Table of contents

An attempt has been made to calculate the isotropic third-order elastic constants of a polycrystalline solid in terms of the single-crystal constants using the methods employed by Voigt and Reuss in the second-order case. The application of these calculations to a variety of different materials shows that the discrepancy between the two methods is much greater than in the case of second-order constants.

A simplified scheme is presented for discussing and calculating estimates of the third-order elastic constants of quasi-isotropic material based on single-crystal data.

The difference between the Voigt and Reuss averages at the third order is shown to depend solely on the second-order single-crystal anisotropy. Only the two smaller constants are affected in this way. Chang's data for niobium are used to compare the methods.

The zero-point vibrational energies of the three cubic lattices are calculated using the Bethe approximation. The harmonic and the quartic anharmonic energies are in good agreement with the results of standard lattice dynamical methods. However, the cubic and some of the non-leading anharmonic terms appear to be considerably overestimated. The results are applied to the inert gas solids to investigate the following models: (i) central nearest-neighbour interactions only; (ii) central all-neighbour interactions using a Lennard-Jones m-6 potential function; (iii) central all-neighbour and non-central interactions (the latter using the potential function given by Axilrod and Teller). Non-nearest-neighbour and non-central interactions were taken into account by using the Einstein approximation while central nearest-neighbour interactions were calculated using the Bethe approximation. It is found that the approximations (ii) and (iii) decrease the harmonic energy and increase the anharmonic energy with respect to approximation (i). The results indicate that the approximation made in (i) is accurate to approximately 5% for calculating the harmonic and 4% for calculating the anharmonic energies of neon. For xenon the accuracy is approximately 11% and 15% respectively. It is therefore concluded that calculations of the properties of the inert gas solids, especially the heavier gases, should include non-central forces and all-neighbour interactions.

ENDOR measurements have been made to determine the transferred hyperfine interaction parameters for the fluorine ions forming the second-nearest anion neighbours to V²⁺ and Mn²⁺ in KMgF₃. As expected, the main contribution to the interaction is due to the dipolar coupling between the fluorine nuclei and the electronic spin on the impurity ion. Contributions to the remainder of the interaction due to overlap effects with the unpaired spin on the intervening first-neighbour fluorines are calculated; the calculated parameters are found to reproduce correctly the form of the interaction, and are sufficiently large to account entirely for it in the vanadium case. The use of free-ion wave functions appears to be a good approximation.

A study is made of the properties of a paramagnetic ion in an E_g orbital state interacting with a lattice. A general formulation is given, appropriate to any spin, and it is shown that, if the ion-lattice interaction is sufficiently great, for the low-lying levels a profound change occurs in the motion of the neighbours. This may properly be described as a dynamic Jahn-Teller effect. The nature of the motion is examined and it is suggested that the ion plus its neighbours acts as a non-linear scattering centre for incident thermal waves, and that there are local vibrations of comparatively large amplitudes at `beat frequencies'. In any situation where this description is applicable it then appears to be a reasonable approximation to replace the ion plus lattice by a `molecular model'.

The collective coordinates appropriate to a trigonally distorted tetrahedron are derived. The changes in the local crystal field produced by such collective displacements of ligand ions relative to a central ion are calculated with particular reference to Cu²⁺ ions in a ZnO lattice.

A detailed theoretical discussion is given of the linear electric field effect occurring in electron paramagnetic resonance spectra from Cu²⁺ ions residing in tetrahedral crystal fields. Contributions from both the electronic effect and ionic effect components to the g shift are derived from basic principles for examples in which the trigonal or tetragonal distortions of the tetrahedron occur. The electronic effect arises because the ground state of Cu²⁺ consists of hybrids of 3d and 4p orbitals, while the ionic effect occurs because the change in crystal field due to the relative displacement of the ligand ions is linear in the applied electric field as a tetrahedron has no centre of symmetry. A measurable g shift is predicted for the trigonally distorted example only. From the theory discussed it is generally expected that the relative amounts of the two effects will vary considerably with both the host lattice and ion.

