Isotope mass-dependent

Collecting all these various contributions together, Watson showed that a more accurate description of the isotopic mass dependence is given by... 

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... 

H. Alawadhi, S. Tsoi, X. Lu, A.K. Ramdas, M. Grimsditch, M. Cardona, R. Lauck, Effect of temperature on isotopic mass dependence of excitonic band gaps in semiconductors ZnO. Phys. Rev.B 75(20), 205207 (2007)... 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

The dissociation energy is unaffected by isotopic substitution because the potential energy curve, and therefore the force constant, is not affected by the number of neutrons in the nucleus. However, the vibrational energy levels are changed by the mass dependence of 03 (proportional to where /r is the reduced mass) resulting in Dq being isotope-... 

When the mass of the tunneling particle is extremely small, it tunnels in the one-dimensional static barrier. With increasing mass, the contribution from the intermolecular vibrations also increases, and this leads to a weaker mass dependence of k, than that predicted by the onedimensional theory. That is why the strong isotope H/D effect is observed along with a weak k m) dependence for heavy transferred particles, as illustrated in fig. 18. It is this circumstance that makes the transfer of heavy reactants (with masses m < 20-30) possible. 

Isotopic molecules will have force fields which are identical to a high degree of accuracy. The vibrational amplitudes, on the other hand, will be mass-dependent, which means that the steric requirements of isotopic molecules will be slightly different. For this reason it is to be expected quite generally that isotopic molecules will respond differently to the change in steric conditions imposed by a chemical reaction, and hence that their reaction rates will differ somewhat. 

The state mixing term, the first in the r.h.s., usually dominates, at least in the presence of avoided crossings. Its determination reduces to a simple problem of interpolation of the Hu matrix elements, according to eq.(16). The second term corresponds, for large R, to the electron translation factor (see for instance [38]). This term depends on the choice of the reference frame that is, for baricentric frames, it depends on the isotopic masses. It contains the Gn matrix, which may be determined by numerical differentiation of the quasi-diabatic wavefunctions [16] this calculation is more demanding, especially in the case of many internal coordinates. It is therefore interesting to adopt the approximation ... 

The mass dependence in this expression comes from the moment of inertia and ratio of the reduced masses for 12CO/13CO. The isotope shift for the 13CO transition is given by ... 

It is clear from figure 6 that the terrestrial data do not cluster about a single point but instead lie along a line of slope 0.5 on the three-isotope diagram, indicating isotopic variation due to mass-dependent fractionation. Since mass fractionation effects in Mg have not been observed in terrestrial materials [30,31], this distribution of observed isotope ratios must be due to fractionation in the ion probe. The physical process which produces the... 

Isotopic fractionation provides illustrative examples of first-order expansions of unknown functions. In general, the mass spectrometric measurement r/ of the ratio between two isotopes of mass m( and m, of the same element, differs from the natural value R/. Only a very small fraction of the original sample produces ions and different processes taking place in different parts of the mass spectrometer act differently on the sensitivity of each isotope. We assume that instrumental isotopic fractionation is mass-dependent. 

Figure 3.5 The domain of the linear law for mass-dependent discrimination between two isotopic ratios.

The important fact that must be remembered is that in the Born-Oppenheimer approximation, Equation 2.8, the potential energy for vibrational motion is Eeiec(S) which is independent of isotopic mass of the atoms. In the adiabatic approximation, the potential energy function is Eeiec(S)+C and this potential will depend on nuclear mass if C depends on nuclear mass. 

The vibrational frequencies of isotopic isotopomers obey certain combining rules (such as the Teller-Redlich product rule which states that the ratio of the products of the frequencies of two isotopic isotopomers depends only on molecular geometry and atomic masses). As a consequence not all of the 2(3N — 6) normal mode frequencies in a given isotopomer pair provide independent information. Even for a simple case like water with only three frequencies and four force constants, it is better to know the frequencies for three or more isotopic isotopomers in order to deduce values of the harmonic force constants. One of the difficulties is that the exact normal mode (harmonic) frequencies need to be determined from spectroscopic information and this process involves some uncertainty. Thus, in the end, there is no isotope independent force field that leads to perfect agreement with experimental results. One looks for a best fit of all the data. At the end of this chapter reference will be made to the extensive literature on the use of vibrational isotope effects to deduce isotope independent harmonic force constants from spectroscopic measurements. 

