Table of contents

For non-cubic solids there is no unique heat capacity at constant volume, although the symbol C_v has often been used to denote the quantity C_p -β²VT/X^T. This quantity is sometimes wrongly identified with C_η, the heat capacity at constant strain. In this paper it is pointed out that C_p -β²VT/X^T is actually the heat capacity defined under the double restraint of constant volume and isotropic stress, here denoted by C_v. The relation of C_v to C_p and C_η is examined, and it is shown that C_p >or= C_v>or= C_η. C_v and C_η are equal if and only if γ₁ = γ₂ = γ₃, γ₄ = γ₅ = γ₆ = 0, where the γ_λ are the generalized Grüneisen functions.

Explicit expressions for C_v - C_η are obtained in terms of experimental thermodynamic properties. With suitable definitions for strain coordinates and elastic constants these expressions are valid at all pressures. Values for C_v - C_η at zero pressure are calculated from experimental data for a variety of non-cubic crystals. For all these crystals the differences between C_v and C_η are of the order of 1% of the heat capacity or less; they are largest for highly anisotropic solids at high temperature.

It is shown that, if thermal phonons are scattered by both processes which conserve and also by those which do not conserve wave number, then the thermal conductivity K may be expressed as a power series of the form

Here τ, C, v and α are respectively the relaxation time for normal processes, the specific heat, the phonon velocity and a numerical constant. K₀ is inversely proportional to the strength of those processes which do not conserve wave number and hence, by measuring experimentally how K varies with the strength of these processes, the value of τ may be found. The method is applied to lithium fluoride giving results which are in reasonable agreement with those obtained in earlier work by more complicated analysis.

A method is described for the determination of strict upper and lower bounds on the frequencies of local modes due to impurities and defects in solids. The physical basis for the method stems from the dynamical properties of a small `defect region' in the solid which surrounds the defect or impurity of interest; mathematically, the method depends upon theorems due, in their original form, to Kato. The method does not depend for its use upon the availability of tabulated data.

The contributions to the zero-point vibrational and static lattice energies from a long-range three-body interaction potential of the form given by Axilrod and Teller are calculated for neon, argon, krypton and xenon. It is shown that for the last three the contribution to the vibrational energy is of the same order as the two-body anharmonic contribution and of opposite sign.

A dispersion relation valid near the exciton pole of the crystal Green operator is derived using perturbation theory. It is evaluated by two routes. The first involves only static interactions while the second introduces retarded lattice sums. The equivalence of the two relations is demonstrated for cubic crystals by calculating the retarded lattice sum as an integral. The divergence in the retarded sum investigated by Avery is isolated and shown to be the exciton-photon coupling constant divided by the free-photon dispersion factor. Some new results for the retarded interaction between molecules in the dipole approximation, and a new formula for the retarded dipole sum over a thick spherical shell are given.

It is pointed out that the steady state can be reached even in the case of a strong electric field, the intensity of which is limited by the applicability of the effective mass approximation.

The statistical (Thomas-Fermi) model is used in a calculation of the potential function of a sodium atom embedded in an electron gas whose density is one electron per atomic cell. Exchange is included.

The self-consistency of a vacancy potential in a model of a spherical metal has been checked numerically in order to estimate the accuracy of such self-consistent calculations. Further calculations on the screening around a magnesium ion core indicate that its effect is weak giving only small changes in the free-electron energy spectrum. The application of this result to alloys is discussed.

The possibility of crystallization and magnetic ordering in an electron gas is examined in the determinantal approximation, using a variational method. A Mott transition is found at r_s similar, equals 10 to a ferromagnetic crystalline state of the Wigner type, while an antiferromagnetic crystalline phase seems to be excluded. A simple way of including correlation effects is proposed, in which a correlation function is introduced into the interaction energy between electrons of antiparallel spin. The possibility then emerges of a crystalline antiferromagnetic phase over the large intermediate density range 10 less, equals r_s less, equals 40.

A formulation of the problem of low-energy electron diffraction from crystal surfaces is given using multiple scattering theory. Only elastic processes are considered. The resultant formulae are straightforward to compute and can be applied to model surfaces of arbitrary complexity, provided they are regular.

In a mixed magnetic system, such as a ferromagnetic alloy, there is a spatial inhomogeneity in the distribution of magnetization and this gives rise to a diffuse scattering of neutrons which is strictly elastic. The observation of this diffuse scattering is capable of giving information about the magnetization distribution on an atomic scale. In practice it is necessary to assume suitable models for the way in which each atom affects the magnetization in its immediate environment. Various models are described and it is shown that, even for concentrated alloys where a number of effects must be allowed for, the cross-section formulae are simple. It follows that neutron experiments on mixed magnetic systems can be interpreted in terms of functions of physical significance.

