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Damping of Molecular Oscillations and Elementary Radiation.
S. Vavilov.
I. Survey of Theoretical Investigations.
For more than 30 years the solid edifice of theoretical optics, created by the 19th century, has lain in ruins after the devastating blows dealt by the Michelson–Morley experiment and Planck’s theory of black radiation. The Michelson experiment destroyed the foundation of the classical optics of a pure ether; Planck’s quanta completely obscured the understanding of the optics of a material medium. On the ruins of the old coherent system, separate supports of the future edifice are very slowly taking shape; its general plan and outlines are not clear to the contemporary physicist.
In our brief outline of the state of the question of the mechanism of elementary radiation, we shall encounter that duality of points of view which is inevitable in contemporary optics until a new theoretical system arises in place of the ruined old one. We shall concentrate chiefly on the problem of the damping of molecular oscillations not subjected to an external perturbing force. This problem has been the subject of a number of major experimental and theoretical investigations in recent years.
§ 1. Radiation of the electron. In the classical electron theory, the cause of radiation, i.e. of an electromagnetic perturbation in the external medium, can be only a change in the state of the charged parts of the molecule, the nucleus and the electrons. If one excludes deformation of the charged particles, then the cause of radiation can prove to be only the acceleration of their motions. The problem of the relation between the acceleration of the motion of an electron and the flow of energy into the external medium has been solved repeatedly, and by different methods¹), always leading to one and the same result²). The slowing of the motion of an electron as a consequence of radiation corresponds to a retarding force:
\[ f=\frac{e^{2}v}{6\pi c^{3}}\ldots\ldots\ldots\ldots (1) \]
where \(e\) is the charge of the electron in rational units \((1/\sqrt{4\pi}\times \text{e.s.u.})\), \(v\) is the second derivative of the velocity vector, \(c\) is the speed of light. If the elec—
¹) A. Liénard. L’éclairage électrique 16, pp. 1, 53, 106, 1898; M. Planck. Wärmestrahlung, p. 106, 1906; M. Abraham. Elektromagnetische Theorie der Strahlung, p. 72, 1905, etc.
²) Th. Wereide (Ann. d. Phys. 52, p. 276, 1917) found that the acceleration in eq. (1) enters only scalarly, but this conclusion is erroneous.
if without radiation the electron would perform harmonic linear oscillations, then in the presence of radiation the oscillations will become damped,
\[ x = A \cdot \varepsilon^{-\delta t}\cdot \cos \omega_0 t \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (2), \]
where \(A\) is the initial amplitude, \(\varepsilon\) is the base of natural logarithms, \(\delta\) is the damping decrement, and \(\omega_0\) is the angular frequency of the oscillations. On the basis of (1) one may find\(^{1}\) that
\[ \delta=\frac{1}{2}\cdot \frac{e^2\cdot \omega_0^2}{6\pi c^3 m} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (3). \]
Substituting for \(\omega_0\) its value \(\dfrac{2\pi \cdot c}{\lambda}\), where \(\lambda\) is the wavelength, we find
\[ \delta=\frac{\pi \cdot e^2}{3m \cdot c \lambda^2}. \]
How general is equation (1), and what assumptions underlie its derivation? This question acquires substantial importance in connection with the widespread quantum theory of Bohr, in which the absence of radiation during the circular uniform motion of an electron about the nucleus is postulated\(^{2}\). The corresponding analysis of equation (1) was carried out by Schott\(^{3}\). Schott points out that in some derivations of (1) two premises are taken: 1) the fundamental equations of the electron theory of Maxwell–Lorentz; 2) Poynting’s theorem on the flow of energy. Poynting’s theorem, however, is not the only and necessary expression for the flux and density of energy\(^{4}\). One can give, for example, such expressions in which the flow of energy is directed along the front of a plane wave, and thus there is no radiation outward. On this basis attempts were made to criticize the postulate of Bohr just mentioned. The derivation of equation (1), however, can also be given independently of Poynting’s theorem\(^{5}\). The only assumptions in the derivation of (1) are Maxwell’s equations for a field free of charges and Lorentz’s equations for an intra-electronic volume. In connection with Bohr’s theory Schott solves the following problem: is it possible to obtain the rotation of an electron in a circle without radiation, if one assumes that Lorentz’s equations inside the electron are not exact?