As written Equation 4.150 applies to the case of a single isotopic substitution in reactant A with light and heavy isotopic masses mi and m2, respectively. Equation 4.150 shows that the first quantum correction (see Section 4.8.2) to the classical rate isotope effect depends on the difference of the diagonal Cartesian force constants at the position of isotopic substitution between the reagent A and the transition state. While Equations 4.149 and 4.150 are valid quantitatively only at high temperature, we believe, as in the case of equilibrium isotope effects, that the claim that isotope effects reflect force constant changes at the position of isotopic substitution is a qualitatively correct statement even at lower temperatures. 

Fig. 5.10 The mass dependence of the critical temperature of the superconducting/resistive transition in isotopically enriched samples of mercury (Triangles Reynolds, C. A., et al. Phys. Rev. 78, 487 (1950). Circles Maxwell, E., Phys. Rev. 78, 477 (1950))...

Fig. 14.5 Three isotope plots of oxygen fractionation in nitrate samples from different locations (After Thiemens, M., Ann. Rev. Earth Planet. Sci. 34, 217 (2006)). For these data, m, Equation 14.31, is 0.9. The expected mass dependent terrestrial fractionation is shown as the... 

Figure 14.6 compares measured and calculated isotope fractionations for all 16 possible ozone isotopomers prepared from an enriched oxygen precursor. In this figure (160160160, 160160170, 160170160, 160160180, etc. are represented as 666, 667, 676, 677, 767, 668, 686, 678, 777, 688, 868, 778, 787, 788, 878, and 888). The calculations are those of Gao and Marcus described in sections below. They are in quantitative agreement with experiment. It is interesting that isotope fractionations observed in product ozone for the totally symmetric isotopomers, 8170 = 1000 ln(777/666) and 8lsO = 1000 ln(888/666), are negative they show the heavy isotope to be depleted. Moreover, these totally symmetric effects lie on the mass dependent fractionation line [ln(777/666)]/[ln(888/666)] 0.5. That... 

The isotope fractionation factor and mass-dependency of fractionation... 

For elements that have three or more isotopes, isotopic fractionations may be defined using two or more isotopic ratios. Assuming that isotopic fractionation occurs through a mass-dependent process, the extent of fractionation will be a function of the relative mass differences of the two isotope ratios. For example, assuming a simple harmonic oscillator for molecular motion, the isotopic fractionation of may be related to as ... 

Equations (8) and (10) are applicable to stable isotope systems where isotopic fractionation occurs through mass-dependent processes which comprise the majority of cases described in this volume. These relations may also be used to identify mass-independent fractionation processes, as discussed in Chapter 2 (Birck 2004). Mass-dependent fractionation laws other than those given above distinguish equilibrium from kinetic fractionation effects, and these are discussed in detail in Chapters 3 and 6 (Schauble 2004 Yormg and Galy 2004). Note that distinction between different mass-dependent fractionation laws will generally require very... 

Figure 4. Illustration of mass-dependent fractionation of Mg isotopes, cast in terms of 5 values. 5 Mg and 5 Mg values based on Mg/ Mg and Mg/ Mg ratios, respectively. A common equilibrium fractionation model, as defined by exponential relations between a values (fractionation factors) for different isotope ratios, is shown in the gray line. A simple linear relation, where the slope is proportional to the mass difference of the isotope pair, is shown in the black line. Additional mass-dependent fractionation laws may be defined, and all are closely convergent over small ranges (a few per mil) in isotope compositions at 5 values that are close to zero.

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