A calculation has been performed by using dipolar lattice sums in α-quartz to interpret the refractivities in terms of the point polarizabilities of silicon and oxygen. The details of the calculations are similar to those in the first two papers of this series which dealt with the TiO₂ structures. If quartz is taken to consist of an isotropic Si⁴⁺ ion and an O^2- ion, then the results of the present calculations show that the oxygen ion can be considered to have an axial polarizability tensor with α₀^parallel = 1.43 ų, α₀^{perpendicular} = 1.32 ų. On the other hand, the analysis does allow an interpretation in terms of neutral oxygen and silicon and it is concluded that further work is needed before the state of the atomic components in the quartz lattice is finally decided.

The decay and growth of F, F-aggregate and colloidal centres in additively coloured crystals have been studied (i) by bleaching the additively coloured crystal with white or F light, (ii) by annealing the optically bleached crystal at room temperature in the dark and (iii) by heating the coloured and `coloured and optically bleached' crystal up to 400 °c. The thermal stability of the centres can be described by the equations colloids, and colloids F. N centres are stable at room temperature and disappear completely on heating the crystal to 100 °c. Correction in the F-band height due to the over-lapping absorption of the excited states of the M centres is evaluated. Peak positions and half-widths of the M, R<sub>2</sub>, N and colloid bands are determined in highly pure and `pure' KBr crystals at room temperature. The variation of the half-width W of the M band in KBr with temperature is found to vary as

The changes in the absorption coefficients at the peaks of the F and F-aggregate bands observed in the optical bleaching experiments, the room-temperature annealing experiments and the thermal bleaching experiments are used to obtain the oscillator strengths of the F-aggregate centres in KBr. By taking the oscillator strength of the F centre as 0.48, the values 0.11, 0.43 and 0.06 are obtained for the oscillator strengths of the M, R and N centres, respectively.

Lithium fluoride samples from two different sources have been irradiated at 81 °K and then warmed to room temperature. Three annealing stages of the F band were found in this temperature range. Stages I and III (at about 120 °K and 250 °K respectively) have also been found by previous authors, whereas stage II (at about 180 °K) has not been reported before. The samples were also warmed to several temperatures between 81 °K and room temperature after irradiation, then recooled to 81 °K and re-irradiated. Both the annealing behaviour and the re-irradiation behaviour of the F band were found to be markedly sample dependent.

An optical absorption band which has not been previously reported was found after irradiation at 81 °K. This band, which has its peak at 415 nm, anneals at about 100 °K.

The angular variations of the electron paramagnetic resonance linewidths of the ΔM = 1 transitions at 35 GHz have been studied in rubies grown by the vapour phase, Verneuil and Czochralski processes and having chromium concentrations in the range 0.015 to 0.20 at.%. Using a strain and mosaic model, parameters characterizing the strain at the Cr³⁺ sites and the axial misorientation have been derived by a least-squares analysis. The spread in zero-field splitting is typically about 5×10^-4 cm^-1, giving Δ(2D)/2D similar, equals 10^-3. The values of the strain components e_xx, e_zz and e_xz lie in the range 10^-4 to 2×10^-2 and appear to be growth dependent.

The saturation behaviour of an electron spin resonance line depends on the rate at which energy, absorbed from the microwave field by a relatively small number of resonant spins, is shared with the remainder of the spin system through the process of spectral diffusion or cross relaxation. This rate, being dependent on the spin-spin interactions, offers a possible means for their study in dilute spin systems. With this object, a general theory of saturation has been developed. It is based on the theory of Provotorov which has been successful in accounting for saturation and related effects in nuclear magnetic resonance. Whereas Provotorov's theory regards the spectral diffusion process as instantaneous (the case of homogeneous broadening), our approach takes account of the finite rate and is therefore applicable to dilute spin systems and inhomogeneously broadened lines. As a result of the generality of our method, the results of Provotorov, Portis, Wolf and Bloembergen et al. occur as special cases.

Measurements are described on a copper complex at 4 °k which support the thermodynamic model on which the theory is based. Where more than two levels are involved, flip-flop transitions connecting one of the saturated levels with a third level transfer local field energy away from the saturated levels. A consequence is that the theory of Bloembergen et al. becomes applicable. Absorption and dispersion saturation measurements made without field modulation on ruby at 4 °k confirm this.

