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Two Bloembergen–Purcell–Pound (BPP) models for analysing nuclear spin relaxation data for translational diffusion in disordered systems are compared with Monte Carlo simulations. One model (the a-BPP model, 'a' standing for average) is commonly used for disordered systems and the other (the Cameron–Sholl BPP model) is more rigorously based and can distinguish between site-and barrier-energy disorder. Simulated relaxation data produced using Gaussian distributions of energy disorder are analysed using the models, and the parameters obtained from the fits are compared with the values used for the simulations. It is found that both models can give reasonable fits to the data. Both models also give reasonable agreement with the simulation parameters provided that the standard deviation of the energy distribution for the a-BPP model is interpreted as the average of the site-and barrier-energy standard deviations. Quantitative estimates are given of the accuracy of the parameters from the fits.

A technique has been developed to calculate the exact diffusion tensor for the lattice-gas model in the low-concentration limit for any complex elementary cell. The analysis takes into account the detailed structure of the elementary cell and the structure of the diffusion barrier. The technique is also applicable to nonMarkoff diffusion and, consequently, can include polaronic and inertia effects. The calculation reduces to simple matrix-algebraic operations which may be carried out analytically in many cases. Applications to some one- and two-dimensional models are described, including disordered systems and non-Markoff diffusion, in structures with complex elementary cells. @S0163-1829~98!04346-X#

An integral expression is derived for the displacement probability of an atom during an atom-vacancy encounter. The analysis is based on a random walk with a one-step memory and is applicable to both self-diffusion and solute diffusion. The integral expression involves parameters which are the probability of a step of the atom occurring in a particular direction relative to the direction of the previous step. These parameters are obtained in terms of lattice generating functions by considering the random walk of a vacancy on an imperfect lattice.

Nuclear spin relaxation rates due to magnetic dipole coupling between spins of a single species undergoing relative diffusion depend on two-spin correlation functions J(p)( omega ). The high- and low-frequency limits of J(p)( omega ) are discussed for liquids, cubic crystals and two- and one-dimensional systems. It is shown that the low-frequency form of J(p) ( omega ) is J(p)(0)-A3 omega 1/2 in liquids and crystals, A2ln omega -1 in two-dimensional systems and A1 omega -1/2 in one-dimensional systems and expressions for the coefficients A1, A2 and A3 are derived which are valid for any diffusive model whose long-range, long-time behaviour may be described by the diffusion equation. The high-frequency form of J(p)( omega ) is derived for a simple-hopping model of spins on one-, two- and three-dimensional lattices and in each case J(p)( omega ) is of the form B omega -2. Expressions are derived for the coefficient B, for each dimension, which correctly include the effect of correlations between the hops of the spins.

The Vernon et al. (1981) theory of spin relaxation of F in PbF2 due to the hyperfine interaction of F with paramagnetic Mn impurities involves the analysis of encounters between the diffusing F- ions and the fixed Mn impurities. Vernon et al. assumed in their analysis that the motion of an ion is that on an empty lattice and that once an ion leaves the vicinity of an impurity the probability of return is negligible. The theory is developed more rigorously to include the effect of return visits of the ions and also for a model in which the ions move by the vacancy mechanism. It is found that the relaxation rates are essentially the same for motion of a single ion on a lattice and for motion by the vacancy mechanism. The effect on the relaxation rates of including return visits to the impurity is significant and best-fit parameters to the experimental data lead to reduced values of the high-temperature hyperfine interaction parameters.

For pt.V see ibid., vol.14, p.447 (1981). The theory of the two-spin correlation functions J(p)( omega ) that determine nuclear spin relaxation rates has been developed for the monovacancy limit of high spin concentrations using the encounter model of Einenstadt and Redfield. The theory is based upon a rate equation for the spin pair probability function and avoids the assumption of a Poisson distribution used by Wolf which is only valid if the spin motions are uncorrelated. In the high-frequency limit omega tau s>>1 and omega tau v >1 the results obtained disagree with those obtained by Feeders (1979) using a moment expansion.

