or dispersed into a
band of colour, a _spectrum_ consisting of all the colours of the
rainbow, red, yellow, green, blue and violet, in the order named, and
with continuous transition stages between neighbouring colours.

[Illustration: FIG. 9.—Prism spectroscope. To the right is
seen the collimator, to the left the telescope, in the foreground a
scheme for illuminating the cross-wire.

(From an old print.)]

In Newton’s original experiment the different wave-lengths were but
imperfectly separated. A spectrum with pure wave-lengths can be
obtained with a _spectroscope_ (cf. Fig. 9). The light to be
investigated illuminates an adjustable vertical slit in one end of a
long tube, called the collimator, with a lens in the other end. If the
slit is in the focal plane of the lens, the light at any point in the
slit goes in parallel rays after meeting the lens. It then meets a
prism, with vertical edges, placed on a little revolving platform. The
rays, refracted by the prism, go in a new direction into a telescope
whose objective lens gives in its focal plane, for every colour, a
clear vertical image of the slit. These images can be examined through
the ocular of the telescope; but since the different colours are not
refracted equally, each coloured image of the slit has its own place.
The totality of the slit images then forms a horizontal spectrum of
the same height as the individual images. By revolving the collimator
different parts of the spectrum can be put in the middle of the field
of view. To facilitate measurements in the spectrum there is in the
focal plane of the collimator a sliding cross-wire with an adjusting
screw or a vertical strand of spider web.

[Illustration: FIG. 10.—The mode of operation of a grating.

_A_, grating; _C, D, E_ ... _H_, slits; _M M_, incident rays. When _D
D′, E E′_ ... are a whole number of wave-lengths, the light waves which
move in the direction indicated by _C N_ and are collected by a lens,
at the focal point will all be in the same phase and therefore will
reinforce each other. In other directions the light action from one
slit is compensated by that from another.]

Instead of using the refraction of light in a prism to separate the
wave-lengths, we can use the interference which arises when a bundle of
parallel light waves passes through a ruled _grating_, consisting
of a great many very fine parallel lines, equidistant from each other;
such a grating can be made by ruling lines with a diamond point on the
metal coating of a silvered plate of glass. From each line there are
sent out light waves in all directions; but if we are considering
light of one definite colour (a given wave-length, _monochromatic
light_), the interference among the waves from all the slits
practically destroys all waves except in the direction of the original
rays and in the directions making certain angles with the former,
dependent upon the wave-length and the distance between two successive
lines (the grating space). Monochromatic light can be obtained by
using as the source of light a spirit flame, coloured yellow with
common salt (sodium chloride). If the slit in a spectroscope is lighted
with a yellow light from such a flame, and if a grating normal to
the direction of the rays is substituted for the prism, then in the
telescope there is seen a yellow image of the slit, and on each side
of it one, two, three or more yellow images. If sunlight is used the
central image is white, since all the colours are here assembled. The
other images become spectra because the different colours are unequally
refracted. In these _grating spectra_, which according to their
distance from the central line are called spectra of the first, second
or third order, the violet part lies nearest to the central line, the
red part farthest away. Since the deflection is the greater the greater
the wave-length, then violet light must have the shortest wave-length
and red the greatest. From the amount of the refraction and the size of
the grating space the wave-length of the light under investigation can
be calculated.

For the yellow light from our spirit flame the wave-length is about
0·000589 mm. or 0·589 μ or 589 μμ. In centimetres the wave-length is
0·0000589 cm.; from the formula ν = _c_/λ, ν = 526 × 10¹². The
frequency is thus almost inconceivably large. For the most distant
red and violet in the spectrum the wave-lengths are respectively
about 800 μμ and 400 μμ, and the frequencies 375 × 10¹² and 750 × 10¹²
oscillations per second.

In scientific experiments a grating of specular metal with parallel
rulings is substituted for the transparent grating. The spectrum is
then given by the reflected light from the parts between the rulings.
Specular gratings can be made by ruling on a concave mirror, which
focuses the rays so that a glass lens is unnecessary. Gratings with
several hundred lines or rulings to the millimetre give excellent
spectra, with strength of light and marked dispersion. The preparation
of the first really good gratings is due to the experimental skill
of the American, Rowland, who in 1870 built a dividing engine from
which the greater part of the good gratings now in use originate. The
contribution which Rowland thereby made to physical science can hardly
be over-estimated.


Spectral Lines.

In the early part of the nineteenth century Wollaston, in England,
and later Fraunhofer in Germany, discovered dark lines in the solar
spectrum, a discovery which meant that certain colours were missing.
The most noticeable of these so-called “Fraunhofer Lines” were named
with the letters A, B, C, D, E, F, G, H, from red to violet. It was
later discovered that some of the lines were double, that the D-line,
for instance, can be resolved into D₁ and D₂; other letters, such
as _b_ and _h_, were introduced to denote new lines. With
improvements in the methods of experiment and research the number
of lines has increased to hundreds and even thousands. The light
from a glowing solid or liquid element forms, on the other hand, a
continuous spectrum, _i.e._ a spectrum which has no dark lines. An
illustration of the solar spectrum with the strongest Fraunhofer lines
is given at the end of the book.

In contrast to the solar spectrum with dark lines on a bright
background are the so-called _line spectra_, which consist of
bright lines on a dark background. The first known line spectrum was
the one given by light from the spirit flame coloured with common
salt, mentioned in connection with monochromatic light. As has been
said, this spectrum had just one yellow line which was later found to
consist of two lines close to each other. It is sodium chloride which
colours the flame yellow. The colour is due, not to the chlorine, but
to the sodium, for the same double yellow line can be produced by using
other sodium salts not compounded with chlorine. The yellow light is
therefore called sodium light. No. 7 in the table of spectra at the end
of the book shows the spectrum produced by sodium vapour in a flame.
(On account of the small scale in the figure it is not shown that the
yellow line is double.)

Another interesting discovery was soon made, namely, that the sodium
line has exactly the same wave-length as the light lacking in the solar
spectrum, where the double D-line is located. About 1860 Kirchhoff and
Bunsen explained this remarkable coincidence as well as others of the
same nature. They showed by direct experiment that if sodium vapour is
at a high temperature it can not only send out the yellow light, but
also absorb light of the same wave-length when rays from a still warmer
glowing body pass through the vapour. This phenomenon is something like
that in the case of sound waves where a resonator absorbs the pitch
which it can emit itself. The existence of the dark D-line in the solar
spectrum must then mean that in the outer layer of the sun there is
sodium vapour present of lower temperature than the white-hot interior
of the sun, and that the light corresponding to the D-line is absorbed
by the vapour. Several ingenious experiments, which cannot be described
here, have given further evidence in favour of this explanation.

In the other line spectra, just as in that from the common salt flame,
definite lines correspond to definite elements and not to chemical
compounds. The emission of these lines is then not a molecular
characteristic, but an atomic one. The line spectra of metals can often
be produced by vaporizing a metallic salt in a spirit flame or in a
hot, colourless gas flame (from a Bunsen burner). It is even better to
use an electric arc or strong electric sparks. The atoms from which
gaseous molecules are formed can also be made to emit light which by
means of the spectroscope is shown to consist of a line spectrum. These
results are obtained by means of electric discharges of various kinds,
arcs, and spark discharges through tubes where the gas is in a rarefied
state.

The other Fraunhofer lines in the solar spectrum correspond to bright
lines in the line spectra of certain elements which exist here on
earth. These Fraunhofer lines must then be assumed to be caused by the
absorption of light by the elements in question. This may be explained
by the presence of these elements as gases in the solar atmosphere,
through which passes the light from the inner layer. This inner surface
would in itself emit a continuous spectrum.

The work of Kirchhoff and Bunsen put at the disposal of science
became a new tool of incalculable scope. First and foremost, spectrum
examinations were taken into the service of chemistry as _spectrum
analysis_. It has thus become possible to analyse quantities of
matter so small that the general methods of chemistry would be quite
powerless to detect them. It is also possible by spectrum analysis
to detect minute traces of an element; several elements were in this
way first discovered by the spectroscope. Moreover, chemical analysis
has been extended to the study of the sun and stars. The spectral
lines have given us answers to many problems of physics—problems which
formerly seemed insoluble. Last but not least spectrum analysis has
given us a key to the deepest secrets of the atom, a key which Niels
Bohr has taught us how to use.

In the discussion of the spectrum we have hitherto restricted ourselves
to the visible spectrum limited on the one side by red and on the other
by violet. But these boundaries are in reality fortuitous, determined
by the human eye. The spectrum can be studied by other methods than
those of direct observation. The more indirect methods include the
effect of the rays on photographic plates and their heating effect on
fine conducting wires for electricity, held in various parts of the
spectrum. It has thus been discovered that beyond the visible violet
end of the spectrum there is an _ultra-violet_ region with strong
photographic activity and an _infra-red_ region producing marked
heat effects. There are both dark and light spectral lines in these
new parts of the spectrum. The fact that glass is not transparent to
ultra-violet or infra-red rays has been an obstacle in the experiments,
but the difficulty can be overcome by using other substances, such as
quartz or rock salt, for the prisms and lenses, or by substituting
concave gratings. By special means it has been possible to detect rays
with wave-lengths as great as 300 μ and as small as about 0·02 μ,
corresponding to frequencies between 10¹², and 15 × 10¹⁵ vibrations per
second, while the wave-lengths of the luminous rays lie between 0·8 and
0·4 μ. The term “light wave” is often used to refer to the ultra-violet
and infra-red rays which can be shown in the spectra produced by prisms
or gratings.

[Illustration: FIG. 11.—Photographic effect of X-rays, which
are passed through the atom grating in a magnesia crystal.]

The electrically produced electromagnetic waves, as already mentioned,
have wave-lengths much greater than 300 μ. In wireless telegraphy
there are generally used wave-lengths of one kilometre or more,
corresponding to frequencies of 300,000 vibrations per second or less.
By direct electrical methods it has, however, not been possible to
obtain wave-lengths less than about one-half a centimetre, a length
differing considerably from the 0·3 millimetre wave of the longest
infra-red rays. Wave-lengths much less than 0·02 μ or 20 μμ exist in
the so-called _Röntgen rays_ or _X-rays_ with wave-lengths
as small as 0·01 μμ corresponding to a frequency of 30 × 10¹⁸. These
rays cannot possibly be studied even with the finest artificially made
gratings, but crystals, on account of the regular arrangement of the
atoms, give a kind of natural grating of extraordinary fineness. With
the use of crystal gratings success has been attained in decomposing
the Röntgen rays into a kind of spectrum, in measuring the wave-lengths
of the X-rays and in studying the interior structure of the crystals.
The German Laue, the discoverer of the peculiar action of crystals on
X-rays (1912), let the X-rays beams pass through the crystal, obtaining
thereby photographs of the kind illustrated in Fig. 11. Later on
essential progress was due to the Englishmen, W. H. and W. L. Bragg,
who worked out a method of investigation by which beams of X-rays are
reflected from crystal faces. The greatest wave-length which it has
been possible to measure for X-rays is about 1·5 μμ, which is still a
long way from the 20 μμ of the furthermost ultra-violet rays.

It may be said that the spectrum since Fraunhofer has been made not
only longer but also finer, for the accuracy of measuring wave-lengths
has been much increased. It is now possible to determine the
wave-length of a line in the spectrum to about 0·001 μμ or even less,
and to measure extraordinarily small changes in wave-lengths, caused by
different physical influences.

In addition to the continuous spectra emitted by glowing solids
or liquids, and to the line spectra emitted by gases, and to the
absorption spectra with dark lines, there are spectra of still another
kind. These are the absorption spectra which are produced by the
passage of white light through coloured glass or coloured fluids. Here
instead of fine dark lines there are broader dark absorption bands,
the spectrum being limited to the individual bright parts. There are
also the band spectra proper, which, like the line spectra, are purely
emission spectra, given by the light from gases under particular
conditions; these seem to consist of a series of bright bands which
follow each other with a certain regularity (cf. Fig. 12). With
stronger dispersion the bands are shown to consist of groups of bright
lines.

[Illustration:

FIG. 12.—Spectra produced by discharges of different character
through a glass tube containing nitrogen at a pressure of ¹/₂₀ that of
the atmosphere. Above, a band spectrum; below, a line spectrum.]

Since the line spectra are most important in the atomic theory, we
shall examine them here more carefully.

The line spectra of the various elements differ very much from each
other with respect to their complexity. While many metals give a great
number of lines (iron, for instance, gives more than five thousand),
others give only a few, at least in a simple spectroscope. With a more
powerful spectroscope the simplicity of structure is lost, since weaker
lines appear and other lines which had seemed single are now seen to
be double or triple. Moreover, the number of lines is increased by
extending the investigation to the ultra-violet and infra-red regions
of the spectrum. The sodium spectrum, at first, seemed to consist of
one single yellow line, but later this was shown to be a double line,
and still later several pairs of weaker double lines were discovered.
The kind and number of lines obtained depends not only upon the
efficiency of the spectroscope, but also upon the physical conditions
under which the spectrum is obtained.

The eager attempts of the physicists to find laws governing the
distribution of the lines have been successful in some spectra. For
instance, the line spectra of lithium, sodium, potassium and other
metals can be arranged into three rows, each consisting of double
lines. The difference between the frequencies of the two “components”
of the double lines was found to be exactly the same for most of
the lines in one of these spectra, and for the spectra of different
elements there was discovered a simple relationship between this
difference in frequency and the atomic weight of the element in
question. But this regularity was but a scrap, so to speak; scientists
were still very far from a law which could exactly account for the
distribution of lines in a single series, not to mention the lines in
an entire spectrum or in all the spectra.

The first important step in this direction was made about 1885 by
the Swiss physicist, Balmer, in his investigations with the hydrogen
spectrum, the simplest of all the spectra. In the visible part
there are just three lines, one red, one green-blue and one violet,
corresponding to the Fraunhofer lines C, F and _h_. These
hydrogen lines are now generally known by the letters Hα, Hᵦ and Hᵧ.
In the ultra-violet region there are many lines also.

Balmer discovered that wave-lengths of the red and of the green
hydrogen line are to each other exactly as two integers, namely, as 27
to 20, and that the wave-lengths of the green and violet lines are to
each other as 28 to 25. Continued reflection on this correspondence led
him to enunciate a rule which can be expressed by a simple formula.
When frequency is substituted for wave-length Balmer’s formula is
written as

       ( 1      1    )
  ν = K(--- - -------),
       ( 4     _n_²  )

where ν is the frequency of a hydrogen line, K a constant equal to 3·29
× 10¹⁵ and _n_ an integer. If _n_ takes on different values,
ν becomes the frequency for the different hydrogen lines. If _n_ =
1 ν is negative, for _n_ = 2 ν is zero. These values of _n_
therefore have no meaning with regard to ν. But if _n_ = 3, then
ν gives the frequency for the red hydrogen line Hα; _n_ = 4 gives
the frequency of the green line Hᵦ and _n_ = 5 that of the violet
line Hᵧ. Gradually more than thirty hydrogen lines have been found,
agreeing accurately with the formula for different values of _n_.
Some of these lines were not found in experiment, but were discovered
in the spectrum of certain stars; the exact agreement of these lines
with Balmer’s formula was strong evidence for the belief that they are
due to hydrogen. The formula thus proved itself valuable in revealing
the secrets of the heavens.

As _n_ increases 1/_n_² approaches zero, and can be made as
close to zero as desired by letting _n_ increase indefinitely. In
mathematical terminology, as _n_ = ∞, 1/_n_² = 0 and ν =
K/4 = 823 × 10¹², corresponding to a wave-length of 365 μμ. Physically
this means that the line spectrum of hydrogen in the ultra-violet is
limited by a line corresponding to that frequency. Near this limit the
hydrogen lines corresponding to Balmer’s formula are tightly packed
together. For _n_ = 20 ν differs but little from K/4, and the
distance between two successive lines corresponding to an increase of
1 in _n_ becomes more and more insignificant. Fig. 13, where the
numbers indicate the wave-lengths in the Ångström unit (0·1 μμ), shows
the crowding of the hydrogen lines towards a definite boundary. The
following table, where K has the accurate value of 3·290364 × 10¹⁵,
shows how exactly the values calculated from the formula agree with
experiment.

[Illustration: FIG. 13.—Lines in the hydrogen spectrum corresponding to
the Balmer series.]


TABLE OF SOME OF THE LINES OF THE BALMER SERIES

   ---------+----------------------------+-------------+-------------+
            |  ν = K(¹⁄₄ - 1/_n_²) = ν   |             |             |
            |       (calculated).        |  ν (found). | λ (found).  |
    --------+----------------------------+-------------+-------------+
    _n_ =  3|K(¼ - ¹/₉  ) = 456,995 bills|456,996 bills|656·460 μμ Hα|
    _n_ =  4|K(¼ - ¹/₁₆ ) = 616,943   “  |616,943   “  |486·268 “  Hᵦ|
    _n_ =  5|K(¼ - ¹/₂₅ ) = 690,976   “  |690,976   “  |434·168 “  Hᵧ|
    _n_ =  6|K(¼ - ¹/₃₆ ) = 731,192   “  |731,193   “  |410·288 “  Hδ|
    _n_ =  7|K(¼ - ¹/₄₉ ) = 755,440   “  |755,441   “  |397·119 “  Hε|
    --------+----------------------------+-------------+-------------+
    _n_ = 20|K(¼ - ¹/₄₀₀) = 814,365   “   |814,361   “  |368·307 “    |
    --------+----------------------------+-------------+-------------+

From arguments in connection with the work of the Swedish scientist,
Rydberg, in the spectra of other elements, Ritz, a fellow countryman of
Balmer’s, has made it seem probable that the hydrogen spectrum contains
other lines besides those corresponding to Balmer’s formula. He assumed
that the hydrogen spectrum, like other spectra, contains several series
of lines and that Balmer’s formula corresponds to only one series. Ritz
then enunciated a more comprehensive formula, the Balmer-Ritz formula:

       (    1        1  )
  ν = K( ------- - -----),
       (  _n″_²    _n′_²)

where K has the same value as before, and both _n′_ and _n″_
are integers which can pass through a series of different values.
For _n″_ = 2, the Balmer series is given; to _n″_ = 1, and
_n′_ = 2, 3 ... ∞ there corresponds a second series which lies
entirely in the ultra-violet region, and to _n″_ = 3, _n′_
= 4, 5 ... ∞ a series lying entirely in the infra-red. Lines have
actually been found belonging to these series.

Formulæ, similar to the Ritz one, have been set up for the line spectra
of other elements, and represent pretty accurately the distribution
of the lines. The frequencies are each represented by the difference
between two terms, each of which contains an integer, which can pass
through a series of values. But while the hydrogen formula, except for
the _n_′s, depends only upon one constant quantity K and its terms
have the simple form K/_n_², the formula is more complicated with
the other elements. The term can often be written, with a high degree
of exactness, as K/(_n_ + α)², where K is, with considerable
accuracy, the same constant as in the hydrogen formula. For a given
element α can assume several different values; therefore the number of
series is greater and the spectrum is even more complicated than that
of hydrogen.

All these formulæ are, however, purely empirical, derived from the
values of wave-lengths and frequencies found in spectrum measurements.
They represent certain more or less simple bookkeeping rules, by which
we can register both old and new lines, enter them in rows, arrange
them according to a definite system. But from the beginning there could
be no doubt that these rules had a deeper physical meaning which it was
not yet possible to know. There was no visible correspondence between
the spectral line formulæ and the other physical characteristics of
the elements which emitted the spectra; not even in their form did the
formulæ show any resemblance to formulæ obtained in other physical
branches.




CHAPTER III

IONS AND ELECTRONS


Early Theories and Laws of Electricity.

The fundamental phenomena of electricity, which were first made the
subject of careful study about two centuries ago, are that certain
substances can be electrified by friction so that somehow they can
attract light bodies, and that the charges of electricity may be
either “positive” or “negative.” Bodies with like charges repel each
other, while those with unlike charges attract each other, and either
partially or entirely neutralize each other when they are brought
close together. Moreover, it had long ago been discovered that in some
substances electricity can move freely from place to place, while in
others there is resistance to the movement. The former bodies are now
called _conductors_ and include metals, while the latter are
called _insulators_, glass, porcelain and air being members of
this class.

In order to explain the phenomena some imagined that there were two
kinds of “electric substances” or “fluids”; and since no change in
weight could be discovered in a body when it was electrified, it was,
in general, assumed that the electric fluids were weightless. In the
normal, neutral body it was believed that these fluids were mixed in
equal quantities, thereby neutralizing each other; on this account they
were supposed to be of opposite characteristics, so one was called
positive and the other negative. According to a second theory, there
was assumed to be just one kind of electricity, which was present in a
normal amount in neutral bodies; positive electricity was caused by a
superfluity of the fluid; negative, by a deficit. In both theories it
was possible to talk of the amount of positive or negative electricity
which a body contained or with which it was “charged,” because the
supporters of the one-fluid idea understood by the terms positive and
negative a superfluity and a deficit, respectively, of the one fluid.
In both theories it was possible to talk about the _direction_
of the electric current in a conductor, since the supporters of the
two-fluid theory understood by “direction” that in which the electric
forces sent the positive electricity, or the opposite to that in which
the negative would be sent. It could not be decided whether positive
electricity went in the one direction or the negative in the other,
or whether each simultaneously moved in its own direction. Both
theories were quite arbitrary in designating the electric charge in
glass, which was rubbed with woollen cloth, as positive. On the whole,
neither theory seemed to have any essential advantage over the other;
the difference between them seemed to lie more in phraseology than in
actual fact.

That the positive and negative states of electricity could not be
taken as “symmetric” seemed, however, to follow from the so-called
discharge phenomena, in which electricity, with the emission of light,
streams out into the air from strongly charged (positive or negative)
bodies, or passes through the air between positive and negative bodies
in sparks, electric arcs or in some other way. In a discharge in air
between a metal point and a metal plate, for instance, a bush-shaped
glow is seen to extend from the point when the charge there is
positive, while only a little star appears when the charge is negative.

Naturally, we cannot discuss here the many electric phenomena and laws,
and must be satisfied with a brief description of those which are of
importance in the atomic theory.

In this latter category belongs _Coulomb’s Law_, formulated about
1785. According to this law, the repulsions or attractions between
two electrically charged bodies are directly as the product of the
charges and inversely as the square of the distance between them (as in
the case of the gravitational attraction between two neutral bodies,
according to Newton’s Law). The unit in measuring electric charges can
be taken as that amount which will repel an equal amount of electricity
of the same kind at unit distance with unit force. If we use the
scientific or “absolute” system, in which the unit of length is one
centimetre, that of time one second and that of mass one gram, then
the unit of force is _one dyne_, which is a little greater than
the earth’s attraction on a milligram weight. Let us suppose that two
small bodies with equal charges of positive (or negative) electricity
are at a distance of one centimetre from each other. If they repel each
other with a charge of one dyne, then the amount of electricity with
which each is charged is called the _absolute electrostatic unit of
electricity_. If one body has a charge three times as great and
the other has a charge four times as great, the repulsion is 3 × 4 =
12 times greater. If the distance between the bodies is increased from
one to five, the repulsion is twenty-five times as small, since 5² =
25. If the charge of one body is substituted by a negative one of same
magnitude the repulsion becomes an attraction of the same magnitude.

In the early part of the nineteenth century methods were found for
producing a steady _electric current_ in metal wires. In 1820, the
Danish physicist, H. C. Ørsted, discovered that an electric current
influences a magnet in a characteristic way, and that, conversely,
the current is affected by the forces emanating from the magnet, by
a magnetic field in other words. The French scientist, Ampère, soon
afterwards formulated exact laws for the _electromagnetic_ forces
between magnets and currents. In 1831, the English physicist, Faraday,
discovered that an electric current is _induced_ in a wire when
currents or magnets in its neighbourhood are moved or change strength.
Faraday’s views on electric and magnetic fields of force around
currents and magnets were further of fundamental importance to the
_electromagnetic wave theory_ as developed by Maxwell. The branch
of physics dealing with all these phenomena is now generally known as
_electrodynamics_.

[Illustration: FIG. 14.—Picture of electrolysis of hydrogen chloride.

_A_, anode; _K_, cathode; _H_, hydrogen atoms; _Cl_, chlorine atoms.]


Electrolysis.

Faraday also studied the chemical effects which an electric current
produces upon being conducted between two metal plates, called
_electrodes_, which are immersed in a solution of salts or acids.
The current separates the salt or acid into two parts which are carried
by the electric forces in two opposite directions. This separation is
called electrolysis. If the liquid is dilute hydrochloric acid (HCl),
the hydrogen goes with the current to the negative electrode, the
_cathode_, and takes the positive electricity with it, while the
chlorine goes against the current and takes the negative electricity to
the positive electrode, the _anode_. We must then assume with the
Swedish scientist, Arrhenius, that, under the influence of the water,
the molecules of hydrogen chloride always are separated into positive
hydrogen atoms and negative chlorine atoms, and that the electric
forces from the anode and the cathode carry these atoms respectively
with and against the current. The electrically charged wandering atoms
are called _ions_, _i.e._ wanderers. The positive electricity
taken by the hydrogen atoms to the cathode goes into the metal
conductor, while the anode must receive from the metal conductor an
equal amount of positive electricity to be given to the chlorine atoms
to neutralize them. The negative charge of a chlorine atom must then be
as large as the positive charge of a hydrogen atom. These assumptions
imply that equal numbers of the two kinds of atoms are present in the
whole quantity of atoms transferred in any period of time.

Faraday found that the quantity of hydrogen which in the above
experiment is transferred to the cathode in a given time is
proportional to the quantity of electricity transferred in the same
time. A gram of hydrogen always takes the same amount of electricity
with it. By experiment this amount of electricity can be determined,
and, since the weight in grams of the hydrogen atom is known, it is
possible to calculate the amount of one atom. In electrostatic units
it is 4·77 × 10⁻¹⁰, _i.e._, 477 billionth[1] parts. A chlorine atom
then carries with it 4·77 × 10⁻¹⁰ electrostatic units of negative
electricity. Since its atomic weight is 35·5, then 35·5 grams of
chlorine will take as much electricity as 1 gram of hydrogen. The ratio
_e_/_m_ between the charge _e_ and the mass _m_ is then 35·5 times as
great for hydrogen as for chlorine.

[1] Billion used here to mean one million million, and trillion to mean
one million billion.

