The Isotope Effect in Band Spectra, Part I

Abstract

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 ${\nu }^e$) of a band spectrum of a diatomic molecule may be written as a quantity independent of mass times some power of $\frac{1}{\mu }$, where, if $M$ and $M^{\prime }$ are the masses of the two nuclei, $\frac{1}{\mu }=\frac{1}{M}+\frac{1}{M^{\prime }}$. The frequency $\nu$ of any line may be considered as the sum of an "electronic," a (positive or negative) "vibrational," and a (positive or negative) "rotational" contribution: $\nu ={\nu }^e+{\nu }^n+{\nu }^m$, where $n=\mathrm{vibrational}$ and $m=\mathrm{rotational}\mathrm{quantum}\mathrm{number}$. From theory and experience in line spectra, ${\nu }^e$ (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: $\nu ={\nu }^e$; ${\nu }^n={\nu }^m=0\mathrm{}$; and the definition of the origin of a band: $\nu ={\nu }^e+{\nu }^n$; ${\nu }^m=0\mathrm{}$. For two isotopes, ${\nu }_2=\nu _2^{}{}_{}{}^e+\nu _2^{}{}_{}{}^n+\nu _2^{}{}_{}{}^m=\nu _1^{}{}_{}{}^e+\rho \nu _1^{}{}_{}{}^n+{\rho }^2\nu _1^{}{}_{}{}^m$, approximately, where ${\rho }^2=\frac{\left(\frac{1}{{\mu }_2}\right)}{\left(\frac{1}{{\mu }_1}\right)}$. The various bands corresponding to various values of ${\nu }^n$ 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 ${\nu }^m$ 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 $n^{\prime }$; 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.