Table of contents

The acoustic impedance Z of liquid ³He has been measured at frequencies of 122 and 124 MHz. The transition from ordinary to zero sound occurs at a mean temperature of 0.035±0.002 °K. The ratio Z/p increases by 13±3 m s^-1, where p is the density of the helium. The theory of zero sound gives a good correlation of this transition temperature with those observed in measurements of the impedance and of the propagation of sound waves at other frequencies. However, the increase in the impedance is greater than is to be expected from the observed change in the velocity of sound. Possible explanations are discussed.

Resonance lines may be broadened by random strains, local electric field or similar perturbations. In simple cases the resonance line shifts linearly with the local strain or electric field; such cases have been treated in an earlier paper. In more complex cases there is no linear shift and the transition energy contains terms quadratic in the perturbation. Such cases occur frequently in spin resonance and nuclear magnetic resonance. These cases cannot be treated by the method used in the linear problem. The present paper describes an approximate method which deals with the quadratic terms. The method is a development of the powerful `statistical method' and can be extended to still higher orders. It is compared with various other approaches to the second-order problem and illustrated by an idealized model and applied to broadening by dislocations. It is shown that dislocation-broadened lines have a width and shift linear in the dislocation density and that the line shape changes as the dislocation density is varied.

The changes in the properties of an Ising lattice caused by various types of defect are related to the spin correlations of the perfect lattice. The critical indices associated with these changes obey the scaling laws with the same gap index Δ as the bulk properties and a susceptibility index which exceeds the bulk index by unity. For planar lattices in zero field an alternative approach leads to a confirmation of the isotropy of the boundary tension near T_c and, incidentally, to an unlimited number of relations between the spin correlations of the normal lattice and its dual.

The Born model of ionic solids has been used to make a complete determination of alkaline earth and chalcogenide ionic radii from solid state data alone. The additivity rule of interionic distances in the NaCl structure alkaline earth chalcogenides is used to relate the first- and second-neighbour contributions to the repulsive energy, the generalized Huggins-Mayer method being employed. The ionic radii found reproduce the interionic distances well, the relative standard deviation over the thirteen salts considered being only 6·5%. In contrast with Fumi and Tosi's results for the alkali halides the radii found are in good agreement with the traditional Goldschmidt radii.

The position and width of the F band in magnesium oxide have been measured in the temperature range 4-400 °K. The data have been analysed in terms of the simplest adequate `configuration coordinate' model. The width results give an effective frequency of 7·8 × 10¹² s^-1, which is close to a peak in the phonon density of states and to the value extrapolated from data for the alkali halides. There is evidence that the effective frequency is reduced by about 5% in the excited state of the F centre. The Huang-Rhys factor is about 39, and luminescence is predicted at about 2·4 eV. The band shape indicates the existence of three small absorption bands on the high-energy side of the main F absorption band. These appear to be associated with the F centre, but their nature is not clear.

The acoustic paramagnetic resonance spectrum of the chromous ion as an impurity in magnesium oxide has tetragonal symmetry. There are three sets of chromous ions each giving a spectrum with its fourfold axis of symmetry along a <100> direction in the magnesium oxide. The three sets of ions can be aligned by the application of external uniaxial compression. The results are consistent with the chromous ion undergoing a dynamic Jahn-Teller effect and there being a distortion from cubic symmetry at each chromous ion site due to random internal strains in the crystal.

A systematic study of the 10 GHz electron paramagnetic resonance spectra of the gadolinium ion (Gd³⁺;⁸S) in isomorphic diamagnetic lattices of hydrated rare earth double-nitrate single crystals is reported. These lattices are of the form M₂triple primeM₃double prime(NO₃)₁₂.24H₂O, where Mtriple prime is bismuth or lanthanum and Mdouble prime is zinc or magnesium. The ratio of Gd³⁺ to Mtriple prime ions was approximately 0·1%. The experimental data for temperatures of 290 °K and 77 °K are fitted to the phenomenological spin Hamiltonian for C_3i symmetry. The angular dependence of the spectra shows substantial variations from that predicted, which have not been reported previously. These variations are not inconsistent with C_3i symmetry but cannot be completely described by any spin Hamiltonian of the above form. This raises doubts as to the usefulness of the spin-Hamiltonian formalism because the intrinsic symmetry of the terms is too high to describe the variations.

Information is presented on the emission and absorption spectra at room and low temperatures of chromium activated MgAl₂O₄ single crystals. These crystals had been grown by a flux-melt technique, and have been shown to be structurally perfect and of high chemical purity. A systematic study of the changes in the spectra with impurity concentration has been made. The appearance of line spectra in absorption for increasing chromium concentration is described. These lines have been assigned to transitions between states of octahedral d³ predicted by crystal field theory. A discussion of their appearance, substantiated by crystallographic data, is given in terms of a modification of the selection rule by the change in the symmetry group from O_h to C_3v.

