Table of contents

In order to understand more fully the way in which the measured structure factor S(k), or the direct correlation function c(k) = 1 - S^-1, depends on the force law, we have investigated these quantities for fluid argon with a Lennard-Jones 6-12 potential by (i) direct use of a virial expansion, and (ii) division of the interaction into a short-range (hard core) part, dealt with almost exactly, and a long-range part which is treated by the random phase approximation.

From (i) it is shown that provided T greater, similar 200 °k, it is even possible at high densities to predict the nodal positions in c(k) as well as the quantitative form at large k. Furthermore, although this theory does not converge for c(0) in dense fluids, it is pointed out that the coefficient of k³ in the small k expansion of c(k) and S(k) depends only on one term of the density expansion and this has encouraged us to calculate the coefficient of k² by this method. The results are quite different from hard spheres (usually opposite in sign) and again reasonable convergence is found over a wide density range.

From (ii), it is shown that c(0) can be obtained. The high-density equation of state of Longuet-Higgins and Widom then follows directly from the random phase approximation. The theory, furthermore, allows the term quadratic in the density to be calculated explicitly in terms of the van der Waals interaction and the hard core diameter. The agreement with experiment is satisfactory. Furthermore, the random phase approximation agrees with the virial expansion results for the r² and and r³ terms in c(r) at small r over a wide density range, though, as expected, deviations occur at the highest densities encountered in fluid argon. The Born-Green theory for this problem does not seem to give a correct account of the small-angle behaviour of c(r) or S(r).

Finally, a rough argument shows that, for liquid metals, the density dependence of the pair potential would convert the term quadratic in the density in the equation of state of Longuet-Higgins and Widom into a `Fermi gas' term proportional to the five-thirds power of the density. However, the separation into hard core plus long-range terms probably has greater uncertainties associated with it in liquid metals than in fluid argon.

A calculation of the linear electro-optical coefficient in NH₄Cl is given which is in qualitative agreement with the observed value. The calculation includes the effects of both the ammonium and the chloride ions, and the chloride ion is found to make a significant contribution to the coefficient of the opposite sign to that of the ammonium ion.

A recent paper by Boyd calculates the superconducting nucleation field of a planar defect such as a grain boundary. We extend Boyd's calculation to the case of two parallel planar defects. Nucleation fields higher than the surface nucleation field H_c3 can appear; we suggest that this result is relevant to the explanation of measurements of anomalous surface nucleation fields in cold rolled foils, and of anomalous supercooling fields in thin films.

Vibrational frequency spectra are presented for vitreous silica, germania and beryllium fluoride; these have been computed from atomic arrangements in physical models described in an earlier paper. The positions of bands in the computed spectra agree well with observed bands in the experimental infra-red and Raman spectra.

It is shown that the evaporation method is effective for discussing the generalized special frequencies of multi-dimensional lattices and, by combining it with Dean's method of obtaining the limits of impurity bands, we can estimate the critical mass ratio of each generalized special frequency. As examples the critical mass ratios of some isotopically disordered two- and three-dimensional lattices with simple structures are actually estimated.

The method of evaporation presented previously by the author is applied to some lattices which are non-isotopically disordered. For the mixed crystal of alkali halide type, the Saxon-Hutner-type statements are again obtained for the common gaps of appropriate constituent systems. For the disordered lattice which contains only the simplest clusters of non-isotopic impurities, some definite conclusions concerning the critical mass ratios of the generalized special frequencies are obtained.

Experiments have been carried out to decide whether impurities inhibit F-centre production by ionizing radiation by acting as centres for electron V_k-centre recombination or by reducing the stability of F centres once formed. The growth of the F band during electron irradiation and its decay following irradiation have been measured, using single radiation pulses and continuously switched beams. It has been found that none of the observed decay processes can account for the inhibition of coloration, and the evidence suggests that impurities act principally as electron V_k recombination centres.

The scattering of 5.4 Å neutrons by defects in single crystals of neutronirradiated magnesium oxide has been measured over the angular range 5-90°. The effects of irradiation dose and subsequent annealing treatments have been studied in the range 5×10¹⁹-2·5×10²¹ neutrons cm^-2 (>1 MeV) and 350-1460 °c respectively. The shapes of the curves of the observed cross section against angle were found to be sensitive to the annealing temperature but were influenced only slightly by the irradiation dose. In all cases the cross section decreases with angle, this decrease becoming more pronounced the higher the annealing temperature.

