Duration of the Stationary States of a Molecule and the Absorption Constant

1) R. Ladenburg. Die quantentheoretische Deutung der Zahl der Dispersionselektronen, ZS. f. Physik, 4, 451, 1921.
2) R. Ladenburg and F. Reiche. Absorption, Zerstreuung und Dispersion in der Bohrschen Atomtheorie. Die Naturwissenschaften, 11, H. 27, 557, 1913.
3) R. Tolman. Duration of molecules in upper quantum states. Physical Review, 23, 693, 1924.

In Bohr’s theory, after absorbing a quantum of energy an atom or molecule remains for some time \(\tau\) in an unchanged stationary state; the mean statistical value of this time for a large number of molecules must be constant, independent of the density of the surrounding monochromatic radiant energy. Collisions with other molecules may, however, cause a “premature” return to the lower stationary state; in this case the energy is sometimes converted into kinetic energy, and no radiation occurs. We propose to give a survey of the experimental methods for determining \(\tau\) in a separate article; here we present one of the methods of theoretical calculation of \(\tau\) on the basis of applying the correspondence principle to the process of absorption.

The calculation is based on the use of the derivation of the law of black radiation proposed by Einstein in 1916–19171. In view of the particular simplicity and significance of this derivation, even apart from the question under consideration, we give it in rather full detail. Suppose there is a molecule capable of interacting with black radiation, existing, for example, in two stationary states with energies \(\varepsilon_1\) and \(\varepsilon_2\), with \(\varepsilon_2 > \varepsilon_1\); let the density of radiation of frequency \(\nu\) be denoted by \(u_\nu\). Let the probability that, at unit density of radiant energy, the molecule will pass from the 1st state into the 2nd be denoted by \(B_{12}\). Then, of the number of molecules \(N_1\) in the 1st stationary state, the number passing into the 2nd per unit time will be

\[ N_1 u_\nu B_{12}, \tag{1} \]

molecules. In the opposite direction, during the same time there will pass, first, some number of molecules \(N_2 A_{21}\), where \(N_2\) is the number of molecules in the 2nd state, and \(A_{21}\) is the probability of reverse return, entirely by spontaneous radioactive decay and simply connected with the lifetime of the atom’s existence in the 2nd state. The quantity \(A_{21}\) does not depend on \(u_\nu\), as we said at the beginning.

But in addition to this number \(N_2 A_{21}\), independently of the density \(u_\nu\), according to Einstein’s hypothesis a further number of molecules \(N_2 u_\nu B_{21}\) must return to the 1st state. External radiation assists not only absorption but also emission, “negative absorption.” Such “negative absorption” must also exist in classical theory, given a suitable relation of the phase of the incident wave and the resonator. If, say, a plane wave is incident, then “negative absorption” must return the accumulated energy specifically into the stock of this plane wave, and not into a spherical wave diverging from the resonator. Thus “negative absorption” is, in a sense, given by classical theory. Consequently, according to Einstein, the total number of molecules returning from the 2nd state to the 1st per unit time is

\[ N_2 A_{21} + N_2 u_\nu B_{21}. \tag{2} \]

Generally speaking, transitions are possible from the 1st state to the 2nd, 3rd, \(\ldots\), \(n\)th in exactly the same way as transitions into the 1st state are possible from the 2nd, 3rd, \(\ldots\), \(n\)th. The condition of thermal equilibrium is, evidently, equivalent to the fact that the number \(N_1\) of atoms in the 1st

the state must be unchanged, as must \(N_2\), etc. The condition for such equilibrium will be a series of equalities of the type:

\[ N_1 \sum u_i B_{1i} - \sum N_i A_{i1} - \sum N_i u_i B_{i1} = 0 . \tag{3} \]

However, in a black body under thermal equilibrium the spectral distribution of the radiant energy must also be unchanged (“equilibrium in the ether”). Every transition, according to the basic postulate of Bohr’s theory, is accompanied by radiation of the characteristic frequency \(\nu = \dfrac{\varepsilon_i-\varepsilon_k}{h}\). Owing to this, instead of the total condition (3), for thermal equilibrium in a black body more simple relations of the following type must be satisfied:

\[ N_1 u_\nu B_{12} - N_2 A_{21} - N_2 u_\nu B_{21} = 0 . \tag{4} \]

By the general statistical principle of Boltzmann–Gibbs:

\[ \frac{N_1}{N_2} = \frac{g_1}{g_2} e^{\frac{\varepsilon_2-\varepsilon_1}{kT}} = \frac{g_1}{g_2} e^{\frac{h\nu}{kT}}, \tag{5} \]

where \(g_1, g_2\) are the a priori probabilities of states 1 and 2. Substituting (5) into (4), we have

\[ u_\nu = \frac{\dfrac{A_{21}}{B_{21}}} {\left(\dfrac{g_1}{g_2}\right) \left(\dfrac{B_{12}}{B_{21}}\right) e^{\frac{h\nu}{kT}} - 1}. \tag{6} \]

Assuming that at \(T=\infty\), \(u_\nu\) also becomes infinitely large, it is easy to find that

\[ \frac{g_1}{g_2}\cdot \frac{B_{12}}{B_{21}} = 1, \tag{7} \]

and (6) is rewritten as:

\[ u_\nu = \frac{A_{21}}{B_{21}} \cdot \frac{1}{e^{\frac{h\nu}{kT}} - 1}. \tag{8} \]

