Table of contents

A formal integral relation is derived relating the triplet correlation function g₃ to the higher g_n for a two-dimensional fluid. The relation is obtained as a power series in the density and is shown to converge absolutely for all realistic fluids. It is also shown that the rule is satisfied to a good approximation by a limited number of the lower g_n. The extension to three dimensions is discussed.

A method is given of calculating the partial structure factors of a binary mixture of inert-gas fluids. The interatomic Lennard-Jones 6-12 potentials are conveniently separated into a hard core and a long-range attractive part, the effect of the core being calculated using a minor modification of the solution of the Percus-Yevick equation for a mixture of rigid spheres while the attractive tail of the potential is taken into acount within the random phase approximation. Results are presented for argon-krypton and argon-xenon mixtures.

Acoustic absorption measurements by the pulse technique have been made in methyl-alcohol-cyclohexane critical mixtures from 0·8-25 MHz and from 37 to 55°C (critical temperature 49·1 °C). The dependence of absorption on frequency and temperature is found to be in only qualitative agreement with the predictions of Fixman's theory.

The importance of accurate high-temperature thermodynamic data for the study of anharmonic effects in solids is discussed, and an adiabatic calorimeter for the measurement of heat capacities in the temperature range 25-500 °C is described. The precision of the heat capacity data is about ±0·1% for T < 400 °C and ±0·2% for T > 400 °C. Measurement of the heat capacity of a Calorimetry Conference sample of Al₂O₃ shows that the accuracy of the calorimetry is probably better than ±0·2% throughout the whole temperature range studied. Results have been obtained for lead to near the melting point and for aluminium to 500 °C. For lead a contribution to C_p is observed for T > 250 °C which is attributed to vacancy formation; a value of 0·5₂ ±0·2₆ ev is derived for the enthalpy of formation of a vacancy.

Methods are described for analysing the high-temperature thermodynamic properties of crystals into quasi-harmonic (qh) and explicit anharmonic (anh) contributions. The most useful functions to investigate are the entropy, the constant volume heat capacity and the Grüneisen parameter γ(T,V) = βV/χ_sC_p. ΔS^anh and ΔC_v^anh may be obtained as functions of temperature, both at the equilibrium crystal volume V_T, and at a fixed volume, for example V₀. The anharmonic Grüneisen parameter γ^anh = (1/ΔC_v^anh)(partial differentialΔS^anh/partial differential ln V)_T gives the volume dependence of ΔS^anh and, to first order, that of ΔC_v^anh also. For metals the contribution from the conduction electrons must be considered, and this is of the form S^el = C_v^el = σT. Theory suggests that σ should have the free-electron value σinfinity at high temperatures (T > capital theta, Greek) although this has not yet received direct experimental confirmation. In analysing the data for lead and aluminium the effects of using both the free-electron value σ_infinity and the experimentally observable low-temperature value σ₀ for the coefficient σ have been investigated.

For both lead and aluminium C_V and γ have been analysed but no attempt was made to analyse S because of the lack of internally consistent C_P data for the whole available temperature range. For lead the data extend to T similar 6capital theta, Greek and C_V^anh is of the form C_V^anh(T, V₀) = AT+2BT² with A negative and B positive. For aluminium, with data up to only T similar 2capital theta, Greek, A is also negative but no higher-order contribution is detectable. In both cases A varies by about a factor of 2 depending on whether σ₀ or σ_infinity is used to compute C_V^el. Analysis of the Grüneisen function yields values for γ_infinity^qh and γ^anh for both metals. Recent calculations of anharmonic effects in closepacked crystals, using simple models, are in surprisingly good agreement with experiment.

The description of anharmonic effects in terms of volume and explicitly temperature-dependent frequency distributions is discussed. The volume dependence may be derived from γ^qh and the temperature dependence from ΔS^anh or ΔC_V^anh. The results obtained for both lead and aluminium are expressed in this way and found to be in excellent agreement with the results of inelastic neutron scattering experiments, provided that σ_infinity is used in the thermodynamic analysis to compute C_V^el.

The elastic constants C₁₁, C₁₂ and C₄₄ of neon, argon, krypton and xenon are calculated at the absolute zero. One calculation is made assuming only two-body interactions. A second calculation is then presented in which the triple-dipole interaction of Axilrod and Teller is included. It is shown that the quantity B = (C₄₄-C₁₂/C₁₂, which is always positive when calculated with only two-body forces, is reduced when the triple-dipole interaction is included. In the cases of argon, krypton and xenon B in fact becomes negative. It should be possible to test this experimentally and if B were found negative this would provide strong evidence for the existence of three-body forces.

