Nuclear masses discussion

The reactions that we discuss in this chapter will be represented by nuclear equations. An equation of this type uses nuclear symbols such as those written above in other respects it resembles an ordinary chemical equation. A nuclear equation must be balanced with respect to nuclear charge (atomic number) and nuclear mass (mass number). To see what that means, consider an equation that we will have a lot more to say about later in this chapter ... 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

Mu is a very light and unstable (ti/2 (half live) = 2 p,s) isotope of hydrogen consisting of a nucleus which is a positive muon (p,+) and an electron. The nuclear mass of Mu is 0.113 amu. For this isotope one would expect much larger KBoele values than those discussed above. 

It is mentioned in passing that the proper masses mA and mB to be used in Equation 3.3 are the atomic masses (nucleus + electrons) rather than the respective nuclear masses as might be expected from a strict Born-Oppenheimer approximation. For further discussion of this point, reference should be made to the reading lists at the end of this chapter and of Chapter 2. The combination of Equations 3.1 and 3.2 corresponds to a classical harmonic oscillator with force constant f and mass p. The harmonic oscillator frequency v is given by the well-known formula... 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

There are several possible ways of introducing the Born-Oppenheimer model " and here the most descriptive way has been chosen. It is worth mentioning, however, that the justification for the validity of the Bom-Oppenheimer approximation, based on the smallness of the ratio of the electronic and nuclear masses used in its original formulation, has been found irrelevant. Actually, Essen started his analysis of the approximate separation of electronic and nuclear motions with the virial theorem for the Coulombic forces among all particles of molecules (nuclei and electrons) treated in the same quantum mechanical way. In general, quantum chemistry is dominated by the Bom-Oppenheimer model of the theoretical description of molecules. However, there is a vivid discussion in the literature which is devoted to problems characterized by, for example, Monkhorst s article of 1987, Chemical Physics without the Bom-Oppenheimer Approximation... ... 

Then, nuclear masses can be replaced by atomic (nuclidic) masses when calculating the binding energy. Whole atom masses can, in fact, be used for mass-difference calculations in all nuclear reaction types discussed in this chapter, except for f>+ processes where there is a resulting annihilation of two electron masses (one /f+ and one fi ). 

In 1955, J.A. Wheeler [1] concluded from a courageous extrapolation of nuclear masses and decay half-lives the existence of nuclei twice as heavy as the heaviest known nuclei he called them superheavy nuclei. Two years later, G. Scharff-Goldhaber [2] mentioned in a discussion of the nuclear shell model, that beyond the well established proton shell at Z=82, lead, the next proton shell should be completed at Z=126 in analogy to the known TV = 126 neutron shell. Together with a new A=184 shell, this shell closure should lead to local region of relative stability. These early speculations remained without impact on contemporary research, however. 

The parameter level splitting in the case of the two-level ammonia-type molecules discussed previously can now be replaced by the molecules nuclear masses. With increasing nuclear masses (Mj, j = 1,2,...,K) one would expect a smaller probability of finding strange states such as superpositions of different isomers (in a thermal situation). In the limit Mj - the nuclei should behave entirely classically. Again, there is no reason to believe that some phase transition takes place at particular values for the nuclear masses. 

Now due to the well-known isotope effect, the change of 7(. with the change of nuclear mass upon isotopic exchange, provides direct evidence for the phonon-mediated mechanism in the BCS superconductor. Since the discovery of high-7). materials, many effects have been shown in isotope effect studies. There exist small isotope exponents, a = — log 7)/log M, 0.12 and 0.04 for LSCO and YBCO7, respectively. The non-zero isotope exponent produces doubts about the exclusion of a phonon-mediated model although the discussion above supports strongly the spin-fluctuation-mediated model. 

The appropriate starting point for a fully relativistic description of the electronic structure of atoms, molecules, clusters and solids is QED. In a fully covariant QED-approach to these systems both the electrons and the nuclei would have to be treated as dynamical degrees of freedom (at least on a classical level in the case of the nuclei). However, in dew of the large difference between the electron mass and the nuclear mass (in particular for heavy nuclei) the Bom-Oppenheimer approximation is usually applied, at least for the discussion of ground state properties. The nuclei are thus treated as fixed external sources,... 

It should be noted that C is in general a function of R since is a function of R. F is the rotational-vibrational wavefunction of the molecule and eq. (9) is the Schroedinger equation for this wavefunction. The potential energy term in eq. (9) is the sum of E (R), the B.O. electronic energy as a function of R, which is independent of nuclear masses, and the adiabatic correction C, which does depend on nuclear masses. It is clear that the subsequent interest of the present discussion centers on C, which is sometimes referred to as the diagonal nuclear motion... 

Here / is the moment of inertia and M is the nuclear mass. The dependence of the shielding on the Mk quantum number of the rotational state KMk) for a linear molecule is discussed in Ref. [3]. 

In which cases is the adiabatic ansatz (15) expected to be accurate As shall become clear in the next subsection and from the discussion in Sec. 5, the adiabatic ansatz is accurate for an electronic state which is well separated energetically from all other electronic states. Under the same electronic conditions, the ansatz improves with increasing nuclear mass. On the other hand, we can expect that the larger the energetical distance of... 

The large mass of a nucleus compared to that of an electron permits an approximate separation of electronic and nuclear motion, which is the basis for the treatments discussed in the preceding sections. Historically, there were several attempts to expand the solutions of the Schrodinger equation, i.e. the energies and the wavefunctions, in powers of a small quantity related to the ratio of electronic and nuclear masses. The breakthrough has been achieved by Born and Oppenheimer in their classic paper in which they have chosen k = (1/M) / as the small quantity.done in the preceding sections, M is some average nuclear mass and we work in atomic units where the electronic mass is 1. 

The Born-Oppenheimer approximation is usually a very good approximation since the nuclear mass is so much greater than the electronmassl 1. Uncoupling the electronic motion from the nuclear motion enables one to solve for the electronic structure for a fixed set of nuclei. The final term, which describes electron-electron repulsion, prevents the direct solution to the electronic structure. The solution requires convergence of the electronic structure via an iterative scheme. This is known as the self-consistent field approximation, which is discussed later " . 

However, other non-electromagnetic effects also influence the electronic structure. Examples are isotope shifts or electro-weak interactions, that lead to parity nonconservation. The latter result from the motion of a nucleus (in an atom relative to the center of mass) and yield a dependence of energy levels on the nuclear mass (and on the finite size of the nucleus, as we have already discussed in chapters 6 and 9). The energy levels are thus shifted, while the electromagnetic perturbations result in a splitting of these levels. 

The particular utility of the first model, which was developed by Fermi and others, is in application to water-metal mixtures. The second model is a rationale for extending these results by similarity techniques, and the third is useful for materials of large nuclear mass. It should be pointed out that each of these methods was designed primarily as a fast hand tool. With the advent of the fast computing machine there is perhaps less interest in these methods nevertheless, the development of the analytical models is helpful for discussing some of the physical features of each system. 

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