We have temporarily restricted ourselves to the electrolysis of
hydrogen chloride. Let us now assume that we have chloride of zinc
(ZnCl₂), which, by electrolysis, is separated into chlorine and zinc.
Each atom of chlorine will, as before, carry 4·77 × 10⁻¹⁰ units of
negative electricity to the anode; but since zinc is divalent (cf. p.
17) and one atom of zinc is joined to two of chlorine, therefore one
atom of zinc must carry a charge of 2 × 4·77 × 10⁻¹⁰ units of positive
electricity to the cathode. An atom or a group of atoms, with valence
of three, in electrolysis carries 3 × 4·77 × 10⁻¹⁰ units, etc.

We see then, that the quantity of electricity which accompanies the
atoms in electrolysis is always 4·77 × 10⁻¹⁰ electrostatic units or an
integral multiple thereof. This suggests the thought that electricity
is atomic and that the quantity 4·77 × 10⁻¹⁰ units is the smallest
amount of electricity which can exist independently, _i.e._, the
_elementary quantum of electricity_ or the “atom of electricity.”
The atom of a monovalent element, when charged or ionized, should
have one atom of electricity; a divalent, two, etc. On the two-fluid
theory it was most reasonable to assume that there were two kinds
of atoms of electricity representing, respectively, positive and
negative electricity. In Fig. 15 there is given, in accordance with the
two-fluid theory, a rough picture of a chlorine ion and a hydrogen ion
and their union into a molecule.

[Illustration: FIG. 15.—Provisional representation (according
to the two-fluid theory) of

A, a hydrogen ion; B, a chlorine ion; and C, a molecule of hydrogen
chloride.]

The atoms of electricity seemed to differ essentially from the
usual atoms of the elements in their apparent inability to live
independently; they seemed to exist only in connection with the atoms
of the elements. They would seem much more real if they could exist
independently. That such existence really is possible, has been
discovered by the study of the motion of electricity in gases.


Vacuum Tube Phenomena.

[Illustration: FIG. 16.—Vacuum tube with cathode rays and a
shadow-producing cross.

P and N, conducting wires for the electric current; _a_, cathode; _b_,
anode and shadow-producer; _c_, _d_, the shadow.]

[Illustration: FIG. 17.—Vacuum tube, where a bundle of cathode rays are
deviated by electric forces.

_A_, anode; _K_, cathode.]

It has previously been said that air is an insulator for electricity,
a statement which is, in general, true; however, as has also been
said, electric sparks and arcs can pass through air. Moreover, it has
been discovered that exhausted air is a very good conductor, so that a
strong current can pass between two metal electrodes in a glass tube
where the air is exhausted, if the electrodes are connected to an outer
conductor by metal wires fused into the glass. In these vacuum tubes
there are produced remarkable light effects, at first inexplicable.
When the air is very much exhausted, to a hundred thousandth of the
atmospheric pressure or less, strong electric forces (large difference
of potential between the electrodes) are needed to produce an electric
discharge. Such a discharge assumes an entirely new character; in the
interior of the glass tube there is hardly any light to be seen, but
the glass wall opposite the negative electrode (the cathode) glows
with a greenish tint (fluorescence). If a small metal plate is put in
the tube between the cathode and the glass wall, a shadow is cast on
the wall, just as if light were produced by rays, emitted from the
cathode at right angles to its surface (cf. Fig. 16). The English
physicist, Crookes, was one of the first to study these cathode rays.
He assumed that they are not ether waves like the light rays, but
that they consist of particles which are hurled from the cathode
with great velocity in straight lines; they light the wall by their
collisions with it. There was soon no doubt as to the correctness of
Crookes’ theory. The cathode rays are evidently particles of negative
electricity, which by repulsions are driven from the cathode (the
negative electrode). A metal plate bombarded by the rays becomes
charged negatively. Let us suppose that we have a small bundle of
cathode rays, obtained by passing the rays from the cathode _K_
(cf. Fig. 17) through two narrow openings _S₁_ and _S_. It
can then be shown that the bundle of rays is deviated not only by
electric forces, but also by magnetic action from a magnet which is
held near the glass. In the figure there is shown a deviation of the
kind mentioned, caused by making the plates at _B_ and _C_
respectively positive and negative; since _B_ attracts the
negative particles and _C_ repels them, the light spot produced by
the bundle of rays is moved from _M_ to _M_₁. The magnetic
deviation is in agreement with Ørsted’s rules for the reciprocal
actions between currents and magnets, if we consider the bundle of
rays produced by moving electric particles as an electric current.
(Since the electric particles travelling in the direction of the rays
are negative, and since it is customary by the expression “direction
of current” to understand the direction opposite to that in which the
negative electricity moves, then, in the case of the cathode rays just
mentioned, the direction of the current must be opposite to that of the
rays.)

From measurements of the magnetic and electric deviations it is
possible to find not only the velocity of the particles, but also the
ratio _e_/_m_ between the charge _e_ of the particle and its mass _m_.
The velocity varies with the potential at the cathode, and may be very
great, 50,000 km. per second, for instance (about one-sixth the speed
of light), or more. It has been found that _e_/_m_ always has the same
value, regardless of the metal of the cathode and of the gas in the
tube; this means that the particles are not atoms of the elements, but
something quite new. It has also been found that _e_/_m_ is about two
thousand times as large as the ratio between the charge and the mass of
the hydrogen atom in electrolysis. If we now assume that _e_ is just
the elementary quantum of electricity 4·77 × 10¹⁰, which in magnitude
amounts to the charge of the hydrogen atom in electrolysis (but is
negative), then _m_ must have about ¹/₂₀₀₀ the mass of the hydrogen
atom. This assumption as to the size of _e_ has been justified by
experiments of more direct nature. The experiments with charge and mass
of electrons which have in particular been carried out by the English
physicist, J. J. Thomson, give reason then to suppose these quite new
and unknown particles to be free atoms of negative electricity; they
have been given the name of _electrons_. Gradually more information
about them has been acquired. Thus it has been possible in various ways
to determine directly the charge on the electron, independently of its
mass. Special mention must be made of the brilliant investigations
of the American, Millikan, on the motion of very small electrified
oil-drops through air under the influence of an electric force. To
Millikan is due the above-mentioned value of _e_, which is accurate to
one part in five hundred. Further, the mass of the electron has been
more exactly calculated as about ¹/₁₈₃₅ that of the hydrogen atom.
Their magnitude has also been learned; the radius of the electron is
estimated as 1·5 × 10⁻¹³ cm. or 1·5 × 10⁻⁶ μμ, an order of magnitude
one ten-thousandth that of the molecule or atom.

After the atom of negative electricity had been isolated, in the form
of cathode rays, the next suggestion was that corresponding positive
electric particles might be discharged from the anode in a vacuum tube.
By special methods success has been attained in showing and studying
rays of positive particles. In order to separate them from the negative
cathode ray particles the German scientist, Goldstein, let the positive
particles pass through canals in the cathode; they are therefore
called _canal rays_. The velocity of the particles is much less
than that of the cathode rays, and the ratio _e_/_m_ between
charge and mass is much smaller and varies according to the gas in the
tube. In experiments where the tube contains hydrogen, rays are always
found for which _e_/_m_, as in electrolysis, is about ¹/₂₀₀₀
of the ratio in the cathode rays. Therefore there can be scarcely any
doubt that these canal rays are made up of charged hydrogen atoms
or hydrogen ions. The values found with other gases indicate that
the particles are atoms (or molecules sometimes) of the elements in
question, with charges one or more times the elementary quantum of
electricity (4·77 × 10⁻¹⁰ electrostatic units). Research in this field
has also been due in particular to J. J. Thomson. From his results, as
well as from those obtained by other methods, it follows that positive
electricity, unlike negative, cannot appear of its own accord, but is
inextricably connected to the atoms of the elements.


The Nature of Electricity.

The earlier conceptions of a one or two-fluid explanation of the
phenomena of electricity appear now in a new light. We are led to think
of a neutral atom as consisting of one mass charged with positive
electricity together with as many electrons negatively charged as are
sufficient to neutralize the positive. If the atom loses one, two or
three electrons, it becomes positive with a charge of one, two or
three elementary quanta of electricity, or for the sake of simplicity
and brevity we say that the atom has one, two or three “charges.”
If, on the other hand, it takes up one, two or three extra electrons
it has one, two or three negative charges. Fig. 18 can give help
in understanding these ideas, but it must not be thought that the
electrons are arranged in the way indicated. The substances, which
appear as electropositive in electrolysis—_i.e._ hydrogen and
metals—should then be such that their atoms easily lose one or more
electrons, while the electronegative elements should, on the other
hand, easily take up extra electrons. Elements should be monovalent or
divalent according as their atoms are apt to lose or to take up one or
two electrons. From investigations with the vacuum tube it appears,
however, that the atoms of the same element can in this respect behave
in more ways than would be expected from electrolysis or chemical
valence.

[Illustration: FIG. 18.—Provisional representation (according
to the electron theory) of

A, a neutral atom; B, the same atom with two positive charges (a
divalent positive ion) and C, the same atom with two negative charges
(a divalent negative ion).]

When an electric current passes through a metal wire, it must be
assumed that the atoms of the metal remain in place, while the
electrical forces carry the electrons in a direction opposite to that
which usually is considered as the direction of the current (cf. p.
70). The motion of the electrons must not be supposed to proceed
without hindrance, but rather as the result of a complicated interplay,
by no means completely understood, whereby the electrons are freed from
and caught by the atoms and travel backwards and forwards, in such a
way that through every section of the metal wire a surplus of electrons
is steadily passing in the direction opposite to the so-called
direction of the current. The number of surplus electrons which in
every second passes through a section of the thin metal wire in an
ordinary twenty-five candle incandescent light, at 220 volts, amounts
to about one trillion (10¹⁸), or 1000 million (10⁹) in 0·000,000,001 of
a second. If the metal conducting wire ends in the cathode of a vacuum
tube, the electrons carried through the wire pass freely into the tube
as cathode rays from the cathode.

This motion of electricity agrees best with the one-fluid theory,
since the electrons, which here alone accomplish the passage of the
electricity, may be considered as the fundamental parts of electricity.
In this respect the choice of the terms positive and negative is very
unfortunate, since a body with a negative charge actually has a surplus
of electrons. Moreover, the electrons really have mass; but since
the mass of a single electron is only ¹/₁₈₃₅ that of the atom of the
lightest element, hydrogen, and since in an electrified body which can
be weighed by scale there is always but an infinitesimal number of
charged atoms, it is easy to understand that, formerly, electricity
seemed to be without weight.

In electrolysis, where the motion of electricity is accomplished by
positive and negative ions, we have a closer connection with the
two-fluid theory. In motions of electricity through air the situation
suggests both the one-fluid and the two-fluid theories, since the
passage of electricity is sometimes carried on exclusively by the
electrons, and sometimes partly by them and partly by larger positive
and negative ions, _i.e._, atoms or molecules with positive and
negative charges.


The Electron Theory.

Proceeding on the assumption that the electric and optical properties
of the elements are determined by the activity of the electric
particles, the Dutch physicist Lorentz and the English physicist
Larmor succeeded in formulating an extraordinarily comprehensive
“electron theory,” by which the electrodynamic laws for the variations
in state of the ether were adapted to the doctrine of ions and
electrons. This Lorentz theory must be recognized as one of the finest
and most significant results of nineteenth century physical research.

It was one of the most suggestive problems of this theory to account
for the emission of light waves from the atom. From the previously
described electromagnetic theory of light (cf. p. 42) it follows that
an electron oscillating in an atom will emit light waves in the ether,
and that the frequency ν of these waves will naturally be equal to
the number of oscillations of the electron in a second. If this last
quantity is designated as ω, then

  ν = ω

It may then be supposed that the electrons in the undisturbed atom
are in a state of rest, comparable with that of a ball in the bottom
of a bowl. When the atom in some way is “shaken,” one or more of the
electrons in the atom begins to oscillate with a definite frequency,
just as the ball might roll back and forth in the bowl if the bowl was
shaken. This means that the atom is emitting light waves, which, for
each individual electron have a definite wave-length corresponding
to the frequency of the oscillations, and that, in the spectrum of
the emitted light, the observed spectral lines correspond to these
wave-lengths.

Strong support for this view was afforded by Zeeman’s discovery of the
influence of a magnetic field upon spectral lines. Zeeman, a Dutch
physicist, discovered, about twenty-five years ago, that when a glowing
vacuum tube is placed between the poles of a strong electromagnet,
the spectral lines in the emitted light are split so that each line
is divided into three components with very little distance between
them. It was one of the great triumphs of the electron theory that
Lorentz was able to show that such an effect was to be expected if
it was assumed that the oscillations of light were produced by small
oscillating electric particles within the atom. From the experiments
and from the known laws concerning the reciprocal actions of a magnet
and an electric current (here the moving particle), the theory enabled
Lorentz to find not only the ratio _e_/_m_ between the electric charge
of each of these particles and its mass, but also the nature of the
charge. He could conclude from Zeeman’s experiment that the charge
is negative and that the ratio _e_/_m_ is the same as that found for
the cathode rays. After this there could not well be doubt that the
electrons in the atoms were the origin of the light which gives the
lines of the spectrum. It seemed, however, quite unfeasible for the
theory to explain the details in a spectrum—to derive, for instance,
Balmer’s formula, or to show why hydrogen has these lines, copper
those, etc. These difficulties, combined with the great number of
lines in the different spectra, seemed to mean that there were many
electrons in an atom and that the structure of an atom was exceedingly
complicated.


Ionization by X-rays and Rays from Radium. Radioactivity.

As has been said, the electrons in a vacuum tube cause its wall to emit
a greenish light when they strike it. Upon meeting the glass wall or a
piece of metal (the anticathode) placed in the tube the electrons cause
also the emission of the peculiar, penetrating rays called Röntgen rays
in honour of their discoverer, or more commonly X-rays. They may be
described as ultra-violet rays with exceedingly small wave-lengths (cf.
p. 54). When, further, the electrons meet gas molecules in the tube
they break them to pieces, separating them into positive and negative
ions (_ionization_). The positive ions are the ones which appear in the
canal rays. The ions set in motion by electrical forces can break other
gas molecules to pieces, thus assisting in the ionization process.
At the same time the gas molecules and atoms are made to produce
disturbances in the ether, and thus to cause the light phenomena which
arise in a tube which is not too strongly exhausted.

The free air can be ionized in various ways; this ionization can be
detected because the air becomes more or less conducting. In fact,
electric forces will drive the positive and negative ions through the
air in opposite directions, thus giving rise to an electric current.
If the ionization process is not steadily continued, the air gradually
loses its conductivity, since the positive and negative ions recombine
into neutral atoms or molecules. Ionization can be produced by flames,
since the air rising from a flame contains ions. A strong ionization
can also be brought about by X-rays and by ultra-violet rays. In the
higher strata of the atmosphere the ultra-violet rays of the sun
exercise an ionizing influence. Most of all, however, the air is
ionized by rays from the so-called _radioactive_ substances which in
very small quantities are distributed about the world.

The characteristic radiation from these substances was discovered in
the last decade of the nineteenth century by the French physicist,
Becquerel, and afterwards studied by M. and Mme. Curie. From the
radioactive uranium mineral, pitchblende, the latter separated the many
times more strongly radioactive element radium. The proper nature of
the rays was later explained, particularly through the investigations
of the English physicists, Rutherford, Soddy and Ramsay. These rays,
which can produce heat effects, photographic effects and ionization,
are of three quite different classes, and accordingly are known as
α-rays, β-rays, and γ-rays. The last named, like the X-rays, are
ultra-violet rays, but they have often even shorter wave-lengths and
a much greater power of penetration than the usual X-rays. The β-rays
are electrons which are ejected with much greater velocity than the
cathode rays; in some cases their velocity goes up to 99·8 per cent.
that of light. The α-rays are positive atomic ions, which move with a
velocity varying according to the emitting radioactive element from
¹/₂₀ to almost ¹/₁₀ that of light. It has further been proved that
the α-particles are atoms of the element helium, which has the atomic
weight 4, and that they possess two positive charges, _i.e._, they
must take up two electrons to produce a neutral helium atom.

There is no doubt that the process which takes place in the emission
of radiation from the radioactive elements is a _transformation of
the element_, an explosion of the atoms accompanied by the emission
either of double-charged helium atoms or of electrons, and the forming
of the atoms of a new element. The energy of the rays is an internal
atomic energy, freed by these transformations. The element uranium,
with the greatest of all known atomic weights (238), passes, by several
intermediate steps, into radium with atomic weight 226; from radium
there comes, after a series of steps, lead, or, in any case, an element
which, in all its chemical properties, behaves like lead. We shall go
no further into this subject, merely remarking that the transformations
are quite independent of the chemical combinations into which the
radioactive elements have entered, and of all external influences.

When α-particles from radium are sent against a screen with a coating
of especially prepared zinc sulphide, on this screen, in the dark,
there can be seen a characteristic light phenomenon, the so-called
scintillation, which consists of many flashes of light. Each individual
flash means that an α-particle, a helium atom, has hit the screen. In
this bombardment by atoms the individual atom-projectiles are made
visible in a manner similar to that in which the individual raindrops
which fall on the surface of a body of water are made visible by the
wave rings which spread from the places where the drops meet the water.
This flash of light was the first effect of the individual atom to
be available for investigation and observation. The incredibility of
anything so small as an atom producing a visible effect is lessened
when, instead of paying attention merely to the small size or mass of
the atom, its kinetic energy is considered; this energy is proportional
to the square of the velocity, which is here of overwhelming magnitude.
For the most rapid α-particles the velocity is 2·26 × 10⁹ cm. per
second; their kinetic energy is then about ⁴/₃₀ of the kinetic
energy of a weight of one milligram of a substance at a velocity of
one centimetre per second. This energy may seem very small, but, at
least, it is not a magnitude of “inconceivable minuteness,” and it is
sufficient under the conditions given above to produce a visible light
effect. We must here also consider the extreme sensitiveness of the eye.

[Illustration: FIG. 19.—Photograph of paths described
by α-particles (positive helium ions) emitted from a radioactive
substance.]

More practical methods of revealing the effects of the individual
α-particles and of counting them are founded on their very strong
ionization power. By strengthening the ionization power of α-particles,
Rutherford and Geiger were able to make the air in a so-called
ionization chamber so good a conductor that an individual α-particle
caused a deflection in an electrical apparatus, an electrometre.

[Illustration: FIG. 20.—Photograph of the path of a β-particle
(an electron).

(Both 19 and 20 are photographs by C. T. R. Wilson.)]

With a more direct method the English scientist C. T. R. Wilson has
shown the paths of the α-particles by making use of the characteristic
property of ions, that in damp air they attract the neutral water
molecules which then form drops of water with the ions as nuclei. In
air which is completely free of dust and ions the water vapour is not
condensed, even if the temperature is decreased so as to give rise to
supersaturation, but as soon as the air is ionized the vapour condenses
into a fog. When Wilson sent α-particles through air, supersaturated
with water vapour, the vapour condensed into small drops on the ions
produced by the particles; the streaks of fog thus obtained could be
photographed. Fig. 19 shows such a photograph of the paths of a number
of atoms. When a streak of fog ends abruptly it does not mean that the
α-particles have suddenly halted, but that their velocity has decreased
so that they can no longer break the molecules of air to pieces,
producing ions. The paths of the β-particles have been photographed
in the same way, although an electron of the β-particles has a mass
about 7000 times smaller than that of a helium atom; the electron has,
however, a far greater velocity than the helium atom. This velocity
causes the ions to be farther apart, so that each drop of water formed
around the individual ions can appear in the photograph by itself (cf.
Fig. 20).




CHAPTER IV

THE NUCLEAR ATOM


Introduction.

We are now brought face to face with the fundamental question, hardly
touched upon at all in the previous part of this work, namely, that of
the construction and mode of operation of the atomic mechanism itself.
In the first place we must ask: What is the “architecture” of the atom,
that is, what positions do the positive and negative particles take up
with respect to each other, and how many are there of each kind? In
the second place, of what sort are the processes which take place in
an atom, and how can we make them interpret the physical and chemical
properties of the elements? In this chapter we shall keep essentially
to the first question, and consider especially the great contribution
which Rutherford made in 1911 to its answer in his discovery of the
positive atomic nucleus and in the development of what is known as the
Rutherford atomic model or nuclear atom.


Rutherford’s Atom Model.

Rutherford’s discovery was the result of an investigation which, in
its main outlines, was carried out as follows: a dense stream of
α-particles from a powerful radium preparation was sent into a highly
exhausted chamber through a little opening. On a zinc sulphide screen,
placed a little distance behind the opening, there was then produced by
this bombardment of atomic projectiles, a small, sharply defined spot
of light. The opening was next covered by a thin metal plate, which can
be considered as a piece of chain mail formed of densely-packed atoms.
The α-particles, working their way through the atoms, easily traversed
this “piece of mail” because of their great velocity. But now it was
seen that the spot of light broadened out a little and was no longer
sharply limited. From this fact one could conclude that the α-particles
in passing among the many atoms in the metal plate suffered countless,
very small deflections, thus producing a slight spreading of the rays.
It could also be seen that some, though comparatively few, of the
α-particles broke utterly away from the stream, and travelled farther
in new directions, some, indeed, glancing back from the metal plate in
the direction in which they had come (cf. Fig. 21). The situation was
approximately as if one had discharged a quantity of small shot through
a wall of butter, and nearly all the pellets had gone through the
wall in an almost unchanged direction, but that one or two individual
ones had in some apparently uncalled for fashion come travelling back
from the interior of the butter. One might naturally conclude from
this circumstance that here and there in the butter were located some
small, hard, heavy objects, for example, some small pellets with which
some of the projectiles by chance had collided. Accordingly, it seemed
as if there were located in the metal sheet some small hard objects.
These could hardly be the electrons of the metal atoms, because
α-particles, as has been stated before, are helium atoms with a mass
over seven thousand times that of a single electron; and if such an
atom collided with an electron, it would easily push the electron
aside without itself being deviated materially in its path. Hardly any
other possibility remained than to assume that what the α-particles
had collided with was the positive part of the atom, whose mass is of
the same order of magnitude as the mass of the helium atom (cf. Fig.
21). A mathematical investigation showed that the large deflections
were produced because the α-particles in question had passed, on their
way, through a tremendously strong electric field of the kind which
will exist about an electric charge concentrated into a very small
space and acting on other charges according to Coulomb’s Law. When, in
the foregoing, the word “collision” is used, it must not be taken to
mean simply a collision of elastic spheres; rather the two particles
(the α-particle and the positive particle of the metal atom) come so
near to each other in the flight of the former that the very great
electrical forces brought into play cause a significant deflection of
the α-particles from their original course.

[Illustration: FIG. 21.—Tracks of α-particles in the interior
of matter. While 1 and 3 undergo small deflections by collisions with
electrons, 2 is sharply deflected by a positive nucleus.]

Rutherford was thus led to the hypothesis that nearly all of the mass
of the atom is concentrated into a positively charged nucleus, which,
like the electrons, is very small in comparison with the size of the
whole atom; while the rest of the mass is apportioned among a number of
negative electrons which must be assumed to rotate about the nucleus
under the attraction of the latter, just as the planets rotate about
the sun. Under this hypothesis the outer limits of the atom must be
regarded as given by the outermost electron orbits. The assumption
of an atom of this structure makes it at once intelligible why, in
general, the α-particles can travel through the atom without being
deflected materially by the nuclear repulsion, and why the very great
deflections occur as seldom as is indicated by experiment. This latter
circumstance has, on the other hand, no explanation in the atomic model
previously suggested by Lord Kelvin and amplified by J. J. Thomson, in
which the positive electricity was assumed to be distributed over the
whole volume of the atom, while the electrons were supposed to move in
rings at varying distances from the centre of the atom.

[Illustration: FIG. 22.—Photograph of the paths of two α-particles
(positive helium ions). One collides with an atomic nucleus.]

The same characteristic phenomenon made evident in the passage of
α-particles through substances by the investigations of Rutherford
appears in a more direct way in Wilson’s researches discussed on
p. 81. His photographs of the paths of α-particles through air
supersaturated with water vapour (see Fig. 22) show pronounced kinks
in the paths of individual particles. Thus in the figure referred
to, there are shown the paths of two α-particles. One of these is
almost a straight line (with a very slight curvature), while the
other shows a very perceptible deflection as it approaches the
immediate neighbourhood of the nucleus of an atom, and finally a very
abrupt kink; at the latter place it is clear that the α-particle has
penetrated very close to the nucleus. If one examines the picture
more closely, there will be seen a very small fork at the place where
the kink is located. Here the path seems to have divided into two
branches, a shorter and a longer. This leads one at once to suspect
that a collision between two bodies has taken place, and that after the
collision each body has travelled its own path, just as if, to return
to the analogy of the bombardment of the butter wall, one had been
able to drive two pellets out of the butter by shooting in only one.
Or, to take perhaps a more familiar example, when a moving billiard
ball collides at random with a stationary one, after the collision
they both move off in different directions. So, when the α-particle
hits at random the atomic nucleus, both particle and nucleus move off
in different directions; though in this case, since the nucleus has
the much greater mass of the two, it moves more slowly, after the
collision, than the α-particle, and has, therefore, a much shorter
range in the air than the lighter, swifter α-particle. Had the gas in
which the collisions took place been hydrogen, for example, the recoil
paths of the hydrogen nuclei would have been longer than those of
the α-particles, because the mass of the hydrogen nucleus is but one
quarter the mass of the α-particle (helium atom).

The collision experiments on which Rutherford’s theory is founded are
of so direct and decisive a character that one can hardly call it a
theory, but rather a fact, founded on observation, showing conclusively
that the atom is built after the fashion indicated. Continued
researches have amassed a quantity of important facts about atoms.
Thus, Rutherford was able to show that the radius of the nucleus is
of the order of magnitude 10⁻¹² to 10⁻¹³cm. This means really that it
is only when an α-particle approaches so near the centre of an atom
that forces come into play which no longer follow Coulomb’s Law for
the repulsion between two point charges of the same sign (in contrast
to the case in the ordinary deflections of α-particles). It should be
remarked, however, that in the case of the hydrogen nucleus theoretical
considerations give foundation for the assumption that its radius is
really many times smaller than the radius of the electron, which is
some 2000 times lighter; experiments by which this assumption can be
tested are not at hand at present.