A theory for the properties of long-wave optical vibration modes in finite ionic crystals of arbitrary shape is given. Neglecting retardation effects, it is found that in finite specimens there exist transverse and longitudinal bulk modes as well as surface modes, which are neither transverse nor longitudinal and which have intermediate frequencies. Explicit solutions are given for diatomic crystal samples with one, two and three dimensions finite. The modifications which arise from more complex unit cells and polarizable ions are given. A detailed calculation of the properties of long-wave optical phonons in finite specimens of strontium titanate is presented.

A theory for the polariton modes (long-wave optical phonons coupled to photons) in finite ionic crystals of arbitrary shape is given. The theory is exemplified by detailed calculations for the case of a spherical specimen. Stationary polariton states of three types are obtained, which, for small enough crystals, are characterized by (i) large amplitudes inside the solid (bulk modes), (ii) large amplitudes near the surface of the solid (surface modes) and (iii) small amplitudes inside the solid (vacuum modes). The absorption spectrum of spherical diatomic samples is calculated and it is found that the polariton modes cause absorption in regions which are forbidden for infinite crystals. The generalization of the theory to the case of complex cubic crystals is given and applied to interpret the absorption spectrum of small SrTiO₃ spheres.

Using group-theoretical methods, the relaxation modes and relaxation times in respect of both electrical and mechanical types have been obtained for the case of a divalent interstitial impurity in alkali halide lattices. These cases involve the simultaneous occurrence of two vacancies in the nearest-neighbour sites.

Measurements of the absolute intensity of the x-ray thermal scattering from sodium fluoride crystals have been made at a representative set of positions in reciprocal space. The distribution of scattered intensity can be interpreted to give the ionic form factor dependence on neighbouring ionic positions during thermal motion. The simplest interpretation is a fractional change of the form factor of a negative ion with relative displacement of a nearest-neighbour positive ion, increasing with the magnitude of the scattering vector K.

The measurements also give transverse and longitudinal phonon dispersion curves for the [1,0,0], [1,1,0] and [1,1,1] directions. The errors in the frequencies are small except for a few higher-frequency phonons. The slopes of the dispersion curves at low wave vectors suggest that the differences between the first- and zero-sound velocities are less than 3%.

Following Wilson's expression for the variance of a diffraction line profile broadened by mistakes, a simple experimental test for randomness is proposed. The same theory applied to the special case of substitutional disorder in spinel suggests a criterion for homogeneity in structures of mixed composition and methods of distinguishing one type of mistake structure from another.

The X-ray diffraction line profiles obtained from the spinels MgFe₂O₄ and Mg(Cr, Al)O₄ have been examined by a step-scanning diffractometer and found to be broadened. An analysis of the variance of the profiles is carried out using the techniques developed by Langford and Wilson, and from this it is concluded that mistakes are the probable origin of broadening in both cases. Further analysis suggests that these mistakes are associated with variable composition. Evidence is presented in support of order in the arrangement of Cr³⁺ ions in Mg(Cr, Al)O₄ while MgFe₂O₄ appears to have a random mistake structure.

The interaction between magnons and neutrons in antiferromagnetically coupled systems is studied by taking into account the mixing of ground and excited orbital states of localized electrons of magnetic atoms due to interaction with neutrons. The magnon-neutron Hamiltonian is obtained by formulating the Heisenberg-type exchange interaction between the magnetic atoms with due cognizance of the perturbation of the electronic orbitals. The scattering cross sections are evaluated for two-magnon Raman processes with the scattering of neutrons. The role of both magnon branches is found to be important in the scattering processes. In the long-wavelength and low-temperature approximation these scattering cross sections show complicated temperature dependence, the dependence being different for different processes considered.

Neutron inelastic scattering and powder diffraction techniques have been used to study the crystal field splitting and magnetic ordering properties of CeAs and CeSb. Powder diffraction data indicate a type 1 face-centred cubic magnetic ordering structure for CeAs with a moment of 0·68 μ_B±10% at 4·2 °K. This value demonstrates that the ground state of Ce³⁺ in CeAs is a doublet. Inelastic scattering measurements give directly a value of the crystal field splitting in this compound of 140±10 °K (12 meV). The powder diffraction data for CeSb could not be interpreted using any of the simple face-centred cubic types of antiferromagnetic ordering. The inelastic data show that the crystal field splitting is smaller than 25 °K (2·1 meV).

A model is proposed to account for the linear change in magnetization which occurs in dysprosium aluminium garnet below 1·7 °K. It is envisaged that the sample exists as a mixture of two phases, the relative amounts of which change with applied field.

