Search

The BPP model for nuclear spin relaxation is modified to account rigorously for jumps of hydrogen within g site hexagons in the C15 AB2 structure. The theory is applied to proton relaxation data for low H concentrations in TaV2Hx and ZrCr2Hx. It is shown that the data are consistent with the results of quantum diffusion calculations, diffusivity measurements and H–H interaction effects deduced from quasielastic neutron scattering experiments.

An exact algebraic expression is derived for the diffusivity for diffusion between interstitial e and g sites in the cubic C15 Laves phase AB2 in terms of four rates of jumping between the interstitial sites. The result is for the low concentration limit of diffusing atoms and is valid for arbitrary positions of the e and g sites. The expression reduces to simple forms in some special cases. Some examples of diffusion of H in C15 intermetallic compounds are discussed.

Experimental data for proton nuclear spin relaxation and diffusion of H in HfTi2Hx are analysed by simultaneously fitting the temperature-dependent relaxation and diffusion data with a common set of parameters. HfTi2Hx has the C15 structure with the H occupying the inequivalent interstitial e and g sites. The fitting of the relaxation data uses a rigorous theory of nuclear spin relaxation between inequivalent sites and makes no assumptions about which types of H jumps are significant for the relaxation. The diffusion data is fitted by developing the theory of diffusion between the inequivalent e and g interstitial sites, which enables the diffusivity to be calculated rigorously as a function of temperature from the H jump rates in the low concentration limit. Monte Carlo simulations are used to estimate the effect of diffusion correlation effects at higher H concentrations. Models for diffusion between inequivalent sites involve a large number of parameters and density functional theory (DFT) calculations are used to provide constraints on them. Good fits to both the relaxation and diffusion data are obtained for energy parameters that are close to those from the DFT calculations. A complete set of jump parameters for H between the interstitial sites is deduced which provides a detailed microscopic description of the diffusion as a function of temperature.

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.

The spectral density functions for the relaxation of nuclear spins due to time-dependent electric quadrupole interactions are studied for the diffusion of atoms on a crystal structure. It is shown that, for an arbitrary concentration c of diffusing atoms, the spectral density functions and consequent relaxation rates scale with concentration as c(1-c). The detailed forms of the functions can be obtained simply from analysis of the low spin concentration limit, unlike the analogous case for magnetic dipolar relaxation. Some applications to metal-hydrogen systems are described.

The first application of a recent theory linking nuclear magnetic resonance spin-lattice relaxation rates to interstitial atom motion in disordered systems is presented. Laboratory and rotating frame relaxation rate data taken as a function of temperature for 1H moving in quasicrystalline Ti45Zr38Ni17H163 are fitted with the new theory, yielding a hydrogen site energy distribution of Gaussian shape and width 47±5 meV. The energy barriers for hydrogen motion show a Gaussian distribution of width 50±5 meV, and the difference between the means of the distributions is 0.42±0.01 eV. This is the first time relaxation rates have been analysed to provide information on both hydrogen site energy and barrier energy distributions simultaneously. The data are also fitted using an approach popular for disordered systems: the integration of the Bloembergen, Purcell, and Pound relaxation theory over a distribution of activation energies. The relative merits of this traditional approach and the recent theory in fitting the relaxation data, and also in fitting measurements of the static and magic angle spinning linewidths, are discussed. Although the traditional approach can fit all the data self-consistently, the theory's unsupported assumptions are undermined by the new approach.

Nuclear spin relaxation rates due to magnetic interactions between diffusing spins in ordered systems are often interpreted in terms of an exponential correlation function (the BPP model). The common extension of this to disordered systems is to average the relaxation rate for a jump rate with energy E over a distribution N(E) of energies in the disordered system. A more rigorous extension of the BPP model to disordered systems is described which incorporates simultaneous barrier- and site-energy distributions. An efficient computational scheme is developed to evaluate the triple integrals required. Results are presented for the correlation functions and relaxation rates for some typical values of parameters for both of the above extensions of the BPP model to disordered systems. There are significant differences between the results for the two models. The model presented here provides a means of analysing experimental relaxation data to deduce information about both barrier- and site-energy distributions.