Nuclear magnetic resonance spectra are calculated for an equilateral triangle of protons tunnelling between equivalent orientations. It is shown that the effect of the motion may be described by introducing an interproton exchange interaction into the nuclear spin Hamiltonian, the coefficient J being related to the height of the potential barrier to rotation. This term has matrix elements equivalent to those of the rotational energy operator. Spectra are obtained for arbitrary J, giving a quantum-mechanical solution of a motional narrowing problem and demonstrating the formal identity of the phenomenon, in this case with exchange narrowing. The limit of large J is in agreement with the classical result of Andrew and Bersohn. Isotropic averages of the spectra with Gaussian broadening to take account of inter-methyl-group interactions are computed for comparison with measurements on powders at low temperatures.

Inelastic neutron scattering measurements are used to determine the wave-vector- and frequency-dependent spin correlation function J(q, ω) within a (110) plane of paramagnetic RbMnF₃ at 3·5T_N. This simple cubic antiferromagnet is known to have a nearest-neighbour Heisenberg exchange Hamiltonian with S = fraction five-twos and J = 0·28 mev, so that a precise comparison of the results with existing theories can be made without unknown parameters. For q vectors between 0.1 and 0·4 Å^-1 (q_max = 0·74 Å^-1), the scattering is isotropic in q with a frequency dependence close to a Lorentzian with depressed wings. From the variation of the Lorentzian widths with wave vector the diffusion constant capital Lambda, Greek = 8·0 + 1·0 mev Ų is deduced, in good agreement with the theory of Bennett and Martin. At larger q vectors the scattering is compared with the Gram-Charlier expansion having the correct second and fourth moments as suggested by Collins and Marshall, and with Fourier transformations of the space- and time-dependent spin correlations calculated by Windsor. Good agreement with these infinite-temperature theories is found throughout the zone except at energy transfers less than 4 mev. The discrepancies at small energies, which are greatest near the antiferromagnetic reciprocal lattice point, are ascribed to the effects of short-range order, and show features in quantitative agreement with the finite-temperature calculations of Sears.

Angular distributions characteristic of the small-angle scattering of neutrons by long-wavelength spin waves have been observed at temperatures up to and above the Curie temperature in iron and nickel.

For temperatures below 0·9T_c the spin-wave stiffness D, the spin-wave scattering cross section and the spin-wave lifetime have been obtained directly from the measured angular distributions, and the results are compared with the predictions of spin-wave interaction theories. For temperatures up to 0·42T_c in iron a good fit to the variation of D can be obtained using an expression of the form D₀-D₁T²-D₂T^5/2 based on the itinerant electron model, but above 0·42T_c the fit becomes progressively poorer and D decreases faster than the fitted expression. The cross section for spin-wave scattering shows a temperature dependence additional to that contained in the thermodynamic factor alone; for iron at temperatures up to 0·7T_c this additional dependence has the form <S^z>{1-(0·30±0·08)T/T_c} and is stronger than the variation with <S^z> predicted by Marshall and Murray in a calculation of the effects of kinematic interactions between spin waves in the Heisenberg model. The observed spin-wave lifetimes have the same form of temperature dependence as is obtained from a nearest-neighbour Heisenberg model calculation by Cooke and Gersch.

At temperatures above 0·9T_c the observed angular distributions were corrected for critical scattering using calculated cross sections for the latter due to Villain. D is found to fall slowly on passing through the Curie temperature in both iron and nickel, and is still of the order of a quarter of the room temperature value when the spin-wave scattering intensity falls below the threshold of observability.

The corrections to the cross section for the diffuse elastic scattering of neutrons from ferromagnetic alloys, due to the inclusion of second-order effects in the fluctuations of short-range-order parameters, are derived for a suitable model. The limiting value of the cross section for forward scattering is obtained by considering the fluctuations of the total magnetization in a general way. The corrections are shown to be small and can be neglected in most cases.