The Nuclear Charge; Atomic Number; Atomic Weight.

It is not necessary to have recourse to a new research to determine the
masses of the nuclei of various atoms, because the mass of the nucleus
is for all practical purposes the mass of the atom. Accordingly, if
the mass of the hydrogen nucleus is taken as unity, the atomic mass
is equal to the atomic weight as previously defined. The individual
electrons which accompany the nucleus are so light that their mass
has relatively little influence (within the limits of experimental
accuracy) on the total mass of the atom.

On the other hand, a problem of the greatest importance which
immediately suggests itself is to determine the magnitude of the
positive charge of the nucleus. This naturally must be an integral
multiple of the fundamental quantum of negative electricity, namely,
4·77 × 10⁻¹⁰ electrostatic units, or if we prefer to call this simply
the “unit” charge, then the nuclear charge must be an integer.
Otherwise a neutral atom could not be formed of a nucleus and
electrons, for in a neutral atom the number of negative electrons which
move about the nucleus must be equal to the number of positive charges
in the nucleus. The determination of this number is, accordingly,
equivalent to the settling of the important question, how many
electrons surround the nucleus in the normal neutral state of the atom
of the element in question.

The answer to the question is easiest in the case of the helium atom.
For when this is expelled as an α-particle, it carries, as Rutherford
was able to show, a positive charge of two units—in other words, two
electrons are necessary to change the positive ion into a neutral
atom. At the same time there is every reason to suppose that the
α-particle is simply a helium nucleus deprived of its electrons; it
follows, therefore, that the electron system of the neutral helium atom
consists of two electrons. Since the atomic weight of helium is four,
the number of electrons is consequently one-half the atomic weight.
Rutherford’s investigation of the deflections of α-particles in passing
through various media had already led him to believe that for many
other elements, to a considerable approximation, the nuclear charge
and hence the number of electrons was equal to half the atomic weight.
Hydrogen, of course, must form an exception, since its atomic weight is
unity. _The positive charge on the hydrogen nucleus is one elementary
quantum, and in the neutral state of the atom, only one electron
rotates about it._ Fig. 23 gives a representation of the structure
of the hydrogen atom, and the structures of the two types of hydrogen
ions formed respectively by the loss and gain of an electron. In the
picture, the position of the electron is, of course, arbitrary, and for
the sake of simplicity its path is supposed to be circular.

[Illustration: FIG. 23.—Schematic representation of the nuclear atom.

A, a neutral hydrogen atom; B, a positive, and C, a negative hydrogen
ion; _K_, atomic nuclei; _E_, electrons.]

As has just been indicated, Rutherford’s rule for the number of
electrons is only an approximation. A Dutch physicist, van den Broek,
conceived in the meantime the idea that the number of electrons in
the atom of an element is equal to its order number in the periodic
table (its “atomic number,” as it is now called). Especially through
a systematic investigation of the X-ray spectra characteristic of the
different elements this has proved to be the correct rule. In fact,
using Bragg’s reflection method of X-rays from crystal surfaces (cf. p.
54), the Englishman, Moseley, made in 1914 the far-reaching discovery
that these spectra possess an exceptionally simple structure, which
made it possible in a simple way to attach an order number to each
element (given on p. 23). On the basis of Bohr’s theory, established a
year before, it could be directly proved that this order number must be
identical with the number of positive elementary charges on the nucleus.

The number which formerly indicated simply the position of an
element in the periodic system has thus obtained a profound physical
significance, and in comparison the atomic weight has come to have
but a secondary meaning. The inversion of argon and potassium in the
periodic system (mentioned on p. 21), which seemed to be an exception
to the regularity displayed by the system as a whole, obtains an easy
explanation on the van den Broek rule; for to explain the inversion
we need only assume that potassium has one electron more than argon,
though its atomic weight is less than that of argon. We see at once
that the atomic weight and number of electrons (or what is the same
thing—the nuclear charge) are not directly correlated to each other.
And since the periodic system based on the atomic number represents
the correct arrangement of the elements according to their respective
properties (especially their chemical properties), we are led naturally
to the conclusion that it is the atomic number and not the atomic
weight that determines chemical characteristics.

The conception of the relatively great importance of the atomic number
as compared with the atomic weight has in recent years received
overwhelming support from the researches of Soddy, Fajans, Russell,
Hevesy and others who have discovered the existence of so-called
_isotope elements_ (from the Greek _isos_ = same, and
_topos_ = place), substances with different nuclear masses (atomic
weights) and different radioactive properties (if there are any),
but with the same nuclear charge, the same number of electrons and,
consequently, occupying the same place in the periodic system. Two such
isotopes are practically equivalent in all their chemical properties as
well as in most of their physical characteristics. One of the oldest
examples of isotopes is provided by ordinary lead with the atomic
weight 207·2 and the substance found in pitchblende with the atomic
weight 206, but identical, chemically, with ordinary lead. This latter
form of lead has already been referred to on p. 79 as the end product
of radioactive disintegrations, and hence it is sometimes called radium
lead.

By his investigations of canal rays the English physicist Aston has
just recently shown that many substances which have always been assumed
to be simple elements, are in reality mixtures of isotopes. The atomic
weight of chlorine determined in the usual way is 35·5, but in the
discharge tube two kinds of chlorine atoms appear, having atomic
weights 35 and 37 respectively; and it must be assumed that these two
kinds of chlorine are present in all the compounds of chlorine known
on the earth in the ratio of, roughly, three to one. To separate such
mixtures into their constituent parts is extremely difficult, precisely
because the constituents have identical properties apart from a small
difference in density, which stands in direct connection with the
atomic weight. Such a separation was first carried out successfully
by the Danish chemist, Brønsted, in collaboration with the Hungarian
chemist, Hevesy (1921). These two scientists were able to separate
a large quantity of mercury of density 13·5955 into two portions of
slightly different densities. All the different isotopes of which
mercury is a mixture were, indeed, not wholly separated; they were
represented in the two portions in different proportions. Thus, in one
of the first attempts, the density of the one part was 13·5986 and of
the other 13·5920 (at 0° C).

It is a perfectly reasonable supposition that it is the electron system
which determines the external properties of the atom, that is, those
properties which depend on the interplay of two or more atoms. For
the electron, rotating about the nucleus at a considerable distance,
separates, so to speak, the nucleus from the surrounding space, and
must therefore be assumed to be the organ which connects the atom with
the rest of the universe. One might also expect the structure of the
electron system to depend wholly on the nuclear charge, _i.e._ on
the atomic number and not on the mass of the nucleus, since it is the
nuclear electrical attraction which holds the electrons in their orbits
and not the relatively insignificant gravitational attraction.

It thus becomes intelligible that the properties of the elements can
be divided into two sharply defined classes, namely: (1) _properties
of the nucleus_, and (2) _properties of the electron system_
in the atom. The credit for first recognizing the sharp distinction
between these two classes, a distinction fundamental for a detailed
study of the atom, is due to Niels Bohr.

The properties of the nucleus determine—(_a_) the radioactive
processes, or explosions of the nucleus, and related processes;
(_b_) collisions, where two nuclei approach extremely near to
each other; and (_c_) weight which, as mentioned above, stands
in direct connection with atomic weight. The properties of the
electron system are, on the other hand, the determining factors in
all other physical and chemical activities, and, as has been stated,
are functions, we may say, of the atomic number of the given element.
The Bohr theory may be said to concern itself with the chemical and
physical properties of the atom with the exception of those which have
to do with the nucleus. We shall consequently devote our attention in
the next chapters to the electron system. But before turning to this we
shall dwell a little further upon the atomic nucleus.


The Structure of the Nucleus.

That the nucleus is not an elementary indivisible particle but a system
of particles, is clearly shown by the radioactive processes in which
α-particles and β-particles (electrons) are shot out of the nuclei of
radioactive elements. Bohr was the first to see clearly that not only
the α-particles emitted in such cases come from the nucleus, but that
the β-particles also have their source there. There is now no doubt
that, in addition to the outer electrons of the atom, which are the
determining factor in the atomic number, there must also be, in the
radioactive substances at any rate, special nuclear electrons which
lead a more hidden existence in the interior of the nucleus. One can
easily understand that isotopes may result as products of radioactive
disintegration. For example, let us suppose that a nucleus emits first
an α-particle (_i.e._ a helium nucleus with two positive charges),
and thereafter sends out two electrons, each with its negative charge,
in two new disintegrations. The nuclear charge in the resultant atom
will then obviously be the same as before, because the loss of the two
electrons exactly neutralizes that of the α-particle. But the atomic
weight will be diminished by four units (i.e. the weight of the helium
nucleus, remembering also that the electrons have but very negligible
masses). Among the radioactive substances are recognized many examples
of isotope elements, with atomic weights differing precisely by four.
The radioactive element uranium is the element with the greatest
atomic weight (238), and atomic number (92), and consequently with the
greatest nuclear charge. Almost all the other radioactive substances
are those with high atomic numbers in the periodic system. The cause
of radioactivity must be sought in the hypothesis that the nuclei
of the radioactive elements are very complicated systems with small
stability, and therefore break down rather easily into less complicated
and more stable systems with the emission of some of their constituent
particles; the corpuscular rays thus produced possess a considerable
amount of kinetic energy.

Accordingly, by analogy, the nuclei of the non-radioactive elements
may be assumed to be composed of nuclear electrons and positive
particles; hydrogen alone excepted. The simplest assumption is that the
hydrogen nucleus is the real quantum or atom of positive electricity,
just as the electron is the atom of negative electricity. On this
theory all substances are built up of two kinds only of fundamental
particles, namely, hydrogen nuclei and electrons. That these particles
may themselves consist of constituent parts is, of course, an open
possibility, but such speculation is beyond our experience up to the
present. In every nucleus there are more positive hydrogen nuclei
than there are negative electrons, so that the nucleus has a residual
positive charge of a magnitude equal to the difference between the
number of hydrogen nuclei and nuclear electrons.

If we now pass from hydrogen which has the atomic weight, atomic number
and nuclear charge of unity, we next encounter helium with the atomic
weight 4, atomic number and nuclear charge 2. The helium nucleus should
therefore consist of 4 hydrogen nuclei, which would together account
for the atomic weight of 4. But since these represent 4 positive
charges, there must also be present in the nucleus 2 negative electrons
to make the resultant nuclear charge equal to 2. We could indeed hardly
conceive of a system composed of 4 positive hydrogen nuclei alone; for
the forces of repulsion would soon drive the separate parts asunder.
The two electrons can, so to speak, serve to hold the system together.
Fig. 24 gives a rough representation of the helium atom. It must be
carefully noted that the picture is purely schematic and the distances
arbitrary. The helium nucleus, composed of 4 hydrogen nuclei and 2
electrons, seems to possess extreme stability, and it is not improbable
that helium nuclei occur as higher units in the structure of the nuclei
of not only the radioactive substances but also the other elements.
We shall perhaps be very near the truth in saying that all nuclei
are built up of combinations of hydrogen nuclei, helium nuclei and
electrons.

[Illustration: FIG. 24.—Schematic representation of a helium atom. _K_,
nuclear system with four hydrogen nuclei and two nuclear electrons;
_E_, electrons in the outer electron system.]

In nitrogen, with the atomic weight 14 and atomic number 7, the nucleus
should consist of 14 hydrogen nuclei (with 12 of them compounded,
perhaps, into 3 helium nuclei) and 7 nuclear electrons, reducing the
resultant positive nuclear charge from 14 to 7. Uranium, with atomic
number 92 and atomic weight 238, should have a nucleus composed of 238
hydrogen nuclei and 146 electrons, and so on for the others. We see at
once that the conception of the nucleus here propounded leads us back
to the old hypothesis of Prout (see p. 15) that all atomic weights
should be integral multiples of that of hydrogen. This hypothesis
apparently disagreed with atomic weight measurements, but the
isotope researches have vanquished this difficulty; thus it has been
mentioned before that chlorine with an atomic weight of 35·5 appears
to be a mixture of isotopes with atomic weights 35 and 37, and other
cases have a similar explanation. Yet the rule cannot be wholly and
completely exact. For, in the first place, the mass of the electrons
must contribute something, though this contribution is far too small
to be measured. But there is also a second matter which plays a part
here. This is the law enunciated by Einstein in his relativity theory,
that every increase or decrease in the energy of a body is correlated
with an increase or decrease in the mass of the body, proportional to
the energy change. We must, therefore, expect that the masses of the
various atomic nuclei will depend not only on the number of hydrogen
nuclei (and electrons), but also on the energy represented in the
attractions and repulsions between the particles of the system, and
in their mutual motions, or the energy which comes into play in the
formation and disintegration of nuclear systems. This is presumably
closely connected, although in a way which is not clearly understood,
with the fact, that if the atomic weights of the elements are to come
out integers, that of hydrogen must not be taken as 1 but as 1·008;
that is, the atomic weight unit must be chosen a little smaller than
the atomic weight of hydrogen (cf. table, p. 23).


Transformation of Elements and Liberation of Atomic Energy.

We shall now treat very briefly two questions which have profoundly
interested many people, because they are concerned with possible
practical applications of our new knowledge of atoms.

The first question is this: Can one not, from this knowledge, bring
about the transformation of one element into another? In answering
this, it can, of course, be said immediately that among the radioactive
substances such transformations are constantly taking place without
human interference, and we certainly have no right to state offhand
that it will be impossible for man ever to bring about such a
transformation artificially. For example, if we could succeed in
getting one hydrogen nucleus loose from the nucleus of mercury, the
latter would thereby be changed into a gold nucleus. Such a thing
is not only conceivable, but in the last few years it has become a
reality, though, to be sure, not with the substances here mentioned.
In 1919 Rutherford, by bombarding nitrogen (N = 14) with α-particles,
was able to knock loose some hydrogen nuclei from the nitrogen nucleus;
perhaps he succeeded thereby in changing the nitrogen nuclei into
carbon nuclei (C = 12) by the breaking off of two hydrogen nuclei from
each nitrogen nucleus. But to disintegrate very few nitrogen nuclei,
Rutherford had to employ a formidable bombardment with hundreds of
thousands of projectiles (α-particles); and even if he had ended with
gold instead of carbon, this would have been, from the economic point
of view, a very foolish way of making gold; and at the present time we
know of no other artificial method for the transformation of elements.
That Rutherford’s investigation has, in any case, extraordinarily great
interest and scientific value is another matter.

The second question is whether one cannot liberate and utilize the
energy latent in the interior of the atom. This question, which was
suggested in the first instance by the discovery of radium, has
recently attracted considerable attention because of reports that,
according to Einstein’s relativity theory, one gram of any substance
by virtue of its mass alone must contain a quantity of energy equal
to that produced by the burning of 3000 tons of coal. The meaning of
this statement is this: it has already been mentioned that according
to the relativity theory a decrease in the energy of a body brings
about a decrease in its mass; it is immaterial in what form the energy
is given up, whether as heat, elastic oscillations, or the like; all
that is said is, that to a certain decrease in mass, will correspond
a perfectly definite emission of energy in some form. If we now could
imagine the whole mass of one gram of a substance to be “destroyed”
(_i.e._ caused to disappear utterly as a physical substance),
and to reappear as heat energy, for example, then we could compute
from the known relation between mass and energy, that the heat energy
thus brought about would be equivalent to that obtained by the burning
of 3000 tons of coal. But in order that all this energy should be
developed, even the hydrogen nuclei and the electrons would have to be
“destroyed,” and no phenomenon is known, supporting the supposition
that such a “destruction” of the fundamental particles of a substance
is possible, or that it is possible to transform these particles into
other types of energy. A thought like this must rather be stamped as
fantasy, the origin of which is to be found in a misunderstanding of a
purely scientific mode of expression.

The case is essentially different with those quantities of energy which
must be assumed to be freed or absorbed in the transformation of one
nuclear system into another, that is, in elemental transformations.
Though these are far smaller in amount, the radioactive processes
indicate that they are not wholly to be despised. For one gram of
radium will upon complete disintegration to non-radioactive material
give off as much energy as is equivalent to 460 kg. of coal. But even
here we must confess that it will take about 1700 years for only half
of the radium to be transformed. It is not at all impossible that
other elemental transformations might lead to just as great energy
developments as appear in the disintegration of radioactive substances.
Let us imagine that four hydrogen nuclei, which together have a mass
of 4 × 1·008 = 4·032, and two electrons could join together to form
a helium nucleus with atomic weight very close to 4. This process
would thus result in a loss of mass which must be assumed to appear in
another form of energy. The amount of energy obtainable in this way
from one gram of hydrogen would be considerably more than that given
off by the disintegration of one gram of radium.

There can hardly exist any doubt that in nature there occur not only
disintegrations, but also (perhaps in the interior of the stars)
building-up processes in which compound nuclei result from simple
ones. It is therefore natural to suppose that by exerting on hydrogen
exceptional conditions of temperature, pressure, electrical changes,
etc., we could succeed by experiments here on earth in forming helium
from it with the development of considerable energy. But at the same
time it is very likely that even under favourable circumstances
such a process would take place with very great slowness, because
the formation of a helium nucleus might well be a very infrequent
occurrence; it would probably be the result of a certain succession
of collisions between hydrogen nuclei and electrons, a combination
whose probability of occurrence in a certain number of collisions is
infinitely less than the probability of winning the largest prize in a
lottery with the same number of chances. Nature has time enough to wait
for “wins,” while mankind unfortunately has not. We know concerning
the disintegration of the radioactive substances that it is of the
character here indicated; of the great number of atoms to be found
in a very small mass of a radioactive substance, now one explodes
and now another. But why fortune should pick out one particular atom
is as difficult to understand as why in a lottery one particular
number should prove to be the lucky one rather than any other. Our
only understanding of the whole matter rests on the law of averages,
or probability as we may call it. We know that of a billion radium
atoms (10¹²) on the average thirteen explode every second; and even
if in any single collection of a billion a few more or a few less may
explode, the average of thirteen per second per billion will always
be maintained in dealing with larger and larger numbers of atoms, as,
for example, with a thousand billion or a million billion. For other
radioactive substances we get wholly different averages for the number
of atoms disintegrating per second; but in no case are we able to
penetrate into the inner character of the process of disintegration
itself. And what holds true of the radioactive substances will also
hold true probably for elemental changes of all kinds; Rutherford with
his hundreds of thousands of α-particle projectiles was able to make
sure of but a few lucky “shots.” The whole matter must at this stage be
looked upon as governed wholly by chance.

One interested in speculating on what would happen if it were possible
to bring about artificially a transformation of elements propagating
itself from atom to atom with the liberation of energy, would find
food for serious thought in the fact that the quantities of energy
which would be liberated in this way would be many, many times greater
than those which we now know of in connection with chemical processes.
There is then offered the possibility of explosions more extensive and
more violent than any which the mind can now conceive. The idea has
been suggested that the world catastrophes represented in the heavens
by the sudden appearance of very bright stars may be the result of
such a release of sub-atomic energy, brought about perhaps by the
“super-wisdom” of the unlucky inhabitants themselves. But this is, of
course, mere fanciful conjecture.

It seems clear, however, that we need have no fear that in
investigating the problem of atomic energy we are releasing forces
which we cannot control, because we can at present see no way to
liberate the energy of atomic nuclei beyond that which Nature herself
provides, to say nothing of a practical solution of the energy problem.
The time has certainly not yet come for the technician to follow in the
theoretical investigator’s footsteps in this branch of science. One
hesitates, however, to predict what the future may bring forth.

Interesting and significant as is the insight which Rutherford and
others have opened up into the inner workings of the nucleus, the
study of the electron system of the atom bears more intimately upon
the various branches of physical and chemical science, and hence
presents greater possibilities of attaining, in a less remote future,
to discoveries of practical significance.




CHAPTER V

THE BOHR THEORY OF THE HYDROGEN SPECTRUM


The Nuclear Atom and Electrodynamics.

Even if Rutherford had not yet succeeded in giving a complete answer to
the first of the questions propounded in the previous chapter, namely,
that concerning the positions of the positive and negative particles of
the atom, one might at any rate hope that his general explanation of
the structure of the atom—that is, the division into the nucleus and
surrounding electrons, and the determination of the number of electrons
in the atoms of the various elements—would furnish a good foundation
for the answer to the second question about the connection between the
atomic processes and the physical and chemical properties of matter.
But in the beginning this seemed so far from being true that it
appeared almost hopeless to find a solution of the problem of the atom
in this way.

We shall best understand the meaning of this if we consider the
simplest of the elemental atoms, namely, the atom of hydrogen with its
positive nucleus and its one electron revolving about the nucleus.
How could it be possible to explain from such a simple structure the
many sharp spectral lines given by the Balmer-Ritz formula (p. 57)?
As has previously been mentioned, the classical electron theory seemed
to demand a very complicated atomic structure for the explanation of
these lines. According to the electron theory, the atoms may be likened
to stringed instruments which are capable of emitting a great number
of tones, and in these atoms the electrons are naturally supposed
to correspond to the “strings.” But the hydrogen atom has only one
electron, and it hardly seems credible that in a mass of hydrogen the
individual atoms would be tuned for different “tones,” with definite
frequencies of vibration.

Now, it certainly cannot be concluded from the analogy with the
stringed instrument that a single electron can emit light of only a
single frequency at one time, corresponding to a single spectral line.
For a plucked string will, as we know, give rise to a simple tone only
if it vibrates in a very definite and particularly simple way; in
general it will emit a compound sound which may be conceived as made up
of a “fundamental” and its so-called “overtones,” or “harmonies” whose
frequencies are 2, 3, ... times that of the fundamental (_i.e._
integral multiples of the latter). These overtones may arise even
separately because the string, instead of vibrating as a whole,
may be divided into 2, 3, ... equally long vibrating parts, giving
respectively 2, 3, ... times as great frequencies of vibration. We call
such vibrations “harmonic oscillations.” The simultaneous existence of
these different modes of oscillation of the string may be thought of in
the same way as the simultaneous existence of wave systems of different
wave-lengths on the surface of water. Corresponding to the possibility
of resolving the motion of the string into its “harmonic components,”
the compound sound waves produced by the string can be resolved by
resonators (cf. p. 44) into tones possessing the frequencies of these
components.

According to the laws of electrodynamics the situation with the
electron revolving about the hydrogen nucleus might be expected to
be somewhat similar to that described above in connection with the
vibrating string. If the orbit of the electron were a circle, it
should emit into the ether electromagnetic waves of a single definite
wave-length and corresponding frequency, ν, equal to ω, the frequency
of rotation of the electron in its orbit; that is, the number of
revolutions per second. But just as a planet under the attraction
of the sun, varying inversely as the square of the distance, moves
in an ellipse with the sun at one focus, so the electron, under the
attraction of the positive nucleus, which also follows the inverse
square law, will in general be able to move in an ellipse with the
nucleus at one focus. The electromagnetic waves which are emitted
from such a moving electron may on the electron theory be considered
as composed of light waves corresponding to a series of harmonic
oscillations with the frequencies:

  ν₁ = ω, ν₂ = 2ω, ν₃ = 3ω ... and so on,

where ω, as before, is the frequency of revolution of the electron.
According as the actual orbit deviates more or less from a circle, the
frequencies ν₂, ν₃ ... will appear stronger or weaker in the compound
light waves emitted. But the actual distribution of spectral lines
in the real hydrogen spectrum presents no likeness whatever to this
distribution of frequencies.

From this it is evident that no agreement can be reached between
the classical electron theory on the one hand and the Rutherford
atom model on the other. Indeed, the disagreement between the two is
really far more fundamental than has just been indicated. According to
Lorentz’s explanation of the emission of light waves, the electrons
in a substance (see again p. 75) should have certain equilibrium
positions, and should oscillate about these when pushed out of them by
some external impulse. The energy which is given to the electron by
such an impulse is expended in the emission of the light waves and is
thus transformed into radiation energy in the emitted light, while the
electrons fall to rest again unless they receive in the meantime a new
impulse. We can get an understanding of what these impulses in various
cases may be by thinking of them, in the case of a glowing solid, for
example, as due to the collisions of the molecules; or in the case of
the glowing gas in a discharge tube, from the collisions of electrons
and ions. The oscillating system represented by the electron (the
“oscillator”) will possess under these circumstances great analogy with
a string which after being set into vibration by a stroke gradually
comes back to rest, while the energy expended in the stroke is emitted
in the form of sound waves. Although the vibrations of the string
become weaker after a while, the period of the vibrations will remain
unchanged; the string vibrations like pendulum oscillations have an
invariable period, and the same will be the case with the frequency
of the electron if the force which pulls it back into its equilibrium
position is directly proportional to the displacement from this
position (the “harmonic motion” force).

Rutherford’s atomic model is, however, a system of a kind wholly
different from the “oscillator” of the electron theory. The one
revolving hydrogen electron will find a position of “rest” or
equilibrium only in the nucleus itself, and if it once becomes united
with the latter it will not easily escape; it will then probably
become a nuclear electron, and such a process would be nothing less
than a transformation of elements (see p. 79). On the other hand,
it follows necessarily from the fundamental laws of electrodynamics
that the revolving electron must emit radiation energy, and, because
of the resultant loss of energy, must gradually shrink its path and
approach nearer the nucleus. But since the nuclear attraction on the
electron is inversely proportional to the square of the distance,
the period of revolution will be gradually decreased and hence the
frequency of revolution ω, and the frequency of the emitted light will
gradually increase. The spectral lines emitted from a great number
of atoms should, accordingly, be distributed evenly from the red end
of the spectrum to the violet, or in other words there should be no
line spectrum at all. It is thus clear that Rutherford’s model was not
only unable to account for the number and distribution of the spectral
lines; but that with the application of the ordinary electrodynamic
laws it was quite impossible to account for the existence even of
spectral lines. Indeed, it had to be admitted that an electrodynamic
system of the kind indicated was mechanically unstable and therefore
an impossible system; and this would apply not merely to the hydrogen
atom, but to all nuclear atoms with positive nuclei and systems of
revolving electrons.

However one looks at the matter, there thus appears to be an
irremediable disagreement between the Rutherford theory of atomic
structure and the fundamental electrodynamic assumptions of Lorentz’s
theory of electrons. As has been emphasized, however, Rutherford
founded his atomic model on such a direct and clear-cut investigation
that any other interpretation of his experiments is hardly possible. If
the result to which he attained could not be reconciled with the theory
of electrodynamics, then, as has been said, this was so much the worse
for the theory.