For the external space Maxwell’s equations hold
\[ \operatorname{rot}\mathbf{H}-\frac{1}{c}\frac{\partial \mathbf{E}}{\partial t}=0 \qquad \operatorname{rot}\mathbf{E}+\frac{1}{c}\frac{\partial \mathbf{H}}{\partial t}=0 \qquad \operatorname{div}\mathbf{E}=0 \qquad \operatorname{div}\mathbf{H}=0 \ . \ . \ (5). \]
\(^{1}\) Cf., for example: W. Mandersloot, Jahrb. d. R. u. E., 13, p. 1, 1916.
\(^{2}\) Cf. P. Einstein. Application of the doctrine of quanta to the theory of spectral lines. Uspekhi fizicheskikh nauk 2, p. 14, 1920.
\(^{3}\) G. A. Schott. Phil. Mag. 36, p. 234, 1918.
\(^{4}\) G. Livens. Phil. Mag. 34, p. 386, 1917.
\(^{5}\) Cf., for example, H. A. Lorentz, The theory of electrons, p. 251, 1909, or G. A. Schott, Electromagnetic radiation, Ch. XI and App. C, D and F.
where $\mathbf E$ and $\mathbf H$ are the electric and magnetic forces, $\operatorname{rot}$ and $\operatorname{div}$ are symbols of vector analysis. For the intra-electronic volume, equation (5) is rewritten in the following entirely general form:
\[ \operatorname{rot}\mathbf H-\frac{1}{c}\frac{\partial \mathbf E}{\partial t}=c\operatorname{rot}\mathbf E+\frac{1}{c}\frac{\partial \mathbf H}{\partial t}=-K \quad \operatorname{div}\mathbf E=\rho \quad \operatorname{div}\mathbf H=\mu \ldots (6). \]
$\rho$ and $\mu$ are the densities of electric and magnetic charges and currents. In Lorentz’s theory:
\[ \mu=0 \quad c=\frac{\rho\cdot v}{c} \quad K=0 \ldots \ldots \ldots (7). \]
In Schott’s theory $\rho,\mu,c$ and $K$ are restricted only by equation (6). The complex analysis carried out by Schott shows that radiation outward does not vanish even for such a completely general form of the intra-electronic equations as (6).
Thus, in order to interpret Bohr’s postulate, one must either abandon Maxwell’s equations (5) for the pure ether, or else assume the existence of sources of energy which compensate the loss of energy through radiation in the atom.
Schott’s analysis in any case allows one to conclude that equation (1) is a completely inevitable and the only expression for the radiation of an electron in the classical theory.
Taking one or another electronic model of the atom as given, we can obtain an entirely exhaustive answer concerning the radiation of such an atom. The frequency of oscillations, amplitude, damping, and polarization can in principle always be found, if the initial conditions and constraints are given and the atom is outside perturbing external forces. Such a classical model of the radiating atom in the simplest case was analyzed by Planck1.
§ 2. The fundamental postulate of quantum theory. The classical atom of electron theory is not capable of explaining the peculiarities of temperature black radiation. The laws of the action of light on a material medium (the photoelectric effect, photochemical reactions, fluorescence), the regularities of spectral lines and bands likewise do not accord with the classical model. The way out, first indicated by Planck, consisted in abandoning the classical atom. In place of the old theory, distinguished by strictness and consistency, it was necessary to erect temporary structures, “quantum theories,” which assumed, for almost every group of experimental facts subject to interpretation, different forms not always compatible with one another. Planck himself gave two variants of quantum theories suitable for understanding the laws of black radiation2. According to the first variant, the emission and absorption of radiant energy by an elementary oscillator can occur only in whole quanta. The second variant
leaves absorption “classical,” i.e. continuous; only emission occurs in whole quanta. Stark1 and Einstein2 create the hypothesis of “light quanta,” i.e. quanta of radiant energy, existing discretely and outside the material medium. This hypothesis beautifully explains the laws of the photoelectric effect, the peculiarities of photochemical reactions and fluorescence, but it stands, for example, in contradiction with the laws of black radiation and with the basic facts of physical and geometrical optics3. Bohr’s theory, elaborated in detail by Sommerfeld, brilliantly explains the spectral regularities, but has it succeeded in embracing the other facts of the optics of a material medium?4 Thus the hypotheses that at the present time bear the name “quantum theory” represent a motley and contradictory conglomerate of ad hoc constructions, whose existence is undoubtedly short-lived and which are raw material for a future harmonious system. In all variants of quantum theory, however, one can single out the following basic postulate: “The transfer of the internal energy of molecules (or of an atom) to the external medium (the ether or other molecules) can occur only in whole quanta \(h\nu_0\),” where \(\nu_0\) corresponds to the proper frequency of oscillations of the atom of the classical theory, and \(h\) is Planck’s constant. The transfer may be effected in various ways—by radiation, by the conversion of internal energy into kinetic energy, etc.