The linear spin-wave theory of a single magnetic impurity in a body-centred Heisenberg antiferromagnet is presented. The impurity-host exchange is assumed positive and hence in the ground state the impurity spin is aligned parallel with the spins on the same sublattice. The calculation consists of deriving the two-time thermal Green functions for the sublattices from their equations of motion. From the poles of the Green functions the criteria for the impurity modes are obtained and calculations of the energies, widths and spin contractions for the modes are given. Because the inelastic neutron cross section can be expressed in terms of spin-spin correlation functions it is readily calculated from the Green functions. Particular attention is paid to the diffuse contribution to the cross section.

In a previous paper we discussed the energy modes of a magnetic impurity with positive impurity-host exchange in an antiferromagnet. This was achieved by deriving the appropriate thermal two-time Green functions from which both the impurity states and inelastic neutron cross section were calculated. In this paper we follow essentially the same scheme to discuss the case of a negative impurity-host exchange coupling. The p-, d- and f-type modes remain the same but only one s-like mode occurs as opposed to two in the previous case. We give the energies and widths of this s mode for a wide range of the perturbation parameters. The diffuse inelastic neutron cross section is also calculated. This has exactly the same form as for an `antiferromagnetic impurity' in a ferromagnet but does not have the same energy behaviour because of the difference in the spin-wave spectra of the ferromagnetic and antiferromagnetic hosts.

The inelastic neutron scattering data of Møller and Houmann for spin waves in terbium at 90 °K are analysed on the basis of a model consisting of Heisenberg exchange and axial and hexagonal crystal field terms. By considering the conditions required to fit the calculated spin-wave energies to the experimental results at points of high symmetry in the Brillouin zone a set of interatomic exchange constants extending to sixth-nearest neighbours is deduced. These provide a reasonably good interpolation scheme for obtaining spin-wave energies throughout the Brillouin zone, as is required for the study of thermodynamic properties and spin-wave renormalization. The question of how well the theory of Ruderman and Kittel, Kasuya and Yosida accounts for the exchange interactions in terbium is also considered. The pertinent quantities S(J_o - J_q) and Svertical barJ_qprime vertical bar are calculated in the free-electron approximation and a striking disagreement is found between the calculated and experimental results for S(J_o - J_q) in the outer part of the `ab plane' of the Brillouin zone.

A variational principle for the free energy is applied to the Hamiltonian for terbium in the ferromagnetic region to study the effect of interactions among spin waves having their origin in the strong crystalline electric fields. At high temperatures this `crystal field renormalization' has the effect of multiplying the crystal field parameters appearing in the spin-wave energies of the non-interacting theory by powers of the reduced magnetization. The powers are found to be close to those associated with the corresponding magnetic anisotropy coefficients. This leads to an appreciable temperature dependence of the spin-wave energies which, after allowing for exchange renormalization, appears to be in reasonable agreement with the inelastic neutron scattering results of Møller and Houmann. The ferromagnetic resonance measurements of Bagguley and Liesegang and of Stanford and Young have also been analysed and found to be consistent with this picture.

The spin-lattice relaxation time T₁ of Tb³⁺ in CaWO₄ has been measured in the temperature range 1·3-9 °K. The dependence of the relaxation rate on temperature can be described by

This suggests that the first excited state lies at 78·8 °K above the ground doublet, which is confirmed by an absorption experiment. These results are compared with those predicted by a set of crystal field parameters suggested by Shekun.

Spin-lattice relaxation experiments using the pulse saturation technique were carried out on crystals of (Cr:Al)K(SO₄)₂. 12H₂O containing 0·1 to 6·0 at.% Cr³⁺. Measurements were made at 9·27 GHz in the temperature range 1·6-4·2 °K. The effect of saturating pulse width on the observed recovery curve was studied and the results analysed using a model involving cross relaxation of characteristic time T_s, through the inhomogeneously broadened line. It was shown that true values of T₁ (similar 3 ms) were only obtained with pulses of width greater than 1 ms or when measurements were made not less than 2 ms after the end of the pulse. Computed values of T_s were 850 and 1000 μs at 4·2 and 1·6 °K respectively.

Measurements were made in [100], [110] and [111] directions and the temperature dependence of T₁ followed the T^-1 law of the direct process. Along [110] and [111] directions, T₁ was concentration independent but, along [100], this independence was not observed.