It could, however, hardly be expected that physicists in general
would be very willing to give up the conceptions of electrodynamics,
even if its basis was being seriously damaged by Rutherford’s atomic
projectiles. Surmounted by its crowning glory—the Lorentz electron
theory—the classical electrodynamics stood at the beginning of the
present century a structure both solid and spacious, uniting in its
construction nearly all the physical knowledge accumulated during the
centuries, optics as well as electricity, thermodynamics as well as
mechanics. With the collapse of such a structure one might well feel
that physics had suddenly become homeless.


The Quantum Theory.

In a field completely different from the above the conclusion
had also been reached that there was something wrong with the
classical electrodynamics. Through his very extended speculations on
thermodynamic equilibrium in the radiation process, Planck (1900) had
reached the point of view expressed in his _quantum theory_, which
was just as irreconcilable with the fundamental electrodynamic laws as
the Rutherford atom.

A complete representation of this theory would lead us too far; we
shall merely give a short account of the foundations on which it rests.

By a black body is generally understood a body which absorbs all the
light falling upon it, and, accordingly, can reflect none. Physicists,
however, denote by the term “perfect black body” in an extended sense,
a body which at all temperatures absorbs all the radiation falling upon
it, whether this be in the form of visible light, or ultra-violet, or
infra-red radiation. From considerations which were developed some
sixty years ago by Kirchhoff, it can be stated that the radiation which
is emitted by such a body when heated does not depend on the nature
of the body but merely on its temperature, and that it is greater
than that emitted by any other body whatever at the same temperature.
Such radiation is called temperature radiation or sometimes “black”
radiation, though the latter term is apt to be misleading, since a
“perfect black body” emitting black radiation may glow at white heat.
It may be of interest to note here the fundamental law deduced by
Kirchhoff, which may best be illustrated by saying that good absorbers
of radiation are also good radiators. An instructive experiment
illustrating this is performed by painting a figure in lampblack on a
piece of white porcelain. The lampblack surface is clearly a better
absorber of radiant energy than the white porcelain. When the whole
is heated in a blast flame, the lampblack figure glows much more
brightly than the surrounding porcelain, thus showing that at the same
temperature it is also the better radiator. Following the same law we
conclude that highly reflecting bodies are not good radiators, a fact
that has practical significance in house heating. The perfect black
body, then, being the best absorber of radiation, is also the best
radiator.

In actual practice no body is absolutely black. Even a body coated
with lampblack reflects about 10 per cent. of the light waves incident
on it. The Danish physicist Christiansen remarked long ago that a
real black body could be produced only if an arrangement could be
made whereby the incident waves could be reflected several times
in succession before finally being emitted. To take the case of
lampblack, three such reflections would reduce the re-emission from
the lampblacked body to only ⁴/₁₀₀₀ of the radiation initially falling
on it. This type of black body was finally realized by making a cavity
in an oven having as its only opening a very small peep-hole, and
keeping the temperature of the wall of the oven uniform. If a ray is
sent into the cavity through the peep-hole, it will become, so to
speak, captured, because, when once inside it will suffer countless
reflections from the walls of the cavity, having more and more of its
energy absorbed at each reflection. Very little of the radiation thus
entering will ever get out again, and consequently such a body will
act very much like a “perfect black body” according to the theoretical
definition above. Accordingly, the radiation which is emitted from such
a glowing cavity through the peep-hole will be black or practically
black radiation.

The cavity itself will be criss-crossed in all directions by radiation
emitted from one part of the inner surface of the cavity and absorbed
by (and partially reflected from) other parts of the surface. When
the walls of the cavity are kept at a fixed and uniform temperature
there will automatically be produced a state of equilibrium in which
every cubic centimetre of the cavity will contain a definite quantity
of radiation energy, dependent only on the temperature of the walls.
Further, in the equilibrium state the radiation energy will be
distributed in a perfectly definite way (dependent as before on the
temperature only) among the various types of radiation corresponding to
different wave-lengths and frequencies. If there is too much radiation
of one kind and too little of another, the walls of the cavity will
absorb more of the first kind than they emit, and emit more of the
second kind than they absorb, and so the state will vary until the
right proportion for equilibrium is attained.

This distribution of the radiation over widely differing wave-lengths
can be investigated by examining spectroscopically the light emitted
from the peep-hole in the cavity. Then, by means of a bolometer or some
other instrument, the heat development in the different portions of the
spectrum can be measured. At a temperature of 1500° C., for example,
one will find that the maximum energy is represented for rays of
wave-length in the close vicinity of 1·8 μ, _i.e._ in the extreme
infra-red. If the temperature is raised, the energy maximum travels off
in the direction of the violet end of the spectrum; if the temperature
is lowered, it will move farther down into the infra-red.

It is also possible to make a theoretical calculation of the
distribution of energy in the spectrum of the black-body radiation
at a given temperature. But the results obtained do not agree with
experiment. The English physicists, Rayleigh and Jeans, developed
on the basis of the classical electrodynamic laws and by apparently
convincing arguments a distribution law according to which actual
radiation equilibrium becomes impossible, since if it were true the
energy in the radiation would tend more and more to go over to the
region of short wave-lengths and high frequencies, and this shifting
would apparently go on indefinitely. The theory thus leads to results
which are not only in disagreement with experiment, but which must be
looked upon as extremely unreasonable in themselves.

Planck, however, had vanquished these difficulties and had obtained a
radiation law in agreement with experiment by introducing an extremely
curious hypothesis. Like Lorentz, he thought of radiation as produced
through the medium of small vibrating systems or oscillators, which
could emit or absorb rays of a definite frequency ν. But while,
according to the Lorentz theory and the classical electrodynamics,
radiation can be emitted in infinitely small quantities (_i.e._
small without limit), Planck assumed that an oscillator can emit and
absorb energy only in certain definite quantities called _quanta_,
where the fundamental quantum of radiation is dependent on the
frequency of the oscillator, varying directly with the latter. If
thus we denote the smallest quantity of energy which an oscillator of
frequency ν can emit or absorb by E, then we can write

  E = _h_ν,

where _h_ is a definite constant fixed for all frequencies.
Accordingly the cavity can receive radiation energy of frequency ν
from the radiating oscillators in its wall, or transfer energy to
these in no smaller quantity than _h_ν. The total energy of that
kind emitted or absorbed at any given time will always be an integral
multiple of _h_ν. Oscillators with a frequency 1½ times as great
will emit energy in quanta which are 1½ times larger, and so on.

The quantity _h_ is independent, not only of the wave-length, but
also of the temperature and nature of the emitting body. This constant,
the so-called _Planck constant_, is thus a universal constant.
If one uses the “absolute” units of length, mass and time (see table,
p. 210), its value comes out as 6·54 × 10⁻²⁷. For the frequency 750
× 10¹² vibrations per second, corresponding to the extreme violet in
the visible spectrum, the Planck energy quantum thus becomes about 5
× 10⁻¹² erg, or 3·69 × 10⁻¹⁹ foot-pounds (note that the “erg” is the
“absolute” unit of work, or the amount of work done when a body is
moved through a distance of 1 cm. by a force of one dyne acting in the
direction of the motion, while the “foot-pound” is the work done when
a force of 1 pound moves a body 1 foot in the direction of the force).
For light belonging to the red end of the spectrum, the energy is about
half as great. If we pass, however, to the highest frequencies and the
shortest wave-lengths which are known, namely those corresponding to
the “hardest” (_i.e._ most penetrating) γ-rays (see p. 78), we
meet with energy quanta which are a million times larger, _i.e._ 2
× 10⁻⁶ erg, although they are still very small compared to any amount
of energy measurable mechanically.

This remarkable theory of quanta, which in the hands of Planck still
possessed a rather abstract character, proved under Einstein’s
ingenious treatment to have the greatest significance in many
problems which, like heat radiation, had provided physicists with
many difficulties. For by assuming that energy in general could only
be given up and taken in in quanta, certain facts about the specific
heats of bodies could be accounted for—facts which the older physics
had proved powerless to explain. The Planck energy quanta, as Einstein
showed, could also explain in a very direct and satisfactory way the
_photoelectric effect_, as it is called. This effect consists
in the freeing of electrons from a metal plate which ultra-violet
rays are allowed to strike. The maximum velocity with which these
electrons are propelled from the plate is found to be independent
of the intensity of the incident light, but dependent simply on the
frequency of the radiation. Careful measurements have indeed shown,
as Einstein predicted, that the incident light really does utilize an
energy quantum _h_ν to free each electron and give it velocity
(cf. p. 172). Of the different methods which nowadays are at hand, the
photoelectric effect constitutes one of the best means of determining
the value of _h_. It has been applied for that purpose by
Millikan, to whose ingenious experiments the most accurate direct
determination of _h_ is actually due.

All this lay completely outside the laws of electrodynamics, and
pointed to the existence of unknown and more fundamental laws. But,
for the time being, physicists had to be satisfied merely with a
recognition of the fact that these mysterious energy quanta play a very
significant part in many phenomena.

A decade ago physics, as regards radiation problems, was in a very
unsettled state; with four separate branches of knowledge, each of
which seemed firm and well-founded enough in itself, but which had no
common connecting link, indeed, were even to some extent inconsistent
with each other. The first of these was the classical electrodynamics
surmounted by the felicitous electron theory of Lorentz and Larmor. The
second was the empirical knowledge of the spectra resting on the work
of Balmer, Ritz and Rydberg. The third was Rutherford’s nuclear atom
model. And the fourth was Planck’s quantum theory of heat radiation.
It was quite evident that progress in the theory of radiation and the
structure of the atom was hopeless as long as these four points of view
remained uncorrelated.


Main Outlines of the Bohr Theory.

Such was the situation when, in 1913, Bohr published his atomic
theory, in which he was able with great ingenuity to unite the
nuclear atom, the Balmer-Ritz formula and the quantum theory. As far
as electrodynamics is concerned, the impossibility of retaining that
in its classical form was presented in a much clearer way than ever
before. But, as will presently be evident, the Bohr theory has a very
definite connection with the classical theory, and Bohr’s attempts to
preserve and develop this connection have proved to be of the greatest
significance for his theory. In spite of the fundamental rupture with
the old ideas, the Bohr theory strives to absorb all that is useful in
the classical point of view.

At the head of the theory appear the two fundamental hypotheses or
postulates on the properties of the atom.

_The first postulate states that for each atom or atomic system
there exists a number of definite states of motion, called “stationary
states,” in which the atom (or atomic system) can exist without
radiating energy. A finite change in the energy content of the atom can
take place only in a process in which the atom passes completely from
one stationary state to another.

The second postulate states that if such a transition takes place with
the emission or absorption of electromagnetic light waves, these waves
will have a definite frequency, the magnitude of which is determined by
the change in the energy content of the atom. If we denote the change
in energy by_ E _and the frequency by ν we may write_

                     E
  E = _h_ν, or ν = -----
                    _h_

_where h is the Planck constant. In consequence of the second
postulate the emission as well as the absorption of energy by the atom
always takes place in quanta._

The two postulates say nothing concerning the nature of the motion in
the stationary states. In the applications, however, a connection with
the Rutherford atomic model is established. Confining our attention
first to the hydrogen atom, the system with which we are concerned
consists, accordingly, of a positive nucleus and one electron revolving
about it. The various states of motion which the electron can assume
in virtue of the first postulate are a series of orbits at different
distances from the nucleus. In each of these “stationary orbits” the
electron follows the general mechanical laws of motion; _i.e._
under the nuclear attraction which is inversely proportional to
the square of the distance, the electron describes an ellipse with
the nucleus at one focus, as has previously been stated; but in
contradiction with the classical electrodynamics it will emit no
radiation while moving in this orbit. Fig. 25 shows a series of these
orbits, to which the numbers 1, 2, 3, 4 have been attached, and which
for simplicity are represented as circular.

[Illustration: FIG. 25.—The Bohr model of the hydrogen atom in
the simplified form (with circles instead of ellipses).]

If the electron passes from an outer orbit to an inner one; for
example, if it goes from number 4 to number 2, or from number 2 to
number 1, the electric force which attracts it to the nucleus will do
work just as the force of gravity does work when a stone falls to the
ground. A part of this work is used to increase the kinetic energy
of the electron, making its velocity in the inner orbit greater than
in the outer, but the rest of the work is transformed into radiation
energy which is emitted from the atom in the form of monochromatic
light. In consequence of the second postulate the frequency of the
emitted radiation is proportional to the energy loss. When the electron
has reached the innermost orbit (the one denoted by 1 in the figure),
it cannot get any nearer the nucleus and hence cannot emit any more
radiation unless it first is impelled to pass from its inner orbit to
an outer orbit again by the absorption of external energy sufficient
to bring about this change. Once in the outer orbit again, it is in a
state to produce radiation by falling in a second time. The innermost
orbit represents thus the electron’s equilibrium state, and corresponds
to the _normal state of the atom_.

If we try to illustrate the matter with an analogy from the theory
of sound, we can do so by comparing the atom not with a stringed
instrument, but with a hypothetical musical instrument of a wholly
different kind. Let us imagine that we have placed one over another
and concentrically a series of circular discs of progressively smaller
radii, and let us suppose that a small sphere can move around any
one of these without friction and without emitting sound. In such a
motion the system may be said to be in a “stationary state.” Sooner
or later the sphere may fall from the first disc on to one lower down
and continue to roll around on the second, having emitted a sound, let
us assume, by its fall. By passing thus from one stationary state to
another it loses a quantity of energy equal to the work which would be
necessary to raise it again to the disc previously occupied, and to
bring it back to the original state of motion. We can assume that the
energy which is lost in the fall reappears in a sound wave emitted by
the instrument, and that the pitch of the sound emitted is proportional
to the energy sent out. If, moreover, we imagine that the lowermost
disc is grooved in such a way that the sphere cannot fall farther, then
this fanciful instrument can provide a very rough analogy with the Bohr
atom. We must beware, however, of stretching the analogy farther than
is here indicated.

It must be specially emphasized here that the frequency of the sound
emitted in the above example has no connection with the frequency of
revolution of the sphere. In the Bohr atom, likewise, the frequency
of revolution ω of the electron in its stationary orbit has no direct
connection with the frequency of the radiation emitted when the
electron passes from this orbit to another. This is a very surprising
break with all previous views on radiation, a break whose revolutionary
character should not be under-estimated. But, however unreasonable it
might seem to give up the direct connection between the revolutional
frequency and the radiation frequency, it was absolutely necessary if
the Rutherford atomic model was to be preserved. And as we shall now
see, the new point of view of the Bohr theory leads naturally to an
interpretation of the Balmer-Ritz formula, which had previously not
been connected with any other physical theory.

The quantity of energy E, which the atom gives up when the electron
passes from an outer to an inner orbit, or which, conversely, is taken
in when the electron passes from an inner to an outer orbit, may, as
has been indicated, be regarded as the difference between the energy
contents of the atom in the two stationary states. This difference
may be expressed in the following way. Let us imagine that we eject
the electron from a given orbit (_e.g._ No. 2 in the diagram)
so that it is sent to “infinity,” or, in other words, is sent so
far away from the nucleus that the attraction of the latter becomes
negligible. To bring about this removal of the electron from the atom
demands a certain amount of energy, which we can call the _ionizing
work_ corresponding to the stationary orbit in question. We may here
designate it as A₂. To eject the electron from the orbit No. 4 will
demand a smaller amount of ionizing work, A₄. The difference A₂ - A₄
is accordingly the work which must be done to transfer the electron
from the orbit No. 2 to the orbit No. 4. This is, however, exactly
equal to the quantity E of energy which will be emitted as light when
the electron passes from orbit No. 4 to orbit No. 2. If we call the
frequency of this light ν, then from the relations E = _h_ν and E
= A₂ - A₄, we have

  _h_ν = A₂ - A₄

If, now, in place of this specific example using the stationary orbits
2 and 4 we take any two orbits designated by the numbers _n″_
(for the inner) and _n′_ (for the outer), we can write for the
frequency of the radiation emitted for a transition between these
arbitrary states

     _h_ν = Aₙ˶ - Aₙˊ or

         Aₙ˶      Aₙˊ
  ν =  ----- -  -----
        _h_      _h_


We have now reached the point where we ought to bring in the
Balmer-Ritz formula for the distribution of the lines in the hydrogen
spectrum. This formula may be written (see p. 59)

          K        K
  _ν_ = ------ - -----
         _n″_²   _n′_²

We can now see very clearly the similarity between the formula derived
from the spectrum investigations and that derived from the two Bohr
postulates. In both formulæ the frequency appears as the difference
between two terms which are characterized in both cases by two integral
numbers, in the first formula, numbers denoting two stationary orbits
in the Bohr model for hydrogen, and in the second the two numbers which
in the Balmer-Ritz formula for the hydrogen spectrum characterize,
respectively, a series and one of the lines of the series. To obtain
complete agreement we have merely to equate the corresponding terms in
the two formulæ. Thus we have for any arbitrary integer _n_

    Aₙ       K
  -----  =  ----   or
   _h_      _n_²

        _h_K
  Aₙ = ------
        _n_²

For the innermost stationary orbit, for which _n_ = 1, the
ionizing work A₁ will accordingly be equal to the product of the
constants _h_ and K of Planck and Balmer respectively; and for the
orbits No. 2, No. 3, No. 4, etc., the values will be respectively ¼,
¹/₉, ¹/₁₆, etc., of this product. From the charges on the nucleus and
the electron, which are both equal to the elementary quantum _e_
of electricity (see p. 90), and from the ionizing energy for a given
orbit we can now find by the use of simple mechanical considerations
the radius of the orbit. If we denote the radii of the orbits 1, 2,
3 ... by _a_₁, _a_₂, _a_₃ ..., we then obtain for the
diameters 2_a_₁, 2_a_₂, 2_a_₃ ... the values 2_a_₁
= 1·056 × 10⁻⁸ cm. (or approximately 2_a_₁ = 10⁻⁸ cm.), 2_a_₂
= 4 × 10⁻⁸ cm., 2_a_₃ = 9 × 10⁻⁸ cm., etc. It is seen that the
radii of the orbits are in the proportion 1, 4, 9 ..., or in other
words the squares of the integers which determine the orbit numbers. It
is in this proportion that the circles in Fig. 25 are drawn. We must
remember, however, that we have here for the moment been thinking of
the orbits as circles, while in reality they must in general be assumed
to be ellipses. The foregoing considerations will, however, still hold
with the single change that 2_a_ₙ will now mean, instead of the
diameter of a circle, the major axis of an ellipse.

Let us return to the formulæ

        Aₙ″    Aₙ′
  ν = ----- - ----
       _h_    _h_

            K       K
  and ν = ------ - -----
           _n″_²   _n′_²

Here _n″_ denotes in the first formula the index number for the
_inner_ of the two orbits between which the transition is supposed
to take place, while in the second formula _n″_ denotes a definite
series in the hydrogen spectrum. If _n″_ is 2 while _n′_
takes on the values 3, 4, 5 ... ∞ then in the Bohr model of the
hydrogen atom this corresponds to a series of transitions _to_ the
orbit No. 2 _from_ the orbits 3, 4, 5 ..., while in the hydrogen
spectrum this corresponds to the lines in the Balmer series, namely,
the red line (Hα) corresponding to the transition 3-2, the blue-green
line (Hβ) to 4-2, the violet line (Hγ) to 5-2 and so on. If we now put
_n″_ = 1 while _n′_ takes the values 2, 3, 4 ..., we get in
the atom transitions to the orbit No. 1 from the orbits No. 2, 3, 4
..., corresponding in the spectrum to what is called the Lyman series
in the ultra-violet (named after the American physicist Lyman, who
has carried on extensive researches in the ultra-violet region of the
spectrum). Thus every line in the hydrogen spectrum is represented by a
transition between two definite stationary states in the hydrogen atom,
since this transition will give the frequency corresponding to the line
in question.

At first sight this would seem perhaps to be such an extraordinary
satisfactory result that it would prove an overwhelming witness in
favour of the Bohr theory. A little more careful thought, on the other
hand, would perhaps cause a complete reversion from enthusiasm and
lead some to say that the whole thing has not the slightest value,
because the stationary states were so chosen that agreement might be
made with the Balmer-Ritz formula. This last consideration, indeed,
states the truth in so far that the agreement between the formula
and the theory, at least as developed here up to this point, is of a
purely formal nature. In the Bohr postulates the frequencies of the
emitted radiation are determined by a difference between two of a
series of energy quantities, characterizing the stationary states,
just as in the Balmer-Ritz formula they appear as a difference between
two of a series of terms (K, K/4, K/9, ...) each characterized by
its integer. Now by characterizing the quantities of energy in the
stationary states by a series of integers (in itself a wholly arbitrary
procedure) complete agreement between the Bohr stationary states idea
and the spectral formulæ can be attained. It is not even necessary to
introduce the Rutherford atomic model to attain this end. By bringing
in this specific model, one might join the new theory to the knowledge
already gained of the atomic structure, and, so to speak, crystallize
the hitherto undefined or only vaguely defined stationary states into
more definite form as revolution in certain concrete orbits. This would
then lead to a more comprehensive conception of atomic structure.
But the theory unfortunately would still be rather arbitrary, since
there would seem to be no justification for picking out certain fixed
orbits with definite diameters or major axes to play a special rôle.
One cannot wonder then that many scientists considered the Bohr theory
unacceptable, or at any rate were inclined to look upon it simply as an
arbitrary, unreasonable conception which really explained nothing.

Naturally, Bohr himself clearly recognized the formal nature of the
agreement between the Balmer-Ritz formula and his postulates. But Bohr
was the first to see that the quantum theory afforded the possibility
of bringing about such an agreement, and he saw, moreover, that the
agreement was not merely fortuitous, but contained within it something
really fundamental, on which one could build further. That atomic
processes on his theory took on an unreasonable character (compared
with the classical theory) was nothing to worry about, for Bohr had
come to the clear recognition that it was completely impossible to
understand from known laws the Planck-Einstein “quantum radiation,”
or to deduce the properties of the spectrum from the Rutherford atom
alone. He therefore saw that his theory was really not introducing new
improbabilities, but was only causing the fundamental nature of the
contradictions which had previously hindered development in this field
to appear in a clearer light.

But in addition to this the choice of the dimensions of the stationary
states was by no means so arbitrary as might appear in the foregoing.
In his first presentation of the theory of the hydrogen spectrum, Bohr
had derived his results from certain considerations connected with the
quantum theory—considerations of a purely formal nature, indeed, just
as those developed in the preceding, but leading to agreement with the
spectral formulæ. He, moreover, called attention to the fact that the
values obtained for the orbital dimensions were of the same order of
magnitude as those which could be expected on wholly different grounds.
The diameter of the innermost orbit, _i.e._, that which defines
the outer limit of the atom in the normal state, was found to be, as
has been noted above, about 10⁻⁸ cm., _i.e._, of the same order
of magnitude as the values obtained for the diameters of molecules
on the kinetic theory of gases (see p. 27). The stationary states
corresponding to very high quantum numbers one could expect to meet
only when hydrogen was very attenuated, for otherwise there could be
no room for the large orbits. We note that the 32nd orbit must have
a diameter 32² (or over 1000 times) as great as the innermost orbit.
Since, now, lines with high number in a hydrogen series correspond on
the Bohr theory to transitions from orbits of high number to an inner
orbit, it became understandable why only comparatively few lines of
the Balmer series are ordinarily observed in the discharge tube, while
many more lines are observed in the spectra of certain stars. For in
such stars the possibility is left open for hydrogen to exist in a
very attenuated state, and yet in such large masses that the lines in
question can become strong enough for observation. In fact, one must
assume that in a great mass of hydrogen a very large number of atoms
send out simultaneously light of the wave-length corresponding to
one line. For the ionizing work, _i.e._, the work necessary to
eject the electron completely from the normal state and thus make the
atom into a positive ion, the Bohr theory gives a value of the same
order of magnitude as the so-called “ionization potentials” which have
been found by experiment for various gases. An exact correspondence
between theory and experiment could for hydrogen not be attained with
certainty, because the hydrogen atoms in hydrogen gas under ordinary
conditions always appear united in molecules.

In his very first paper, however, Bohr had studied Balmer’s formula
also from another point of view, and had derived in this way an
expression for the Rydberg constant K which agreed with experiment.
These considerations have reference to the above-mentioned connection
of the theory with the classical theory of electrodynamics.

Such a connection had previously been known to exist in the fact
that, for long wave-lengths, the radiation formula of Planck reduces
practically to the Rayleigh Jeans Law which can be derived from
electrodynamics. This is related to the fact that when ν is small (long
wave-lengths), the energy quantum _h_ν is very small, and hence
the character of the radiation emitted will approach more and more
nearly to a continuous “unquantized” radiation. One might then expect
that the Bohr theory also should lead in the limit of long wave-lengths
and small frequencies to results resembling those of the ordinary
electrodynamic theory of the radiation process. On the Bohr theory
we get the long wave-lengths for transitions between two stationary
states of high numbers (numbers which also differ little from each
other). Thus suppose _n_ is a very large number. Then the
transition from the orbit _n_ to the orbit _n_ - 1 will give
rise to radiation of great wave-length. For in this case Aₙ and Aₙ₋₁
differ very little, and accordingly _h_ν is very small, as must
ν be also. According to the electrodynamic theory of radiation, the
revolving electron should emit radiation whose frequency is equal to
the electron’s frequency of revolution. According to the Bohr theory it
is impossible to fulfil this condition exactly, since radiation results
from a transition between two stationary orbits in each of which the
electron has a distinct revolutional frequency. But if _n_ is a
large number, the difference between the frequencies of revolution ωₙ
and ωₙ₋₁ for the two orbits _n_ and _n_ - 1, respectively,
becomes very small; for example, for _n_ = 100, it is only 3 per
cent. For a certain high value of _n_, then, the frequency of the
emitted radiation can therefore be _approximately_ equal to the
frequency of revolution of the electron in both the two orbits, between
which the transition takes place. But even if this proved correct for
values of n about 100, one could not be sure beforehand whether it
would work out right for still larger values of _n_, for example,
1000.