Let us see how firm are the direct experimental foundations of this fundamental postulate of the new optics? We leave aside all the numerous cases of indirect proofs of the postulate, whose number is growing almost every day, but in which the appearance of the quantity \(h\nu_0\) is not always clear5. We regard the following as direct experimental proof of the postulate. Suppose that in some physical phenomenon a group of molecules \(N\) over a time \(t\) has transferred its internal energy to the external medium, the magnitude of the total transferred energy being \(E\). If the conditions of the experiment are such that each molecule that has carried out the transfer of energy is, so to speak, “put out of action,” i.e. is destroyed, then, having measured experimentally \(N\) and \(E\), on the basis of the postulate we must, with sufficient approximation, obtain for the quantum:
\[ \frac{E}{N}=h\nu_0 \ldots\ldots\ldots\ldots\ldots (8). \]
We have carefully added “with sufficient approximation” because the postulate allows the transfer not only of one, but also of several quanta
at once, as, for example, must be assumed in order to obtain the formula for black radiation1. In the general case postulate (8) will be rewritten as follows:
\[ \frac{E}{N} \geq h\nu_0 \ldots\ldots\ldots\ldots\ldots (9). \]
There are a sufficient number of occasions for an experimental verification of (9), indicated, for example, by chemical and photochemical reactions, radioactivity, chemiluminescence, etc. In all the indicated cases \(E\) and \(N\) can be determined directly or indirectly. Relation (9) has in fact been tested, however, in very few cases: rather systematically in the study of photochemical processes2 and in individual cases of chemical reactions3. In all those cases where the physicochemical conditions were sufficiently clear, relation (9) was confirmed. Consideration of the data already available and the setting up of new measurements for testing postulate (9) are in any case extremely important. As an example we shall consider the radioactive decay of \(Ra\,C\); the decay is accompanied by the emission of \(\alpha\)-, \(\beta\)- and \(\gamma\)-rays, while the total energy released in the decay of \(Ra\,C\), which is in equilibrium with the other decay products in \(1\ \mathrm{gr.}\) of \(Ra\), is equal to \(50.2\ \mathrm{cal}\) per hour4. The number of atoms of \(Ra\,C\) decaying per hour is \(1.22 \cdot 10^{14}\)5. Hence:
\[ \frac{E}{N}=\frac{50.2}{1.22\cdot 10^{14}}\ \mathrm{cal}=1.74\cdot 10^{-5}\ \mathrm{erg}. \]
On the basis of postulate (9)
\[ h\nu_0 \leq 1.74\cdot 10^{-5}\ \mathrm{erg}, \]
\[ \nu_0 \leq 2.65\cdot 10^{21}. \]
Such a result does not contradict experiment. It is most probable to suppose that the proper frequency of \(Ra\,C\) coincides with one of the frequencies of the characteristic \(\gamma\)-radiation of \(Ra\,C\), but according to Rutherford’s estimate6 the limiting frequency of the \(\gamma\)-rays emitted by \(RaC\) is \(4.28\cdot 10^{20}\), i.e. less than \(2.65\cdot 10^{21}\), in agreement with postulate (9). The existing factual material is too scanty to make a final conclusion about the correctness of the postulate, but in any case there are as yet no data contradicting it. Let us note, however, that experimental confirmation of (9) still does not make it possible to assert categorically the quantum character of every separate, elementary act of radiation. In experiment we always obtain an average statistical result, valid for an enormous number of molecules, but not obligatory
for each separately. In any case, postulate (9) leaves the mechanism of elementary radiation entirely unexplained, determining only the total energy of the mean act of radiation.