The average jump rate for particles undergoing diffusion in disordered systems is calculated for arbitrary concentrations of particles. The disorder model considered is a combination of site-energy and barrier-energy disorder. Uniform and Gaussian distribution functions are used as examples. The average jump rate does not show Arrhenius behaviour in general, but can in particular cases. The tracer diffusion coefficient is deduced from the average jump rate in the low-concentration limit for the site-energy disorder model for which there are no spatial correlations in successive jumps. For models where spatial correlations do occur, expressions for the diffusion coefficient and the average jump rate can be used to obtain information about the correlations.

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#

It was shown by Jaroszkiewicz and Strange in 1985 that nuclear spin relaxation due to dipolar coupling between like spins diffusing on inequivalent sites involves magnetization recoveries which are linear combinations of exponentials. An alternative approach to obtaining the relevant rate equations is described, which starts from the well known expressions for dipolar relaxation of pairs of spins. The method can be extended easily to other inequivalent-site systems involving both like and unlike diffusing spins. The results for the multiexponential magnetization recoveries can be calculated by routine numerical methods.

Previous calculations of spectral density functions for like-spin magnetic dipolar interactions between spins undergoing two-dimensional diffusion are extended to the hexagonal lattice and the honeycomb structure. The diffusion models used are the random-walk and mean-field models. Analytic approximations to the functions are obtained. It is shown that the simple function proposed by Richards cannot provide good accuracy over the whole range of frequency and temperature, and a modified function is suggested.

Expressions are obtained for the continuous time development of several properties associated with an atom-vacancy encounter due to atomic diffusion at low vacancy concentrations. These are the waiting-time distributions for the next jump of an atom following a given jump, the probability of eventual return of a vacancy, the mean number of atom jumps in an encounter and the atom displacement probability. All these quantities are expressed as summations or integrals which can be computed numerically using as input the probabilities P k(I) of a random walker being displaced by I in k steps for I the origin and a nearest-neighbour site. The probability of return of the vacancy and the mean number of atom jumps in an encounter have the asymptotic form a - b/t 1/2. Numerical results are presented for the sc lattice.

The form of the transverse depolarization function of an I spin due to dipolar interaction with a system of S spins is analysed for local field interactions without using the high-temperature approximation. This analysis is valid in the absence of diffusion and in the presence of a large, static external magnetic field. It is relevant to muons implanted in metals and to the free induction decay of dilute I nuclei in a system of S nuclei, for which the local field model is valid for short times. An analytic expression for the transverse depolarization function is obtained using the Boltzmann distribution for the occupation of Zeeman states of the S spins at a temperature T. The behaviour of the depolarization function depends on the ratio of the external magnetic field B to the temperature T and three regimes are identified. For small B/T the familiar high-temperature result is independent of B/T and sample shape. For intermediate values of B/T the depolarization function depends on B/T and has an envelope that is independent of sample shape and a time-dependent phase that is dependent on sample shape. For larger values of B/T the depolarization function is again independent of B/T with the phase remaining dependent on sample shape. The results indicate the effects that could be observed at values of B/T larger than those corresponding to the high-temperature approximation and provide the values of B/T at which these would be apparent.

The theory of nuclear spin relaxation is developed for a random walk model of H diffusing between sites that form a non-Bravais lattice structure, including multiple rates of jumps between sites in a unit cell. The results are applied to H diffusing by first and second nearest neighbour jumps between interstitial d sites in the A15 compound Nb(3)AlH(x). The random walk model is exact for H-metal dipolar relaxation in the low H concentration limit and provides a good approximation at arbitrary concentrations when average site-blocking of jumps is included. This model can show the high temperature [Formula: see text] frequency dependence of the relaxation rates for one-dimensional diffusion possible in this structure for nearest neighbour jumps along the crystal axes. The low and high H concentration proton-relaxation data for Nb(3)AlH(x), as a function of temperature and resonance frequency, are fitted well at high temperatures using a simple set of H jump parameters. The theory provides a more rigorous approach and enables a more complete analysis than the simple Bloembergen-Purcell-Pound model used in previous work.

