Rotational Spectra and Isotopes.

A. Haas. Rotationsspektrum und Isotopie, Zeitschrift für Physik, 4, 68, 1921.

The twofold application of quantum theory to the frequency of the emitted light and to the angular momentum of a molecule leads, in the case of a rotating molecule, to the theory of the so-called rotational spectra, which is well confirmed by experiment. Three types of rotational spectra may be distinguished: 1) the case of a rotating rigid dipole, for example the molecule \(HCl\), etc. (a pure rotational spectrum). The theory leads to the following formula for the oscillation frequencies of this spectrum:

\[ \nu=\frac{h}{8\pi^{2}J}\left(\pm 2n-1\right) \tag{1} \]

where \(h\) is Planck’s constant, \(J\) is the moment of inertia of the molecule, and \(n\) is an integer. Such spectra are observed in the far infrared part of the spectrum; 2) the case of a nonrigid dipole which, in addition to rotation, also executes its own oscillations with frequency \(\nu_0\) (a rotational-vibrational spectrum). The frequencies of the lines of this spectrum are determined by the formula

\[ \nu=\nu_{0}+\frac{h}{8\pi^{2}J}\left(\pm 2n-1\right) \tag{2} \]

3) If one adopts the standpoint of Bohr’s radiation theory and assumes that radiation is accompanied by the “transition” of an electron from one “stationary state” to another, then a change in the moment of inertia during “transitions” must be considered possible. This view leads to the following formula for a rotational spectrum [a band spectrum (Bandenspektrum)]:

\[ \nu=\nu_{0}+\frac{h}{8\pi^{2}}\left\{\frac{n^{2}}{J_{1}}-\frac{(n\mp 1)^{2}}{J_{2}}\right\} \tag{3} \]

\(J_{1}\) and \(J_{2}\) are the values of the moment of inertia before and after the transition of the electron. All three cases of rotational spectra therefore depend on the moment of inertia of the molecule. The latter, in turn, depends on the masses of the atoms composing the molecule. If, for example, we are dealing with a diatomic molecule, the masses of which are \(m_{1}\) and \(m_{2}\), and the distance between the centers of gravity is \(a\), then

\[ J=a^{2}\frac{m_{1}m_{2}}{m_{1}+m_{2}} \tag{4} \]

Let us imagine that at least one of the atoms of the molecule has isotopes (for example, one). There are, consequently, two modifications of atoms of the first type with masses

\[ m_{1},\ m_{1}+\mu \]

where \(\mu\) is a number small in comparison with \(m_{1}\). The moment of inertia of the molecule into which the atom of mass \(m_{1}+\mu\) enters will be:

\[ J'=a^{2}\frac{(m_{1}+\mu)m_{2}}{m_{1}+\mu+m_{2}} \tag{5} \]

Whence:

\[ \frac{J'}{J}=1+x \]

where

\[ x=\frac{\mu m_2}{m_1(m_1+m_2+\mu)}. \tag{6} \]

The presence of isotopy must show itself in the fact that every line of the rotational spectrum will be double (a doublet). The distance between the components of the doublet for spectrum (1) is expressed as follows:

\[ \Delta \nu = ax \tag{7} \]

for spectra (2) and (3)

\[ \Delta \nu = a(\nu-\nu_0). \tag{8} \]

If both molecules have isotopes, then we obtain a more complex structure for each member of the rotational spectrum, depending on the number of isotopes (triplet, quadruplet, etc.).

The absolute values of \(\Delta \nu\) must be very small, of the order of \(0.1\ \mathrm{\AA}\). This magnitude, however, lies within the limits of observation. Larger values may be expected in the infrared spectrum. The quantitative study of rotational spectra has begun only very recently. There are as yet no reliable materials for testing the above direct consequence of the theory1.

S. Vavilov.

1. The work of Kratzer, which appeared soon after the article reviewed here, confirms Haas’s proposals. Details will be given in the next issue of the journal. — Ed. ↩