Isotope theoretical development

With the improvement of analytical techniques, it is now possible to measure oxygen isotopic profiles in minerals. By analogy to the concepts of bulk mineral closure temperature versus closure temperature at every point in the interior of a mineral for the case of thermochronology, future theoretical development will need to clarify the concepts and meanings of Tc and Tae based on interior isotopic or elemental compositions of two contacting minerals. Furthermore, the principles of the above discussion may also be applied to elemental exchange reactions, such as Fe-Mg exchange between minerals. 

Theoretical Development. Separating isotopes on an industrial scale, where the components are chemically identical and differ physically only in mass, represented a new chemical engineering challenge. As shown in Table I, the challenge was made more difScult in that the desired isotopes deuterium, lithium , boron , and uranium were of low concentration in their respective naturally occurring parent elements. 

Because of the selection rule AK = 0, the rotational constant about the symmetry axis, A (C for an oblate top), cannot be evaluated from the rotational spectrum of a symmetric top. This selection rule follows because there is no dipole moment component perpendicular to the symmetry axis, and applies rigorously for a rigid molecule. Recent theoretical developments have shown that actually a very small dipole moment perpendicular to the symmetry axis can be induced through the effects of centrifugal distortion. This lifts the A AT = 0 selection rule and allows the observation of AK = 3, 6,... transitions with a sensitive spectrometer. This has enabled the structures of pyramidal XY3 molecules, which have only two structural parameters (see Table II), to be evaluated directly from and h without the need for isotopic information. The observation of so-called forbidden transitions often results... 

The full quantum mechanical study of nuclear dynamics in molecules has received considerable attention in recent years. An important example of such developments is the work carried out on the prototypical systems H3 [1-5] and its isotopic variant HD2 [5-8], Li3 [9-12], Na3 [13,14], and HO2 [15-18], In particular, for the alkali metal trimers, the possibility of a conical intersection between the two lowest doublet potential energy surfaces introduces a complication that makes their theoretical study fairly challenging. Thus, alkali metal trimers have recently emerged as ideal systems to study molecular vibronic dynamics, especially the so-called geometric phase (GP) effect [13,19,20] (often referred to as the molecular Aharonov-Bohm effect [19] or Berry s phase effect [21]) for further discussion on this topic see [22-25], and references cited therein. The same features also turn out to be present in the case of HO2, and their exact treatment assumes even further complexity [18],... 

In this chapter we discuss improvements documented in the literature over the past decade in these areas and others. Chemical procedures, decay-counting spectroscopy, and mass spectrometric techniques published prior to 1992 were previously discussed by Lally (1992), Ivanovich and Murray (1992), and Chen et al. (1992). Because ICPMS methods were not discussed in preceding reviews and have become more commonly used in the past decade, we also include some theoretical discussion of ICPMS techniques and their variants. We also primarily focus our discussion of analytical developments on the longer-lived isotopes of uranium, thorium, protactinium, and radium in the uranium and thorium decay series, as these have been more widely applied in geochemistry and geochronology. 

Since the discovery of the deuterium isotope in 1931 [44], chemists have long recognized that kinetic deuterium isotope effects could be employed as an indicator for reaction mechanism. However, the development of a mechanism is predicated upon analysis of the kinetic isotope effect within the context of a theoretical model. Thus, it was in 1946 that Bigeleisen advanced a theory for the relative reaction velocities of isotopic molecules that was based on the theory of absolute rate —that is, transition state theory as formulated by Eyring as well as Evans and Polanyi in 1935 [44,45]. The rate expression for reaction is given by... 

Obviously, there is an isotope effect on the vibrational frequency v . For het-eroatomic molecules (e.g. HC1 and DC1), infrared spectroscopy permits the experimental observation of the molecular frequencies for two isotopomers. What does one learn from the experimental observation of the diatomic molecule frequencies of HC1 and DC1 To the extent that the theoretical consequences of the Born-Oppenheimer Approximation have been correctly developed here, one can deduce the diatomic molecule force constant f from either observation and the force constant will be independent of whether HC1 or DC1 was employed and, for that matter, which isotope of chlorine corresponded to the measurement as long as the masses of the relevant isotopes are known. Thus, from the point of view of isotope effects, the study of vibrational frequencies of isotopic isomers of diatomic molecules is a study involving the confirmation of the Born-Oppenheimer Approximation. 

The use of reduced isotopic partition function ratios to study kinetic isotope effects was first undertaken by Bigeleisen this work was corrected and elaborated by Bigeleisen and Wolfsberg. References are cited at the end of this chapter. Application of the equations developed above to specific chemical reactions will be found in Chapter 10, where other theoretical approaches will also be presented. 

To begin we are reminded that the basic theory of kinetic isotope effects (see Chapter 4) is based on the transition state model of reaction kinetics developed in the 1930s by Polanyi, Eyring and others. In spite of its many successes, however, modern theoretical approaches have shown that simple TST is inadequate for the proper description of reaction kinetics and KIE s. In this chapter we describe a more sophisticated approach known as variational transition state theory (VTST). Before continuing it should be pointed out that it is customary in publications in this area to use an assortment of alphabetical symbols (e.g. TST and VTST) as a short hand tool of notation for various theoretical methodologies. 

Criss et al. (1987) and Gregory et al. (1989) developed a theoretical framework that describes the kinetics of oxygen isotope exchange between minerals and coexisting fluids. Figure 2.17 shows characteristic patterns in 8-5 plots for some hy-drothermally altered granitic and gabbroic rocks. The arrays displayed on... 

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