By Walsh J.L.
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As a result, mass-independent fractionation effects were discovered later and are still less understood. 4 Natural Isotopic Composition of the Elements mass-dependent or mass-independent, it is implicitly assumed to be massdependent. Because the changes in the isotopic composition due to isotope fractionation are small and a difference in isotope ratios relative to one another can be determined more easily than an absolute isotope ratio, an isotope ratio for a sample is usually expressed as the difference between the speciﬁc isotope ratio and that of a selected standard: d¼ Rsample À Rstandard Â 1000ð%Þ Rstandard The difference is multiplied by 1000, and thus expressed in units of permil (m), to obtain values that can be dealt with easily.
A, 89, 1–20. , The mass-spectra of chemical elements. Philos. , 39, 611–625. D. E. , Chichester. 9 Faure, G. M. , Chichester. 10 Lodders, K. (2003) Solar system abundances and condensation temperatures of the elements. Astrophys. , 591, 1220–1247. P. (2005) Isotopic compositions of the elements, 2001. J. Phys. Ref. , 34, 57–67. Faure, G. M. (2007) Introduction to Planetary Science – the Geological Perspective, Springer, Berlin. A. (2008) Natural fractionation of 238 U/235U. Earth Planet. Sci. , 72, 345–359.
6 Potential energy as a function of interatomic distance in a diatomic molecule (Morse curve) showing vibrational energy levels for molecules containing the heavier (H) and lighter (L) of the two isotopes considered. The dissociation energies D0 for these two types of molecules are also indicated. Reproduced with permission of Springer from . 4 Natural Isotopic Composition of the Elements center can only adopt speciﬁc values. As a result, the diatomic molecule considered can only occupy discrete vibrational energy levels, described by 1 Evib ¼ n þ hn 2 where n is the vibrational quantum number, h Planck’s constant, and n the vibrational frequency, which is determined by the force constant k (corresponding to the bond strength within the molecule) and the reduced mass of the diatomic molecule m: sﬃﬃﬃ 1 k n¼ 2p m The reduced mass is deﬁned as 1 1 1 þ ¼ m m1 m2 or m¼ m1 m2 m1 þ m2 where m1 and m2 are the masses of the constituent atoms.
Boundary values of an analytic function and the Tchebycheff method of approximation by Walsh J.L.