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_³. 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.