In order to investigate this latter point we must look into the
formulæ for the revolutional frequency ω in a stationary orbit and for
the radiation frequency ν. Since, according to the Bohr theory, we
can apply the usual laws of mechanics to revolution in a stationary
orbit, it is an easy matter to find an expression for ω. From a short
mathematical calculation we can deduce that ω = R/_n_³, where R is
the frequency of revolution for the first orbit (_n_ = 1). We find
ν, on the other hand, by substituting in the Balmer-Ritz formula the
numbers _n_ and _n_ - 1, and a simple calculation shows that
for great values of _n_, the expression for ν will approach in the
limit the simple form ν = 2K/_n_&sup3. For large orbit numbers,
ν accordingly varies as ω, _i.e._, inversely proportional to the
third power of n, and by equating R and 2K, we find that the values for
ν and ω tend more and more to become equal.

In this way the value of K, the Balmer constant, may be computed. It is
found that

              _m_
  K = 2π²_e_⁴-----
              _h_³

where _e_ is the charge on the electron, _m_ the mass of the
electron, and _h_ is Planck’s constant. Upon the substitution
of the experimental values for these quantities, a value of K is
determined which agrees with the experimental value (from the spectral
lines investigation) of 3·29 × 10¹⁵ within the accuracy to which
_e_, _m_ and _h_ are obtainable. This agreement has
from the very first been a significant support for the Bohr theory.

One might now object that we have here considered radiation due to a
transition between two successive stationary states, _e.g._, No.
100 and No. 99, or the like (a “single jump” we might call it). On the
other hand, for transitions between states whose numbers differ by 2,
3, 4 or more (as in a double jump, or a triple jump) the agreement
found above will wholly disappear, and doubt be cast on its value.
For in such cases of high orbit numbers the frequency of revolution
will remain approximately the same even for a difference of 2, 3, 4 or
more in orbit number; but the radiation frequency for a double jump
will be nearly twice that for a single jump, while that for a triple
jump will be nearly three times, etc. Accordingly, for approximately
the same revolutional frequency ω we shall have in these cases for
the radiation frequency very nearly ν₁ = ω, ν₂ = 2ω, ν₃ = 3ω, etc. We
must, however, remember that when the orbit in the stationary states is
not a circle, but an ellipse (as must in general be assumed to be the
case), the classical electrodynamics require that the electron emits
besides the “fundamental” radiation of frequency ν₁ = ω, the overtones
of frequencies ν₂ = 2ω, ν₃ = 3ω .... We then also here see the outward
similarity between the Bohr theory and the classical electrodynamics.
We may say that the radiation of frequency ν, produced by a single
jump, _corresponds_ to the fundamental harmonic component in the
motion of the electron, while the radiation of frequency ν₂, emitted by
a double jump, corresponds to the first overtone, etc.

The similarity is, however, only of a formal nature, since the
processes of radiation, according to the Bohr theory, are of
quite different nature than would be expected from the laws of
electrodynamics. In order to show how fundamental is the difference,
even where the similarity seems greatest, let us assume that we have
a mass of hydrogen with a very large number of atoms in orbits,
corresponding to very high numbers, and that the revolutional frequency
can practically be set equal to the same quantity ω. There may take
place transitions between orbits with the difference 1, 2, 3 ... in
number, and as the result of these different transitions we shall
find, by spectrum analysis, in the emitted radiation frequencies which
are practically ω, 2ω, 3ω, etc. According to the radiation theory of
electrodynamics we should also get these frequencies and the spectral
lines corresponding to them. It must, however, be assumed that they are
produced by the simultaneous emission from every individual radiating
atom of a fundamental and a series of overtones. According to the Bohr
theory, on the other hand, each individual radiating atom at a given
time emits only one definite line corresponding to a definite frequency
(monochromatic radiation).

We can now realize that the Bohr theory takes us into unknown regions,
that it points towards fundamental laws of nature about which we
previously had no ideas. The fundamental postulates of electrodynamics,
which for a long time seemed to be the fundamental laws of the physical
world itself, by which there was hope of explaining the laws of
mechanics and of light and of everything else, were disclosed by the
Bohr theory as merely superficial and only applicable to large-scale
phenomena. The apparently exact account of the activities of nature,
obtained by the formulæ of electrodynamics, often veiled processes of
a nature entirely different from those the formulæ were supposed to
describe.

One might then express some surprise that the laws of electrodynamics
could have been obtained at all and interpreted as the most fundamental
of all laws. It must, however, be remembered that the Bohr theory for
large wave-lengths, _i.e._, the slow oscillations, leads to a
formal agreement with electrodynamics. It must, moreover, be remembered
that the laws of electrodynamics are established on the basis of
large-scale electric and magnetic processes which do not refer to
the activities of separate atoms, but in which very great numbers of
electrons are carried in a certain direction in the electric conductors
or vibrate in oscillations which are extremely slow compared with light
oscillations. Moreover, the observed laws, even if they can account
for many phenomena in light, early showed their inability to explain
the nature of the spectrum and many other problems connected with
the detailed structure of matter. Indeed the more this structure was
studied, the greater became the difficulties, the stronger the evidence
that the solution cannot be obtained in the classical way.

If we ask whether Bohr has succeeded in setting up new fundamental
laws, which can be quantitatively formulated, to replace the laws of
electrodynamics and to be used in the derivation of everything that
happens in the atom and so in all nature, this question must receive
a negative answer. The motion of the electron in a given stationary
state may, at any rate to a considerable extent, be calculated by the
laws of mechanics. We do not know, however, why certain orbits are,
in this way, preferred over others, nor why the electrons jump from
outer to inner orbits, nor why they sometimes go from one stationary
orbit to the next and sometimes jump over one or more orbits, nor why
they cannot come any closer to the nucleus than the innermost orbit,
nor why, in these transitions, they emit radiation of a frequency
determined according to the rules mentioned.

It must not be forgotten that in science we must always be patient
with the question “Why?” We can never get to the bottom of things.
On account of the nature of the problem, answers cannot be given
to the questions why the smallest material particles (for the
time being hydrogen nuclei and electrons)—the elementary physical
individuals—exist, or why the fundamental laws for their mutual
relationships—the most elementary relationships existing between
them—are of this or that nature; a satisfactory answer would
necessarily refer to something even more elementary. We cannot claim
more than a complete _description_ of the relative positions and
motions of the fundamental particles and of the laws governing their
mutual action and their interplay with the ether.

If we examine our knowledge of the atomic processes in the light
of this ideal we are tempted, however, to consider it as boundless
ignorance. We are inconceivably far from being able to give a
description of the atomic mechanism, such as would enable us to follow,
for example, an electron from place to place during its entire motion,
or to consider the stationary states as links in the whole instead of
isolated “gifts from above.” During the transition from one stationary
state to another we have no knowledge at all of the existence of the
electron, indeed we do not even know whether it exists at that time
or whether it perhaps is dissolved in the ether to be re-formed
in a new stationary state. But even if we turn aside from such a
paradoxical consideration, it must be recognized that we do not know
what path the electron follows between two stationary orbits nor how
long a time the transition takes. As has been done in this book, the
transition is often denoted as a jump, and many are inclined to believe
that the electron in its entire journey from a distant outer orbit to
the innermost spends the greatest part of the time in the stationary
orbits, while each transition takes but an infinitesimally short time.
This, however, in itself does not follow from the theory, nor is it
implied in the expression “the stationary states.” These states may in
a certain sense be considered as way stations; but when we ask whether
an electron stays long in the station, or whether the stationary state
is simply a transfer point where the electron changes its method of
travelling so that the frequency of its radiation is changed, these are
other matters, and we cannot here go into the considerations connected
with them.

To get an idea of some of the difficulties inherent in the attempt to
make concrete pictures of the nature of the processes, let us again
consider the analogy between the Bohr atom of hydrogen and a special
kind of musical instrument in which sounds are produced by the fall
of a small sphere between discs at various heights (see p. 120). It
will be most natural here to think of the sounds as developed by the
sphere when it hits the lower disc, and to think of the tones of higher
pitch as given by the harder blows, corresponding to the larger energy
(determinative of the pitch) released by the fall. We can, however,
by no means transfer such a picture to the atomic model. For in the
latter we cannot think of the stationary state as a material thing
which the electron can hit, and it is also unreasonable to imagine
that the radiation is not emitted until the moment when the transition
is over and the electron has arrived in its new stationary state. We
must, on the contrary, assume that the emission of radiation takes
place during the _whole_ transition, whether the latter consumes
a shorter or longer time. If it were the case that a transition always
took place between two successive stationary states, it would then be
possible to use the musical instrument to illustrate the matter. Let
us denote the discs from the lowest one up with the numbers 1, 2, 3,
... corresponding to the stationary states 1, 2, 3, ... and for the
moment consider a fall from disc 6 to disc 5. We can now imagine that
the space between the two discs is in some way tuned for a definite
note. Thus we might place between the discs a series of sheets of paper
having such intervals between them that the sphere in its fall strikes
their edges at equal intervals of time, _e.g._, ¹/₁₀₀ second.
The disturbance then set up will produce a sound with the frequency
100 vibrations per second. If the distance between the discs 5 and 4
is double that between 6 and 5, the sphere in the fall from 5 to 4
will lose double the energy lost in the descent from 6 to 5, and will
therefore emit a note of double frequency. The sheets of paper in the
space between 5 and 4 must then be packed more tightly than between
6 and 5. And so the space between any two discs may thus be said to
have its own particular classification or “tuning.” In analogy with
this we might think of the space about a hydrogen nucleus divided by
the stationary states into sections each with its own “tuning.” But
apart from the intrinsic peculiarity of such an arrangement and the
particular difficulties it will meet in trying to explain the more
complicated phenomena to be mentioned later, the one fact that the
electron in a transition from one stationary state to another can
jump over one or more intervening stationary orbits, makes such a
representation impossible. If the sphere in the given example could
fall from disc 6 to disc 4, it should during the whole descent emit
a note of higher pitch than in the descent from 6 to 5. But this
could not possibly take place, if the space from 6 to 5, which must
be traversed _en route_ to 4, is tuned for a lower note. The
same consideration applies to the hydrogen atom. Naturally it is not
impossible to continue the effort to illustrate the matter in some
concrete manner (one might, for example, imagine separate channels each
with its own particular tuning between the same two discs). But in all
these attempts the situation must become more and more complicated
rather than more simple.

On the whole it is very difficult to understand how a hydrogen atom,
where the electron makes a transition from orbit 6 to orbit 4, can
during the entire transition emit a radiation with a frequency
different from that when the electron goes from orbit 6 to orbit 5.
Although it seems as if the two electrons in making the transition are
at first under identical conditions, still, nevertheless, the one which
is going to orbit 4 emits from the first a radiation different from
that emitted by the one going to orbit 5. Even from the very beginning
the electron seems to arrange its conduct according to the goal of its
motion and also according to future events. But such a gift is wont to
be the privilege of thinking beings that can anticipate certain future
occurrences. The inanimate objects of physics should observe causal
laws in a more direct manner, _i.e._, allow their conduct to be
determined by their previous states and the contemporaneous influences
on them.

There is a difficulty of a similar nature in the fact that from the
same stationary orbit the electron sometimes starts for a single jump,
another time for a double jump, and so on. From certain considerations
it is often possible to propound laws for the probability of the
different jumps, so that for a great quantity of atoms it is possible
to calculate the strengths (intensities) of the corresponding spectral
lines. But we can no more give the reason why one given electron at a
given time determines to make a double jump while another decides to
make a single jump or not to jump at all, than we can say why a certain
radium nucleus among many explodes at a given moment (cf. p. 102). This
similarity between the occurrence of radiation processes on the Bohr
theory and of the radioactive processes has especially been emphasized
by Einstein.

It must, by no means, be said that the causal laws do not hold for the
atomic processes, but the hints given here indicate how difficult it
will be to reach an understanding—in the usual sense—of these processes
and consequently of the processes of physics in general. There is much
that might indicate that, on the whole, it is impossible to obtain a
consistent picture of atomic processes in space and time with the help
of the motions of the nuclei and the electrons and the variations in
the state of the ether, and with the application of such fundamental
conceptions of physics as mass, electric charge and energy.

Even if this were the case, it does not follow that a comprehensive
description in time and space of the physical processes is impossible
in principle; but the hope of attaining such a description must perhaps
be allied to the representation of “physical individuals” or material
particles of an even lower order of magnitude than the smallest
particles now known—electrons and hydrogen nuclei—and to ideas of
more fundamental nature than those now known; we are here outside our
present sphere of experience.

From all the above remarks it would be very easy to get the impression
that the Bohr theory, while it gives us a glimpse into depths
previously unsuspected, at the same time leads us into a fog, where it
is impossible to find the way. This is very far from being the case. On
the contrary, it has thrown new light on a host of physical phenomena
of different kinds so that they now appear in a coherence previously
unattainable. That this light is not deceptive follows from the fact
that the theory, which has been gradually developed by Bohr and many
other investigators, has made it possible to predict and to account
for many phenomena with remarkable accuracy and in complete agreement
with experimental observation. The fundamental concepts are, on the
one hand, the stationary states, where the usual laws of mechanics can
be applied (although only within certain limits), and, on the other,
the “quantum rule” for transitions between the states. But at the very
beginning it has been necessary in many respects to grope in the dark,
guided in part by the experimental results and in part by various
assumptions, often very arbitrary.

For Bohr himself, a most important guide has been the so-called
_correspondence principle_, which expresses the previously
mentioned connection with the classical electrodynamics. It is
difficult to explain in what it consists, because it cannot be
expressed in exact quantitative laws, and it is, on this account,
also difficult to apply. In Bohr’s hands it has been extraordinarily
fruitful in the most varied fields; while other more definite and more
easily applicable rules of guidance have indeed given important results
in individual cases, they have shown their limitations by failing
in other cases. We can here merely indicate what the correspondence
principle is.

As has been said (cf. p. 130), it has been found that in the limiting
region (sufficiently low frequencies) where the Bohr theory and the
classical electrodynamics are merged in their outward features, a
series of frequencies ν₁, ν₂, ν₃ for monochromatic radiation, emitted
by different atoms in the single jumps, double jumps, etc., of the
electrons, are equal to the frequencies ω, 2ω, 3ω ... which, according
to the laws of electrodynamics, are contained in each of these atoms
respectively as fundamental and the first, second ... overtones in the
motion of the electron. Farther away from this region the two sets of
frequencies are no longer equally large, but it is easy to understand,
from the foregoing, the meaning of the statement that, for example,
the radiation of a triple jump with the frequency ν₃ “corresponds”
to the second overtone 3ω in the revolution of the electron. It is
this correspondence which Bohr traces back to the regions where
there is even a great difference in two successive orbits and where
the frequency produced by a transition between these orbits is very
different from the frequencies of revolution in the two orbits or
their overtones. He expresses himself as follows: “The probability for
the occurrence of single, double, triple jumps, etc., is conditioned
by the presence in the motion of the atom of the different constituent
harmonic vibrations having the frequency of the fundamental, first
overtone, second overtone, etc., respectively.”

In order to understand how this “correspondence,” apparently so
indefinite, can be used to derive important results, we shall give
an illustration. Let us assume that the mechanical theory for the
revolution of an electron in the hydrogen atom had led to the result
that the orbits of the electrons always had to be circles. According to
the laws of electrodynamics, the motion of the electron would in this
case never give any overtones, and, according to the correspondence
principle, there could not appear among the frequencies emitted by
hydrogen any which would correspond to the overtones, _i.e._,
there would not be any double jumps, triple jumps, etc., produced,
but the only transitions would be those between successive stationary
orbits. The investigation of the spectrum shows, however, that multiple
jumps occur as well as single jumps, and this fact may be taken as
evidence that the orbits in the hydrogen atom are not usually circles.
Let us next assume that, instead, we had obtained the result that the
orbits of the electrons are always ellipses of a certain quite definite
eccentricity, corresponding to certain definite ratios in intensity
between the overtones and the fundamentals; that, for instance, the
intensity of the classical radiation due to the first overtone is in
all states of motion always one-half that due to the fundamental, the
intensity due to the second overtone always one-third that due to the
fundamental, etc. Then the radiation actually emitted should, according
to the correspondence principle, be such that the intensities of the
lines corresponding to the double and triple jumps, which start from a
given stationary state, are respectively one-half and one-third of the
intensity corresponding to a single jump from the same state.

By these examples we can obtain an idea of how the correspondence
principle may in certain cases account for various facts, as to what
spectral lines cannot be expected to appear at all, although they would
be given by a particular transition, and concerning the distribution
of intensities in those which really appear. The illustration given
above, however, has really nothing much to do with actual problems, and
objections may be raised to the rough way in which the illustration has
been handled. The correspondence principle has its particular province
in more complicated electron motions than those which appear in the
unperturbed hydrogen atom—motions which, unlike the simple elliptical
motion, are not composed of a series of harmonic oscillations (ω, 2ω,
3ω ...) but may be considered as compounded of oscillations whose
frequencies have other ratios. The correspondence principle has, in
such cases, given rise to important discoveries and predictions which
agree completely with the observations.

We have dwelt thus long upon the difficult correspondence principle,
because it is one of Bohr’s deepest thoughts and chief guides. It has
made possible a more consistent presentation of the whole theory, and
it bids fair to remain the keystone of its future development. But from
these general considerations we shall now proceed to more special
phases of the problem and examine one of the first great triumphs in
which the theory showed its ability to lead the way where previously
there had been no path.


The False Hydrogen Spectrum.

In 1897 the American astronomer, Pickering, discovered in the spectrum
of a star, in addition to the usual lines given by the Balmer series, a
series of lines each of which lay about midway between two lines of the
Balmer series; the frequencies of these lines could be represented by a
formula which was very similar to the Balmer formula; it was necessary
merely to substitute _n_ = 3½, 4½, 5½, etc., in the formula on
p. 57 instead of _n_ = 3, 4, 5, etc. It was later discovered
that in many stars there was a line corresponding to _n″_ = ³/₂
or _n′_ = 2 in the usual Balmer-Ritz formula (p. 59). It was
considered that these must be hydrogen lines, and that the spectral
formula for this element should properly be written

              1            1
  ν = K ----------  -  ---------
        ( _n″_/2 )²    ( _n′_/2 )²

where _n″_ and _n′_ can assume integral values. This was
done since it was not to be believed that the spectral properties of
chemically different elements could be so similar. This view was very
much strengthened when Fowler, in 1912, discovered the Pickering lines
in the light from a vacuum tube containing a mixture of hydrogen and
helium. It could not quite be understood, however, why the new lines
did not in general appear in the hydrogen spectrum.

According to the Bohr theory for the hydrogen spectrum it was
impossible—except by giving up the agreement (cf. p. 129) with
electrodynamics in the region of high orbit numbers—to attribute to the
hydrogen atom the emission of lines corresponding to a formula where
the whole numbers were halved. The formula given above might, however,
also be written as

        (  1        1   )
  ν = 4K(------ - ------).
        ( _n″_²   _n′_² )

If the earlier calculations had been carried out a little more
generally, _i.e._, if instead of equating the nuclear charge with
1 elementary electric quantum _e_, as in hydrogen, it had been
equated with N_e_ where N is an integer, then the frequency might
have been written as

         (   1      1    )
  ν = N²K(------ - ------).
         ( _n″_²   _n′_² )

This formula is evidently the same as that just given when N equals
2. Now we know that helium has the atomic number and nuclear charge 2
(cf. p. 90); a normal neutral helium atom has two electrons and it is,
therefore, very different from a hydrogen atom. If, however, a helium
atom has lost one electron and therefore has become a positive ion
with one charge, it is a system like the hydrogen atom with only one
single electron moving about the nucleus. It differs in its “outer”
characteristics from the hydrogen atom only in having a nuclear charge
twice as great, _i.e._ its spectral formula must be given with
N = 2, or N² = 4. The formula for the supposed hydrogen lines would
consequently fit the case of a helium atom which has lost an electron.
Bohr was aware of this, and he therefore suggested that the lines in
question were due, not to hydrogen, but to helium.

At first all the authorities in the field of spectroscopy were against
this view; but most of the doubt was dispelled when Evans showed that
the lines could be produced in a vacuum tube where there was only
helium with not a trace of hydrogen.

In a letter to _Nature_ in September 1913, Fowler objected to
the Bohr theory on the ground that the disputed line-formula did not
exactly correspond to the formula with 4K, but that there was a slight
disagreement. Bohr’s answer was immediate. He called attention to the
fact that—since temporarily he had sought only a first approximation—in
his calculations he had taken the mass of the nucleus to be infinite
in comparison to the mass of the electron, so that the nucleus could
be considered exactly at the focus of the ellipse described by the
electron. In reality, he said, it must be assumed that nucleus and
electron move about their common centre of gravity, just as in the
motion in the solar system it must be assumed that not the centre of
the sun, but the centre of gravity of the entire system remains fixed.
This motion of the nucleus leads to the introduction of a factor M/(M +
_m_) in the expression for the constant K given on p. 129, where
M is the mass of the nucleus and _m_ that of the electron, which
in hydrogen is ¹⁄₁₈₃₅ that of the nucleus. In helium, M is four times
as large as in hydrogen, so that the given factor here has a slightly
different value. The difference in the values for K for the hydrogen
and for the helium spectrum which was found by Fowler, is 0·04 per
cent., which agrees exactly with the theoretical difference.

Bohr thus turned Fowler’s objection into a strong argument in favour of
the theory.


The Introduction of more than one Quantum Number.

During the first years after 1913, Bohr was practically alone in
working out his theory, at that time still assailed by many, and
in showing its application to many problems. In 1916, however, the
theorists in other countries, led by the well-known Munich professor,
Sommerfeld, began to associate themselves with the Bohr theory, and
their investigations gave rise to much essential progress. We shall
here mention some of the most important contributions.

In the theory for the hydrogen spectrum propounded above, it was
assumed that we had to do with a single series of stationary orbits,
each characterized by its quantum number. But as shown by theoretical
investigations each of the stationary orbits must, when more detail is
asked for, also be indicated by an additional quantum number.

[Illustration: FIG. 26.—A compound electron motion produced by the very
rapid rotation of an elliptical orbit.]

This is closely connected with the fact that the motion of the electron
is not quite so simple as previously assumed. We have assumed that
the electron moves about the nucleus just as a planet moves about the
sun (according to Kepler’s Laws), in an ellipse with the sun at one
focus, since the electron is influenced by an attraction inversely
proportional to the square of the distance, just as the planets are
attracted by the sun according to Newton’s Law. We must, however,
remember that we are here concerned with the electric attraction which
at a given distance is determined, not by mass, but by the electric
charges in question. If the latter remain unchanged, while the mass
of the electron varies, the motion will be changed, because the same
force has less effect upon a greater mass. According to the Einstein
principle of relativity, the mass of the electrons, in accordance with
ideas expounded long ago by J. J. Thomson, will not be constant, but
to a certain extent depend upon the velocity, which will vary from
place to place, when the orbit is an ellipse. As a result of this, the
motion becomes a central motion of more general nature than a Kepler
ellipse. Since the influence of the change of mass is very small, the
orbit can still be considered as an approximate Kepler ellipse; but the
major axis will slowly rotate in the plane of the orbit. In reality,
the orbit will not therefore be closed, but will have the character
which is shown in Fig. 26; this, however, corresponds to a much more
rapid rotation of the major axis than that which actually takes place
in the hydrogen atom, where—even in case of the swiftest rotation—the
electron will revolve about 40,000 times round the nucleus at the same
time as the major axis turns round once.

If the electron moves in a fixed Kepler ellipse, the energy content of
the atom will be determined by the major axis of the ellipse only. If
these axes for the stationary states with quantum numbers 1, 2, 3 ...
are respectively denoted by 2_a_₁, 2_a_₂, 2_a_₃ ... the
frequency, for instance, in the transition from No. 3 to No. 2—since
it is determined by the loss of energy—will be the same whether the
orbits are circles or ellipses. If, on the other hand, the electron
moves in an ellipse which itself rotates slowly, the energy content, as
can be shown mathematically, will depend not only upon the major axis
of the ellipse, but also to a slight degree upon its eccentricity, or,
in other words, on its minor axis. Then in the transition 3-2 we shall
get different energy losses and consequently different frequencies,
according as the ellipse is more or less elongated. If it were the case
that the eccentricity of the ellipses for a certain quantum number
could take arbitrary values, then in the transition between two numbers
we could get frequencies which may take any value within a certain
small interval, _i.e._, a mass of hydrogen with its great quantity
of atoms would give diffuse spectral lines, _i.e._ lines which are
broadened over a small continuous spectral interval. This is, however,
not the case; but long before the appearance of the Bohr theory it
had been discovered that the hydrogen lines, which we hitherto have
considered as single, possess what is called a _fine structure_.
With a spectroscopic apparatus of high resolving power each line can be
separated into two lying very close to each other. This fine structure
can now be explained by the fact that in a stationary state with
quantum number 3 and major axis of the orbit 2_a_₃, for instance,
the eccentricity of the orbit has neither one single definite value
nor all possible values, but, on the contrary, it has several discrete
values of definite magnitude, to which there correspond slightly
different but definite values of the energy content of the atom. It is
now possible to designate the series of stationary orbits, which have
the major axis 2_a_₃ with the principal quantum number 3, with
subscripts giving the _auxiliary quantum number_ for stationary
orbits corresponding to the different eccentricities, so that the
series is known as 3₁, 3₂, 3₃. Instead of a single line corresponding
to the transition 3-2, there are then obtained several spectral lines
lying closely together and corresponding to transitions such as 3₃-2₂,
3₂-2₁, etc. By theoretical considerations, requiring considerable
mathematical qualifications, but of essentially the same formal nature
as those Bohr had originally applied to the determination of the
stationary orbits in hydrogen, Sommerfeld was led to certain formal
quantum rules which permit the fixing of the stationary states of the
hydrogen atom corresponding to such a double set of quantum numbers.
The results he obtained as regards the fine structure of the hydrogen
lines agree with observation inside the limit of experimental error.

Although Sommerfeld’s methods have also been very fruitful when applied
to the spectra of other elements, they were still of a purely formal
and rather arbitrary nature; it is, therefore, of great importance that
the Leiden professor, Ehrenfest, and Bohr succeeded later in handling
the problem from a more fundamental point of view, Bohr making use of
the correspondence principle previously mentioned. It should be said
here, by way of suggestion, that Bohr used the fact that the motions of
the electrons are not simple periodic but “multiple periodic.” We see
this most simply if we think of the revolution of the electrons in the
elliptical orbit as representing one period, and the rotation for the
major axis of the ellipse as representing a second period.

[Illustration: FIG. 27.—The model of the hydrogen atom with stationary
orbits corresponding to principal quantum numbers and auxiliary quantum
numbers.]

