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.