The dynamical theory of x-ray diffraction is adapted to the case of nuclear resonant diffraction of Mössbauer radiation. It is found that the dispersion surfaces for the Laue and Bragg conditions of incidence are markedly different and that anomalous transmission occurs in both cases. The effect of the accompanying electronic scattering is to increase the anomalous absorption coefficient. The integrated reflecting power in both Laue and Bragg cases is found to be roughly proportional to the modulus of the total scattering amplitude.

The characteristic energy loss spectra of electrons reflected from freshly evaporated Al and Cu specimens have been measured for various primary electron energies in the range 9-265 ev. Contrary to expectation the ratio of the intensity of the 10 ev surface plasma loss to that of the 15 ev volume loss in Al, for electrons scattered through 90°, is found to decrease with decreasing primary energy in the range from about 240 ev to 80 ev. This appears to be due to the behaviour of the surface loss, which decreases rapidly in intensity with decreasing primary energy. Both the surface and volume plasma losses of Al cease to be excited when the primary energy is less than about 26-29 ev, the surface loss cutting off at a slightly lower primary energy than the volume loss. The modified surface loss of Al resulting from oxidation of the specimen surface is observed at 7.1 ev, and the half-width of the 10 ev surface loss increases from 3.8 ev to 5.3 ev with increasing surface oxidation, indicating that the surface plasma oscillations are damped by the contamination.

The energy losses of Cu have been investigated by observing their intensity dependence on (i) the primary electron energy for a fixed scattering angle of 90° and (ii) the electron scattering angle for various fixed primary energies, scattering angles of 90° and 20° being employed. The results indicate that a loss observed at 3.5 ev is possibly due to excitation of surface plasma oscillations.

Spin-lattice relaxation times have been measured at frequencies from 9·5 to 24 GHz for isolated (IrCl₆)^2- complexes in (NH₄)₂PtCl₆ and K₂PtCl₆. The relaxation rate for the ammonium salt can be expressed as a sum of terms corresponding to a direct process, a Raman process and a concentration-dependent process as follows:

for the ratio Ir:Pt = c less than about three per cent (f in Hz, T in °K). The value of Δ(similar 6·5 °K) agrees with the previously measured exchange interaction between nearest-neighbour (IrCl₆)^2- complexes. At higher values of c, a faster variation than c² is found. Data for the potassium salt are similar except for the strongly enhanced Raman process, 1/T₁ = 3·5 × 10^-2T⁹.

Theoretical estimates for the direct and Raman states indicate that a simple Debye model is not sufficient to explain all experimental data. All data regarding the concentration-dependent process agree with a mechanism in which single complexes cross relax to exchange-coupled clusters of three or more complexes, and relaxation of such clusters is discussed briefly. In particular, exchange-coupled pairs relax much too slowly to affect relaxation of single complexes (see also II).

Measurements of the spin-lattice relaxation times at X-band and K-band frequencies have been carried out for transitions within the excited T = 1 state of nearest-neighbour exchange-coupled pairs of (IrCl₆)^2- complexes in crystals of (NH₄)₂PtCl₆. The temperature dependence of the relaxation time indicates that a two-phonon process dominates at the low concentrations (Ir:Pt = 1.2% and 2.3%) and a calculation is made of the two-phonon relaxation rate occurring through modulation of the crystal field. This gives the observed order of magnitude but predicts larger anisotropy in the relaxation rate than was observed. Other possible mechanisms are also discussed.

At higher iridium concentrations (Ir:Pt = 5.0% and 8.5%) we observed a concentration-dependent process with a temperature variation 1/T₁ similar exp(-Δ/kT). This is consistent with cross relaxation to larger clusters.

The theory of low-spin d⁵ octahedral complexes is re-examined, and it is found that excited states can contribute significantly to the magnetic properties of the ground state. Electron spin resonance experiments have been performed on (IrX₆)^2- complexes (X = F, Cl, Br), and analysis of the data using the more detailed theory shows that these complexes are much more covalent (in terms of the magnitude of the admixture parameter fπ) than had hitherto been supposed.