The general theory outlined in previous papers for the vibrational properties of finite specimens is applied to ionic crystals of cylindrical shape. Unlike the previously studied cases, the polariton modes are mixtures of magnetic- and electric-type oscillations, except for normal incidence. Our calculated absorption curves agree well with those observed on small fibres.

An alternative formulation of the polariton absorption process in finite crystals is given in terms of the quantal Green function theory in the spirit of Hopfield's treatment for infinite crystals.

The localization of `classical electrons' in disordered materials (e.g. liquid metals) is studied with the aid of percolation theory. It is shown that the transition from localized to delocalized states occurs over a fairly narrow range of energy, but is not mathematically discontinuous except in the one-dimensional case. Some implications for the theory of this transition for real quantum-mechanical electrons are suggested.

It is shown that by considering the electrostatics of finite charge distributions one can derive a generalized form of the traditional Ewald method for point charges.

The results of measurements of the spin-lattice relaxation properties of mixed single crystals of titanium and aluminium acetylacetonate are reported. Measurements were made at microwave frequencies (9·3 GHz) in the range of liquid helium temperatures. The relaxation times are found to be strongly dependent on the paramagnetic concentration, and these results are discussed in terms of two relaxation mechanisms which have been proposed to account for concentration-dependent relaxation. Only in the low-concentration limit is relaxation behaviour found which is in accord with the established theory. In this limit the observed spin-lattice relaxation time can be fitted by a sum of direct and Raman terms of the form T₁^-1 = 1·2 × 10^-2T + 1·5 × 10^-6T⁹ s^-1

The spin-lattice relaxation time by the direct process of the ground state of the ferrous ion in ruby is estimated using a crystal-field treatment and a method which does not use normal coordinates for the ion and its immediate neighbours. For a regular octahedron the transition probability is shown to be identical with that using the normal mode approach of Van Vleck. In contrast with the corresponding situation in MgO the relaxation time is found to be long, which suggests that the ion is not a strong candidate for the role of a fast-relaxing impurity in ruby. Since, in addition, magnetic dipole transitions are forbidden for a non-Kramers doublet, the problem of detecting ferrous ions in Al₂O₃ is discussed.

The frequency transform of the time-dependent two-spin autocorrelation function is evaluated using the modified paramagnetic neutron scattering function introduced by Collins and Marshall. The dimensional effects on one-, two- and three-dimensional crystals are demonstrated, and the physical basis for such dimensional effects is duscussed. For the one-dimensional Heisenberg paramagnet comparison with exact calculations by Carboni and Richards shows a marked improvement over the Gaussian distribution commonly assumed in the description of paramagnetic neutron scattering.

Green function methods are used to include the effects of exciton-magnon interaction in a calculation of the shapes of magnon sidebands on narrow exciton lines in antiferromagnets. The general theory is applied to RbMnF₃, for which crystal numerical results are presented. It is found that good agreement with experimental results for the ⁶A₁ to ⁴T₁ exciton transition can be obtained if the exchange constant between a neighbouring ⁶A₁-⁴T₁ pair of manganese ions is about one-half of the exchange constant for a ⁶A₁-⁶A₁ pair.

We discuss the dependence of the free energy on the shape of the external boundary of a sample for a Heisenberg model with dipolar forces. We prove the theorem that enables us to perform a restricted renormalization of the vertices of time-ordered diagrams, and find that the molecular field rigorously contains all the shape-dependent terms in the free energy.

The magnetization exponent β_D of the Syozi model of dilute ferro-magnetism is estimated for the four most common three-dimensional lattices using the Padé approximant method. It is found in all cases that 0·354 <or= β_D <or= 0·380. Using the previously established relation αprime = 1 - β/β_D between the specific heat exponent αprime and the magnetization exponent β for the Ising model below T_c and assuming β = fraction five-sixteens, the above result implies 0·117 <or= αprime <or= 0·178. On the grounds that previous estimates of αprime have been much lower, it is concluded that most probably αprime = fraction one-eighth = 0·125 with the consequence β_D = fraction five-fourteens = 0·357.... The possibility αprime = fraction one-sixteens is well excluded by this method.

The specific heat amplitude of the Ising model below T_c is deduced by computation of the magnetization amplitude of the Syozi model.