Density operator theory of nuclear spin relaxation due to fluctuating magnetic dipole or electric quadrupole interactions is based on a differential equation for the density operator which, in the weak collision limit, depends on the spectral density functions of the dipolar or quadrupolar fluctuations. It is shown that the differential equation for relaxation in the rotating frame may be expressed in a similar form to that for relaxation in the laboratory frame. Expressions for magnetization recoveries in the rotating frame can then be simply deduced from the laboratory frame expressions by replacing the spectral density functions in the laboratory frame results by linear combinations of spectral density functions. The method is applied to single-exponential relaxation for the dipolar mechanism and to single- and multiple-exponential relaxation for the quadrupole mechanism for both longitudinal and transverse magnetization recoveries.

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.

The theory of nuclear spin relaxation is developed for a random walk model of hydrogen diffusing between interstitial g sites in the cubic C15 AB2 structure. The theory is applied in analysing the proton relaxation data for low H concentration in TaV2Hx. The random walk theory results are used to assess the accuracy of a simpler modified Bloembergen–Purcell–Pound model. The temperature dependences of the two rates of jumps between g sites are deduced. The temperature dependence of the rate of jumps between g sites on a hexagon shows Arrhenius behaviour above ~50 K and a non-Arrhenius form at lower temperatures. The values of this jump rate are incompatible with the results from the analysis of quasielastic neutron scattering experiments. Possible reasons for this discrepancy are suggested.

The theory of nuclear spin relaxation due to paramagnetic ions in crystals can be treated rigorously in the rapid diffusion and dimensionless limits for all concentrations of a random distribution of paramagnetic ions if the ion-nucleus interaction is known. Some approximations commonly made in the literature to evaluate the relaxation functions in these limits are critically examined for the dipole-dipole and indirect RKKY relaxation mechanisms. If spherically averaging the dipole-dipole mechanism is avoided the resulting relaxation rates are anisotropic. The anisotropy is calculated for cubic crystals and is large enough to be observed experimentally. It is shown that the usual approximations of evaluating the lattice summation in the rapid diffusion limit as an integral and of averaging the RKKY range functions can introduce appreciable errors. The form of the magnetisation recovery in the diffusionless limit is calculated for the dipole-dipole mechanism and the free electron form of the Benoit-de Gennes-Silhouette (BGS) RKKY mechanism without making these approximations. Difficulties arise for the Giovannini-Heeger (GH) RKKY mechanism in that it can yield negative results for the relaxation rates if the free electron form is used.

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.

We report nuclear magnetic resonance (NMR) measurements of the proton (/sup 1/H) spin-lattice relaxation rate (R/sub 1/) in the hexagonal-close-packed (hcp) solid solution (..cap alpha..) phase of the Sc-H system over the temperature range 10--300 K in which hydrogen pairs are known to form. At low temperatures (10--120 K), fast localized motion of hydrogen between closely spaced tetrahedral interstitial sites in the lattice gives rise to a peak in the relaxation rate. Both the temperature and frequency dependences of the relaxation rate peak exhibit characteristics typical of amorphous and disordered systems, suggesting the formation of hydrogen pairs with little long-range order results effectively in a ''proton glass'' within the metal lattice. The measurements reveal an electronic structure transition near 170 K where the unpaired electron spin density at the proton sites decreases with increasing temperature.

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.

The theory of the effect of the ions on the magnetic susceptibility of conduction electrons has been extended to liquid metal alloys. Detailed calculations have been performed for the noble metal-tin systems. It is concluded that the non linear variation of the susceptibility with concentration of these alloys can be explained in terms of conventional pseudopotential theory.

