The Number of Dispersion Electrons According to Quantum Theory

R. Ladenburg. Die quantentheoretische Deutung der Zahl der Dispersionselektronen.
Zeitschr. f. Phys. 4, p. 451, 1921.

The author considers the interpretation which, from the standpoint of the new quantum conceptions, is given to the dispersion constant \(\mathfrak N\), which, according to the classical electron theory, determines—where it has the meaning of the number of oscillating electrons per unit volume—the course of a whole series of optical phenomena, such as absorption, emission, anomalous dispersion, and magnetic rotation.

For this purpose, on the one hand, following Planck, he forms the expression for the energy emitted per second by oscillating electrons in equilibrium with their radiation (the energy absorbed). This expression has the form

\[ J_1 = A_1 = \frac{\alpha}{m} e^2 \mathfrak N u_0, \]

where \(u_0\) is the radiation density, and \(\mathfrak N\) is the number of electrons.

On the other hand, he makes use of the treatment of the processes of quantum emission and absorption which, in the most general initial assumptions, was given by Einstein (Phys. Zeitschr. 18, p. 121, 1917).

These general conceptions of the processes of molecular radiation emission are as follows: molecules situated in the field of their own radiation can assume only a discrete series of states, characterized by the energy values \(E_1, E_2, \ldots\). In thermal equilibrium with the radiation, the distribution of molecules over energy values will be canonical; hence the ratio of the number of molecules in state \(k\) to the number of molecules in state \(i\) will be

\[ \frac{N_k}{N_i} = \frac{G_k}{G_i} \frac{e^{-E_k/kT}}{e^{-E_i/kT}}, \]

where \(G_i\), a constant independent of \(T\), denotes the statistical “weight” of state \(i\). Further, the following intramolecular processes are possible:

1) the spontaneous transition of a molecule from \((k)\) to \((i)\) \((E_k > E_i)\), characterized by the probability \(a_{ki}\);

2) the same transition, but under the influence of the radiation field present—the probability \(b_{ki} u_{ik}\), where \(u_{ik}\) is the density of the monochromatic radiation under consideration, of frequency \(\nu_{ik}\);

3) transition from \((i)\) to \((k)\)—probability \(b_{ki} u_{ik}\).

The first two processes give emission, the last—absorption. Hence, assuming temperature equilibrium, and equating the number of elementary processes 1) and 2) to the number of processes 3) occurring per unit time, Einstein obtained the formula:

\[ u_{ik}=\frac{a_{ki}/b_{ki}}{e^{\frac{E_k-E_i}{kT}}-1}, \]

comparison of which with Planck’s formula gives:

\[ \frac{a_{ki}}{b_{ki}}=\frac{8\pi h}{c^3}\nu_{ik}^{3} \quad \text{and} \quad E_k-E_i=h\nu_{ik}. \]

Using these results of Einstein, the author directly obtains an expression for the entire monochromatic energy emitted per second, equal at equilibrium to that absorbed:

\[ J_2=A_2=N_i \frac{g_k}{g_i} a_{ki}\frac{c^3}{8\pi \nu_{ik}^{2}}\,u_{ik}. \]

Equating the expression \(J_2\) to the expression \(J_1\), the author obtains the following relation between the experimentally determined constant \(\mathfrak{N}\) and the constants characterizing the quantum processes \((u_{ik}=u_0)\):

\[ \mathfrak{N}=N_i \frac{g_k}{g_i} a_{ki}\frac{m}{8\pi^2}\frac{c^3}{e^2\nu_{ik}^{2}}. \]

The remainder of the article consists in applying the interpretation obtained for the previously experimentally determined constant \(\mathfrak{N}\) to estimate the various probabilities that determine atomic emission and absorption, although the legitimacy of such a transfer of results, obtained for equilibrium thermal radiation, to the processes of gaseous emission and absorption in an ordinary experimental arrangement appears doubtful.

The results of these considerations are as follows:

In alkali-metal atoms the mean lifetime of the initial state \(2P\) (measured as \(1/a_{ki}\)) is approximately equal to

\[ \frac{3m}{8\pi^2}\frac{c^3}{e^2\nu^2}, \]

i.e. to the damping time of a classical resonator of the frequency corresponding to the transition \(1S\text{–}2P\) (the first member of the principal series).

The decrease of \(\mathfrak{N}\) along the series signifies a decrease of the probability \(a_{ki}\), i.e. the probability of spontaneous decay of the initial state (with transition to one and the same given state). The author connects this with the increase, along the series, in the number of possible paths for decay of the initial state, alongside the given path leading to emission of a member of the series.

The experimentally established ratio \(\frac{\mathfrak{N}_{D_2}}{\mathfrak{N}_{D_1}}=2\) for the sodium doublet \(D\) means that the state \(2P_2\) is twice as probable as the state \(2P_1\), and accordingly occurs more often.

A. Terenin.