Spin-wave theory, renormalized to take account of spin-wave interactions, is used to calculate the temperature and field dependence of the magnetization of terbium and the temperature dependence of the magnetic contribution to the specific heat. The calculations are based on an interpolation scheme for the spin-wave energies as a function of wave vector and of temperature which gives an approximate fit to the available data obtained by inelastic neutron scattering. Good agreement is found with the magnetization measurements of Hegland, Legvold and Spedding up to about 180 °K. The question of what is the best expression for the internal energy from which to calculate the magnetic contribution to the specific heat is discussed, and calculations are reported based on an expression proposed by Radcliffe. A crude comparison with experimental data of Lounasmaa and Sundström and of Jennings, Stanton and Spedding shows reasonable agreement in view of the approximate nature of the calculations and the method of analysing the data.

Measurements of magnetization at temperatures down to 4·2 °K and in fields up to 17 kOe have been made on three samples of ZrZn₂ having Fe contaminations of 25, 55 and 500 parts per million; Curie temperatures have also been determined. All three samples are ferromagnetic and have Curie temperatures of 22 °K, 21 °K and 17 °K respectively. It is suggested that ZrZn₂ is intrinsically ferromagnetic.

A further exploration is made of the theory of Falicov, Pippard and Sievert explaining Stark's magnetoresistance data for zinc with a magnetic field aligned along the hexad axis, where the effects of spin splitting on the triangular orbits were observed. An alternative model for the spin splitting is suggested in terms of a spin-dependent phase shift upon Bragg reflection. Because the triangles are very small it is checked that the Pippard network model is in good agreement with a calculation using the effective Hamiltonian. The formal theory of spin precession along any arc of an orbit is also reconsidered along lines suggested by Roth. It seems that spin flip caused by spin-orbit coupling is unlikely in zinc, but a calculation of the effects of such a hypothetical spin flip on the magnetoresistance is made and the agreement with experiment is substantially improved.

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.

Cellular eigenvalues have been computed for copper metal on using exactly the same potential as Burdick and they agree with the augmented plane wave method results within 0·002 ryd. Empty-lattice eigenvalues for all irreducible representations of all points of symmetry are correct (with one exception) to 0·000 02 ryd, the exceptional error being 0·0001 ryd. For general k values most errors are below 0·005 ryd. These results have been obtained both for face-centred and body-centred cubic lattices.

An improved version of the approximate method of calculating electronic band structures proposed earlier is presented and used to calculate the band structures of copper and iron. The results obtained are compared with the accurate calculations of Burdick and Wood and found to agree within 0·02 ryd for most levels.

An approximate expression has been derived for the wave number and frequency-dependent spin susceptibility χ(Q, ω) of the paramagnetic phase of a system of interacting electrons in a narrow s band. This expression has the property of reducing to the random phase result in the weak-interaction (band) limit and gives a Curie law for the ordinary susceptibility in the strong-interaction (atomic) limit. When the number of electrons per atom is small, χ(Q, ω) is expressible in terms of an effective electron-electron interaction similar to that of Kanamori. The stability of the paramagnetic phase relative to ferromagnetic ordering is discussed.

We have investigated the K-absorption edge of copper in the metal and its compounds. It is found that owing to chemical combination the K-absorption edge of copper is, in general, shifted towards the high-energy side by 2-10 ev. However, in the case of Cu₂O the shift is small and towards the low-energy side. A white line at the absorption edge has been detected for CuO, CuCO₃, Cu[CH₃COO]₂ and Cu₃[Fe(CN)₆]₂.

The Mössbauer effect has been used to study the 2⁺ -> 0⁺ gamma-ray transitions in ^176,178,180Hf. Using absorbers of Hf metal, HfO₂ and HfB₂, the ratios of the quadrupole moments of the first excited 2⁺ levels of the three isotopes were found to be Q¹⁷⁶/Q¹⁷⁸ = 1·040±0·008, Q¹⁷⁸/Q¹⁸⁰ = 1·014±0·013 and Q¹⁷⁶/Q¹⁸⁰ = 1·053±0·015. The isomer shifts between Hf metal and HfO₂ for the three isotopes were found to be ΔE(¹⁷⁶Hf) = (0·21±0·10) mm s^-1, ΔE(¹⁷⁸Hf) = (0·09±0·03) mm s^-1 and ΔE(¹⁸⁰Hf) = (0·10±0·06) mm s^-1. Based on estimates of electronic wave functions, these isomer shifts are shown to correspond to changes in the mean square radius between the first excited and ground states which are a factor of 3-5 lower than predicted by the self-consistent cranking model.

Quadrupole splittings were observed at the hafnium sites in Hf(NO₃)₄, HfOCl₂. 8H₂O and HfCl₄. Evidence was found for an effective magnetic field at the hafnium sites in the intermetallic compounds HfCo₂, Co₂HfAl and Zr_0.5Hf_0.5Fe₂.