A representation for the specific heat below T_c of a three-dimensional Ising ferromagnet (or fluid) is constructed. On expansion, it yields the first twelve terms of the exact low-temperature series for the tetrahedral lattice, and behaves asymptotically as

where αprime = fraction one-eighth, A^- = 0·20 and B^- = 0·13. However, such behaviour is only becoming evident at t identical with 1 - T/T_c similar, equals 10^-4. Indeed, from t similar, equals 4 × 10^-2 to 4 × 10^-4, C_H appears to be increasing roughly logarithmically.

The specific heat of the Ising model is compared directly with experimental results for four `classical' fluids and three Ising-like antiferromagnets. Difficulties in establishing experimental values for the critical exponent αprime are discussed.

A simple explanation is proposed of why the `rounding-off', observed in most magnetic transitions, is often more pronounced below T_c.

We discuss the additional damping to which magnetic domain wall motion is subject due to the replacement of a small proportion of the yttrium in yttrium iron garnet by erbium. Published ferromagnetic resonance data for a specimen of composition Er_0·078Y_2·922Fe₅O₁₂ are used to predict the wall mobility in this material assuming that the erbium doping, acting according to the longitudinal relaxation model, is the sole cause of damping. Measurements are reported of the time taken to reverse the flux in single crystal toroids of pure yttrium iron garnet and of the above composition when reversal takes place by domain wall motion, following the sudden application of a magnetic field. Comparison of theory and experiment shows that the proposed mechanism controls the reversal time for the doped material between 3 °K and 60 °K, where the predicted damping is strong. At higher and lower temperatures the controlling mechanism, which appears to be the same in the doped and pure samples, is different and not understood.

The ordinary Hall coefficient R₀ for body-centred cubic iron at temperatures below 100 °K is analysed using a two-band model and it is shown that many of the experimental observations of Volkenshtein and Fedorov and of Dheer may be explained. These include strong dependences of R₀ on temperature and on small amounts of impurity, and changes in sign of R₀. The several different scattering mechanisms of importance in pure Fe at low temperatures will each dominate in a particular region, leading to large changes in R₀ in a relatively small temperature range. Parameters are introduced to describe the effects of anisotropies in the Fermi surfaces and in the relaxation times, and these are explicitly estimated for the case of scattering by impurities, using a technique whereby the wave number and the relaxation time are expanded in a series of cubic harmonics up to fourth order. The value of R₀ as the temperature T -> 0 °K is estimated to be (0·3 ± 0·15) × 10^-13 Ω cm G^-1 which is consistent with experimental data.

A new examination of the theory of pseudo-potentials, necessitated by the discovery that smoothness of the pseudo-wave function is not a relevant criterion for their effectiveness, provides an understanding of the analytic properties of pseudo-potentials in terms of their pseudo-core energies E_cprime. In particular, it is found for Hermitian pseudo-potentials that, if E_cprime greater, similar <c|H₀|c>, the Born series cannot have good convergence properties, but, for E_cprime = 0, the series has good asymptotic convergence. The new pseudo-potential, with E_cprime = 0, is the correct form to use at all energies, but differs substantially from other forms only at higher energies. Its effectiveness is demonstrated by a model calculation.

The structure factors for four liquid metals (Zn, Tl, Sn and Bi) at temperatures close to the melting point have been determined by means of neutron diffraction and the data used to calculate the total correlation functions. In addition a detailed study has been made of liquid Pb as a function of temperature. Data are given for the structure factors, the total correlation functions and the direct correlation functions at four temperatures in the range 340 °c to 1100 °c. The total and direct correlation functions were used in the Percus-Yevick and hyper-netted chain equations to calculate the interionic potential. The structure factors were applied to the Ziman theory of electron transport in liquid metals. It is shown that the predicted temperature dependence of resistivity and thermoelectric power is in good agreement with experiment.

Critical opalescence in mixtures of Bi-Zn, Pb-Ga and Bi-Ga has been observed in the scattering of 5·3 Å neutrons. Separation of quasi-elastic from inelastic scattering has been used to separate the long-range concentration fluctuation part of the cross section from other effects. To a first approximation, this cross section agrees with the Ornstein-Zernike theory of critical scattering.