The intensities of optical absorption spectra of Er³⁺ in single crystals of yttrium gallium garnet at 700 °K have been shown to fit Judd's theory of 4f-4f transitions with the following parameters: T₂ = 1·44 × 10^-21 s, T₄ = 1·12 × 10^-21 s, T₆ = 1·42 × 10^-21 s. These have been used to estimate Einstein coefficients useful in a discussion of the quantum efficiency of the fluorescence from the ⁴S_3/2 state, and the efficiency of infra-red quantum counter action in the material. For pure spontaneous radiation an efficiency of 71% for transitions to the ⁴I_15/2 ground state is expected. The quantum efficiency has also been directly measured and a value of 12% was obtained. This is lower owing to competing phonon processes. Using the experimental value for the quantum efficiency, and the above values of T, a theoretical photon gain for quantum counter action of 0·55 × 10^-8 has been calculated.

High-resolution optical absorption studies on GdAlO₃ containing nominally 2 wt% Er³⁺ reveal a splitting of Er lines that is ascribable to magnetic interaction with the ordered Gd³⁺ system below T_N (3·7 °K). These lines are studied as a function of external fields up to 90 kOe applied along the axis of the antiferromagnetic Gd³⁺ system. Assuming the Er-Gd nearest-neighbour interaction to be dominant and equal for all six nearest-neighbour Gd³⁺, it may be concluded that the antiferromagnetic Gd³⁺ system has the simplest possible two-sublattice structure.

Millimetre wave antiferromagnetic resonance experiments have been made in α-Fe₂O₃ using both d.c. and pulsed magnetic fields. Both the weak-ferromagnetic modes above the Morin transition temperature and the usual low-temperature antiferromagnetic resonance were examined. The theoretical expression for the frequency dependence of the low-frequency weak-ferromagnetic mode has been verified up to 40 kG. The angular dependence of this mode at 68·7 Gc/s shows a departure from theory, however, although lower-frequency measurements do not. The angular dependence of the high-frequency weak-ferromagnetic mode observed at 142 Gc/s and 273 °K is in satisfactory agreement with theory. The low-temperature antiferromagnetic resonance for H₀ along the c axis is similar to previous results; however, off-axis the resonance is observed to greater angles than expected, possibly as a result of a non-zero canting field when the spins deviate from the c axis.

Near the Morin transition temperature metamagnetic behaviour is observed and the results are discussed with reference to previous measurements.

The magnetic structure of goethite (α-FeOOH) has been determined by Mössbauer and neutron powder diffraction techniques using both natural and synthetic samples. The antiferromagnetic structure, which exists below the Néel point of (130±2) °c, is collinear and the magnetic and chemical unit cells are the same; a = 4·587, b = 9·937 and c = 3·015 Å, space group Pbnm. The iron atoms related by the b-glide plane are coupled ferromagnetically and are antiferromagnetically coupled to the remaining iron atoms. The spins lie parallel to the c axis, and the major axis of the electric field gradient was found to lie in the ab plane, making an angle of about 40° with the b axis. No evidence was found for the existence of more than a single hyperfine field in the temperature range between 4·2 °K and room temperature: the value of 504 kG at low temperatures is discussed in relation to the covalency of a number of ferric compounds. The exchange interactions in the oxyhydroxide are related to the magnetic structure; the observed behaviour of the susceptibility above the Néel point is qualitatively explained by the existence of residual magnetic order.

The phase transitions found in the lower oxides of vanadium are discussed in terms of a model recently proposed by Fröhlich, in which Coulomb correlations play a dominant role: below the temperature of the phase transition, T_t, they suppress the band motion of the d electrons in favour of localization, whilst as T -> T_t they cause the thermal ionization of these electrons (out of their localized states into delocalized band states) to become a co-operative process, through screening effects; at T_t, the population of the d band increases catastrophically leading to a first-order phase transition into the metallic state. Experimental evidence, collected from VO, V₂O₃ and VO₂, which tends to support the mechanisms of the model is presented.

The VO₂ phase transition is investigated in detail and the following conclusions reached: the crystallographic transformation which occurs at T_t(similar, equals 340 °K) can be regarded as a consequence of the a priori electronic transition; for T < T_t, crystallographic data together with the observed magnetic behaviour strongly suggest that the d electrons are trapped in homopolar bands between pairs of vanadium ions - accordingly, the system behaves as a low-mobility semiconductor and exhibits Van Vleck temperature-independent paramagnetism; for T > T_t, all the d electrons move with very low mobility (similar 0·1 cm² v^-1 s^-1) in a degenerate t_2g sub-band, of width about 1 ev, which is split off from the fivefold degenerate d band by the internal crystal field of the anion sublattice - the system is now metallic and the observed magnetic susceptibility attributed to Kubo temperature-independent paramagnetism and spin paramagnetism of the d conduction electrons, which because of the high effective mass (m^* similar, equals 40m) of the latter is temperature dependent. Finally, the thermodynamics of the transition is briefly discussed.