Axe and Burns have discussed the effects of covalency in Tm²⁺ in calcium fluoride, with particular emphasis on its contribution to the crystalline-field splittings of the energy levels. This theory has been developed to give a consistent interpretation of the transferred hyperfine interaction with the neighbouring fluorine ions and the orbital reduction factor, as well as the crystal-field splittings. The contribution to the transferred hyperfine structure from polarization of the 5p electrons of the rare earth ion is extrapolated from the configuration 4f⁷ and is shown to be fairly small. The treatment is extended to the isoelectronic ion Yb³⁺ where the covalency is larger. The larger crystal-field splitting is predicted by the theory, as well as larger transferred hyperfine structure and orbital reduction factor, and the quantitative estimates of the crystal field splittings are as good as those of Axe and Burns for Tm²⁺. No attempt is made at exact calculation, but it is demonstrated that the fairly crude covalent calculation gives a consistent interpretation of all of the available experimental information for the two isoelectronic ions.

Manganese-rich γ-phase alloys of manganese-palladium and manganese-nickel have been investigated with x-ray and neutron diffraction, and the Néel and structure-transformation temperatures fixed with the aid of Young's modulus, resistivity and susceptibility measurements.

High manganese content alloys distort from a face-centred cubic to a face-centred tetragonal structure at the Néel temperature. Alloys containing more than 12 at.% Pd and 15 at.% Ni are cubic below the Néel temperature (1 gt-or-equal, slanted c/a greater-than sign R: 0.999), and do not distort till a much lower temperature. In the more solute-rich alloys the structure transformation is suppressed entirely.

A mechanism for the distortion is suggested, and it is pointed out that the resistivity and susceptibility results are not consistent with a localized model of the d electrons in these alloys.

Structure due to nuclear quadrupole interaction has been observed in the ²⁷Al magnetic resonance from dilute alloys of zinc in aluminium. Analysis gives the following values for the quadrupole coupling constants at the first- and second-nearest-neighbour positions around a zinc atom: e²qQ/h = (923 ± 10)kHz and (176 ± 8)kHz with asymmetry factors η = 0.249 ± 0.010 and η = 0 respectively. The results support the existence of oscillations of conduction electron density around the zinc atoms but it is necessary to consider the departure of the screening-charge distribution from spherical symmetry. This requires some modification of theory.

The spin-wave stiffness D of iron-palladium alloys containing 1·0, 2·2, 4·0 and 10·0 at.% iron has been measured by the neutron small-angle-scattering technique. The values of D are consistent with those derived from low-temperature saturation magnetization and specific heat measurements on alloys with iron concentrations ranging from 0·15 to 1·5 at.%. In the range up to 1 at.% D varies linearly with fractional iron concentration c, with an equation of best fit of the form D = (5040±290)c mev Ų. At higher concentrations of iron D increases more slowly.

The constant of proportionality for D in the linear region is greater by a factor of about 4 than that obtained by Doniach and Wohlfarth in a calculation based on the s-d interaction model. The results are also discussed in terms of a simple molecular field model of a dilute iron-palladium alloy, in which the giant moments are assumed to interact through a long-range Heisenberg exchange interaction J(r). The exchange field due to a giant moment is assumed to have the same spatial variation as the polarization distribution in the palladium matrix, and values of the stiffness are obtained which agree within a factor of less than 2 with the measured values.

It is suggested that the non-linear behaviour of D at higher concentrations arises from close overlap between three or more giant moments.

The conduction electron spin polarization is calculated by treating the static part of the s-d interaction exactly. It is shown that in addition to the contribution from the scattering states there is a contribution from the bound states. The contribution from the latter has been calculated in detail.

A systematic survey has been made with epitaxial films of n-type GaAs of the dependence of the position and amplitude of magnetophonon resistance oscillations on the experimental parameters of magnetic field, temperature and sample mobility. A number of corroborative experiments have also been performed with n-type InSb and InAs. With the magnetic field applied perpendicular to the current direction, a dependence of magnetophonon peak position on both temperature and mobility was found. A change in the curvature of the band edge can be deduced from the temperature dependence, and the low-frequency effective mass at the bottom of the conduction band of GaAs is found to be 0·065₃m at 280 °K and 0·067₅m at 70 °K. With InSb the band-edge mass of the electrons is found to be 0·0127m at 160 °K and 0·0134m at 60 °K. The dependence of the peak position on mobility is thought to be related to an expression describing the magnetic field dependence of the amplitude of the oscillatory terms. The amplitudes of the oscillatory terms do not depend on the mobility in a simple manner and appear to be a function of the number of ionized impurity sites rather than of the actual mobility of the sample.

