The Isotope Effect in Band Spectra, Part I

Theory of the isotope effect in band spectra of diatomic molecules. --- Generalizing previous discussions by Loomis and Kratzer, it is shown that each coefficient in the detailed expressions for the spectral terms (except those involved in ${\ensuremath{\nu}}^{e}$) of a band spectrum of a diatomic molecule may be written as a quantity independent of mass times some power of $\frac{1}{\ensuremath{\mu}}$, where, if $M$ and ${M}^{\ensuremath{'}}$ are the masses of the two nuclei, $\frac{1}{\ensuremath{\mu}}=\frac{1}{M}+\frac{1}{{M}^{\ensuremath{'}}}$. The frequency $\ensuremath{\nu}$ of any line may be considered as the sum of an "electronic," a (positive or negative) "vibrational," and a (positive or negative) "rotational" contribution: $\ensuremath{\nu}={\ensuremath{\nu}}^{e}+{\ensuremath{\nu}}^{n}+{\ensuremath{\nu}}^{m}$, where $n=\mathrm{vibrational}$ and $m=\mathrm{rotational}\mathrm{quantum}\mathrm{number}$. From theory and experience in line spectra, ${\ensuremath{\nu}}^{e}$ (if present) should be substantially identical for corresponding band systems of two or more isotopes, so that these should have a common "origin." This leads to the definition of the origin of a band-system: $\ensuremath{\nu}={\ensuremath{\nu}}^{e}$; ${\ensuremath{\nu}}^{n}={\ensuremath{\nu}}^{m}=0$; and the definition of the origin of a band: $\ensuremath{\nu}={\ensuremath{\nu}}^{e}+{\ensuremath{\nu}}^{n}$; ${\ensuremath{\nu}}^{m}=0$. For two isotopes, ${\ensuremath{\nu}}_{2}=\ensuremath{\nu}_{2}^{}{}_{}{}^{e}+\ensuremath{\nu}_{2}^{}{}_{}{}^{n}+\ensuremath{\nu}_{2}^{}{}_{}{}^{m}=\ensuremath{\nu}_{1}^{}{}_{}{}^{e}+\ensuremath{\rho}\ensuremath{\nu}_{1}^{}{}_{}{}^{n}+{\ensuremath{\rho}}^{2}\ensuremath{\nu}_{1}^{}{}_{}{}^{m}$, approximately, where ${\ensuremath{\rho}}^{2}=\frac{(\frac{1}{{\ensuremath{\mu}}_{2}})}{(\frac{1}{{\ensuremath{\mu}}_{1}})}$. The various bands corresponding to various values of ${\ensuremath{\nu}}^{n}$ should form for each isotope an essentially identical pattern about the origin. The scale of this pattern should, however, be greater for a lighter isotope (vibrational isotope effect), and large separations between corresponding bands of isotopes may result for bands remote from the origin. Two typical band-patterns for a mixture of isotopes are given in Fig. 1. Analogous relations hold with respect to the ${\ensuremath{\nu}}^{m}$ terms (rotational isotope effect) (cf. Fig. 2). An explicit equation for band-l eads is obtained; this contains important mixed and higher power terms in $n$ and ${n}^{\ensuremath{'}}$; the corresponding isotope effect is discussed. In studying isotope effects in band spectra, compounds of isotopic elements with other elements of high atomic weight are favorable; a low-temperature light source is very desirable; other practical factors are discussed. Quantitative and semi-quantitative confirmations of the isotope effect already obtained (BO, SiN, CuH, CuCl, CuBr, CuI) lend powerful support to the general quantum theory of band spectra.Applications of the isotope effect to the study of isotopy and the interpretation of band spectra.---Electronic band spectra give a valuable new method for the determination of the existence of isotopes. An advantage over the positive ray method is that characteristic band patterns associate without question isotopes belonging to a given element; a disadvantage is that accurate comparison of mass between different elements is less easy. Photometric intensity comparisons give a means of measuring relative abundance of isotopes. Infra-red spectra are much less favorable than electronic for the study of isotopy, largely bacause of the fewness of the bands. The vibrational isotope effect in electronic band spectra furnishes the first really reliable, and a remarkably simple, criterion for the location of the origin of a band-system. Simultaneously, it gives the first direct mode of attack on the correct absolute numbering of vibrational energy levels. Thus in BO the vibrational quantum numbers are probably 1/2, 3/2, 5/2..., not 0, 1, 2,... The rotational isotope effect should likewise aid in the analysis of band structure. The very strong dependence of the isotope coefficient ($\ensuremath{\rho}\ensuremath{-}1$) on the mass ${M}^{\ensuremath{'}}$ of the non-isotopic or second element in the molecule supplies a new and reliable method of identifying the emitters of band spectra.Isotope effects in the specific heat of gases are a necessary theoretical consequence of the success of the foregoing theory.