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.

The theory developed by Riedinger (1971) to calculate the Green function and electron density of states for a metal-impurity system in terms of a model Hamiltonian description of the pure metal bandstructure is extended to the case of interstitial impurities in transition metals. The theory is applied to dilute concentrations of hydrogen in palladium. The perturbing potential is assumed to be a screened Coulomb potential together with an exchange and correlation contribution and the screening parameter is determined from the Friedel sum rule. Calculations of the perturbed electronic density of states are performed for the hydrogen occupying tetrahedral and octahedral sites in the palladium crystal structure. While there is strong perturbation of the palladium valence band, no bound states are formed below the bottom of the band. Such states have resulted from some previous calculations on palladium hydride systems.

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.III see ibid., vol.8, p.775-84 (1978). Calculations of the energy, Delta E, of hydrogen in the transition metals Pd, Rh and Ni previously reported were in error because of an inaccurate approximation used for the exchange and correlation energy contribution xi xc. Corrected results are presented and now disagree with the experimental results. The theory has therefore been critically reexamined and several refinements considered in detail. It is shown that the original theory correctly included the non-orthogonality of the basis functions. Further, the inclusions of second-neighbour sites of the impurity in the Vdd' matrix and gradient corrections to the exchange and correlation potential are shown to have little effect. The original perturbation treatment of the Vss' matrix has been tested by treating the g=0 orthogonal plane waves more rigorously through a Wannier-function representation. Significant changes occur at low energies in the electron density of states but the Delta E values are little affected. It is possible that the g not=0 Wannier functions could produce significant effects. A model calculation for a non-spherical impurity potential shows that a spherically symmetric screened Coulomb potential may be inadequate to describe hydrogen in transition metals.

The theory of nuclear quadrupolar relaxation in liquid metals is formulated in terms of the Van Hove functions and two approximations (A and B) are suggested for the correlation functions of the field gradient that are involved. The ion-ion potentials required are taken as those due to Appapillai and Williams (see abstr. A36507 of 1973). Detailed calculations are performed for Rb, Hg, Ga, In, Sb and Bi and the experimental data for Na, Cu and Al are also discussed. It is argued that approximation A underestimates and approximation B overestimates the cancellation between the two and three particle terms in the correlation functions and that the use of the Appapillai-Williams potential coupled with the ionic Sternheimer antishielding factor underestimates the field gradients involved. The temperature dependence of the relaxation rates is predicted to be somewhat less than that of D-1 where D is the diffusion constant, which agrees qualitatively with the experimental data.

For pt.II see ibid., vol.8, p.1737 (1975). The theory of nuclear spin relaxation by translational diffusion in solids previously developed by Sholl is extended to the case of single crystals. Numerical results are presented over the entire temperature range for FCC, BCC and SC crystal structures enabling the relaxation rates to be determined for polycrystals and for any orientation of magnetic field in single crystals. Some errors in previous work are corrected. The magnitude of the field dependence is quite different for the three crystal structures considered. The validity of the approximation of averaging the relaxation rate rather than the magnetization is examined.

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.

The theory of the electron density of states due to an interstitial impurity in a transition metal developed previously (see ibid., vol.7, p.799 (1977)) is used to determine the energy of the impurity at tetrahedral and octahedral sites in FCC transition metals. The contributions to the energy are a bandstructure term due to the perturbed density of electron states, an electrostatic term to include the additional electrostatic interactions and a term arising from electron exchange and correlation. Detailed calculations are performed for the case of hydrogen in palladium and it is found that the octahedral site has lower energy than the tetrahedral site in agreement with experiment. The calculated values of the energy and heat of formation are also found to be in reasonable agreement with experiment. The accuracy of the host metal bandstructure calculation is shown to be significant. It is concluded that the theory should be sufficiently accurate to determine interstitial site energies for other transition metals.