Condition (9) is the only one common to all variants of the quantum theory. The detailing of the problem of radiation has been carried out in all directions only in Bohr’s theory; therefore in what follows we shall have to speak exclusively about this variant of quantum theory, while referring the reader who is little acquainted with the foundations of Bohr’s theory to Einstein’s survey.¹)
§ 3. Rubinowicz’s selection principle (Auswahlprinzip).
In Bohr’s theory the natural frequency of vibrations of an atom \(\nu_0\) loses its definite meaning, since the atom itself may have infinitely various dimensions depending on whether the electrons are situated on one or another stationary orbit. With every emission of radiation by an atom, i.e. with the transition of electrons from some orbits to others, according to postulate (8) energy \(h\nu\) appears or disappears in the external medium, but \(\nu\) itself is determined by the magnitude of the quantum of energy, and not conversely. Bohr formulates this postulate as follows:
\[ h\nu = W_1 - W_2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots (10), \]
where \(W_1\) and \(W_2\) are the energy of the system corresponding to the initial and the final stationary orbit. Postulate (10), borrowed by Bohr from Einstein’s theory of the photoelectric effect, is considered the starting point of the theory. According to Stark’s remark²) postulate (10) amounts to this: an event subsequent in time determines the character of the preceding one.
Among the few classical laws of electrodynamics that remain unviolated in Bohr’s theory is the law of conservation of energy. Mechanically this law is derived from Newton’s second law, which is broader in content: the principle of conservation of quantity of motion, or momentum. Newton’s equations of motion fell away, if only because relativistic mechanics—Einstein’s³)—is being implanted in its place. The momentum of an atom in Bohr’s theory is
\[ p = \frac{nh}{2\pi} \ldots \ldots \ldots \ldots \ldots \ldots \ldots (11), \]
where \(n\) is an integer. In classical electrodynamics the momentum of the radiation of an atom must pass entirely into the external medium. Can we require that this principle be fulfilled in Bohr’s theory, and what limitations do we obtain thereby? This question was posed and solved by Rubinowicz.⁴) What is the magnitude of the momentum of radiation—
¹) See Einstein, loc. cit.
²) J. Stark, Jahrb. d. R. u. E. 17, p. 150, 1920.
³) Cf., e.g., F. Stade. Phil. Mag. 40, p. 31, 1920.
⁴) A. Rubinowicz. Phys. Ztschr. 19, pp. 441, 465, 1918.
during emission in the classical theory? For simplicity let us consider the case of a wave circularly polarized1. If such a wave falls upon a material medium consisting of charged nuclei and electrons, then the electrons will begin to rotate in a circle with angular velocity \(\frac{2\pi}{T}\). A rotating electron radiates, as we saw in § 1; consequently, in order to maintain it in its former orbit a certain amount of work is expended, drawn from the incident light; the magnitude of this work is \(M \cdot \frac{2\pi}{T}\), where \(M\) is the moment of rotation. If we denote by \(W\) the light energy absorbed per unit time, then
\[ W = M \cdot \frac{2\pi}{T}, \]
whence:
\[ M=\frac{W}{2\pi \nu}\ldots\ldots\ldots\ldots\ldots (12). \]
This expression obviously also gives the angular momentum of the wave emitted by an electron rotating in a circle. For the general case of an elliptically polarized wave, Abraham2 derived the following expression for the angular momentum of the wave:
\[ M=\frac{W}{2\pi \nu}\cdot \frac{2ab\sin\gamma}{a^2+b^2}\ldots\ldots\ldots\ldots (13). \]
Here \(a,b\) are the semiaxes of the ellipse of oscillation, and \(\gamma\) is the phase difference of the component rectilinear oscillations along \(a\) and \(b\). For a circularly polarized ray, \(a=b\), \(\gamma=\frac{\pi}{2}\), i.e. (12) is satisfied. Rubinowicz assumes that equation (13) is also valid for Bohr’s atom. On the basis of (8)
\[ W=h\nu. \]
Consequently,
\[ M=\frac{h}{2\pi}\cdot \frac{2ab\sin\gamma}{a^2+b^2}\ldots\ldots\ldots\ldots (14). \]
According to Bohr’s theory, the change of the angular momentum of the atom (11) can occur only in such a way that \(n\) remains an integer. Let us denote the new value of the integer in (11), after the transfer of the angular momentum (14) to the external medium, by \(n'\). Then, on the basis of the principle of conservation of angular momentum:
\[ \frac{nh}{2\pi}-\frac{n'h}{2\pi} = \frac{h}{2\pi}\cdot \frac{2ab\sin\gamma}{a^2+b^2} \]
or:
\[ n-n'=\frac{2ab\sin\gamma}{a^2+b^2}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (15) \]
\(a\) and \(b\) enter into this expression symmetrically; let, for example, \(a>b\); in that case \((a-b)^2>0\), i.e. \(a^2+b^2-2ab>0\)
or:
\[ a^2+b^2>2ab, \]
and a fortiori
\[ a^2+b^2>2ab\sin\gamma\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (16). \]
Consequently,
\[ n-n'\leq 1\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (17). \]
The numerator and denominator of equality (15) are equal only in the case when \(a=b,\ \gamma=\pm\frac{\pi}{2}\). But according to Bohr’s theory \(n-n'\) can only be an integer.
