Isotope second-order terms

A deviation from the ideal fractionation law will appear as a residual correlation between the ratios corrected for mass bias using the exponential law. It can be verified that even very small 8M/M in Equation (40) produces a potentially important isotopic effect on the order of 1 + P(5M,. - 5Mt)/Mj. The alignment of the correlation between the corrected ratios x = hir and y = In rj produced by sloping peaks in a log-log plot has a slope of 5M,/8M,. For a rormd peak, the second-order term should be included. 

For the nuclear electric quadrupole interaction, the situation is rather different. Corrections similar to those just discussed, but arising from second-order terms involving the square of the magnetic hfs constants, change the value of P by about 2% for isotope 147, and 5% for 149. After these corrections, a ratio (147/149) of... 

This is a relatively small contribution when compared with V R) but it is nevertheless important when considering the isotopic dependence of the electronic energy. The second term, vlf iR), describes the contribution from spin-spin coupling (in first order) and spin-orbit coupling (in second order) to the electronic energy ... 

The potential constants k, a, and h are independent of isotopic substitution (within the framework of the Born-Oppenheimer approximation). The isotopic mass dependence is completely situated in the reduced mass /X. It is physically reasonable to assume for the diatomic molecule-oscillator that a and h are sufficiently small so that V can be regarded as a perturbation to Equation 3 and that it is necessary to consider no terms in the perturbation energy higher than the second power in a and higher than the first power in b. The term only yields a non-vanishing contribution in second order while the term yields a first-order contribution to the energy. One obtains by standard methods... 

The Aa(192 —188) was measured in 10 lines with isotopically highly enriched osmium by Aufmuth and Wobker [32] and analyzed by means of the parametric method. Due to the influence of crossed-second-order effects they have obtained for the difference between the field shifts of the 56 6s f and terms AT(5F)-AT(3F) = (-h53.4 + 9.6) x 10- cm- in contrast to the value —35x10 cm obtained for the same terms by taking into account configuration mixing and intermediate coupling. 

Frequently, one of the isotopes is present only in trace quantities (usually the heavier isotope) and the overall fraction of reaction, f, reduces to fi, to good approximation. If this is not the case Equations7.17 and 7.18 can be easily rewritten in terms of f or fq instead of fL. By the same token, with one isotopomer present in trace amounts (the commonly occurring case) the probability of reaction between two labeled molecules is very low, and formal second and higher order kinetics can be approximated using the first-order kinetic expressions developed above. 

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