Table of contents

A method is developed for determining the critical point and the critical exponent from terms in the series expansion of a function. For low temperature Ising model series the method provides an alternative technique to the method of Padé approximants. Application to some high and low temperature series for the Ising model on a simple cubic and a diamond lattice suggest the following exponents: γ = 1·2500 ± 0·0002, γprime = 1·29 ± 0·04, β = 0·312 ± 0·002 and αprime = 0·13 ± 0·07 for the high and low temperature susceptibility, the spontaneous magnetization and the low temperature specific heat respectively. These results are largely in agreement with previous analyses.

Low temperature Ising model series in the usual variable z = exp(−4J/kT) are analysed. The positions of the singularities in the disk|z| <or=|z_c| and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0·312 ± 0·002 for the simple cubic and face-centred cubic lattices, and β = 0·312 ± 0·004 for the body-centred cubic lattice. γprime = 1·31 ± 0·02 for the simple cubic lattice, γprime = 1·28 ± 0·04 for the body-centred cubic lattice, and γprime = 1·3 ± 0·1 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.

This paper considers the mechanisms whereby the addition of Yb³⁺ enhances the efficiency of quantum counter action in Er³⁺ doped LiYF₄. The particular scheme chosen for this investigation is the doubling scheme in which two photons of wavelength close to 9800 Å excite green fluorescence from the Er³⁺ ion. The important factors which have been identified are that the influence of an increase of Yb³⁺ concentration on the quantum efficiency of the ⁴S_3/2 to ⁴I_15/2 fluorescence is very much less than an equivalent increase in the Er³⁺ concentration, the absorption coefficient of the Yb³⁺ in the 9800 Å region is over an order of magnitude larger than that for Er³⁺ and the efficiency of the transfer of energy from Yb³⁺ to Er³⁺ is sufficiently high to ensure the energy absorbed by the Yb³⁺ is transferred to the Er³⁺. The optimum concentration has been found to be in the region of 2% Er³⁺: 18% Yb³⁺ doped LiYF₄.

The frequencies of normal modes of vibration propagating in high symmetry directions in tin telluride at 100 °K have been measured by means of inelastic neutron scattering techniques. The shape of the longitudinal optic branches is dominated by the screening effects of the free carriers in the crystal. We have developed a modification of the shell model in which this screening is included in a consistent way, and we are able to obtain a satisfactory description of the dispersion curves using this model with an ionic charge of 1·34e, and a slightly worse fit using a non-ionic model. Both these models were based on the Hartree dielectric function, and were found to be superior to a model based on the Thomas-Fermi dielectric function.

The domain-wall switching properties of pure yttrium iron garnet, YIG, and Yb-doped YIG are investigated from 1·5 to 300 °K. The measured domain-wall velocity ν depends on the applied field H according to the relation ν = R(H - H₀) for walls parallel to the (110) and (211) crystallographic planes in YIG and to the (211) plane in Yb_0·15 Y_2·85Fe₅O₁₂. For YIG negative values are deduced for H₀, the threshold field; this is puzzling. The domain-wall mobility R is in order-of-magnitude agreement with the measurements of previous workers, though the detailed agreement is not good. We confirm the report that domain-wall motion in pure YIG is much more heavily damped than would be predicted, using a phenomenological model, from ferromagnetic resonance damping. The mechanism is not elucidated but extensive experimental measurements are recorded. For Yb-doped YIG a large additional damping of the wall motion is found at low temperatures, with a minimum in R at about 10 °K, and its magnitude is predicted by the longitudinal relaxation model applied to the anisotropic single-ion energy levels of the Yb³⁺ ion. Best agreement is found here, as in earlier work on ferromagnetic resonance, when the thermal relaxation time of the populations of Yb³⁺ levels is assumed to have contributions from the single-magnon process of spin-magnon relaxation and from Raman processes.

An investigation is made of the properties of paramagnetic ions which have orbital triplets interacting with lattice vibrations. The linear coupling between the orbital levels and local distortions which have E_g symmetry are eliminated by a sequence of transformations on the Hamiltonian. These lead to an equivalent Hamiltonian in which the Jahn-Teller energy appears explicitly and most of the other operators have modified forms. The Hamiltonian of the lattice is discussed in more detail than usual by the introduction of symmetry adapted coordinates and momenta. Explicit formulae are obtained for the Jahn-Teller energy in terms of elastic constants and coupling coefficients, and for the displacements of lattice points, which are a consequence of the Jahn-Teller effect. It is also shown how a continuum model of the lattice can be introduced, and used to derive a cosθ/R² dependence for the distortion associated with the Jahn-Teller effect. The analysis throws a good deal of light on the reason why the cluster model works so well.