Such a condition is compatible with expression (17), obtained from the principle of conservation of momentum, only for three cases:
\[ n-n'= \begin{cases} +1\\ 0\\ -1 \end{cases} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (18). \]
We thus arrive at a remarkable rule showing that not all spectral lines possible on the basis of (10) and Ritz’s combination principle1 are compatible with the principle of conservation of angular momentum; this is the selection rule discovered by Rubinowicz.
Rule (18) also makes it possible to draw further, no less important, conclusions concerning the polarization of the emitted wave. Indeed, for \(n-n'=\pm 1,\ a=b,\ \gamma=\pm\frac{\pi}{2}\), i.e. the wave is necessarily polarized circularly to the right or to the left; for \(n-n'=0\), either \(b=0\), or \(\gamma=0\); in all cases the wave is linearly polarized. Under the usual conditions of observation of spectral lines, the emitting atoms have no definite orientation and therefore the polarization of the lines escapes observation; it begins to manifest itself definitely only when strong magnetic or electric fields are imposed, orienting the emitting molecules (the Zeeman and Stark phenomena). Rubinowicz extended his theory to the case of oscillators with several degrees of freedom, and also to the case when the molecules are under the action of an external field. It is precisely this part of the theory that provides abundant experimental material for testing the new principle (the fine structure of spectral lines, the splitting of lines in a magnetic and an electric field). The theory was irreproachably confirmed by experiment.
Thus, restricting Bohr’s theory to the principle of conservation of the moment of momentum made it possible at once to determine the essential characteristic of the emitted wave, its polarization, and at the same time to exclude the enormous number of spectral lines possible according to Bohr’s theory and Ritz’s combination principle, as incompatible with Newton’s second law.
Almost simultaneously with the appearance of Rubinowicz’s work, Bohr published¹) an extremely interesting investigation in which he formulates a principle of correspondence between the frequencies of the radiation and the times of revolution of the electron in the corresponding orbit. The reader will find an exposition of this principle in the article by Yu. A. Krutkov²). In contrast to Rubinowicz’s selection principle, the correspondence principle is of a completely empirical character; the justification of this principle is so far only in experience. In Sommerfeld’s words, “Bohr found in his correspondence principle the magic wand that makes it possible to use the results of classical wave theory for the theory of quanta without removing the fundamental contradictions.” The new empirical principle of Bohr at once resolves the question of the intensity and polarization of radiation, including, as a part, all the results of Rubinowicz’s principle. The theoretical essence of the correspondence principle is not yet clear³). The strange correspondence between classical theory and the principles of quantum theory, revealed by Bohr’s work, gives hope of approaching more closely the riddle of quanta, posed almost 20 years ago and still unsolved. For the sake of a proper historical perspective, let us note, however, that Planck, in his theory of black radiation, while solving the problem completely, i.e. determining at once the energy, frequency, and polarization of the radiation, was compelled, considerably earlier than Bohr, to establish a principle of correspondence, equating the results of the quantum and classical theories⁴).