Comparison with the formula for black radiation leads to the conclusion that:

\[ \frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}, \tag{9} \]

or, using (7), we have:

\[ A_{21} = \frac{8\pi h\nu^3}{c^3} \cdot \frac{g_1}{g_2} B_{12}. \tag{10} \]

In the last lines we have considered only the transition \(1 \rightleftarrows 2\); if we were speaking of the transition \(i \rightleftarrows k\), we would obtain several expressions of the type (10):

\[ \left. \begin{aligned} A_{ki} &= \frac{8\pi h\nu_{ik}^{3}}{c^{3}} \cdot \frac{g_i}{g_k} B_{ik},\\[4pt] A_{ik} &= \frac{8\pi h\nu_{ik}^{3}}{c^{3}} \cdot \frac{g_k}{g_i} B_{ki}. \end{aligned} \right\} \tag{10'} \]

\[ \text{[[unclear: partially cut-off footnote at bottom of page]]} \]

The probabilities \(A_{k1}, A_{k2}, A_{k3}, \ldots\) correspond to mutually exclusive transitions; the probability that the molecule will leave the state \(k\) at all for any one of the lower states will be:

\[ A_k=A_{k1}+A_{k2}+A_{k3}+\cdots \tag{11} \]

We said that the return to one of the lower states occurs according to the law of radioactive decay:

\[ \frac{dN_k x}{dt}=A_k N_k(1-x), \tag{12} \]

where \(x\) is the fraction of returning molecules. Hence:

\[ 1-x=e^{-A_k t}, \]

and, consequently, the reciprocal quantity \(A_k\) is equivalent to the “mean” lifetime \(\tau\) of decay.

The quantity \(B_{12}\) in formula (10) is the probability that a quantum \(h\nu\) will be absorbed by the given molecule when the density of radiant energy is equal to unity. It is not difficult to understand that \(B_{12}\), according to the quantum theory, must be connected with the absorption constant \(a\). If the intensity of the radiation is \(I_\nu\), then the number of molecules that absorb the quantum \(h\nu\) will be:

\[ \frac{N_1\cdot B_{12}}{c}\cdot I_\nu\,d\nu=\frac{I_\nu\cdot a\,d\nu}{h\nu} \]

in the denominator of the left-hand side the velocity of light \(c\) has appeared because \(B_{12}\) is referred to the density of radiant energy, whereas in our case the intensity per unit area figures. Hence

\[ B_{12}=\frac{c}{N_1 h\nu}\cdot a. \tag{13} \]

If the absorption band is finite, but not especially broad, then \(\nu\) will change little within its limits. Making the hypothesis that \(B_{12}\) is constant within the given band, it is necessary to replace \(a\) in (13) by the integral \(\int a\,d\nu\), extended over the limits of the absorption band, i.e.

\[ B_{12}=\frac{c}{N_1 h\nu}\int a\,d\nu. \tag{14} \]

Substituting (14) into (10), we have:

\[ A_{21}=\frac{8\pi h\nu^3}{c^3}\frac{g_1}{g_2}\cdot \frac{c}{N_1 h\nu}\int a\,d\nu = \frac{8\pi \nu^2}{c^2 N_1}\frac{g_1}{g_2}\int a\,d\nu. \tag{15} \]

Correspondingly to (10) the expressions \(A_{k1}, A_{k2}\), etc. are formed; moreover:

\[ A_k=\sum_{i=1}^{i=k-1} A_{ki}=\frac{1}{\tau}. \tag{16} \]

In the classical theory\(^1\),

\[ \int a\,d\nu=\frac{N_1 e^2\pi}{m\cdot c}, \]

where \(e, m\) are the charge and mass of the electron; substituting this into (16), we find

\[ A_k=\frac{1}{\tau}=\frac{8\pi^3 e^2}{mc^3}\sum \nu^2\cdot\frac{g_i}{g_k}, \tag{17} \]

\(^1\) See M. Planck. Wärmestrahlung, 2nd edition, 155, 1913.

if the ratio \(\dfrac{g_l}{g_k}\) is of the order of unity and the frequency \(\nu\) obtained in the transition from the \(k\)-th state to the \(l\)-th is considerably greater than the others, then, approximately,

\[ \frac{1}{\tau}=\frac{8\pi^2 e^2\nu^2}{mc^3}\,\frac{g_l}{g_k}, \]

or

\[ \tau=\frac{g_k}{g_l}\,\frac{mc^3}{8\pi^2 e^2\nu^2}. \tag{18} \]

On the other hand, in the classical theory the damping constant of free oscillations \(\tau_0\) in the formula \(I_0\cdot e^{-\frac{t}{\tau_0}}\) is

\[ \tau_0=\frac{3mc^3}{8\pi^2 e^2\nu^2}. \tag{19} \]

Experimentally, formulas (18) and (19) are not satisfied exactly, but they give the correct order of magnitude.

The derivation of (17) was made in 1921 by Ladenburg. Apparently quite independently, in 1924 Tolman derives equation (15), but in doing so falls into error by not taking account of condition (11), or (16), and simply assuming for the quantity

\[ A_{ki}=\frac{1}{\tau}. \]

As a result, applying this derivation to rotational spectra, Tolman obtains for individual components of the spectrum \(\tau\) of the order of 1 sec.!

S. Vavilov.

1. A. Einstein, Verh. d. d. phys. G., 1916, and Phys. Zeitschr. 18, 121, 1917. ↩