Calculations are made of the total moment associated with transition impurities in ferromagnetic or paramagnetic transition metal hosts allowing for exchange enhancement on the host sites neighbouring an impurity. The model correctly predicts positive dμ/dc values when transition impurities to the right of iron in the periodic table are dissolved in Fe and the `giant moments' of PdFe, PdCo; it also explains the behaviour of PdCr as that of a `dwarf moment' with analogies to a Kondo bound state.

The magnetic contributions to the specific heats of erbium and holmium ethyl sulphates at low temperatures have been measured by the magnetic relaxation method. The results are compared with values calculated from optical and paramagnetic resonance spectra, and it is shown that the observed specific heat of holmium ethyl sulphate contains no contribution from the Schottky anomaly arising from the excitation of the Ho³⁺ ions to the low-lying excited state of the Ho³⁺ ion in the ethyl sulphate lattice. The contrast with earlier results on cerium ethyl sulphate is discussed.

We have made torque magnetometer studies of superconducting films of up to 6 at.% indium in tin. The general structure of the magnetization curves is similar to that observed earlier for tin and indium films; in particular, the curves are highly irreversible.

In the parallel field case the measurement is of the perpendicular moment M_{perpendicular} induced by a small perpendicular field H_{perpendicular}. We suggest that the appropriate way to describe the parallel field curves is by the Bean model with a critical current decreasing as the applied field increases. This seems to agree with the general shape of our experimental curves. Near to the critical field H_cparallel Deltour and Tinkham have proposed J_c is proportional to 1 -(H_parallel/H_cparallel)² for films which have a second-order transition on to the normal state at H_cparallel. Not all of our data agree with this form for J_c, and we use a simple model of a pinning centre (a local thin spot) to suggest that J_c may decrease more rapidly near H_cparallel.

The behaviour of the film critical fields is generally in agreement with theory except that the perpendicular critical fields are about 20% too high.

The general flux line distribution in a type II superconductor in the mixed state is curved in three dimensions. In this paper flux line motion is treated using the methods of irreversible thermodynamics; a general condition for the flux line distribution to be stationary is obtained. The effects of pinning centres are included and forces acting in the mixed state are analysed. The macroscopic critical state equation in three dimensions is derived and its solution in some special cases discussed.

We discuss the application of electron tunnelling to the study of super conductors and describe the results of tunnelling experiments involving superconducting thallium. We show that by measuring the first and second derivatives of the tunnel junction characteristics it is possible to determine the superconducting density-of-states variation in thallium and to draw some conclusions about the shape of its phonon spectrum. The principle peaks in the phonon spectrum (corresponding to the transverse and longitudinal modes) are found to occur at 4·0 mev and 9·45 mev, respectively. The end of the phonon spectrum appears to be located at 10·6 mev and evidence for this and other critical points is presented in this paper. The average value of the full gap for thallium was found to be 2Δ(0)_Tl = (0.74±0.02) mev (i.e. 2Δ(0)_Tl = (3.55±0.1) kT_c) although considerable anisotropy of the gap was observed. Values of 2Δ(Tl) ranging from 0·59 mev to at least 0·97 mev, and possibly 1·29 mev, were measured.

The thermal conductivity of the intermediate state of a superconductor is derived assuming that (i) the structure is lamellar, (ii) the electrons carry the heat, (iii) the electronic mean free path is much less than the width of the laminae, and (iv) there is thermal resistance at the boundary between normal and superconducting laminae due to the reflection of the electrons. The theory is applied to the experimental results of Mendelssohn and Shiffman, and it is shown that a fit is possible only if the width of the laminae is very much less than that normally assumed.

An investigation is presented of the low-temperature relaxation of nuclear spins in ferromagnetic heavy rare earth metals by the Suhl-Nakamura mechanism of virtual magnon exchange. Numerical estimates of the r.m.s. homogeneous linewidth and Bloch decay time T₂ are made in ¹⁵⁹Tb, ¹⁶¹Dy, ¹⁶³Dy, using the 90 °K Tb dispersion data of Møller and Houmann.

We find T₂ values of approximately 2½, 25, 12 μs respectively.

Sample inhomogeneous fields increase T₂ slightly and thus these values are in reasonable agreement with the measurements of Kobayashi, Sano and Itoh, who find values about twice as large.

The band energies of copper and gold have been calculated by the composite wave variational method at the symmetry points Γ, X and L of the Brillouin zone and the results for all the noble metals have been compared with the estimations of Cohen and Heine. It is found that at the zone face L of the Brillouin zone the p states are below the s states for copper and silver, but for gold the reverse is the case. The energy gaps in all cases, however, are not sufficient to show any contact between the Fermi surface and the zone face L. But the values of the effective mass are nearer to the experimental values than those of Cohen and Heine.