§ 4. The problem of the damping of oscillations in Bohr’s theory. The characterization of radiation given by all the above-stated formal postulates and principles of Bohr’s theory is still incomplete. What is lacking is an equation determining the course of radiation in time—its damping. After the appearance of Rubinowicz’s work and Bohr’s correspondence principle, it was quite natural to try to proceed in the same direction of seeking a correspondence between the classical and quantum theories also for the damping of radiation. Therefore, almost simultaneously there appeared the works of Epstein⁵), Reiche⁶), and Wien⁷), which attempt to find a solution of the problem of damping by extending the correspondence principle—
¹) N. Bohr. Kopenhagen Academie, 1918.
²) Ю. А. Крутков. The principle of Bohr’s analogy in quantum theory, see p. 272.
³) A. Sommerfeld. Atombau, p. 527. 1921.
⁴) M. Planck. Wärmestrahlung. p. 159. 1913.
⁵) P. S. Epstein. Sitzb. d. math.-phys. Klasse der Bayer Ak. d. W. zu München, 1919, Heft 1, p. 73.
⁶) F. Reiche. Phys. Zeitschr. 20, p. 296. 1919.
⁷) W. Wien. Ann. d. Phys. 60, p. 587. 1919.
...of action. The theory developed in the works indicated is identical; in our survey we adhere to Einstein’s exposition.
The difference between the points of view of the classical and the quantum theories on the mechanism of radiation is as follows. The classical electron radiates continuously, with a continuous change of orbit. The quantum electron, being on definite orbits, as we have seen, does not radiate. Radiation occurs only during the limited interval of time of the electron’s jump from one orbit to another.
In his correspondence principle Bohr states that the classical and quantum theories lead to practically identical results with respect to the frequency, polarization, and also the intensity of the emitted spectral lines for long waves, and postulates the same correspondence for waves of any length. Einstein extends Bohr’s proposition as follows: “In the limiting case of infinitely large orbits, from the beginning of the electron’s transition from one orbit to another the laws of classical electrodynamics come into force, and the motion of the electron is accompanied by classical radiation until the nearest stationary orbit is reached.” The further mode of reasoning is as follows. Let us take, for generality, an oscillator with 3 degrees of freedom; let the initial and final orbits of the electron be characterized by the quantum numbers
\[ m_1,\ m_2,\ m_3, \]
\[ n_1,\ n_2,\ n_3, \]
In the case of long waves the differences of the quantum numbers \(m_1-n_1,\ m_2-n_2,\ m_3-n_3\) are very small in comparison with the numbers themselves; the initial and final orbits practically coincide. The matter proceeds, therefore, as though the electron, remaining on the same orbit, at some instant \(t_1\) begins to radiate, and radiates until it has lost the whole quantum \(h\nu\). By the method employed by Bohr, in the correspondence principle this same representation is quite formally and empirically extrapolated to the case of waves of any length.
Let us consider a Bohr atom of hydrogen type, and, for simplification, neglect the motion of the atomic nucleus, change the mass of the electron owing to the motion, and suppose the orbit to be circular. Denoting by \(a\) the radius of the orbit, by \(\varphi\) the azimuth, for a circular orbit we find:
\[ \left. \begin{aligned} x&=a\cdot \cos \varphi=a\cdot \cos \Omega(t-t_0)\\ y&=a\cdot \sin \varphi=a\cdot \sin \Omega(t-t_0) \end{aligned} \right\} \qquad\ldots\ldots (19). \]
Let \(p\) denote the angular momentum \(\dfrac{nh}{2\pi}\), \(k\) the number of charges of the nucleus, and \(\mu\) the mass of the electron. According to Bohr’s theory\(^1\) the energy of the system \(A\) and the radius of the orbit \(a\) are expressed thus:
\[ A=-\frac{\mu k^2 e^4}{2p^2}\qquad a=\frac{p^2}{\mu k e^2}\qquad\ldots\ldots (20) \]
\(^1\) P. Einstein, loc. cit.