The general form of the angular dependence of the nuclear spin-lattice relaxation rate is calculated for relaxation due to the various magnetic interactions between the nuclei and conduction electrons in crystalline metallic materials. The forms of angular dependence for each crystal symmetry class are identical with those of the relaxation rate, in the high-temperature (fast-diffusion) limit, due to nuclear-nuclear dipolar coupling, when at least one of the interacting nuclear species takes part in translational diffusion.

Torrey has developed a theory of dipolar relaxation by atomic diffusion in polycrystalline solids for a model of nearest-neighbour jumps in which the average over crystal orientations was taken in an approximate way. The theory is developed here rigorously in a form which could be readily generalized to a single crystal and to more complicated jump models. As an example it is shown that the Torrey theory for a FCC lattice gives results for T1, T1 rho and T2 that are 2.5% too large for high temperatures and 1.4% too small for low temperatures. The errors would be larger for a BCC lattice and lattices with low coordination numbers.

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.

The Green function treatment of Riedinger has recently been successfully applied to the problem of dilute substitutional and interstitial impurities in noble and transition metal hosts (Riedinger 1971, Cook and Smith 1974, Sholl and Smith 1977). This approach is limited in its generality, however, by its approximate treatment of the s-s interaction terms. As a result, it is not applicable to systems such as CuZn, CuGa and CuGe, or CuH and AgH, where the s electrons play a dominant role. An exact expression for the density of states in such dilute impurity systems is derived starting from the Dyson equation and employing a matrix method. This expression is identical to Riedinger's when the s-s interaction terms are set to zero.

The authors introduce a new model for the screened Coulomb potential appropriate to dilute interstitial hydrogen impurities in a metallic host. This new potential provides an accurate representation of the self-consistent potential of a proton in an electron gas and is more suitable than the customary Thomas-Fermi potential in that it yields a charge density which is finite at the impurity site. Calculations have been performed using this new potential and the screened Coulomb potential employed in earlier work, for the case of hydrogen in palladium. Neither the change in the electronic structure, Delta N(E), nor the change in the total energy, Delta E, are found to be very dependent on the exact form of the screened Coulomb potential. The theoretical formalism has also been extended to the case of dilute hydrogen interstitial impurities in the FCC transition metals Ni and Rh. In both metals Delta E is found to be approximately -1.1 Ryd, and the octahedral site is predicted to have lower energy in agreement with experiment.

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.

Previous calculations of the field gradient in BeO on the basis of an ionic model have shown that it is doubtful whether such a model is adequate. Measurements of the temperature dependence of the quadrupole splitting in BeO are presented and it is shown that the very small observed temperature dependence is inconsistent with the ionic model. An alternative covalent model is proposed, based on a description of the valence electron wave functions in terms of molecular orbitals. The difference between the variational parameters used in describing the molecular orbitals in the inequivalent bonds in BeO is unknown, and so it is not possible to deduce the field gradient uniquely. The covalent model, however, is consistent with the small temperature variation of the field gradient.

The calculation of the electric field gradient in ionic crystals assumed to consist of discrete deformable ions is discussed, the ions being characterized by their multipolar polarizabilities. A set of linear self-consistent equations for calculating the electric field and field gradient components at all inequivalent lattice sites is derived with particular reference to long-range effects, and including terms up to the quadrupole moment. The calculation of antishielding factors is not considered. Some of the lattice sums involved are conditionally convergent and the implications of this on their evaluation are discussed. It is shown that the conditional convergence and the existence of Lorentz factors is intimately related. The lattice sums are transformed into a rapidly convergent form by the method of plane-wise summation used by de Wette in 1961. The field gradient at the site of 127I in hexagonal AgI is calculated and found to be critically dependent on the relative displacement of the Ag+ and I- sublattices, its value changing sign over the experimental range of this crystal parameter. An upper bound of 1.7 Mc/s for the nuclear quadrupole coupling constant is deduced on the basis of the ionic model.