Fig. 27 shows a number of the possible stationary orbits in the
hydrogen atom according to Sommerfeld’s theory; for the sake of
simplicity the orbits are drawn as completely closed ellipses. If we
examine, for instance, the orbits with principal quantum number 4,
we have here three more or less elongated ellipses, 4₁, 4₂, 4₃, and
the circle 4₄; in all of them the major axis has the same length,
and the length of the major axis is to that of the minor axis as the
principal quantum number is to the auxiliary quantum number (for the
circle 4 : 4 = 1). On the whole, to a principal quantum number _n_
there correspond the auxiliary quantum numbers 1, 2 ... _n_, and
the orbit for which the auxiliary quantum number equals the principal
quantum number is a circle. We see that in the more complicated
hydrogen atom model there is possibility for a much greater number of
different transitions than in the simple model (Fig. 25, p. 119). Some
of the transitions are indicated by arrows. Since the energy content of
the atom is almost the same for orbits with the same principal quantum
number and different auxiliary quantum numbers, three transitions
like 3₃-2₂, 3₂-2₁ and 3₁-2₂ will give about the same frequency, and
therefore spectral lines which lie very close together. In a transition
like 4₄-4₃ the emitted energy quantum _h_ν, and also ν, will be so
extremely small that the corresponding line will be too far out in the
infra-red for any possibility of observing it.

It must be pointed out that the above considerations only hold if
the hydrogen atoms, strictly speaking, are undisturbed. Thus, very
small external forces, which may be due to the neighbourhood of other
atoms, etc., will be sufficient to cause changes in the eccentricity
of the stationary orbits. In such a case the above definition of the
auxiliary quantum number becomes obviously illusory, and the original
character of the fine structure disappears. This is in agreement with
the experiments, since the Sommerfeld fine structure can be found only
when the conditions in the discharge tube are especially quiet and
favourable.


Influence of Magnetic and Electric Fields on the Hydrogen Lines.

[Illustration: FIG. 28.—The splitting of three hydrogen lines under the
influence of a strong electric field.]

As previously mentioned (p. 76), the spectral lines are split into
three components when the atoms emitting lights are exposed to magnetic
forces. The agreement found here between observation and the Lorentz
electron theory was considered as strong evidence of the correctness
of the latter. According to the Bohr theory, the picture upon which
this explanation rested must be abandoned entirely; but fortunately it
has been shown that the Bohr theory leads to the same results; and,
moreover, Bohr, with the assistance of the correspondence principle,
has been able to set forth the more fundamental reason for this
agreement.

The German scientist, Stark, showed, in 1912, that hydrogen lines are
also split by electric fields of force. In Fig. 28 it is shown how
very complicated this phenomenon is; here the classical electron
theory could not at all explain what happened. This phenomenon
could also be accounted for by the extended Bohr theory (with the
introduction of more than one quantum number), as it was shown
independently by Epstein and by Schwarzschild in 1916; further, the
correspondence principle has again shown its superiority, since
it makes possible an approximate determination of the different
intensities of the different lines. A calculation carried out by H. A.
Kramers has shown that the theory gives a remarkably good agreement
with the experiments.

Not until we think of the extraordinary accuracy of the measurements
which are obtained by spectrum analysis, can we thoroughly appreciate
the importance of the quantitative agreement between theory and
observation in the hydrogen spectrum that has just been mentioned.
Moreover, we must remember how completely helpless we previously were
in the strange puzzles offered even by the simplest of all spectra,
that of hydrogen.




CHAPTER VI

VARIOUS APPLICATIONS OF THE BOHR THEORY


Introduction.

We have dwelt at length upon the theory of the hydrogen spectrum
because it was particularly in this relatively simple spectrum that
the Bohr theory first showed its fertility. Moreover, by studying
the case of the hydrogen atom with its one electron, it is easier to
gain insight into the fundamental ideas of the Bohr theory and its
revolutionary character. Naturally, the theory is limited neither
to the hydrogen atom nor to spectral phenomena, but has a much more
general application. As has already been said, it takes, as its
problem, the explanation of every one of the physical and chemical
properties of all the elements, with the exception of those properties
known to be nuclear (cf. p. 94). This very comprehensive problem can
naturally, even in its main outlines, be solved but gradually and by
the co-operation of many scientists, and it is quite impossible to go
very deeply into the great work which has already been accomplished,
and into the difficulties which Bohr and the others working on the
problem have overcome. We must be content with showing some especially
significant features.


Different Emission Spectra.

While the neutral hydrogen atom consists simply of a positive nucleus
and one electron revolving about the nucleus, the other elements, in
the neutral state, have from two up to 92 electrons in the system
of electrons revolving around the nucleus. Even 2 electrons, as in
the helium atom, make the situation far more complicated, since we
have in this case a system of 3 bodies which mutually attract or
repel each other. We are thus confronted with what, in astronomy, is
known as the three-body problem, a problem considered with respect
by all mathematicians on account of its difficulties. In astronomy,
the difficulties are restricted very much when the mass of one body
is many times greater than that of the others, as in the case of the
mass of the sun in relation to that of the other planets. Here, by
comparatively simple methods, it is possible to calculate the motions
inside a finite time-interval with a high degree of approximation even
when there are not two but many planets involved.

We might now be tempted to believe that in the atom we had to deal with
comparatively simple systems—solar systems on small scale—since the
mass of the nucelus is many times greater than that of the electrons.
But even if the suggested comparison illustrates the position of the
nucleus as the central body which holds the electrons together by its
power of attraction, the comparison in other respects is misleading.
While the orbits of the planets in the solar system may be at any
distance whatsoever from the sun, and the motions of the planets are
everywhere governed by the laws of mechanics, the atomic processes,
according to the Bohr theory, are characterized by certain stationary
states, and only in these can the laws of mechanics possibly be
applied. But in addition, the forces between nucleus and electrons
are determined not at all by the masses, but rather by the electric
charges. In the helium atom the nuclear charge is only double that
of an electron, and the attraction of the nucleus for an electron
will therefore be only twice as large as the repulsions between two
electrons at the same distance apart. This repulsion under these
circumstances will, therefore, also have great influence on the ensuing
motion. In elements with higher atomic numbers the nuclear charge has
greater predominance over the electron charges; but, on the other hand,
there are then more electrons. The situation is in each case more
complicated than in the hydrogen atom.

Nevertheless, the line spectra of the elements of higher atomic number
show how the lines, as in the hydrogen spectrum, are arranged in series
although in a more complicated manner (cf. p. 59); in any case in
many instances there is great similarity between the radiation from
the hydrogen atom and that from the more complicated atoms. Thus in
the line spectra of many elements, just as in that of hydrogen, the
frequency ν of every line can be expressed as a difference between
two _terms_, involving certain integers which can pass through
a series of values. From the combinations of terms, two at a time,
the values of ν corresponding to the different spectral lines can be
derived. This so-called _combination principle_ enunciated by the
Swiss physicist, Ritz, can evidently be directly interpreted on the
basis of Bohr’s postulates, since the different combinations may be
assumed to correspond to definite atomic processes, in which there is
a transition between two stationary states, each of which corresponds
to a spectral term.

Moreover, the terms (cf. p. 59) may often be approximately given by the
Rydberg formula

       K
  ----------
  (_n_ + α)²

where K has about the same value as in hydrogen, and α can take on a
series of values α₁, α₂ ... αₖ, while _n_ takes on integer values.
Since we thus determine the different lines by assigning values to
the two integers _n_ and _k_ in each term, we have in this
respect something like the fine structure in the hydrogen spectrum,
where the stationary states are determined by a principal quantum
number and an auxiliary quantum number. The spectra of which we are
speaking here, and for which the terms have the form given above, are
often called _arc spectra_, because they are emitted particularly
in the light from the electric arc or from the vacuum tube. We must
expect that the similarity which exists in the law for the distribution
of spectral lines will correspond to a similarity in the atomic
processes of hydrogen and the other elements.

The hydrogen atom emits radiation corresponding to the different
spectral lines when an electron from an outer stationary orbit jumps,
with a spring of varying size, to an orbit with lower number, and at
last finds rest in the innermost orbit in a normal state, where the
energy of the atom is as small as possible. Similarly, we must assume
that the electrons in other atoms, during processes of radiation, may
proceed in towards the nucleus until they are collected as tightly
as possible about the nucleus, corresponding to the normal state of
the atom, where its energy content is as small as possible: “capture”
of electrons by the nucleus. The region in space which, in the normal
state, includes the entire electron system, must be assumed to be of
the same order of magnitude as the dimensions of the atom and molecule
which are derived from the kinetic theory of gases. This normal state
may be called a “quiescent” state, since the atom cannot emit radiation
until it has been excited by the introduction of energy from without.
This excitation process consists of freeing one (or more) electrons,
in some way or other, from the normal state and either removing it
out to a stationary orbit farther away from the nucleus or ejecting
it completely from the atom. Not all electrons can be equally easily
removed from the quiescent state. Those moving in small orbits near
the nucleus will be tighter bound than those moving in larger orbits
farther from the nucleus. The arc spectrum is now caused by driving one
of the most loosely bound electrons out into an orbit farther from the
nucleus or removing it completely from the atom. In the latter case
the rest of the atom, which with the loss of the negative electron
becomes a positive ion, easily binds another electron, which, with the
emission of radiation, corresponding to lines of the series spectrum,
can approach closer to the nucleus.

Let us now assume, first, that this radiating electron moves at so
great a distance from the nucleus and the other electrons that the
entire inner system can be considered as concentrated in one point;
then the situation is quite as if we had to deal with a hydrogen
atom. If the atomic number is as high as 29 (copper), for instance,
the nuclear charge will consist of twenty-nine elementary quanta of
positive electricity; but since there are twenty-eight electrons in
the inner system, the resultant effect is that of only one elementary
quantum of positive electricity, as in the case of a hydrogen nucleus.
The spectral lines which are emitted in the jumps between the more
distant paths will be practically the same as hydrogen lines.
But, since in the jumps between these distant orbits, very small
energy quanta will be emitted, the frequencies are very small, the
wave-lengths very great, _i.e._, the lines in question lie far out
in the infra-red.

[Illustration: FIG. 29.—Different stationary orbits which the outermost
(11th) electron of sodium may describe.]

When the electron has come in so close to the nucleus that the
distances in the inner system cannot be assumed to be small in
comparison to the distance of the outer electron from the nucleus,
the situation is changed. The force with which the nucleus and the
inner electrons together will work upon the outer electron will be
appreciably different from the inverse square law of attraction of a
point charge. The consequence of this difference is that the major
axis in the ellipse of the electron rotates slowly in the plane of the
orbit as described in case of the theory of the fine structure of the
hydrogen lines (cf. p. 146), and even if the cause is different the
result is the same; the orbit of the outer electron in the stationary
states will be characterized by a quantum number _n_ and an
auxiliary quantum number _k_. If the electron comes still closer
to the nucleus, its motion is even more complicated. When the electron
in its revolution is nearest the nucleus it will be able to dive into
the region of the inner electrons, and we can get motions like those
shown in Fig. 29 for one of the eleven sodium electrons. The inner
dotted circle is the boundary of the inner system which is given by the
nucleus and the ten electrons remaining in the “quiescent” state—little
disturbed by the restless No. 11. In the figure we can see greater or
smaller parts of No. 11’s different stationary orbits with principal
quantum numbers 3 and 4. We shall not account further for the different
orbits and the spectral lines produced by the transitions between
orbits, but shall merely remark that the yellow sodium line, which
corresponds to the Fraunhofer D-line (cf. p. 49), is produced by the
transition 3₂-3₁, between two orbits with the same principal quantum
number. The sketch shows to a certain degree how fully many details of
the atomic processes can already be explained. The theory can even give
a natural explanation of why the D-line is double.

We have restricted ourselves to the case where only one electron is
removed from the normal state of the neutral atom. It may, however,
happen that two electrons are ejected from the atom so that it becomes
a positive ion with two charges. When an electron from the outside is
approaching this doubly charged ion it will, at a distance, be acted
upon as if the ion were a helium nucleus with two positive charges. The
situation, in other words, will be as in the case of the false hydrogen
spectrum (cf. p. 142), where the constant K in the formula for the
hydrogen spectrum is replaced by another which is very close to 4K. But
if the atom is not one of helium, but one with a higher atomic number,
the stationary orbits of the outer electron which approach closely to
the nucleus will not coincide exactly with those in the ionized helium
atom, corresponding to the fact that the terms in the formula for
the spectrum, instead of the simple form 4K/_n_², have the more
complicated form 4K/(_n_ + α)². Spectra of this nature are often
called _spark spectra_, since they appear especially strong in
electric sparks; they appear also in light from vacuum tubes, when an
interruptor is placed in the circuit, making the discharge intermittent
and more intense.

An atom with several electrons can, however, be much more violently
excited from its quiescent state when an electron in the inner region
of the atom is ejected by a swiftly moving electron (a cathode ray
particle or a β-particle from radium) which travels through the atom.
Such an invasion produces a serious disturbance in the stability of
the electron system; a reconstruction follows, in which one of the
outer, more loosely bound electrons takes the vacant position. In the
transitions, in which these outer electrons come in, rather large
energy quanta are emitted. The emitted radiation has therefore a very
high frequency; monochromatic X-rays are thus emitted. Since these have
their origin in processes far within the atom, it can be understood
that the different elements have different characteristic X-ray
spectra, which can give very valuable information about the structure
of the electron system (cf. p. 91).

Between these X-ray spectra and the series spectra previously mentioned
there lie, as connecting links, those spectra which are produced when
electrons are ejected from a group in the atom which does not belong
to the innermost group, but does not, on the other hand, belong in the
outermost group in the normal atom. We have very little experimental
knowledge about such spectra, because the spectral lines involved have
wave-lengths lying between about 1·5 μμ and 100 μμ. Rays with these
wave-lengths are absorbed very easily by all possible substances; they
have very little effect on photographic plates, where they are absorbed
by the gelatine coating before they have an opportunity to influence
the molecules susceptible to light. But there can be scarcely any doubt
that, in the course of a few years, experimental technique will have
reached such efficiency that this domain of the spectrum, so important
for the atomic theory, will also become accessible to experiment. In
individual cases, wave-lengths as small as 20 μμ have already been
obtained by Millikan.

Of entirely different character from these spectra are the _band
spectra_. They are in general produced by electric discharges
through gases which are not very highly attenuated (cf. p. 55); they
are not due to purely atomic processes, but can be designated as
_molecular spectra_. Their special character is due to motions in
the molecule, not only motions of the electrons, but also oscillations
and rotations of the nuclei about each other. We shall not go into
these problems here; in what follows we shall investigate a certain
type of band spectra somewhat more closely in connection with the
absorption of radiation.

While the band spectra with a spectroscope of high resolving power
can be more or less completely resolved into lines, this is not the
case with the _continuous spectra_. They are emitted not only by
glowing solids (cf. p. 54), but also by many gaseous substances. When
such gases are exposed to electric discharges they emit, in addition to
the line spectra and band spectra, continuous spectra which in certain
parts of the spectrum furnish a background for bright lines which come
out more strongly. It might seem impossible to correlate these with the
Bohr theory; but in reality a spectrum does not always have to consist
of sharp lines. This can at once be seen from the correspondence
principle. If the motions in the stationary states are of such
nature that they can be resolved into a number of discrete harmonic
oscillations each with its own period (for instance the orbit of an
electron in a rotating ellipse; cf. p. 149), then, according to the
correspondence principle, in the transition between two such stationary
states there are produced sharp spectral lines “corresponding” to these
harmonic components. But not all motions of atomic systems can be thus
resolved into a number of definite harmonic oscillations. When this
cannot be done, the stationary states cannot be expected to be such
that transitions between them produce radiation which can be resolved
into sharp lines.

A simple example, where it is easily intelligible that the Bohr theory
will not lead to sharp lines, is obtained in a simple consideration
of the hydrogen atom. Let us examine the lines belonging to the
Balmer series which are produced when an electron passes to the No. 2
orbit from an orbit with higher orbit number, which is farther from
the nucleus. As has been said, we obtain here an upper limit for the
frequency corresponding to a value of the outer orbit number which is
infinite; this means, in reality, that the electron in one jump comes
in from a distance so great that the attraction of the nucleus is
infinitely small. The energy released by such a jump is the same as the
ionizing energy A₂ which is required to eject the electron from the
orbit No. 2 and drive it from the atom. It is here assumed, however,
that the electron out in the distance was practically at rest. If the
captured electron has a certain initial velocity outside, it will have
a corresponding kinetic energy A. When in one jump this electron comes
from the outside into orbit No. 2, the energy lost by the electron
and emitted in the form of radiation will be the sum of the ionizing
energy A₂ and the original kinetic energy A. The frequency ν will then
become greater than that corresponding to A₂; and since the velocity
of the electron before it is captured is not restricted to certain
definite values, neither is the value of ν. The radiation from a great
quantity of hydrogen atoms which are binding electrons in this way
will, in the spectrum, not be concentrated in certain lines, but will
be distributed over a region in the ultra-violet which lies outside
of the limit calculated from the Balmer formula; still in a certain
sense this continuous spectrum is correlated with the Balmer series.
In the spectra from certain stars there has actually been discovered a
continuous spectrum, which lies beyond the limits of the Balmer series
and may be said to continue it.

Also the X-rays, which are generally used in medicine, have varying
frequencies; this is caused by the fact that some of the electrons
which, in an X-ray tube, strike the atoms of the anticathode and
travel far into it at a high speed, lose a part or all of their
velocity without ejecting inner electrons. The lost kinetic energy then
appears directly as radiation. These remarks ought to be sufficient
to show that the radiation, for instance, from a glowing body, where
the interplay of atoms and molecules is very complicated, can give a
continuous spectrum.


Electron Collisions.

The excitation of an atom in the normal state (cf. p. 157), by which
one of its electrons is removed to an outer stationary orbit, may
be caused by a foreign electron which strikes the atom. A study of
collisions between atoms and free electrons is therefore of the
greatest importance when investigating more closely the conditions by
which series spectra are produced.

These investigations can be carried out by giving free electrons
definite velocities by letting them pass through an electric field,
where the “difference of potential” is known in the path traversed
by the electrons. When an electron moves through a region with a
difference of potential of one volt (the usual technical unit), the
kinetic energy of the electron will be increased by a definite amount
(of 1·6 × 10⁻¹² erg). If its initial velocity is zero, its passage
through this field will make the velocity 600 km. per second; if the
potential difference were 4 volts, 9 volts, etc., the velocity obtained
by the electron would be 2, 3, etc., times larger. For the sake of
brevity we shall say that the kinetic energy of an electron is, for
instance, 15 volts, when we mean that the kinetic energy is as great as
would be given by a difference of potential of 15 volts.

In 1913 the German physicist Franck began a series of experiments by
methods which made it possible to regulate accurately the velocity
of the electrons, and to determine the kinetic energy before and
after collisions with atoms. He first applied the methods to mercury
vapour, where the conditions are particularly simple, since the mercury
molecules consist of only one atom. Franck bombarded mercury vapour
with electrons all of which had the same velocity. He then showed that
if the kinetic energy of the electrons was less than 4·9 volts the
collisions with the atoms were completely “elastic,” _i.e._, the
direction of the electron could be changed by the collision, but not
its velocity. If, however, the velocity of the impinging electrons was
increased so much that it was somewhat larger than 4·9 volts, there
was an abrupt change in the situation, since many of the collisions
became completely inelastic, _i.e._, the colliding electron lost
its entire velocity and gave up its entire kinetic energy to the atom.
If the initial velocity was even greater, so that the kinetic energy
of the colliding electron was 6 volts, for instance, then when the
collision took place there would always be lost a kinetic energy of 4·9
volts, since the electrons would either preserve their kinetic energy
intact or have it reduced to 1·1 volt (cf. Fig. 30).

[Illustration: FIG. 30.—Schematic drawing of Franck’s
experiment with electron collisions. _G_ is a glowing metal wire
which emits electrons. If between _G_ and the wire net _T_
there is a difference of potential of 6 volts, the electrons will pass
through the holes of the net with great velocity out into the space
_R_, where there is mercury vapour. _a_ represents a free
electron _F_ and a mercury atom _Hg_ before the collision,
while _b_ represents them after the collision; with the collision
_F_ loses a kinetic energy corresponding to 4·9 volts; at the
same time a bound electron _B_ in the atom goes over to a larger
stationary orbit.]

This remarkable phenomenon can be understood from the Bohr theory if we
assume that to send the most loosely bound electron in the mercury atom
out to the nearest outer stationary orbit there is required an energy
of 4·9 volts, since in that case, according to the first postulate,
an energy of less than this magnitude cannot be absorbed by the atom.
The use of the word “understanding” must here be qualified; if the
forces which influence the free electron as it comes into the electron
system of the mercury atom are no other than the usual repulsion from
the electrons and the attraction from the nucleus, the conduct of the
colliding electron can in no way be explained by the laws of mechanics.
But what happens is in agreement with the characteristic stability of
the stationary states, and Bohr had prophesied how it would happen.
Curiously enough Franck believed in the beginning that his experiment
disagreed with the Bohr theory because he made the mistake of supposing
that what happened was merely ionization, _i.e._, complete
disruption of a bound electron from a mercury atom.

Franck’s experiments showed, moreover, that mercury vapour, as soon as
the inelastic collisions appeared, began to emit ultra-violet light of
a definite wave-length, namely, 253·7 μμ. The product of the frequency
ν of this light and Planck’s constant _h_ agrees exactly with the
energy quantum possessed by an electron which has passed a potential
difference of 4·9 volts; but this also agrees with what might be
expected, according to the Bohr theory, from the radiation the removed
electron would emit upon returning to the normal state. The energy
which is respectively absorbed and emitted in the two transitions must
be indeed _h_ν.

Since an electron can not only be driven out to the next stationary
orbit, but also to an even more distant one (or entirely ejected) and
thence can come in again in one or more jumps, it is evident that
a far more complicated situation may arise. The Franck experiment,
which now has been extended to many other elements, clearly gives
extraordinarily valuable information in such cases. In mercury it has
been found that the energy a free electron must have in order to
eject an electron from an atom and turn the atom into a positive ion,
corresponds to a difference of potential of 10·8 volts, a value which
Bohr had predicted. At the same time that Franck’s experiments, in
this respect and in others, have strengthened the Bohr theory in the
most satisfactory way, they have also advanced its development very
much. Indeed it may be said that they have been of the greatest help in
atomic research. Even if the spectroscope has greater importance, the
investigations on electron collisions make the realities in the Bohr
theory accessible to study in a more direct and palpable manner.

[Illustration: FIG. 31.—Stratification of light in a vacuum tube.]

The peculiarities in the electron collisions appear most clearly in
an old and well-known phenomenon of light, namely, the stratification
of the light in a vacuum tube (Fig. 31). This stratification, which
previously seemed so incomprehensible, agrees exactly with the feature
so fundamental in the atomic theory that a free electron cannot give
energy under a certain quantum to an atom. We can imagine that, in the
non-illuminated central space between the bright strata, the electrons
each time under the influence of the outer electric field obtain the
amount of kinetic energy which must serve to excite the atoms of the
attenuated vapour.

As has been said (p. 161), electron collisions may cause the emission
of characteristic X-rays; but to produce them very great energy is
required. Therefore the electrons which are to produce this effect must
have an opportunity to pass freely through a certain region under the
influence of a proportionately strong electric field (with potential of
from 1000 to 100,000 volts and more). The electrons find such a field
in a highly exhausted X-ray tube, where the electrons under strong
potential are driven from the cathode against the anticathode, into
which they penetrate deeply.


Absorption.

In the experiments previously described it was the electron collisions
which furnished the energy required to excite the atoms, _i.e._,
to carry them from the normal state over into a stationary state
with greater energy. This “excitation energy” may, however, also be
furnished to the atoms in the form of radiation energy; we shall now
examine this case more closely.

Let us assume that to transfer an atom from the normal state to
another stationary state, or, in other words, to transfer one of the
electrons to an outer stationary orbit, a certain quantity of energy E
is demanded; then the radiation emitted by the atom when it returns to
the normal state will have a frequency ν depending upon the relation
E = _h_ν or ν = E/_h_, where _h_, as usual, is the Planck
constant. But just as the atom in the transition from the stationary
state to the normal state can emit radiation only with the definite
frequency ν, then the opposite transition can only be performed by
absorption of radiation with the same frequency; when this happens the
absorbed radiation energy has exactly the value E=_h_ν.

This reciprocity, which may be considered as a direct consequence
of the Bohr postulates, agrees with what has been said (cf. p. 50)
about the correspondence between the lines in the line spectrum of an
element and the dark absorption lines of that element—_e.g._, the
Fraunhofer lines in the solar spectrum. Let us examine, as an example,
the yellow sodium line, the D-line. Light with the corresponding
frequency, 526 × 10¹² vibrations per second, is emitted by a sodium
atom, when the loosest bound electron goes over from a stationary orbit
with quantum numbers 3₂ to the orbit 3₁, which belongs to the normal
state of the sodium atom. The transition in the opposite direction,
3₁ to 3₂, can take place under absorption of radiation only when in
the light from some other source of light, which passes the sodium
atoms, there are found rays with the frequency 526 × 10¹². Even if
there is present radiation energy with some other frequency, the sodium
atoms take no notice of this energy; they absorb only rays with the
frequency stated, and every time an atom absorbs energy from a ray the
energy taken is always an energy quantum of the magnitude _h_ν,
_i.e._ about 6·54 × 10⁻²⁷ × 526 × 10¹² = 3·44 × 10⁻¹² ergs (1 erg
is the unit of energy used in the determination of _h_). When
there are present a large number of sodium atoms (as, for instance,
in the previously mentioned common salt flame), the transition 3₁ to
3₂ can take place in some atoms, the transition 3₂ to 3₁ in others;
therefore, at the same time there can be absorption and radiation of
the light in question. Whether absorption or radiation at any given
time has the upper hand depends upon various conditions (temperature,
etc.).