The magnetic properties of a Ti³⁺ ion in an Al₂O₃ lattice are discussed using a dynamic Jahn-Teller model in which an orbital triplet interacts with the vibrational E modes of the ligands. Calculations of the effect of applied electric field and strain on the electron paramagnetic resonance spectra are made. It is found that second-order vibronic effects are very important in both cases. The linear electric field effect gives rise to large g shifts of the order 10^-4 for an applied electric field of 1 kv cm^-1 while the strain tensors are of order 10² per unit strain. Comparison is made with experimental results.

The effect of an applied electric field on the electron paramagnetic resonance spectrum of the ground state of the substitutional ion Ti³⁺ in Al₂O₃ has been measured. Two parameters characterize the coupling of the spins of ½ to the applied electric field. These are obtained by measuring the dependence of the integrated absorption intensity on the orientation of the applied magnetic field in two separate experiments. In the first of these the transition was observed by using a radio-frequency magnetic field as is normally done (magnetic dipole transitions), and in the second by using a radio-frequency electric field (electric dipole transitions). The measured values of the coupling parameters thus obtained are shown to agree well with those obtained theoretically by Bates and Bentley.

The spin-lattice relaxation time T₁ of the Cr³⁺ ion in the Al₂O₃ lattice has been measured in zero magnetic field. The theoretical value of T₁ in this particularly simple case is shown to agree well with the measured value for all types of crystal growth at low Cr³⁺ concentration. The anomalous concentration dependence observed in the range 0·02-0·1% Cr³⁺ for all samples except those grown by the vapour phase modification of the Verneuil process is ascribed to the inhomogeneous Cr³⁺ ion distribution in most crystals.

Cr³⁺, Mn²⁺, Fe²⁺ and Fe³⁺ have been crystallized in PbF₂ from the melt. A preponderance of Cr³⁺ occupied substitutional sites with the excess positive charge compensated by an excess fluoride ion in the next-nearest-neighbour interstitial position, yielding the electron spin resonance characteristics g = 1·984±0·005 and fine splitting D = 0·334±0·005 cm^-1. Mn²⁺, for which g = 2·002±0·002 and ⁵⁵A = 0·0090±0·0002 cm^-1, substitutionally replaced Pb²⁺. Poorly resolved ¹⁹F superhyperfine structure was detected. Fe²⁺, Fe³⁺ and Li⁺ produced broad electron spin resonance signals and it was concluded that these ions induced conduction centres in the crystal.

The structure factors F of niobium monocarbide have been determined from the integrated intensities of Bragg reflections. The intensities from nearly stoichiometric powder samples were measured with monochromatized Cu Kα radiation, and they were put on an absolute scale with respect to the absolute value F (111) of aluminium. The electron distributions of the atoms were analysed in terms of cubic harmonic expansions. It was found that, in comparison with the free-atom model, the charge density of the niobium atom has extended radially and the carbon atom has contracted. Traces of a non-spherical component of the lowest order, which would indicate covalent bond formation, were observed in the carbon atom but not in niobium. A charge transfer of 1·0±0·2 electrons from the niobium Wigner-Seitz cell to that of carbon was found. This is in agreement with the results of recent spectroscopic measurements and band-structure calculations, which indicate that in refractory carbides the metal atoms act as electron donors.

Qualitative phase analysis is carried out for a series of pyroceramics of different chemical composition. The degree of crystallinity, the values of the micro-deformations, and of regions of coherent scattering are also determined.

X-ray analysis is studied to determine the dependence of coherent scattering regions (c.s.r.) on chemical composition.

Measurements of the Mössbauer spectra of γ FeOOH show that it is an antiferromagnet with a Néel temperature of 73 °K. The hyperfine field at the ⁵⁷Fe nuclei is 460±5 kG. By applying an external magnetic field to a single crystal the Fe³⁺ magnetic moments are shown to be collinear in the antiferromagnetic state, with the alignment along the c axis. From measurements in the paramagnetic state using polarized γ radiation the electric field gradient is shown to be negative in sign, and its major axis lies in the ac plane making an angle of about 55° with the c axis.

The theory of defects in antiferromagnetic insulators is applied to the perovskite structure and the Green functions calculated. The conditions imposed by symmetry on the interaction of light with such defects are also investigated. A comparison is made with experiments on Eu³⁺ and Ni²⁺ impurities in KMnF₃ and RbMnF₃.

The magnetic structure of K₂NiF₄, K₂MnF₄ and Rb₂MnF₄ can be approximately regarded as a two-dimensional square lattice, which enables the magnetic Green functions to be expressed in terms of elliptic integrals. Using these, the theory of a magnon sideband on an exciton absorption line is compared with the experimental measurements on Rb₂MnF₄, giving reasonable agreement. A prediction is made for the two-magnon Raman lineshape, and the energies of localized modes associated with impurities calculated.