When the magnetic field is applied parallel to the direction of current flow, resistance minima are always observed at magnetic fields close to the fields at which maxima are observed in the transverse orientation, but these minima cannot be used to determine band parameters as they are shifted with respect to the transverse maxima by amounts which can be as high as 16% in the case of InAs. Additional series of minima were observed at high temperatures with both GaAs and InAs in the longitudinal configuration. These extra series are attributed to the presence of indirect scattering processes involving two optical phonons.

For group II-VI and group III-V compounds a qualitative agreement has been found between thermodynamic estimates of the lattice bond energy and atomic displacement energies determined by electron irradiation. This correlation will be useful in the identification of atomic species displaced at electron threshold energies yet to be observed.

By combining low temperature luminescence measurements on electron-irradiated samples with reported luminescence data on non-stoichiometric specimens it has been possible to identify the luminescence role and energy levels of a number of defects in cadmium telluride. Cadmium and tellurium vacancies are shown to produce energy levels 0·036 ev and 0·46 ev above the valence band respectively. Using electrons from a Van de Graaff accelerator and observing photoluminescence and cathodoluminescence emission spectra at liquid helium temperatures the displacement energies of tellurium and cadmium have been found to be 7·8 ev and 5·6 ev respectively. The defects introduced by low temperature electron irradiation have been shown to produce emission characteristics which are not observed in specimens grown from the melt. It is suggested that this is the result of the strong perturbation of the electronic wave functions of a cadmium vacancy by the displaced cadmium atom at a nearby interstitial site.

Some of the outstanding difficulties of the interpretation of the banded emission spectra (1·3-2·2 μm range) in zinc and cadmium sulphide in terms of internal transitions within the Cu²⁺ ion are discussed. Determinations of the decay times for the infra-red emission in cadmium sulphide for various exciting conditions at different temperatures confirmed that these difficulties exist. It is proposed that the cation vacancy may constitute the luminescence centre and an LCAO-MO treatment is developed to provide quantitative evidence for this argument. By combining experimental and theoretical results it was possible to derive two models for the vacancy which adequately explain all the infra-red emission and absorption properties of this centre. A description of energy level models in terms of the many-electron wave function approach provides a much more realistic account of luminescence centres than the one-electron functions normally adopted. The vacancy models are self-consistent and are flexible enough to explain many of the observations of infra-red stimulation and quenching effects of visible luminescence in zinc sulphide.

Cathodoluminescence spectra have been obtained from undoped and n-type gallium phosphide crystals at temperatures between 80 °k and 300 °k. The gallium phosphide was grown epitaxially by vapour transport on GaAs substrates, the n-type specimens being doped with either sulphur or tellurium. The crystals showed only weak edge luminescence, and this is interpreted as the radiative recombination of free excitons and free holes at neutral donor centres. The free-exciton recombination bands show a weak no-phonon component and other bands that give evidence of the expected phonon co-operation for indirect band to band transitions. The phonon energies deduced from these bands are in good agreement with those observed in the optical absorption edge and lattice vibration absorption spectra. The luminescence attributed to free-hole recombination at donor centres has enabled values for E_D, the donor electron binding energy, to be obtained for the two impurities involved (sulphur, 105±5 mev; tellurium, 90±5 mev). The temperature dependence of the luminescence features is shown to be consistent with the proposed interpretation.

The isotopic dependence of the nearest-neighbour distance of an impurity appears to be important in determining the isotope shift of the R lines in ruby and MgO. It is suggested that anharmonic effects ought always to be considered along with quadratic effects in the electron-phonon interaction in analysing isotopic properties of impurities.

The anisotropy of the electric field-induced, one-phonon transverse-optical absorption band in diamond, and the quadratic dependence of the integrated absorption on the applied field E are found to agree with theoretical predictions. The band is centred at (1332·2±0·2) cm^-1 irrespective of|E| up to 1·14 × 10⁵ v cm^-1. The results are compared with some previous measurements by other authors.

Recent work by other authors has led them to identify the acceptor centres in natural semiconducting diamond as substitutional aluminium impurity atoms. It is shown here that current evidence does not warrant an unambiguous identification.

Using the available results of Hartree calculations, a potential valid for the vicinity of a P impurity in Si is set up. This potential is treated as a perturbation (acting over a small region) on the effective mass ground state and the first-order energy correction is calculated to be -6·3 mev.