For the sake of simplicity we have here tacitly understood that
there can be but one definite transition (from the normal state)
corresponding to the assumption that the sodium spectrum had no other
lines than the D-line. In reality this is not the case, and there can
equally occur absorption of rays with larger frequencies belonging to
other spectral lines in the sodium atom and corresponding to other
possible transitions between stationary states in the sodium atom. If
the temperature of the sodium vapour is sufficiently low, in which
case almost all the atoms are in the normal state, it is evident that
in the absorption only those lines will appear which correspond to
transitions from the normal state, and which therefore form only a part
of all the lines of the sodium spectrum. We thus obtain an explanation
of the previously enigmatical circumstance that not all spectral lines
which can appear in emission will be found in absorption. At the same
time we get, in absorption experiments, valuable information about the
structure of the atom beyond what the observations in the emission
spectra are able to give.

Interesting phenomena may arise owing to the fact that the jumps
between the stationary states of the atom sometimes, as we know, take
place in single jumps, sometimes in double or multiple jumps, so that
the intermediate stationary states are jumped over. There is then
evidently a possibility that absorption can take place, for instance,
with a double jump of an electron, which may later return to the
original stationary orbit in two single jumps. The absorbed radiation
energy will then appear in emission with two frequencies which are
entirely different from the frequency of the absorbed rays (this latter
in this case will be the sum of the other two). When an element is
illuminated with a certain kind of rays, it can, in other words, emit
in return rays of a different nature. Such changes of frequencies
have also been observed in experiment; they contain, in principle, an
explanation of the characteristic phenomenon called _fluorescence_.

We shall not go further into this problem, but dwell for a time on
the characteristic phenomenon of absorption which is known as _the
photoelectric effect_. In this phenomenon (cf. p. 116) a metal
plate, by illumination with ultra-violet light, is made to send out
electrons with velocities the maximum value of which is independent of
the strength of the illumination, but depends only on the frequency
of the rays. What happens is that some of the electrons in the metal
which otherwise have, as their function, the conduction of the electric
current, by absorbing radiation energy, free themselves from the metal
and leave it with a certain velocity. The reason why the rays for most
metals must be ultra-violet (_i.e._ have a high frequency and
consequently correspond to a proportionately large energy quantum)
depends upon the fact that the energy quantum absorbed by the electrons
must be large enough to carry out the work of freeing the electrons.
But as long as the frequency of the rays (and therefore their energy
quantum) is no less than what is needed for the freeing process, it
does not need to have certain fixed values. If the energy quantum
_h_ν which the rays can give off is greater than is required
to free the electrons, the surplus becomes kinetic energy in the
electrons, which thus acquire a velocity which is the greater the
greater the frequency ν, and which coincides with the maximum velocity
observed in the experiments. What happens here is evidently something
which can be considered as the reverse of the process which leads
to the production of the continuous hydrogen spectrum (described on
p. 163). In the latter case, electrons with different velocities are
bound by the hydrogen atoms, which thus emit rays with frequencies
increasing with increasing velocity, while, _vice versa_, in the
photoelectric effect rays with different frequencies free the electrons
and give them velocities increasing with increasing frequencies.

It must be acknowledged that there is something very curious in
this effect. If the electromagnetic waves, as has been assumed, are
distributed evenly over the field of radiation, it is not easy to
understand why they give energy to some atoms and not to others, and
why the selected ones always—with a given frequency—acquire a definite
energy quantum, independent of the intensity of the radiation. For
small intensities of the incident radiation, the atom, in order to
acquire the proper quantum, must absorb energy from a greater part
of the field of radiation (or for a longer time) than for large
intensities. When the atoms acquire energy in electron collisions,
the situation is apparently easier to understand, since in this case
the colliding electrons give their kinetic energy to definite atoms,
namely, those which they strike.

Einstein, in 1905, when there was not yet any talk of the nuclear atom
or the Bohr theory, enunciated his theory of _light quanta_,
according to which the energy of radiation is not only emitted and
absorbed by the atoms in certain quanta, with magnitudes determined
by the frequency ν, but is also present in the field of radiation in
such quanta. When an atom emits an energy quantum _h_ν, this
energy will not spread out in waves on all sides, but will travel in a
definite direction—like a little lump of energy, we might say. These
light quanta, as they are called, can, like the electrons, hit certain
atoms.

But even if in this theory the difficulties mentioned are, apparently,
overcome, far greater difficulties are introduced; indeed it may
be said that the whole wave theory becomes shrouded in darkness.
The very number ν which characterizes the different kinds of
rays loses its significance as a frequency and the phenomena of
interference—reflection, dispersion, diffraction, and so on—which are
so fundamental in the wave theory of the propagation of light, and on
which, for instance, the mechanism of the human eye is based, receive
no explanation in the theory of light quanta.

For instance, in order to understand that grating spectra can be
produced at all, we must think of a co-operation of the light from all
the rulings (cf. Fig. 10, p. 47), and this co-operation cannot arise
if all the slits at a given moment do not receive light emitted from
the same atom. In a bundle of rays which comes in at right angles to a
grating, we must, in order to explain the interference, assume that the
state of oscillation at a given moment is the same in all slits, that,
for instance, there are wave crests in all at the same time, if we
borrow a picture from the representation of water waves. Only in this
case there can behind the grating at certain fixed places—for which the
difference in the wave-length of the distances from successive slits is
a whole number of wave-lengths—steadily come wave crests from all the
slits at one moment and wave troughs from all at another moment (the
classical explanation of the “mechanism” of a grating). If we imagine,
however, that some slits are hit by light quanta from one atom and
others from a second atom, it is pure chance if there are wave crests
simultaneously in all slits, because the different atoms in a source
of light emit light at different times, depending purely on chance. An
understanding of the observed effect of a grating on light seems then
out of question.

The theory of light quanta may thus be compared with medicine which
will cause the disease to vanish but kill the patient. When Einstein,
who has made so many essential contributions in the field of the
quantum theory, advocated these remarkable representations about the
propagation of radiation energy he was naturally not blind to the
great difficulties just indicated. His apprehension of the mysterious
light in which the phenomena of interference appear on his theory is
shown in the fact that in his considerations he introduces something
which he calls a “ghost” field of radiation to help to account for the
observed facts. But he has evidently wished to follow the paradoxical
in the phenomena of radiation out to the end in the hope of making some
advance in our knowledge.

This matter is introduced here because the Einstein light quanta have
played an important part in discussions about the quantum theory,
and some readers may have heard about them without being clear as to
the real standing of the theory of light quanta. The fact must be
emphasized that this theory in no way has sprung from the Bohr theory,
to say nothing of its being a necessary consequence of it.

In the Bohr theory, absorption and radiation must be said to be
completely reciprocal processes, _i.e._ processes of essentially
the same nature, but proceeding in opposite directions. In itself it
cannot be said to be more incomprehensible that an atom absorbs energy
from a field of radiation in agreement with the Bohr postulates than
that it emits energy into the field; but in both cases we naturally
encounter the great difficulties mentioned in Chap. V.

We have hitherto restricted ourselves to the purely atomic processes.
But just as in the emission of radiation we meet spectra which owe
their characteristics to _molecular processes_ (band spectra, cf.
p. 162), we have also absorption spectra with characteristics depending
essentially upon motions of the atomic nuclei in the molecules. A
particularly interesting and instructive example of this nature is
met with in the infra-red region of the spectrum in certain broad
absorption lines or absorption bands, which are due to gases having
molecules containing several atoms. In hydrogen chloride, for instance,
there is found, in the region of the spectrum which corresponds to a
wave-length of about 3·5 μ, such an absorption band, which by more
accurate investigation has been shown to consist of a great number of
absorption lines.

The explanation of this collection of lines must be sought in the
motions which the hydrogen nucleus and the chlorine nucleus perform,
as they in part _vibrate_ with respect to each other and in part
_rotate_ about their common centre of gravity. Just as in the case
of the motions of the electrons in the atom, there are also certain
stationary states for the nuclear motions. When the molecule absorbs
radiation energy it will go from one of these states to another, where
the energy content is greater. This absorption of energy proceeds
according to the quantum rule, _i.e._, the product of the Planck
constant _h_ and the frequency ν for the absorbed radiation must
be equal to the difference in energy between the two stationary states;
only those rays which have frequencies fulfilling this condition are
absorbed.

[Illustration: FIG. 32.—Schematic representation of possible
motions in a molecule of hydrogen chloride. _O_ is the centre
of gravity of the molecule. The black circles give the states of
equilibrium of the nuclei, the circles _s_ their outer positions
in oscillating, and the circles _r_ positions during the rotation
of the nuclei.]

In hydrogen chloride, at standard temperature, the molecules will be
in different stationary states of rotation (cf. the remarks on p. 27),
corresponding to different definite values of the rotation frequency,
while the nuclei, on the other hand, must be assumed to be at rest
with reference to each other, _i.e._, they preserve their mutual
distance. In Fig. 32, _H_ and _Cl_ indicate the circles
which the two nuclei will describe about the centre of gravity; here,
however, it must be remarked that the hydrogen circle is drawn too
small in comparison with that of chlorine. If heat rays with all
possible wave-lengths around 3·5 μ are sent through the hydrogen
chloride, that radiation energy will be absorbed which can in part set
the nuclei in oscillation and in part change the state of rotation.
Let us for a moment assume that only the former change could happen.
Then a ray with wave-length 3·46 μ would be absorbed, this frequency
corresponding to the energy in the stationary state of oscillation
into which the molecule goes; this frequency is very nearly equal to
the frequency with which the nuclei vibrate relatively to each other.
In reality, at the same time that the nucleus is set in oscillation,
there will always be a change in the state of rotation—consisting
either in an increase or in a decrease in the velocity of rotation.
The energy absorbed, and therefore the frequency for the radiation
absorbed, is thereby changed a little, so that in the spectrum of the
rays sent through we do not obtain an absorption line corresponding
to 3·46 μ, but a line somewhat removed from that. Since there are,
however, many stationary states of rotation to start from, and since in
some molecules there is one transition, in others another, we get many
absorption lines on each side of 3·46 μ.

Even before Bohr propounded his theory, at a time when the quantum
theory did not yet have a clarified form, the Danish chemist, Niels
Bjerrum, had predicted that the infra-red absorption lines ought to
have such a structure. This structure must be interpreted in the above
way which differs somewhat from Bjerrum’s ideas, but his prediction was
essentially strengthened by investigations, and it was one of the most
significant features in the development of the quantum theory prior to
1913. The first to detect the structure of the infra-red absorption
bands was the Swedish physicist, Eva von Bahr. Her experiments were
later extended in a most significant way by the work of Imes and
other American investigators. They enable us to calculate exactly the
distance between the two nuclei in the molecule.

It may be asked what becomes of the energy which the hydrogen chloride
molecule thus absorbs, and whether it necessarily after a longer or
shorter time must be re-emitted as radiation. The latter is not the
case. In a collision between molecules or atoms, the energy which
one molecule (or atom) has absorbed by radiation can undoubtedly be
transferred to another molecule, the velocity of which is thereby
increased. The theoretical necessity of the occurrence of such
collisions was clearly shown for the first time in a very significant
investigation by two of Bohr’s students, Klein and Rosseland. Without
collisions of this nature the radiation energy absorbed could never
be transformed into heat energy. Here we come to a very great and
important field, which has a very close connection with the theory of
the chemical processes and to a better explanation of which the more
recent experiments of Franck and his co-workers have made important
contributions.




CHAPTER VII

THE STRUCTURE OF THE ATOM AND THE CHEMICAL PROPERTIES OF THE ELEMENTS


Introduction.

We have hitherto restricted ourselves mainly to those applications of
the Bohr theory which have a direct connection with the processes of
radiation. We have shown how fertile the theory has proved to be, how
many problems, previously inexplicable, have been solved, and what
exact agreement has been established between experiment and theory in
this comprehensive field. We may now ask how the theory accounts for
the chemical behaviour of the different elements. As early as 1913,
Bohr, in connection with his researches on spectral phenomena, had
considered the chemical properties of the elements and had pointed out
interesting possibilities.

[Illustration: FIG. 33.—Early representation of the formation of a
hydrogen molecule (Bohr, 1913).]


The Combination of Atoms into Molecules.

In his discussion of hydrogen, Bohr suggested a model for the structure
of its molecule, which we shall give here, because, by a simple
example, it illustrates how two neutral atoms may form a molecule
(cf. p. 13). In Fig. 33, _a_, _K₁E₁_ and _K₂E₂_ are
two neutral hydrogen atoms which are approaching each other with the
orbits of the two electrons parallel. The nucleus _K_₁ and the
electron _E_₂ then attract each other as do the nucleus _K_₂
and the electron _E_₁. The two electrons repel each other as do
the two nuclei; but when the electrons are in opposite positions of
their orbits, the forces of attraction outweigh the effect of the
forces of repulsion. Calculation shows that when the atoms are allowed
to approach each other the positions of the atoms will be as is shown
in Fig. 33, _b_, where the orbits are closer to each other than
the nuclei. Finally, for small distances of the two nuclei, the two
orbits will be merged into one, as is shown in Fig. 33, _c_.
This orbit will be slightly larger than the original ones. The two
hydrogen atoms may, in this way, combine into one molecule. In fact,
an equilibrium position can be found for which the nuclei are held
together, in spite of their mutual repulsion, by the attractive forces
existing between them and the electrons which are moving in their
common orbit. It must, however, be assumed, for reasons which cannot
be given here, that the hydrogen molecule is, in reality, constructed
somewhat differently; probably the orbits of the electrons make an
angle with each other.

The formation of a hydrogen molecule may also be supposed to occur
when a positive hydrogen ion, _i.e._, a hydrogen nucleus, and
a negative hydrogen ion, _i.e._, a hydrogen nucleus with two
electrons, are drawn together by their mutual attractions. The forces
of attraction would be much stronger than in the first example given,
and the formation of a neutral molecule would not take place in the
same way. More energy would also be released, but the final result will
be the same.

Just as in the case of the hydrogen molecule, other molecules may be
formed from atoms belonging to the same or to different elements. The
method of formation of molecules varies according as it is a union
of neutral molecules in the normal state, or a union of positive and
negative ions. Conversely, by chemical decomposition a molecule can be
separated either into neutral parts or into ions. If, for instance,
common salt (sodium chloride, NaCl) is dissolved in water, the salt
molecules are, under the influence of the water molecules, decomposed
in Na-ions with one positive charge and Cl-ions with one negative
charge, corresponding to the monovalent electropositive character of
sodium and the monovalent electronegative character of chlorine.

The possibilities are, however, far from being exhausted by these two
methods of composition and decomposition. An atom may exist not only
in the normal state where it has its complete number of electrons
collected as tightly as possible about the nucleus, and in the ionized
state with one or more too many or too few electrons; but in a neutral
atom one or more of the outer electrons may be in a stationary orbit
at a greater distance from the nucleus than corresponds to the normal
stationary state. It is easy to understand that an atom in such an
“excited” or (as it is called in chemistry) _active_ state often
finds it easier to act in concert with other atoms than when it is in
the normal state; in this latter state the atom is often more like a
little compact lump of neutral substance than in the active state.

It will, in any case, be understood that the interplay between the
atoms, which reveals itself in the chemical processes or reactions
between different elements, offers many opportunities for the Bohr
theory to give in the future a more detailed explanation than was
possible to the earlier theories of chemistry. We must also mention
the fact that it has become possible to elucidate in main features
the phenomena, hitherto unexplained, of the chemical effects of
light, as on a photographic plate (_photochemistry_) and of
_catalysis_, which consists in bringing about, or accelerating,
the chemical interaction between two substances by the presence of a
third substance which does not itself enter in the compound, and often
needs only to be present in very small quantities. It must, however, be
emphasized that at present we do not yet possess a detailed theory of
molecular constitution comparable with our knowledge of the structure
of the atoms.


The Periodic System.

Instead of inquiring how the chemical processes may take place, we
shall now study the general correlation between the chemical properties
and the atomic numbers of the elements, a correlation which has found
its empirical expression in the natural or periodic system of the
elements (cf. p. 23). The explanation of the puzzles of this system
must be said to be one of the finest results which Bohr has obtained,
and it constitutes a striking evidence in favour of the quantum theory
of atoms.

There is nothing new in the idea of connecting the arrangement of
the elements in the periodic system with an arrangement of particles
in the atom in regular _groups_, the character of which varies,
so to say, periodically with increasing number of particles. In the
atom model of Lord Kelvin and J. J. Thomson (cf. p. 86), with the
positive electricity distributed over the volume of the whole atom,
Thomson tried to explain certain leading characteristics of the
periodic system by imagining the electrons as arranged in several
circular rings about the centre of the atom. He pointed out that the
stability of the electronic configurations of this type varied in a
remarkable periodic way with the number of electrons in the atom. By
considerations of this nature Thomson was able to enunciate a series of
analogies to the behaviour of the elements in the periodic system as
regards the tendency of the neutral atoms to lose one or more electrons
(electropositive elements) or to take up one or more electrons
(electronegative elements). But, setting aside possible objections to
his considerations and calculations, the connection with the system
was very loose and general, and his theory lost its fundamental
support when his atomic model had to give way to Rutherford’s. With
Bohr’s theory the demand for a stable system of electrons was placed in
an entirely new light.

In his treatise of 1913, Bohr tried to give an explanation of the
structure of the atom, by thinking of the electrons as moving in a
larger or smaller number of circular rings about the nucleus. His
theory did not exclude the possibility of orbits of electrons having
different directions in space instead of lying in one plane or being
parallel. The tendency of the considerations was to attain a definite,
unique determination of the structure of the atom, as is demanded by
the pronounced stability of the chemical and physical properties of
the elements. The results were, however, rather unsatisfactory, and it
became more and more clear that the bases of the quantum theory were
not sufficiently developed to lead in an unambiguous way to a definite
picture of the atom. Nowadays the simple conception of the electrons
moving in circular rings in the field of the nucleus is definitely
abandoned, and replaced by a picture of atomic constitution of which we
shall speak presently.

In the following years the general conception of the group distribution
of the electrons in the atom formed the basis of many theoretical
investigations, which in various respects have led to a closer
understanding of chemical and physical facts. The German physicist,
Kossel, showed that the characteristic X-ray spectra of the elements,
which are due to a process of reconstruction of the atom subsequent to
the removal of one or more of the innermost electrons (cf. p. 161),
give a most striking support to the assumption that the electrons are
distributed in different groups in which they are bound with different
strength to the atom.

The connection between the electron groups and the chemical valence
properties of the atoms, to which Thomson had first drawn attention and
which also played an important part in Bohr’s early considerations,
was further developed in a significant way by Kossel, as well as
by Lewis and by Langmuir in America. These chemical theories had,
however, little or no connection with the quantum theory of atomic
processes; even the special features of the Rutherford atom, which are
of essential importance in the theory of the hydrogen spectrum and of
other spectra, played only a subordinate part.

In 1920 Bohr showed how, by the development of the quantum theory
which had taken place in the meantime, and the main features of which
consisted in the introduction of more than one quantum number for the
determination of the stationary states and in the establishment of the
correspondence principle, the problem of the structure of the atom had
appeared in a new light. In fact, he outlined a general picture of
atomic constitution, based on the quantum theory, which in a remarkable
way accounted for the properties of the elements. In order to decide
doubtful questions, he has often had to call to his aid the observed
properties of elements, and it must be readily admitted that the
finishing touches of the theory are still lacking. But from his general
starting-point he has been able to outline the architecture of even
the most complex atomic structures and to explain, not only the known
regularities, but also the apparent irregularities of the periodic
system of the elements.

The method Bohr used in his attempt to solve the problem was to study
_how a neutral normal atom may gradually be formed by the successive
capturing and binding of the individual electrons in the field of
force about the nucleus of the atom_. He began by assuming that he
had a solitary nucleus with a positive charge of a given magnitude.
To this nucleus free electrons are now added, one after the other,
until the nucleus has taken on the number sufficient to neutralize the
nuclear charge. Each individual electron undergoes a “binding” process,
_i.e._ it can move in different possible stationary orbits about
the nucleus and the electrons already bound. With the emission of
radiation it can go from stationary states with greater energy to
others with less energy, ending its journey by remaining in the orbit
which corresponds to the least possible energy. We may designate
this state of least energy as the normal state of the system, which,
however, is only a positive atomic ion, so long as all the electrons
needed for neutralization are not yet captured.

From the exposition in the preceding chapter it will be seen that
the ordinary series spectra (arc spectra) may be considered as
corresponding to the last stage in this formation process, since the
emission of each line in such a spectrum is due to a transition between
two stationary states in each of which N-1 electrons are bound in their
normal state, _i.e._ as tight as possible, by the nucleus, while
the Nth electron moves in an orbit mainly outside the region of the
other electrons. In the same way the spark spectra give witness of the
last stage but one of the formation process of the atom, since here
N-2 electrons are bound in their normal state while an N-1th electron
moves in an orbit large compared with the dimensions of the orbits
of the inner electrons. From these remarks it will be clear that the
study of the series spectra is of great importance for the closer
investigation of the process of formation of the atom outlined above.
Furthermore, the general ideas of the correspondence principle, which
directly connects the possibility of transition from one stationary
state to another with the motion of the electron, has been very useful
in throwing light on the individual capturing processes and on the
stability of the electronic configurations formed by these. In what
follows we cannot, however, reproduce Bohr’s arguments at length; we
must satisfy ourselves with some hints here and there, and for the rest
restrict ourselves to giving some of the principal results.

Before going farther we shall recall what has previously been said
about the quantum numbers. In the undisturbed hydrogen atom, the
stationary orbits can be numbered with the principal quantum numbers
1, 2, 3 ... _n_. But to each principal quantum number there
corresponds not one but several states, each with its auxiliary quantum
number 1, 2, 3 ... _k_, _k_ at the most being equal to the
principal quantum number. In a similar way, the stationary orbits of
the electrons in an atom containing several electrons can be indicated
by two quantum numbers, the 3₂ orbit, for instance, being that with
principal quantum number 3 and auxiliary quantum number 2. But while
in the hydrogen atom the principal quantum number _n_, in the
stationary orbits which are slowly rotating ellipses, is very simply
connected with the length of the major axis of the ellipse, and
_k_: _n_ is the ratio between the minor and major axes,
still in other atoms with complex systems of electrons the significance
of the principal quantum number is not so simple and the orbit of
an electron consists of a sequence of loops of more complicated
form (cf. Fig. 29). We must satisfy ourselves with the statement
that a definition of their significance can be given, but only by
mathematical-physical considerations which we cannot enter into here.
It may, however, be stated that, if we restrict ourselves to a definite
atom, the rule will hold that, among a series of orbits with the same
auxiliary quantum number but different principal quantum numbers, that
orbit in which the electron attains a greater distance from the nucleus
has the higher number. Another rule which holds is, that an orbit with
a small auxiliary quantum number in comparison with its principal
quantum number (as 4₁ for instance, cf. Fig. 29), will consist of very
oblong loops with a very great difference between the greatest and
least distances of the electron from the nucleus, while the orbit will
be a circle when the two quantum numbers are the same as for 1₁, 2₂,
3₃. Although each orbit has two quantum numbers, we often speak simply
of the 1-, 2-, 3- ... _n_-quantum orbits, meaning here the orbits
with the principal quantum numbers 1, 2, 3 ... _n_.

The one electron of hydrogen will, upon being captured, first be at
“rest” when it reaches the 1₁-path, and we might perhaps be led to
expect that in the atoms with greater nuclear charges the electrons in
the normal state also would be in the one quantum orbit 1₁, because
to this corresponds the least energy in hydrogen. This assumption
formed actually the basis of Bohr’s work of 1913 on the structure of
the heavier atoms. It cannot be maintained, however. Considerations
of theoretical and empirical nature lead to the assumption that
the electrons which already are gathered about the nucleus can make
room only to a certain extent for new ones, moving in orbits of the
same principal quantum number. Those electrons which are captured
later are kept at an appropriate distance; they are, for instance,
prevented from passing from a 3-quantum orbit to a 2-quantum one,
if the number of electrons moving in 2-quantum orbits has reached a
certain maximum value. When it is said that the captured electrons
end in the stationary state which corresponds to the least energy,
it must, therefore, mean, not the 1-quantum orbit, but the innermost
possible under the existing circumstances. _The final result will be
that the electrons are distributed in groups, which are characterized
each by their quantum numbers in such a way that passing from the
nucleus to the surface of the atom, the successive groups correspond to
successive integer values of the quantum number, the innermost group
being characterized by the quantum number one. Moreover, each group is
subdivided into sub-groups corresponding to the different values which
the auxiliary quantum number may take._

That the electrons first collected keep the latecomers at an
appropriate distance must be understood with reservations; a new
electron moving in an elongated orbit can very well come into the
territory already occupied; in fact, it may come closer to the nucleus
than some of the innermost groups of electrons. In case an outer
electron thus dives into the inner groups, it makes a very short visit,
travelling about the nucleus like a comet which at one time on its
elongated orbit comes in among the planets and perhaps draws closer to
the sun than the innermost planet, but during the greater part of its
travelling time moves in distant regions beyond the boundaries of the
planetary system. It is a very important characteristic of the Bohr
theory of atomic architecture that the outer electrons thus penetrate
far into the interior of the atom and thus chain the whole system
together.

Such a “comet electron” has, however, a motion of a very different
nature from that of a comet in the solar system. Let us suppose that
the nuclear charge is 55 (Caesium), that there already are fifty-four
electrons gathered tightly about the nucleus, and that No. 55 in an
orbit consisting of oblong loops moves far away from the nucleus, but
at certain times comes in close to it. Then, for the greater part of
its orbit, this electron will be subject to approximately the same
attraction as the attraction towards one single charge, as a hydrogen
nucleus; but when No. 55 comes within the fifty-four electrons it
will for a very short time be influenced by the entire nuclear charge
55. Together with the nearness of the nucleus, this will cause No. 55
to acquire a remarkably high velocity and to move in an orbit quite
different from the elliptical one it followed outside. Moreover, the
great velocity of the electron during its short visit to the nucleus is
in a considerable degree determinative of its principal quantum number;
this will be higher than would be expected from the dimensions of the
outer part of the orbit if we supposed the motion to take place about a
hydrogen nucleus (cf. Figs. 27 and 29).

After these general remarks we shall try in a few lines to sketch the
Bohr theory of the structure of the atomic systems from the simplest to
the most complicated. We shall not examine the entire periodic system
with its ninety-two elements, but here and there we shall bring to
light a trait which will illustrate the problem—partly in connection
with the schematic representations in the atomic diagrams at the end of
the book.