The critical spin fluctuations in MnF₂ have been studied near the critical temperature T_N using a triple axis neutron spectrometer with high resolution. Due to the anisotropy in MnF₂, the longitudinal and the transverse component of the spin fluctuations can be separated experimentally.

The longitudinal component of the static susceptibility diverges at T_N, while the time fluctuations undergo the expected thermodynamic slowing down. The spectral width above T_N varies in proportion to the temperature power law {(T-T_N)/T_N}^0·62±0·02. The temperature variation of the static susceptibility and the inverse correlation range of the longitudinal fluctuations are in agreement with the predictions of the Heisenberg model.

The divergence of the transverse component is suppressed by the anisotropy, and this component is therefore finite at T_N. The variation above T_N was interpreted in terms of the transverse transition temperature T_{perpendicular}, introduced by Moriya. In relation to this temperature, the static properties behave in an almost classical manner, that is in agreement with the molecular field theory. The transverse spectral width above T_N varies as {(T-T_{perpendicular})/T_{perpendicular}}^0·53±0·05. Below T_N the transverse fluctuations consist of two spin waves. The spin-wave renormalization with temperature was measured in the long wavelength limit, and was found to vary in proportion to the staggered magnetization.

The dynamical scaling law prediction for antiferromagnets of a q^3/2 dependence of the spectral width at T_N was verified for the transverse component.

The sign of the nuclear hyperfine field at a Sn nucleus in the ordered ferromagnetic Heusler-type alloy Co₂MnSn has been determined using polarized Mössbauer γ-rays. Unlike all other fields measured at various nuclei in Heusler alloys, the field at Sn in Co₂MnSn is shown to be positive. The measured value of + 107 ± 1 kOe at 4·2 °κ is compared with the theoretically predicted value obtained by assuming the field to be due to the oscillating conduction electron spin polarization induced by localized moments on the Mn atoms. This model, which yields the correct negative sign for all other measured fields, is also shown to yield the correct positive sign for the field at Sn in Co₂MnSn.

The time-of-flight technique has been used to measure the energy spectra of initially cold neutrons after scattering by liquid sodium at 115, 149 and 197 °c. Nearly-elastic peaks were observed, and have been compared with predictions from the model of Egelstaff and Schofield for atomic translational motion. Peak shapes and intensities are accurately fitted, and a disposable parameter c which describes the approach to the long-time behaviour given by Einstein's stochastic description has been determined at each temperature. The peak widths are in disagreement with the model of Singwi and Sjolander, but in accord with an extension of this model by Oskotskii. An essential feature of Oskotskii's extension is the concept of the collective diffusion of groups of atoms, and this type of motion is here found to account for 40% of the total diffusion coefficient of sodium. The parameter c of the Egelstaff-Schofield model may be interpreted as giving the mean number of atoms in a diffusing group; the present data yield a number of 30±1 atoms which is essentially constant over the temperature range studied. A comparison of these results with less extensive data for liquid lead and tin is made, and it is suggested that diffusive motion of groups of several tens of atoms could be a common feature of liquid metals. By contrast, a comparable analysis of data for liquid argon gives a number of atoms less than four for the size of any such group.

A description is given of a simple semi-empirical model of the emission of secondary electrons from metals by primary electrons with energies in the range 100-1000 eV. An expression is derived for p(i), the probability of emission of i secondary electrons by one primary electron. The values of some of the constants in this expression are found by matching its first moments to the corresponding experimental yields of secondary-electron emission. The properties of the p(i) distribution are similar to those of the few experimental distributions with which a tentative comparison is possible.

Expressions for the exchange and correlation corrections to the model potential form factor w_q(k) and the normalized energy-wavenumber characteristic F_N(q), which were derived in a recent paper by Shaw, are evaluated for eight simple metals. An expression for the second derivative of the Kohn-Sham density functional for the exchange and correlation energy is proposed and then is used in these calculations. The energy-wavenumber characteristic computed from the optimum model potential, corrected for exchange and correlation, is used to calculate the phonon spectrum of magnesium. The predicted spectrum is found to agree very well with the experimental data, and the exchange and correlation corrections are shown to be important in obtaining this agreement. The corrections to F_N(q) and w_q(k) which are tabulated for eight metals can be used to determine the importance of exchange and correlation in the prediction of other metallic properties.

A virtual crystal model has been used to determine energy bands in silver-palladium alloys. Results for densities of states as a function of concentration are calculated and compared directly with experimental measurements of electronic specific heat and magnetic susceptibility. The range of validity of the energy band results is estimated by considering the effects of disorder.

The resistance due to impurity potential scattering and spin-disorder scattering are additive in simple Born approximation. If the impurity scattering is calculated generally by a Green function or phase shift analysis it is also found to affect the spin-disorder scattering because of the modification of the local charge density. This is calculated using the previous Green function method when the spin scattering is derived to second order both for spin-ordered and spin-disordered systems. It is also calculated for spin effects in third order (Kondo effect) in the spin-disordered case. It is found that the spin resistivity is markedly affected in magnitude and in sign if a resonance is caused by the impurity potential near the Fermi energy.