On the other hand, the angular velocity:
\[ \omega=\frac{\mu k^{2} e^{4}}{p^{3}}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (21). \]
From (20) and (21) we have:
\[ a=\left(\frac{k e^{2}}{\mu \Omega^{2}}\right)^{1/3}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (22). \]
In classical theory, on the basis of (1), one can derive that the energy emitted by an electron moving with acceleration \(V\) is equal to:
\[ -\frac{2e^{2}}{3c^{3}}V^{2}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (1). \]
But in our case of an electron moving in a circle, on the basis of (19):
\[ V^{2}=\ddot{x}^{2}+\ddot{y}^{2}=a^{2}\Omega^{4}, \]
where \(\ddot{x}\) and \(\ddot{y}\) are the second derivatives of \(x\) and \(y\) with respect to time. Thus the change in the electron’s energy owing to radiation in 1 second is:
\[ \frac{dA}{dt}=-\frac{2}{3}\frac{e^{2}a^{2}\Omega^{4}}{c^{3}}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (23). \]
According to the initial hypothesis of Einstein, the electron begins to behave “classically” from the moment \(t_{1}\) of departure from the initial orbit until the moment \(t_{2}\) of arrival at the final orbit. During the interval of time \(t_{2}-t_{1}\) the electron “classically” emits the energy \(h\nu\). Thus, on the basis of (23) and of the postulate (8),
\[ \frac{2}{3}\frac{e^{2}a^{2}\Omega^{4}}{c^{3}}(t_{2}-t_{1})=h\nu\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (24). \]
For very long waves
\[ \nu=\frac{\omega}{2\pi}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (25). \]
From (21), (22), (24), (25) we have
\[ t_{2}-t_{1}=\frac{3}{2}\left(\frac{h}{2\pi}\right)^{6}\frac{c^{3}}{\mu e^{10}k^{4}}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (26). \]
Thus the radiation time for long waves is calculated. This time, in contrast to the classical model, is, of course, finite, and there is really no question of damping—in particular, of the decrement of damping of oscillations—in Bohr’s theory there exists only a duration of radiation, which occurs throughout with the same amplitude. Just as in classical theory the presence of damping converts a monochromatic spectral line into a line of finite (or, strictly speaking, infinite) width, so also in Bohr’s quantum theory the finite duration of the oscillations, i.e. the finite
the number of waves still capable of interfering must be led to a spectral line of finite width¹). Einstein calculates this quantity on the basis of (25) and (26), passing from frequencies to wavelengths. The width of the spectral line \(\Delta l\) is approximately equal to²):
\[ \Delta l=\frac{l}{N}, \]
where \(N\) is the number of oscillations of wavelength \(l\) occurring during the time \(t_2-t_1\). Denoting
\[ a=\frac{3h}{16\pi^3}\left(\frac{2\pi c^3\mu^2}{e^{10}}\right)=2.27\cdot 10^9 \qquad\cdots\cdots (27). \]
Einstein finds
\[ N=\frac{a}{k^{2/3}}\,l^{2/3},\qquad \Delta l=\frac{k^{2/3}}{a}\,l^{4/3} \qquad\cdots\cdots (28). \]
The results obtained, (26) and (28), do not, however, as yet have practical significance, since they apply to infinitely long waves. The bold and, in essence, wholly unfounded step that Einstein then takes consists in the extrapolation of formulas (26) and (28) to waves of any length—following the example of Bohr, who made an equally risky a priori, but justified a posteriori, extrapolation. Einstein extends his theory also to a system with several degrees of freedom, and as a result of rather complicated calculations, not free from arbitrariness, arrives at formulas formally similar to (26) and (28). We do not consider it necessary to expose all the perplexities involved in extending formulas (26) and (28) to the region of short waves, since it is evident that neither the classical theory nor the quantum theory, when carried through consistently, is capable of giving such results. As in the correspondence principle of Bohr, in these formulas we are still encountering a purely intuitive guessing of the true state of affairs.
Einstein’s theory admits of experimental verification in two directions: 1) by direct measurement of the duration of the atom’s luminescence and 2) by measurement of the width of spectral lines. In the second, experimental part of our survey we shall describe the rather numerous methods by which the duration of atomic luminescence can be measured. We shall see that the corresponding measurements of Stark³) and Wien⁴) on the lines of hydrogen and other substances led to completely contradictory results. From this side the question of the experimental justification of Einstein’s equations must for the present be considered open. Measurement of the width of spectral lines in
¹) Cf. P. Drude, Lehrbuch d. Optik, p. 143, 1906.
²) Ibid.
³) J. Stark, Ann. d. Phys., 49, p. 731, 1916.