_Description of the Atomic Diagrams._

The curves drawn represent parts of the orbital loops of the electrons
in the neutral atoms of different elements. Although the attempt has
been made to give a true picture of these orbits as regards their
dimensions, the drawings must still be considered as largely symbolic.
Thus in reality the orbits do not lie in the same plane, but are
oriented in different ways in space. It would have been impracticable
to show the different planes of the orbits in the figure. Moreover,
there is still a good deal of uncertainty as to the relative positions
of these planes. On this account the orbits belonging to the same
sub-group, _i.e._, designated by the same quantum numbers, are
placed in a symmetric scheme in the sketch. For groups of circular
orbits the rule has been followed to draw only one of them as a
circle, while the others in the simpler atoms are drawn in projection
as ellipses within the circle, and in the more complicated atoms are
omitted entirely. The two circular orbits of the helium atom are both
drawn in projection as ellipses. Further, for the sake of clearness,
no attempt has been made to draw the inner loops of the non-circular
orbits of electrons which dive into the interior of the atom. In
lithium only, the inner loop of the orbit of the 2₁ electron has been
shown by dotted lines.

In order to distinguish the groups of orbits with different principal
quantum numbers two colours have been used, red and black, the red
indicating the orbits with uneven quantum numbers, as 1, 3, 5, the
black those with even quantum numbers, as 2, 4, 6. Wherever possible
the nucleus is indicated by a black dot; but in the sketches of atoms
with higher atomic numbers the 1-quantum orbits are merged into one
little cross and the nucleus has been omitted. It should be noticed
that the radium atom is drawn on a scale twice as great as that for the
other atoms.

We shall begin with the capture of the first electron. If the nucleus
is a hydrogen nucleus the _hydrogen atom_ is completed when the
electron has come into the 1₁ orbit, a circle with diameter of about
10⁻⁸ cm. (cf. the diagram). If the nucleus had had a greater nuclear
charge the No. 1 electron would have behaved in the same way, but the
radius of its orbit would have been less in the same ratio as the
nuclear charge was greater. For a lead nucleus, with charge (atomic
number) 82, the radius of the 1₁ orbit is ¹/₈₂ that of the hydrogen 1₁
orbit. Since atoms with high atomic numbers thus collect the electrons
more tightly about them it is understandable that, in spite of their
greater number of electrons, they can be of the same order of magnitude
as the simpler atoms.

Let us now examine the _helium atom_. The first electron, which
its nucleus (charge 2) catches, moves as shown in a circle 1₁, but
with a smaller radius than in the case of the hydrogen atom. Electron
No. 2 can be caught in different ways, and the closer study of the
conditions prevailing here, which are still comparatively simple since
there are only two electrons, has been of greatest importance in the
further development of the whole theory. We cannot go into it here, but
must content ourselves with saying that the stable final result of the
binding of the second electron consists in the two electrons moving in
circular 1-quantum orbits of the same size with their planes making
an angle with each other (cf. the diagram). This state has a very
stable character, and the helium atom is therefore very disinclined to
interplay with other atoms, with other helium atoms as well as with
those of other elements. Helium is therefore monatomic and a chemically
inactive gas.

In all atomic nuclei with higher charges than the helium nucleus the
first two orbits are also bound into two 1-quantum circular orbits at
an angle with each other; this group cannot take up any new electron
having the same principal quantum number. It takes on an independent
existence and forms the innermost electron group in all atoms of atomic
number higher than 2.

Electron No. 3 will accordingly not be bound in the same group with
1 and 2. It must be satisfied with a 2-quantum orbit, 2₁, which
consists of oblong loops, and, when nearest the nucleus, comes into the
territory of the 1-quantum orbits. It is but loosely bound compared
to the first two electrons, and the lithium atom, which has only
three electrons, can therefore easily let No. 3 loose so that the
atom becomes a positive ion. _Lithium_ is therefore a strongly
electropositive monovalent metal. The element _beryllium_ (No. 4)
will probably have two electrons in the orbits 2₁; it will therefore be
divalent. But during the short visit of these electrons to the nucleus
they are subject to a greater nuclear charge than in lithium. The 2₁
electrons are therefore, in beryllium, more firmly bound than in
lithium, and the electropositive character of beryllium is therefore
less marked.

We have something essentially new in the _boron_ atom (atomic No.
5) where the two electrons No. 3 and No. 4 are taken into 2₁ orbits,
but where No. 5 will very probably be bound in a circular 2₂ orbit.
How the conditions will be in the normal state of the following atoms
preceding neon is not known with certainty. We only know that the
electrons coming after the first two will be captured in 2-quantum
orbits, the dimensions of which get smaller, according as the atomic
number increases.

The _neon_ atom (compare the diagram) has a particularly stable
structure, both “closed” and symmetric, which besides two 1₁-orbits
contains four electrons in 2₁ orbits and four electrons in 2₂
orbits. As regards the four electrons in 2₁ orbits, they do not have
symmetrical positions at every moment or move simultaneously either
towards or away from the nucleus; on the contrary, it must be assumed
that the electrons come closest to the nucleus at different moments at
equal intervals of time.

The name of inert or inactive gases is given to the entire series
of helium (2), neon (10), argon (18), krypton (36), xenon (54) and
niton (86), the O-column in the periodic system given in the table
on p. 23. All these elements are monatomic and quite unwilling to
enter into chemical compounds with other elements (although there is
about 1 per cent. of argon in the air about us this element has, on
this account, escaped the observation of chemists until about 1895,
when it was discovered by the English chemist, Ramsay). This complete
chemical inactivity is explained by the fact that the atoms of all
these elements have a nicely finished “architecture” with all the
electrons firmly bound in symmetrical configurations. They may be said
to form the mile posts of the periodic system, and to be the ideals
towards which the other atoms aspire. The table shows how the electrons
in the atoms of these gases are divided among the types of orbits
corresponding to the different quantum numbers.

  TABLE SHOWING THE DISTRIBUTION OF THE ELECTRONS OF
      DIFFERENT ORBITAL TYPES IN THE NEUTRAL ATOMS OF THE
      INACTIVE GASES.

  -------+------+---------------------------------------+
         |      |          Quantum Numbers.             |
         |Atomic+---+---+---+---+---+---+---+---+---+---+
         |Number| 1₁| 2₁| 2₂ | 3₁| 3₂| 3₃| 4₁| 4₂ | 4₃| 4₄|
  -------+------+---+---+---+---+---+---+---+---+---+---+
  Helium |   2  | 2 |   |   |   |   |   |   |   |   |   |
  Neon   |  10  | 2 | 4 | 4 |   |   |   |   |   |   |   |
  Argon  |  18  | 2 | 4 | 4 | 4 | 4 | - |   |   |   |   |
  Krypton|  36  | 2 | 4 | 4 | 6 | 6 | 6 | 4 | 4 | - | - |
  Xenon  |  54  | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | - |
  Niton  |  86  | 2 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 |
  ?      | 118  | 2 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 |
  -------+------+---+---+---+---+---+---+---+---+---+---+

  -------+------+------------------------------------------------------+
         |      |                  Quantum Numbers.                    |
         |Atomic+---+---+---+---+---+---+---+---+---+---+---+---+---+--+
         |Number| 5₁| 5₂| 5₃| 5₄ | 5₅| 6₁| 6₂| 6₃| 6₄| 6₅| 6₆| 7₁ | 7₂| 7₃|
  -------+------+---+---+---+---+---+---+---+---+---+---+---+---+---+--+
  Helium |   2  |   |   |   |   |   |   |   |   |   |   |   |   |   |  |
  Neon   |  10  |   |   |   |   |   |   |   |   |   |   |   |   |   |  |
  Argon  |  18  |   |   |   |   |   |   |   |   |   |   |   |   |   |  |
  Krypton|  36  |   |   |   |   |   |   |   |   |   |   |   |   |   |  |
  Xenon  |  54  | 4 | 4 | - | - | - |   |   |   |   |   |   |   |   |  |
  Niton  |  86  | 6 | 6 | 6 | - | - | 4 | 4 | - | - | - | - |   |   |  |
  ?      | 118  | 8 | 8 | 8 | 8 | - | 6 | 6 | 6 | - | - | - | 4 | 4 | -|
  -------+------+---+---+---+---+---+---+---+---+---+---+---+---+---+--+

The elements _fluorine_, _oxygen_ and _nitrogen_ can
attain the ideal neon-architecture by binding respectively one, two
and three additional electrons. Naturally they do not become neon
atoms, but merely negative atomic ions with single, double or triple
charge; and their tendency in this direction appears in their character
of monovalent, divalent and trivalent electronegative elements
respectively. If we return to carbon it can probably not become a
tetravalent negative ion by binding four free electrons; but in the
typical carbon compound, methane (CH₄), the neon ideal is realized
in another manner. In fact, it is reasonable to assume that the four
electrons of the hydrogen atoms together with the six of the carbon
atom give approximately a neon-architecture. The four hydrogen nuclei
naturally cannot be combined with the carbon nucleus; the mutual
repulsions keep them at a distance. They will probably assume very
symmetrical positions within the electron system which holds them
together. The nitrogen atom may in a similar way find completion in
a neutral molecule with neon-architecture, if it unites with three
hydrogen atoms to form ammonia NH₃; but the three hydrogen nuclei,
although having symmetrical positions, will not lie in the same plane
as the nitrogen nucleus. The electric centre of gravity for the
positive nuclei will therefore not coincide with the centre of gravity
for the negative electron system. The molecule obtains thus what might
be called a positive and a negative pole, and this dipolar character
appears in the electrical action of ammonia (its dielectric constant).
Something similar holds true for the water molecule, where, in a
neon-architecture of electrons, in addition to the oxygen nucleus in
the centre there are two hydrogen nuclei which are not co-linear with
the oxygen nucleus.

If we go on from neon in the periodic system we come to _sodium_
(11). When the sodium nucleus captures electron No. 11, this cannot
find room in the neon-architecture formed by the first ten electrons.
Since the eleventh electron thus cannot find a place in either a 2₁ or
a 2₂ orbit, it is bound in a 3₁ orbit (cf. Fig. 29 and diagram at the
end). The atom then has a character like that of the lithium atom, and
we can therefore understand the chemical relationship between the two
elements, which are both monovalent electropositive metals.

We shall not dwell longer upon the individual elements of the atomic
series. If we pass from neon through sodium (11), magnesium (12),
aluminium (13), etc., to _argon_ (18), we get what is essentially
a repetition of the situation in the series from lithium to neon. We
first get two orbits of the 3₁ type in magnesium, a 3₂ orbit is for
the first time added in aluminium, and for the atomic number 18, eight
3-quantum orbits, together with the eight orbits of the inner 2-quantum
group and the two of the innermost 1-quantum group, give the symmetric
architecture of _argon_ (cf. table on p. 196, and diagram at the
end).

The architecture of the argon atom is in a certain sense less complete
than that of the neon atom. In argon there are indeed four orbits of
the 3₁ type and four of the 3₂ type, but the third kind of 3-quantum
orbit, the circular 3₃ one, is missing. Nor does it appear in the next
element, _potassium_ (19). The electron No. 19 prefers, instead
of the 3₃ orbit, the 4₁ orbit, which consists of oblong loops and
which gives a firmer binding because it dives in among the electrons
bound earlier, while the circle 3₃ would lie outside them all. We
thus obtain an atom of type similar to the lithium and sodium atoms.
But the slighted 3₃ path lies, so to speak, on the watch to steal a
place for itself in the neutral atom, and this has grave results for
the subsequent development. Even in _calcium_ (20), after the
first eighteen electrons are bound in the argon architecture, both the
nineteenth and the twentieth go into a 4₁ orbit, and the behaviour of
calcium is like that of magnesium. But since the increasing nuclear
charge means for the electron No. 19 a decrease in the dimensions and
an increase in the binding of the orbits corresponding to the quantum
number 3₃, a point will finally be attained where the 3₃ orbit of the
nineteenth electron lies within the boundaries of what may be called
the argon system, _i.e._, the architecture corresponding to
the first eighteen electrons, and corresponds to a stronger binding
than a 4₁ orbit would do. In _scandium_ (21) the 3₃-type orbit
occurs for the first time in the neutral atom and will not only come
into competition with the 4₁ type, but will also cause a disturbance
in the 3-quantum groups, which in the following elements must undergo
reconstruction. As long as this lasts the situation is very complicated
and uncertain. When the reorganization is almost completed, we come to
the blotting out of chemical differences, particularly known from the
triad, _iron_, _cobalt_ and _nickel_. Moreover, there
comes a fluctuation in the valency of the elements. Iron can, as has
been said, be divalent, trivalent or hexavalent. This oscillation in
valency begins in _titanium_.

We should perhaps expect that the reconstruction would be completed
long before nickel (28) is reached, because even with twenty-two
electrons we could get four orbits of each of the 3-quantum types
(3₁, 3₂ and 3₃); but from the chemical facts we are led to assume
that in a completed group of 3-quantum orbits there can be room for
six electrons in each sub-group. At first sight we should, then,
expect the end of the reconstruction with nickel, which has indeed
eighteen electrons more than neon where the group of 2-quantum orbits
was completed. We might expect that nickel would be an inert element
in the series with helium, neon, and argon. On the contrary, nickel
merely imitates cobalt. This is explained by the fact that the group of
eighteen 3-quantum orbits, although it has a symmetric architecture,
is weakly constructed if the nuclear charge is not sufficiently large.
The binding of this group is too weak for it to exist as the outer
group in a neutral atom. In _nickel_ the electrons, in a less
symmetrical manner, will probably arrange themselves with seventeen
3-quantum orbits and one 4-quantum orbit.

The group of eighteen 3-quantum orbits becomes stable, however, when
the nuclear charge is equal to or larger than 29, in which case it
can become the outer group in a positive ion. In this we find the
explanation of the properties of the atom of _copper_. The
neutral copper atom has its twenty-ninth electron bound in a 4₁ orbit
consisting of oblong loops (cf. diagram at the end); this electron
can easily be freed and leaves a positive monovalent copper ion with
a symmetrical architecture. Even under these circumstances, although
possessing a certain stability, the ion is not very firmly constructed.
Thus the fact that copper can be both monovalent and divalent, must be
explained by the assumption that for a nuclear charge 29, the 3-quantum
group still easily loses an electron.

When we come to _zinc_ (30) the group of eighteen is more firmly
bound; zinc is a pronounced divalent metal which in its properties
reminds us of calcium and magnesium. From zinc (30) to krypton (36) we
have a series of elements which in a certain way repeat the series from
magnesium (12) to argon (18).

In Fig. 34 is shown Bohr’s arrangement of the periodic system in which
the systematic correlation of the properties of the element appears
somewhat clearer than in the usual plan (cf. p. 23). It shows great
similarity with an arrangement proposed nearly thirty years ago by the
Danish chemist, Julius Thomsen. The elements from scandium to nickel,
where, in the neutral atom, the electron group of 3-quantum orbits is
in a state of reconstruction, are placed in a frame; the neutral
oblique lines connect elements which are “homologous,” _i.e._,
similar in chemical and physical (spectral) respect.

[Illustration: FIG. 34.—The periodic system of the elements.
The elements where an inner group of orbits is in a stage of
reconstruction are framed. The oblique lines connect elements which in
physical and chemical respects have similar properties.]

In _krypton_ (36) we again have a stable architecture with an
outer group of eight electrons, four in 4₁ orbits and four in 4₂
orbits. Owing to the appearance of 4₃ orbits in the normal state of the
atoms of elements with atomic number higher than 38, there follows in
the fifth period of the natural system a reconstruction and provisional
completion of the 4-quantum orbits to a group of eighteen electrons,
which shows a great simplicity with the completion of the 3-quantum
group in the fourth period. In Fig. 34 the elements where the 4-quantum
group is in a state of reconstruction are framed. The 4-quantum group
with eighteen electrons is of more stable construction than the group
of eighteen 3-quantum orbits in the elements with an atomic number
lower by eighteen. This is due to the fact that all the orbits in the
first-mentioned group are oblong and therefore moored, so to say, in
the inner groups, while in the complete group of 3-quantum orbits there
are six circular orbits. This is the reason why silver, in contrast to
copper, is monovalent.

The next inactive gas is _xenon_ (54), which outside of the
4-quantum group has a group of eight electrons in 5-quantum orbits,
four in 5₁ orbits and four in 5₂ orbits. We notice that in xenon the
group of 4-quantum orbits still lacks the 4₄ orbits. On the theory
we must, therefore, expect to meet a new process of completion and
reconstruction when proceeding in the system of the elements. The
theoretical argument is similar to that which applies in the case
of the completion of the 3-quantum group which takes place in the
fourth period of the natural system. In fact, in the formation of
the normal atoms of the elements next after xenon, caesium, 55,
and barium, 56, the fifty-four electrons first captured will form a
xenon configuration, while the fifty-fifth electron will be bound in
a 6₁ orbit, consisting of very oblong loops, which represents a much
stronger binding than a circular 4₄ orbit. Calculation shows, however,
that with increasing nuclear charge there must soon appear an element
for which a 4₄ orbit will represent a stronger binding than any other
orbit. This is actually the case in _cerium_ (58), and starting
from this element we meet a series of elements where, in the normal
neutral atom, the 4-quantum group is in a state of reconstruction.
This reconstruction must occur far within the atom, since the group of
eighteen 4-quantum orbits in xenon is already covered by an outer group
of eight 5-quantum orbits. The result is a whole series of elements
with very slight outward differences between their neutral atoms, and
therefore with very similar properties. This is the _rare earths_
group, which in such a strange way seemed to break down the order
of the natural system (cf. p. 21), but which thus finds its natural
explanation in the quantum theory of the structure of the atom.

The elements in which the 4-quantum group is in a state of
reconstruction are, in Fig. 34, enclosed in the inner frame in the
sixth period. Moreover, in the outer frame all elements are enclosed
where the group of 5-quantum orbits is in a state of reconstruction,
which started, even before cerium in lanthanum (57), where the
fifty-fifth electron in the normal state is bound in a 5₃ orbit. The
element _cassiopeium_, with atomic number 71, which is the last
of the rare earths, stands just outside the inner frame, because
in the normal neutral atom of this element the 4-quantum group is
just completed; this group, instead of eighteen electrons with six
electrons in each sub-group, consists now of thirty-two electrons with
eight electrons in each sub-group. The theory was able to predict that
the element with atomic number 72, which until a short time ago had
never been found, and the properties of which had been the subject of
some discussion, must in its chemical properties differ considerably
from the trivalent rare earths and show a resemblance to the
tetravalent elements zirconium (40), and thorium (90). This expectation
has recently been confirmed by the work of Hevesy and Coster in
Copenhagen, who have observed, by means of X-ray investigations,
that most zircon minerals contain considerable quantities (1 to 10
per cent.) of an element of atomic number 72, which has chemical
properties resembling very much those of zirconium, and which on this
account had hitherto not been detected by chemical investigation. A
preliminary investigation of the atomic weight of this new element, for
which its discoverers have proposed the name _hafnium_ (Hafnia =
Copenhagen), gave values lying between 178-180, in accordance with what
might be expected from the atomic weight of the elements (71) and (73).
(Cf. p. 23.)

The further completion of the groups of 5- and 6-quantum orbits, which
in the rare earths had temporarily come to a standstill, is resumed
in hafnium and goes on in a way very similar to that in which the 4-
and 5-quantum groups in the fifth, and the 3- and 4-quantum groups in
the fourth period underwent completion. Thus the reconstruction of
the 5-quantum group which began in lanthanum, and which receives a
characteristic expression in the triad of the platinum metals, has come
to a provisional conclusion in _gold_ (79), gold being the first
element outside the two frames which, in Fig. 34, appear in the sixth
period. The neutral gold atom possesses, in its normal state—besides
two 1-quantum orbits, eight 2-quantum orbits, eighteen 3-quantum
orbits, thirty-two 4-quantum orbits and eighteen 5-quantum orbits—one
loosely bound electron in a 6₁ orbit.

In _niton_ (86), finally, we meet again an inactive gas, the
structure of the atom of which is indicated in the table on p. 196.

In this element the difference between nuclear and electron properties
appears very conspicuously, since the structure of the electron system
is particularly stable, while that of the nucleus is unstable. Niton,
in fact, is a radioactive element which is known in three isotopic
forms; one of these is the disintegration product of radium, the
so-called radium emanation; it then has a very brief life. In the
course of four days over half of the nuclei in a given quantity of
radium emanation will explode.

In the diagram at the end of the book, as an example of an atom with
very complicated structure, there is given a schematic representation
of the atom of the famous element _radium_, on a scale twice
as large as the one used in the other atoms. It follows clearly
enough, from what has been said in Chap. IV., that the structure of
the electron system has nothing to do with the radioactivity. All
the remarkable radiation activities are due to the nucleus itself.
There has not even been room in the figure to draw the nucleus; the
1-quantum orbits consist only of a small cross, and in the other groups
we have contented ourselves with summary indications. The electron
system, with its eighty-eight electrons, is, however, in itself very
interesting, with its symmetry in the number of electrons in the
different groups. In the different quantum groups from 1 to 7 there are
found respectively two, eight, eighteen, thirty-two, eighteen, eight
and two electrons. The last group is naturally of a very different
nature from the first; they are “valence electrons,” which easily
get loose and leave behind a positive radium ion with stable niton
architecture. Radium then belongs to the family of the divalent metals,
magnesium, calcium, strontium and barium.

Four places from radium is _uranium_ (92) and the end of the
journey, if we restrict ourselves to the elements which are known to
exist. One could very well continue the building-up process still
further and discuss what structure would have to be assumed for the
atoms of the elements with higher atomic numbers. That they cannot
exist is not the fault of the electron system but of the nuclei, which
would become too complicated and too large to be stable. In the table
on p. 196 there is shown the probable structure of the atom of the
inert gas following niton; it must be assumed to have one hundred and
eighteen electrons distributed in groups of two, eight, eighteen,
thirty-two, thirty-two, eighteen and eight among the quantum groups
from 1 to 7.

As has been said, in all this symmetrical structure of the atoms
of the elements, Bohr has in many cases had to rely upon general
considerations of the information that observation gives about the
properties of the individual elements. It must, however, not be
forgotten that the backbone of the theory is and remains the general
laws of the quantum theory, applied to the nucleus atom in the same
way as they originally were applied to the hydrogen atom, leading
thereby to the interpretation of the hydrogen spectrum.

We have, further, a most striking evidence as to the correctness of
Bohr’s ideas in the fact that not only do the pictures of the atoms
which he has drawn agree with the known chemical facts about the
elements, but they are also able to explain in the most satisfactory
manner possible the most essential features of the characteristic X-ray
spectra of the different elements, a field we shall not enter upon here.

In all that has been said above we have been considering the Bohr
theory simply as a means of gaining a deeper understanding of the laws
which determine activities in the atomic world. Perhaps we shall now be
asked if we can “utilize” the theory, or, in other words, if it can be
put to practical use.

To this natural and not unwarranted question we may first give the
very general answer, that progress in our knowledge of the laws of
nature always contributes sooner or later, directly or indirectly, to
increase our mastery over nature. But the connection between science
and practical application may be more or less conspicuous, the path
from science to practical application more or less smooth. It must be
admitted that the Bohr theory, in its present state of development,
hardly leads to results of direct practical application. But since
it shows the way to a more thorough understanding of the details
in a great number of physical and chemical processes, where the
peculiar properties of the different elements play parts of decided
importance, then in reality it offers a wealth of possibilities for
making predictions about the course of the processes—predictions
which undoubtedly in the course of time will be of practical use in
many ways. In this connection the discovery of the element hafnium,
discussed on p. 204, may be mentioned. It must be for the future to
show what the Bohr theory can do for technical practice.

Below is given an explanation of the different symbols which occur
at various places in the book; also the values of important physical
constants.

1 m. = 1 metre; 1 cm. = 1 centimetre = 0·394 inches.

1 μ = 1 micron = 1/1000 of a millimetre = 0·0001 cm. = 10⁻⁴ cm.

1 μμ = 1/1,000,000 of a millimetre = 10⁻⁷ cm.

1 cm.³ = 1 cubic centimetre.

1 g. = 1 gram; 1 kg. = 1 kilogram = 2·2 pounds.

1 kgm. = 1 kilogrammetre (the work or the energy required to lift 1 kg.
1 m.).

1 erg = 1·02 × 10⁻⁸ kgm. = 7·48 × 10⁻⁸ foot-pounds.

λ represents wave-length.

ν represents frequency (number of oscillations in 1 second).

ω represents frequency of rotation (number of rotations in 1 second).

_n_ represents an integer (particularly the Bohr quantum numbers).

The velocity of light is _c_ = 3 × 10¹⁰ cm. per second = 9·9 × 10⁸
feet per second.

The wave-length of yellow sodium light is 0·589 μ = 589 μμ = 2·32 ×
10⁻⁵ inches.

The frequency of yellow sodium light is 526 × 10¹² vibrations per
second.

The number of molecules per cm.³ at 0° C. and atmospheric pressure is
about 27 × 10¹⁸.

The number of hydrogen atoms in 1 g. is about 6·10²³.

The mass of a hydrogen atom is 1·65 × 10⁻²⁴ g.

The elementary quantum of electricity is 4·77 × 10⁻¹⁰ “electrostatic
units.”

The negative electric charge of an electron is 1 elementary quantum (1
negative charge).

The positive electric charge of a hydrogen nucleus is 1 elementary
quantum (1 positive charge).

The mass of an electron is ¹⁄₁₈₃₅ that of the hydrogen atom.

The diameter of an electron is estimated to be about 3 × 10⁻¹³ cm.

The diameter of the atomic nucleus is of the order of magnitude 10⁻¹³
to 10⁻¹² cm.

The diameter of a hydrogen atom in the normal state (the diameter of
the first stationary orbit in Bohr’s model) is 1·056 × 10⁻⁸ cm.

The Balmer constant K = 3·29 × 10¹⁵.

The Planck constant _h_ = 6·54 × 10⁻²⁷.

An energy quantum is E = _h_ν.

The Balmer-Ritz formula for the frequencies of the lines in the hydrogen
spectrum is

        (   1        1    )
  ν = K ( ------ - -------)
        (  _n″_²    _n′_² )


[Illustration: DIAGRAMS OF SPECTRA AFTER BUNSEN AND KIRCHHOFF]

[Illustration: STRUCTURE OF THE RADIUM ATOM]

[Illustration: MAIN LINES OF THE ATOMIC STRUCTURE OF SOME ELEMENTS]