Measurements on the ultrasonic attenuation and velocity are presented for several Invar-type iron-nickel alloys in the temperature range 1·5-77°K. The results are interpreted in terms of the presence of a relaxation mechanism within the alloys, and a simple relation is derived between the attenuation and the rate of change of the elastic modulus with temperature.

Measurements are reported of the ice-point thermopowers and resistivities for five series of silver-palladium-cadmium alloys in which the electron density e/a is held constant within each series. A Nordheim-Gorter type of relation has been shown to be applicable to these ternary alloys and the characteristic thermopower S_o of the solvent (Ag) has been obtained as a function of e/a for 0·95 <e/a<1·0. The values of S_o were found to be positive and constant as e/a was changed despite the drastic changes of Fermi surface topology expected in the above electron density interval. It is concluded that the diffusion thermop ower of pure silver (and probably the other noble metals) is insensitive to the details of the Fermi surface and that the energy derivative of the mean free path must be negative to account for its positive thermopower.

X-ray measurements on iron-palladium-hydrogen alloys containing up to 12·3 at.% Fe have enabled a phase diagram for these alloys to be constructed. The boundary between the α (low hydrogen content) and β (high hydrogen content) phases disappears for iron contents over 11·5 at.%. Mössbauer experiments confirm the existence of two phases up to this composition. Curie temperatures for the α-and β-phase hydrides and for hydrogen-free alloys have been measured by the Mössbauer technique and are compared with results obtained by other methods. Curie point measurements from Mössbauer spectra on equilibrium β-phase Fe-Pd-H show a change from a long- to short-range magnetic-ordering mechanism in the region 4 to 6 at.% Fe as the iron content is increased.

The field dependence of oscillatory photoconductivity is discussed on the basis of an analytical model and detailed Monte Carlo calculations. The occurrence of negative photoconductivity is investigated with particular reference to the spatial distribution of electric field and the instability of the carrier system.

Measurements of the thermal conductivity and Righi-Leduc coefficient of Cd₃As₂ have been made in a transverse magnetic field of up to about 2 T, at temperatures between 68 °K and 300 °K. For one sample the measurements have been extended down to 8 °K. The measurements have allowed the separation of the lattice and electronic components of the thermal conductivity. It is found that, owing to inelastic scattering of the electrons, the Lorenz number is usually much less than the normal value. The lattice conductivity falls continuously with increasing temperature, reaching a value of about 1 wm⁻¹ deg⁻¹ at 300 °K for most of the samples studied. An anomalously high value of the thermal conductivity was observed for one sample that had an exceptionally high carrier concentration.

We have measured infra-red cyclotron resonance in epitaxial thin films of indium arsenide on semi-insulating gallium arsenide and bulk indium arsenide substrates. A novel feature is that the crystal quality and perfection of thin films is found to be better than the best available bulk specimens as indicated by the observation of clear, narrow cyclotron absorptions.

By a study of the magnetic absorption at different temperatures we have separated effects due to cyclotron resonance spin up, spin down and impurity resonance. Theoretical extrapolation of results yields a low field, zero k-vector effective mass for conduction band electrons in InAs of m^* = (0·0230±0·0003)m₀. Information on the impurity binding energy is presented.

Our two-phonon absorption measurements also provide accurate values of ω_TO = 215·5±3 cm⁻¹ ω_LO = 241·8±3 cm⁻¹.

Evidence of resonant electron-phonon (polaron) coupling in InAs, of similar nature to that previously reported for InSb, is shown by comparison with theoretically predicted absorption peak positions, effective masses and half widths and has yielded an estimate of Fröhlich coupling constant of α = 0·05±0·01.

The use of such epitaxial layers for optical measurements has been extended to GaAs. Microwave 2 mm cyclotron resonance has been observed in a thin epitaxial film of GaAs yielding an electron effective mass of m^* = (0·068±0.002)m₀.

Intercalated layers of alkali metals in C₂₄Rb and C₂₄Cs are disordered (liquid-like) at room temperature but undergo a reversible transformation to an ordered state (freeze) at −114 °c and −110 °c respectively. The origin of structural differences between the ordered form II of these compounds and that of the corresponding potassium compound, C₂₄K, is discussed.

The idea of forming invariants from the critical amplitudes of ferromagnets is extended to include the specific-heat amplitudes. Hence for planar Ising lattices it is shown that current estimates of the bulk susceptibility amplitudes are of great consistency, and that surface properties all become isotropic in the critical region. A hypothesis of Fisk and Widom about the equation of state is confirmed.