⁴) W. Wien, l.c.
visible, or ultraviolet, part of the spectrum cannot serve for verifying formula (28), because a considerably greater broadening is caused by entirely different causes: the Doppler effect, or the influence of some molecules on the vibrations of others1. The width of the spectral line, i.e. the distance in wavelengths between those two points of the intensity-distribution curve where the intensity is reduced by half in comparison with the maximum, will be as follows (if only the Doppler effect is operative):
\[ \Delta \lambda_D = 0.82 \cdot 10^{-6}\lambda \sqrt{\frac{T}{M}} \qquad \ldots \ldots \ldots \quad (29) \]
here \(T\) is the absolute temperature, \(M\) the molecular weight. Only for very short waves and low temperatures may one hope to detect the pure effect. Epstein calculates the following table for the Schumann spectrum:
| \(\lambda\) | \(\dfrac{\Delta\lambda_D}{\Delta\lambda}\) |
|---|---|
| \(1.87 \cdot 10^{-5}\ \mathrm{cm.}\) | 60 |
| \(6.56 \cdot 10^{-6}\) | 30 |
| \(1.20 \cdot 10^{-6}\) | 10 |
Thus, in the region of the X-ray spectrum a pure damping effect due to radiation should be observed. For the \(\alpha\)-lines of the X-ray series \(K, L, M\), the \(\Delta\lambda\)’s calculated by formula (28) are as follows:
| \(K_\alpha\) | \(0.0001\,\mu\mu\) |
| \(L_\alpha\) | \(0.0002\,\mu\mu\) |
| \(M_\alpha\) | \(0.0003\,\mu\mu\) |
Analyzing the only work of Stenström2 devoted to the question of the width of X-ray lines, Epstein finds in it at least qualitative confirmation of (28).
Thus, unlike the other postulates of Bohr’s theory, Epstein’s formulas, being in essence an arbitrary extrapolation of theory, still have no adequate experimental support.
§ 5. Conclusion. The classical atom is not capable of explaining all the actually observed features of radiation, but one cannot fail to note that all the conclusions of § 1 refer essentially to the electron, and not to the atom. Since there are no direct experimental data on the radiation of an isolated electron situated outside an atom, we have as yet no grounds for denying the classical—
... conclusion for the electron. Only for electrons bound to the nucleus in real atoms does doubt arise as to the applicability of classical laws. But one must not forget that the very concept of the quantum \(h\nu\) refers exclusively to internal energy, to the energy of a system, which the isolated electron does not possess. In this respect the study of the laws of the continuous X-ray spectrum—obtained, according to the usual conceptions, when the motion of free (i.e., non-atomic) electrons is accelerated by their impact upon the anticathode—is extremely important. The experimental data in this respect are still insufficiently definite and too contradictory.
There can hardly be any doubt as to the correctness of the basic postulate of quantum theory (9); however, this postulate gives very little for the understanding and description of atomic radiation in all respects. This postulate determines neither the duration, nor the polarization, nor the amplitude of elementary radiation. Moreover, with equal justification it may be interpreted either as statistical, valid for a large number of molecules, or as elementary.
The derivation of further properties of elementary radiation is given by Bohr’s theory with astonishing accuracy, describing all the complex regularities of spectra. The imposition of the principle of conservation of angular momentum automatically leads, as we have seen, to the solution of the problem of the polarization of the emitted wave and at the same time to the selection of real lines from among those possible in principle. The formal solution of the question of the intensity and damping of elementary radiation has been found in the correspondence principle of classical and quantum theory, formulated by Bohr and extended by Einstein and others.
The empirical correctness of the correspondence principle is, however, at the same time a blow both to classical theory and to Bohr’s theory. A logical analysis of either theory leads to results that approach one another asymptotically only in the transition to long waves1. Empirically, coincidence has been found in many respects for waves of any length. The only conclusion under such circumstances is the conclusion of the incorrectness of each theory taken separately.
At the beginning of the article we spoke of the collapse of the classical theory of the atom of Planck after the blow dealt to it by Planck himself in the analysis of the peculiarities of black-body radiation. In conclusion of the article we must speak of the collapse of Bohr’s theory after the formulation of the empirical “correspondence principle” by Bohr himself and Einstein. This principle, being empirically correct, undoubtedly conceals within itself paths also toward the theoretical solution of the problem of radiation.