Transition probability relaxation theory

The relationship between spin-lattice relaxation time (T,) and J.(o>0), can be derived through the time dependent perturbation theory by calculating transition probabilities... 

The essential difference between the two transition probability densities lies in the fact that for the gaussian distribution pw r, ) the different moments E[Xm], m = 1, 2,. . . , n, exist, while for the Cauchy distribution pc(j, x) they do not exist. The Levy distributions characterized by p(t, k) = exp -a k qT) with 0< <2U 127 128 play a prominent role in the theory of relaxation processes.129 133... 

In the realm of theory also, greater demands will be made. As such studies (37—39) as those of Cu—Ni (Fig. 13) and Ag—Pd (Fig. 14) have shown, the d levels of the two species in transition metal alloys tend to maintain their atomic identities, at least when the levels in the pure components are sufficiently well separated in energy. However, neither calculation nor experiment has been done with refinement sufficient for quantitative testing of a theory, such as the coherent potential approximation, designed to describe the d band behavior. In pure metals and intermetallic compounds, band calculations can be compared directly with experiment if transition probabilities and relaxation effects are understood. With care they can be used also in evaluation of the effective interelectronic terms which enter equations such as (18a). Unfortunately, one cannot, by definition, produce a set of selfconsistent band calculation results for a matrix of specific valence electron snpmdl.. . configurations thus, direct estimates for I of Eq. (18a) or F of Eq. (18b) cannot be made. However, band calculations for a set of systems can indicate whether or not it is reasonable to factor level shifts into volume and electron count terms, in the manner of Eqs. (18a) and (23). When this cannot be done, one must revert to a more general expression for a level shift, such as Eq. (1). 

Consider a closed system characterized by a constant temperature T. The system is prepared in such a way that molecules in energy levels are distributed in departure from their equilibrium distribution. Transitions of molecules among energy levels take place by collisional excitation or deexcitation. The redistribution of molecular population is described by the rate equation or the Pauli master equation. The values for the microscopic transition probability kfj for transition from ith level toyth level are, in principle, calculable from quantum theory of collisions. Let the set of numbers vr be vibrational quantum numbers of the reactant molecule and vp be those of the product molecule. The collisional transitions or intermolecular relaxation processes will be described by ... 

The first microscopic theory for the phenomenon of nuclear spin relaxation was presented by Bloembergen, Purcell and Pound (BPP) in 1948 [2]. They related the spin-lattice relaxation rate to the transition probabilities between the nuclear spin energy levels. The BPP paper constitutes the foundation on which most of the subsequent theory has been built, but contains some faults which were corrected by Solomon in 1955... 

The first pubUshed criticism of the binary collision model was due to Fixman he retained the approximation that the relaxation rate is the product of a collision rate and a transition probabihty, but argued that the transition probability should be density dependent due to the interactions of the colliding pair with surrounding molecules. He took the force on the relaxing molecule to be the sum of the force from the neighbor with which it is undergoing a hard binary collision, and a random force mA t). This latter force was taken to be the random force of Brownian motion theory, with a delta-function time correlation ... 

An early paper by Sun and Rice considered the relaxation rate of a diatomic molecule in a one-dimensional monatomic chain. An analysis similar to the Slater theoiy of unimolecular reaction was used to obtain the frequency of hard repulsive core-core collisions, and then (in the spirit of the IBC model) this was multiplied by the transition probability per collision from perturbation theory and averaged over the velocity distribution to obtain the population relaxation rate. This was apparently the first prediction that condensed-phase relaxation could occur on a time scale as long as seconds. 

A surprising amount of insight concerning nuclear spin relaxation can be obtained simply by treating the various available spin states as analogous to chemical states linked by kinetics. Although the spin transition probabilities such as W+ remain to be determined either by experiment or by quantum mechanical theory, as do kinetic rate constants, nevertheless the flow of spins obeys essentially the same kinetic laws as does any other equilibrating system. 

Any coUision theory necessarily consists of two essentially different parts, the first being the determination of the cross-sections and transition probabilities for the inelastic molecular colhsions which occur, and the second being the statistical part, in which characteristic relaxation and reaction rates are expressed in terms of the assumed known transition probabilities. This article is concerned almost entirely with this second aspect of the theory, in which it proves possible to obtain many explicit results using only very general features of the transition probabilities. Detailed studies of the inelastic collisions themselves, in which the cross-sections are derived from postulated interactions, have not yet led to a consistent and easily summarizable body of results, and are therefore not included in this review. 

This section is mainly concerned with the formal theory of relaxation processes, so it will be supposed that the molecules A undergo transitions among a set of states due to collisions with X, that the binary collision conditions are satisfied, and that over this set of states the transition probability is complete and satisfies detailed balance. An integral written without indicated limits will be understood to extend over all the states in question. 

If em adsorbed molecule is moving in electronically adiabatic potential well the model of two-dimensional anharmonic oscillator can be used for the description of its vibrational spectrum. This spectrum E(,n) (n = (ni,Ti2)) can be calculated by the numerical or analytical integration of the two-dimensional Schrodinger equation. The calculations of the transition probabilities e E, E) (or e(n, n)) for such oscillator have been performed with the help of the pertiu-bation theory or more sophisticated approaches. Provided these transition probabiUties e(n, n) are known for dense enough energy spectrum (AE ksT ), a diffusion model of energy relaxation may be used... 

They can serve therefore as a test for Ti dispersion. In Fig. 12 the relaxation results are shown for D-RADP-15. The solid lines are a fit of the theory [19] to the data. Above Tc the lit is excellent, whereas below Tc it probably suffers from the fact that the phase transition is already diffuse and only nearly of second order. This proves that a soft mode component is needed to explain the data. Furthermore, the fact that the ratio ti/t2 remains unchanged above and below Tc proves that the order parameter fluctuations are in the fast motion regime on both sides of the transition. 

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