Recent developments in the d resonance theory are used to discuss the structure of the transition metals and to present an explanation for the occurrence of the body-centred cubic phase.

The interpretation of X-ray band emission spectra in terms of densities of states, and the role of transition probabilities and the Auger effect are discussed. The shapes of the aluminium K and L₂₃ spectra are calculated showing the relatively large part played by the latter two effects. Other metal spectra are discussed in the light of this, and extensions of the calculations are suggested.

Accurate L₂₃ emission spectra from Na, Mg and Al have been obtained using improved counting techniques. Various possible corrections to the Al spectrum are discussed, and it was found that the spectrum compared favourably with a band structure calculation. In order to derive effective potentials from the spectrum, methods of determining the position of the bottom of the band are discussed. It is shown that the effects of exchange and correlation can be neglected. The effective potentials are derived, and compared with various theoretical estimates and the potentials derived by Ashcroft from de Haas-van Alphen measurements.

Finally, methods of extending this work are discussed.

The determination of the structure factor for liquid metals at temperatures up to 1300 °C by means of neutron diffraction techniques is described. Particular attention is given to the corrections which must be applied if reliable structure factors are to be extracted from the measurements, including those for absorption, multiple scattering and inelastic effects. It is concluded that for the heavier elements the neutron method can yield reliable data which are accurate to a few per cent over most of the range of momentum transfer.

The experimental behaviour of the electronic specific heat coefficient γ of face-centred cubic disordered Cu-Zn alloys, which shows positive initial slopes against zinc concentrations, is explained by taking into account the level broadening of the d band which results from the presence of impurity potentials. As a result of level broadening, the d band has the effect of increasing the density of states at the Fermi level of an alloy. This contribution increases as zinc concentrations are increased, while the contribution from the s band is in the opposite direction. The density of states of the d band calculated by Burdick is used, and that of the s band is calculated by use of the nearly free-electron approximation by Ziman. The position of the d band relative to the Fermi level in an alloy is determined from the absorption edge of Cu-Zn which has been measured by Biondi and Rayne. Then only one unknown parameter, which represents the level breadth of the d state, appears in the expression for γ. It is shown that when the value of this parameter is chosen appropriately the present theory agrees fairly well with experiment. The experimental data for γ in face-centred cubic disordered Cu_1-2xZn_xNi_x, which increases with x, can be explained using the values of the level breadth parameter found from analysis of Cu-Zn and Cu-Ni. This fact shows that the behaviour of γ for copper-based alloys can be interpreted consistently on the basis of the present model.

Measurements have been made of the reflectivity of solid solutions of Zn_xHg_1-xTe in the spectral range 2200-6500 Å. Reflection peaks appeared for almost the whole domain of solubility. The various peaks are ascribed to transitions at particular points in the Brillouin zone. A band structure model for Zn_xHg_1-xTe alloys is proposed.

Theoretical calculations for the model of Birman, in which the infra-red emission from copper-doped zinc sulphide is attributed to an internal transition of the Cu²⁺ ion, indicate that configurational d-p mixing produced by the odd harmonics of the tetrahedral crystal field cannot account for the observed absorption (or emission) intensities for transitions internal to the Cu²⁺ centre, in either copper-doped zinc sulphide or copper-doped zinc oxide. Calculations show that the intensity must depend on a strong interaction between the Cu²⁺ ion and the zinc sulphide (or zinc oxide) lattice. Such a strong interaction with the lattice would mean g factors significantly reduced from the free-electron value, and probably a short spin-lattice relaxation time. These two effects, taken in conjunction with one another, are probably the reason why the electron spin resonance signal of Cu²⁺ in zinc sulphide is difficult to detect. Finally certain shortcomings of the Birman model are discussed.

There are difficulties in accounting for the intensity of the emission, and the photo-response of the spin resonance signal, in vanadium-doped phosphors assuming only V³⁺ is present. It is suggested that the assumption that V²⁺ is also present can account for these discrepancies, and two experiments are described which should provide a test of the latter possibility.

Magnetophonon resistance oscillations have been observed with samples of high purity n-type CdTe. The positions of the resistance extrema are shifted away from the magnetic fields predicted from the known values of the frequency of the q = 0 longitudinal optical phonons and of the low-frequency effective mass. These deviations, which amount to 13% in the transverse orientation and to 6% in the longitudinal orientation, are provisionally attributed to the perturbation of the electronic Landau levels by the interaction with all longitudinal optical modes (the polaron effect).

A microwave method was used for investigating the change of the impedance of a thin copper wire which takes place due to a short current pulse with a density of 4 × 10⁶ A cm^-2. The excitation of the longitudinal oscillation in the wire and delay in change of its resistivity with respect to time of the current pulse were obtained.