For pt.X see ibid., vol.19, no.21, p.4115 (1986). The spectral density functions J(p)( omega ) relevant to nuclear spin-relaxation rates for dipolar coupling between diffusing spins in solids are calculated for the encounter model in cubic crystals. The calculations are based on the analysis of rigorous equations and the high-frequency results are accurate to better than 1%. An extrapolation scheme based on the known low-frequency analytic form is used to obtain numerical results at lower frequencies.

For pt.VI see ibid., vol.14, p.447 (1981). In crystals rates of nuclear spin relaxation due to dipolar coupling between diffusing spins depend on the mean jump rate of the spins. Interactions between the diffusing spins may alter the local jump rate of a spin and Feeders (1982) has recently shown that such interaction effects could have a significant effect on the spin-lattice relaxation rates at low spin concentrations in a simple cubic lattice. The interaction model of Feeders is applied to the high-frequency low-concentration limit of the nuclear spin relaxation rate correlation functions for both like- and unlike-spin dipolar coupling in cubic crystals and in one- and two-dimensional systems. Expressions are derived for the field-averaged spin-lattice relaxation rate as a function of concentration, which could be used to obtain information on the interaction effects from experimental relaxation rates.

For pt.XI see J. Less-Common Met., vol.129, p.335 (1987). The theory of nuclear spin relaxation due to dipolar coupling between diffusing spins in crystals requires the calculation of spectral density functions for particular jump models of the diffusion. The results of such calculations have always been presented in a tabular form that is inconvenient to use compared with the simpler, but much less accurate, Bloembergen-Purcell-Pound theory. An approximate analytic function is proposed which, when fitted to the results of detailed calculations, has comparable accuracy to these calculations but has the simplicity of use of the BPP theory.

For pt.I see abstr. A79623 of 1974. The theory of nuclear spin relaxation due to dipole-dipole interactions between diffusive species on a crystal lattice has been formulated exactly and applied to a FCC lattice in a previous publication. The results are presented here for BCC and SC lattices and it is found that the differences between the exact theory and the approximate Torrey theory are larger in these lattices than in the FCC lattice.

Some expressions for the tracer correlation factor f for diffusion by the vacancy mechanism are reviewed and their equivalence justified. In the case of the low-vacancy-concentration limit it is proved that the probability of the second jump of an atom being the reverse of the first is 2/Z where Z is the coordination number of the lattice. This result provides a simple derivation of f in some cases and gives a physical interpretation of the approximation f=1-2/Z. The Monte Carlo evaluation of f is discussed and quantitative estimates made of the number of atoms to be sampled to achieve a given accuracy. Calculated values of the averaged cosines of angles between the direction of an initial jump and the direction of the (k+1)th jump of an atom are presented for the cubic lattices and for the honeycomb lattice. It is deduced that the nature of the correlations in the low-vacancy-concentration limit are different to those at other vacancy concentrations.

The spectral density functions J(p)( omega ) relevant to nuclear spin relaxation rates due to magnetic dipole coupling between diffusing spins in cubic crystals are calculated, for the simple hopping model, by a Monte Carlo method. The analysis makes use of known exact results in various limits. The statistical fluctuations in the Monte Carlo calculations become more significant at high frequencies but accurate analytic results are known in this regime. Numerical results are therefore obtained over the entire frequency range for all spin concentrations except the monovacancy limit.

The form of the double Fourier transform S(k, omega ) of the occupancy correlation function of an atom diffusion in a cubic crystal by the vacancy mechanism has recently been derived by Tahir-Kheli and Elliott (1982). It is shown that their results may be obtained more simply from the encounter model using known values of the atom displacement probability due to an atom-vacancy encounter. Results for the full width at half maximum of the spherically averaged S(k, omega ) are calculated for the mean field theory and for the encounter model.