We have studied the K absorption edge of arsenic metal and its four compounds. It is found that there is a shift in the K absorption edge due to chemical combination.

An expression is derived for the structure-dependent part of the total energy of a simple metal, which differs from previous formulations in that it incorporates corrections arising from k and E dependence of diagonal elements of the pseudopotential. This is applied to the study of the divalent simple metals using the model potential of Heine and Abarenkov, and calculations are presented which support and augment the qualitative discussion of these structures by Heine and Weaire. The variation of c/a ratio in the hexagonal metals Be, Mg, Zn, Cd is reproduced quite well. In particular, the large values of c/a for Zn and Cd are seen to result from an effect closely related to the occurrence of the rhombohedral structure for α-Hg. The peculiar variation of c/a with composition in disordered Cd-Mg alloys is also accounted for by this model. Calculations of the variation of energy of Hg with distortion of the face-centred cubic structure show it to be unstable with respect to rhombohedral distortion and metastable with respect to tetragonal distortion, as suggested by Heine and Weaire to account for the distorted α- and β-Hg structures. The calculated magnitude of distortion is in both cases much too large. The general conclusion is that the model provides a good qualitative description of the structures of these metals, although detailed agreement with experiment is at present not achievable.

The crystal structure of Ga is too complex for one to attempt to `derive' it theoretically in any meaningful sense. However, one can test that the observed structure `makes sense' from various points of view. We find that it does, as regards number of near neighbours, their distances, covalency and the axial ratios which make Ga pseudohexagonal about the a axis as well as pseudotetragonal about the c axis. Our considerations are based on the expression for the total energy in terms of the pseudopotential v(q). The situation is dominated by the position of q₀ where v(q) passes through zero and by several sets of Brillouin zone planes which just envelope the Fermi sphere. The band gaps across the latter immediately explain qualitatively the marked anisotropy of the resistivity tensor.

The distance of the first minimum in the interatomic potential favours a structure with about seven nearest neighbours. Covalency amounts to about 27 kcal (g-atom)^-1.

Through the choice of a single parameter (an effective core radius R_core) a simple model potential for the electron-ion interaction in a metal yields form factors V(K) similar to those of the Heine-Abarenkov model potentials. Values of R_core near to the usual ionic radii give V(K) curves which are shown to be useful for momentum transfers K in the transport range and nearby, the region of interest normally encountered in the theory of metals. The potentials are applied in the calculation of resistivities, phonon spectra and phase shifts. Energy-dependent properties of the potentials are inferred from thermopower data.

The saturation magnetizations of body-centred cubic solid-solution alloys of iron with titanium, molybdenum, tungsten, technetium and rhenium have been measured between room temperature and 8 °K. Mean atomic magnetic moments (μB/atom) calculated from the data decrease linearly with concentration between 2 and 10 at.% solute. The rates of decrease for the solutes listed above are -3·392, -2·105, -2·044, -1·405 and -1·557 μB per unit of concentration respectively. When the data are plotted as a function of electron concentration they approximately superimpose on an extension of the Slater-Pauling curve. Combination of the magnetization results with the neutron diffraction measurements of Collins and Low shows that, while the solute atoms carry moments antiparallel to those on the iron atoms, the average moment per iron atom is either unchanged or increases slightly.

The changes in electronic specific heat, residual resistivity and thermoelectric power caused by the introduction of different impurities into transition metals are calculated in the low impurity concentration limit by a localized perturbation method. The electronic structure of the pure transition metals is described by a tight-binding d band and an overlapping free-electron s band. It turns out that, in this model, simple relations connect the three quantities to the d-band structure and to the charge difference between host and impurity. The predictions of the model are shown to be frequently very different from those of the rigid band model.

A quantum-mechanical description of the low-temperature motion of a methyl group explains the observations, incompatible on the classical theory, of a narrow nuclear magnetic resonance spectrum allied to weak spin-lattice relaxation.

The elastic constants C₁₁, C₁₂ and C₄₄ of the inert gas solids are calculated including the contributions from a non-central force. It is shown that the quantity (C₄₄-C₁₂) C₁₂^-1 provides a critical test for the importance of non-central forces.

It is proved that, in the critical region above T_c, the boundary susceptibility of a lattice is approximately (T-T_c)^-γ-v. It is argued that the boundary tension is approximately (T_c-T)^μ as T -> T_c^-, with μ = 2-(α_sprime+vprime). (α_sprime, γ, v, vprime are the usual bulk indices of specific heat, susceptibility and correlation length.) The appropriate index in Macleod's law is briefly discussed.