It is shown that simple screening theory will explain the change in solvent Knight shift on alloying for every dilute solution of an alkali in an alkali on which observations have so far been made. No recourse is made to semi-empirical potentials or to size effects. On the present interpretation the experiments give direct information on the sign and magnitude of the displacement of charge in the d component around a dissolved ion.

The saturation magnetizations of body-centred cubic, iron-rich, solid solution alloys with copper, gold, germanium, tin and antimony have been measured between room temperature and 8 °K. Mean atomic magnetic moments calculated from the data decrease linearly with solute concentration. The rates of decrease of moment with solute concentration for the solutes listed are -2·0, -1·1, -1·4, -1·0 and -1·0, respectively. Only iron-copper alloys come close to fitting the dilution model usually considered appropriate for iron alloys with non-transition element solutes. The other solutes apparently cause an increase in the average moment per iron atom. The exact mechanism by which this increase occurs is not clear.

The magnetic properties of the series of pseudo-binary compounds YFe_uCo_2-u have been examined in an attempt to obtain further information concerning the nature of the transition-metal moment in the cubic Laves-phase rare-earth - transition-metal compounds. The variation of the observed moment throughout the series can be understood on the basis of a non-localized moment associated with the transition-metal ion. The variation of the Curie temperature may be associated with the variation of the lattice parameter between the terminal compounds.

Hall effect measurements by Putley and other workers on n-type InSb in the temperature range 2-5 °K and in magnetic fields up to 15 kG have revealed a rapid increase in the Hall coefficient R with decreasing temperature, with an activation energy depending on magnetic field.

In order to study more carefully the isolated donor levels as a function of magnetic field, the theory of the shielding of a charged centre by conduction electrons, in which the cylindrically symmetrical screening about the field direction is accounted for from the outset, is considered. We have then carried out variational calculations of the two lowest impurity levels, as a function of magnetic field. While the individual levels are appreciably changed by screening, the energy gap is very insensitive to the detailed choice of electron density over a wide range of densities and has its hydrogenic value to high accuracy. The magnitude of the observed activation energy accords well with this energy gap over a substantial range of magnetic fields. The model on which the Hall effect measurements are to be understood is then seen to be one in which the impurity states higher than the ground state are broadened by overlap and merge into a quasi-continuum with the InSb conduction band.

The residual concentration dependence of the activation energy, as well as its magnetic field dependence, suggests that the measured activation energy will eventually reflect energy gaps between the ground and higher excited states of the donors as initially broadened impurity levels cease to conduct in higher fields.

This paper presents an analytical investigation of the generation of the second harmonic component in the current density in a non-degenerate semi-conductor due to an electric vector and gradient of electron density and/or temperature; ionized impurity scattering has been considered to be the sole mechanism of scattering. Using the expression obtained for the second harmonic component of the current density, expressions for the magnitude of the transmitted and reflected second harmonic component of the electric vector have been obtained when a plane-polarized electromagnetic wave is normally incident on a semiconducting slab with gradients of electron density and/or temperature. For a typical case of a germanium slab the amplitudes of reflected and transmitted second harmonic components are seen to be 1·3% and 6·8% respectively of that of the fundamental when an electromagnetic wave of amplitude 0·1 e.s.u. cm^-1 is incident on it.

Satisfactory solutions have been obtained for the phase transitions of alkali halides by employing the Born-Mayer model with the recently revised van der Waals coefficients. These new coefficients for the most part eliminate the use of high van der Waals term multiplicants K > 1 as required earlier by Rao and co-workers when Mayer's coefficients were employed; there appears to be need for increased van der Waals terms in the Born-Mayer description of alkali halides.

It is proposed that the line spectrum of nitrogen-nitrogen pairs in gallium phosphide arises from the recombination of an exciton situated at one nitrogen atom, its energy being perturbed by the strain field of a second nitrogen atom. The model predicts a series of lines with energies displaced from a series limit by an amount βprimeγ_NN^-3, with βprime a constant and γ_NN the nitrogen-nitrogen separation.

The low-temperature limit of the effective Grüneisen parameter of calcium, strontium and barium fluorides has been calculated on Axe's shell model and compared with the values derived from thermal expansion by Bailey and Yates.

Objections are made against a recently found concentration dependence of the exchange coupling constant J.