At low vacancy concentrations in crystals it is possible to define an encounter of a given pair of atoms and a vacancy as the set of displacements of the pair of atoms caused by a randomly walking vacancy. The relative displacement probability of a pair of atoms caused by an encounter with a vacancy is calculated from random walk theory applied to the vacancy motion. The resulting displacement probability is used to calculate the high-frequency limit of the correlation functions relevant in nuclear spin relaxation by translational diffusion due to the vacancy mechanism in cubic crystals.

For pt.IV see ibid., vol.11, p.4405 (1978). Describes efficient methods for evaluating the lattice and Brillouin-zone summations involved in the reciprocal-space formalism for any spin-jumping model. The mean-field theory expressions of Feeders and Sankey (1978) are extended to the BCC lattice and numerical results are presented over the entire temperature range for SC, FCC and BCC lattices. Formulae are also derived for the low-temperature limit which exactly include the effects of spatial correlations.

For pt.III see ibid., vol.9, p.4315 (1976). The two-spin systems describe the dipole-dipole interactions between stationary host spins and spins migrating amongst either the tetrahedral or the octahedral interstitial sites. The tetrahedral sites in a FCC crystal form a SC lattice and two models, the symmetric and asymmetric jump models, are considered for third-nearest-neighbour jumps on this lattice. Numerical results for the correlation functions relevant for single crystals and polycrystals are presented over the entire temperature range. It is found that the simpler, but unphysical, symmetric jump model is a good approximation to the more complicated asymmetric jump model. Discrepancies between the numerical results of Barton and Sholl and those of Wolf (1975, 1977) are discussed.

If the static lattice Green function of a crystal is known then the distortions produced in the crystal by a point defect may be obtained simply from the Green function and the forces acting on the crystal due to the defect. The numerical evaluation of the Green function for a given model of the crystal force constants involves Brillouin zone summation of a function which is singular at the origin and this singularity can cause considerable numerical convergence problems. It is shown that a useful calculational procedure involves evaluating the Brillouin zone summation by using the Chadi and Cohen sets (1973) of special points in the Brillouin zone together with an efficient extrapolation procedure. Green functions are calculated for some FCC and BCC metals using the force constants obtained from Born-von Karman fits to experimental lattice vibration spectra.

Nuclear spin relaxation rates due to magnetic dipole coupling between diffusing spins in a crystal are linear combinations of correlation functions J(p)( omega ). The high-frequency (low-temperature) form of these correlation functions is obtained for the simple hopping model by solving iteratively the rate equations for the occupancy probability functions describing the relative motion of a pair of spins on a lattice. The resulting high-frequency expansion of J(p)( omega ) is exact for all spin concentrations and gives higher-order terms beyond the familiar asymptotic limit. Numerical results are calculated for cubic lattices, two-dimensional hexagonal and square lattices and for the linear lattice.

Nuclear spin rates of relaxation due to magnetic dipole coupling between spins undergoing relative translational diffusion in single crystals will depend on the orientation of the applied static magnetic field with respect to the crystal axes. The general form of the orientation dependence is calculated for the nuclear spin correlation functions J(q)( omega ) and hence for the relaxation rates and the second moment of a rigid-lattice absorption line broadened by dipolar interactions. The only assumption concerning the diffusional motion is the principle of detailed balance and the orientation dependence is obtained using the transformation properties of spherical harmonics in terms of rotation matrices. Explicit results are obtained for the correlation functions and relaxation rates for each point group and the form of the relaxation rates in the low-frequency (high-temperature) limit is deduced.

In the modern theory of simple metals a large number of physical properties, such as the cohesive energy, lattice constants, lattice structure, elastic constants and vibration spectra are calculated from a self-consistent pseudopotential expansion of the total energy of the ions and conduction electrons. This expansion, which is actually a structural expansion of the cohesive energy valid for an arbitrary arrangement of the ions, has only been known to second order. In the present paper the third-order term in this expansion is derived.

The Madelung constants for the transition metal dichalcogenide layer compounds have previously been evaluated by Shen and Liang (1983) by a direct summation method. It is shown that the use of the two-dimensional Poisson summation formula gives a much more efficient method of evaluating the lattice sums.