Evidence is presented which suggests that the defect scattering occurs predominantly from vacancies. The results were compared with the scattering expected from a number of vacancy aggregate models and correlations were obtained for aggregates containing between 1 and about 100 vacancies per cluster in the temperature range 150 °c (during irradiation) to 1460 °c respectively.

At the two extremities of this size range it is shown that the present conclusions are in reasonable agreement with the results obtained by other workers from electron spin resonance and electron microscope observations respectively.

After a 4 MeV electron irradiation at 500 °c to a dose of 1·4×10¹⁹ electrons cm^-2 no defect scattering could be observed within the accuracy of the cross-section measurements, ±0·5 mbn sr^-1 molecule^-1.

A new approach to the problem of calculating the phonon component of the thermal conductivity of structurally disordered solids is presented. The basis of the method is the introduction of collective variables U(k) defined by

where u_α(j) denotes a Cartesian component of the displacement from equilibrium of the jth atom whose equilibrium site is at r_j. For simplicity only monatomic solids are considered.

The plane waves are chosen to satisfy periodic boundary conditions within a volume containing N atoms and the sum over k contains just N terms. In the case of amorphous solids the wave vectors are chosen to be within the Debye sphere. It is assumed that the waves are linearly independent. In the case of solids containing a large amount of crystalline order, for example when dislocations are present, it is not possible to give a precise definition of the upper limit in the sum over plane waves. For many practical purposes this is fairly unimportant, especially at low temperatures where only long-wavelength modes are excited.

The merit of this approach is that it avoids describing a solid as a disordered lattice, which is very convenient in the case of amorphous solids and also when defects such as dislocations and interstitials are present in a lattice. The Boltzmann equation in this representation has precisely the same form as that of the standard theory of thermal conductivity. The collision terms have been calculated to first order in perturbation theory.

The three-terminal capacitor method has been used to measure linear thermal expansion coefficients of single crystals of CoCl₂.6H₂O in the temperature range 1·5-4·2 °K. λ-type behaviour was found at the Néel temperature 2·28 ± 0·01 °K. The form of the expansion has been accounted for qualitatively in terms of superexchange interactions. The estimated precision of the measurements is ±6% for the principal values and ±1° for the orientation of the principal axes of the expansion tensor.

Graphical methods are used to separate up to four different contributions to the heat capacities of the rare earth metals at low temperatures (T <4.2 °K). Since the proportions of the different contributions vary from metal to metal and with temperature, the flexibility of graphical analysis gives it an advantage over standardized (computer) analysis.

For each of the metals, we obtain the coefficient of the electronic contribution and the coefficient of the leading term in a series expansion for the nuclear hyperfine contribution, both to an accuracy comparable with that of the experimental results (a few per cent at worst). Estimates of the lattice contributions are obtained only for ytterbium and lutetium. The magnitude of the magnetic contribution to the heat capacity is large enough for four metals (praseodymium, neodymium, europium and thulium) for its general form to be established.

The main difference between the graphical and standardized (computer) analyses appears in the estimates derived for the electronic contribution. For four of the metals, the two estimates differ by factors of 2 to 5.

The excitonic optical absorption in the presence of a static electric field has been evaluated by solving the effective mass equation numerically. The results are compared with a previous approximate calculation and some experimental measurements. A certain degree of agreement has been found with experiment.

The reported measurements show that AlF₃ is a suitable host lattice for the study of trivalent iron group ions. Single crystals suitable for electron paramagnetic resonance measurements can be grown from the vapour phase. The hyperfine interaction of the fluorine nuclei with the Cr³⁺ ion has been investigated by the ENDOR technique, the values of A_s and A_p being determined as -10.08±0.05 MHz and -9.24±0.07 MHz respectively. These measurements show that the impurity ion is in the symmetrical position in the fluorine octahedron, unlike Al³⁺ which is not in the centre of this octahedron.

Previous series expansion studies of the magnetization and susceptibility of the Ising model are extended to the higher derivatives of the free energy with respect to the magnetic field. It is found that just above the critical temperature the 2nth derivative is given by

where in two dimensions γ = 1¾, Δ = 1fraction seven-eighths and in three dimensions γ = 1¼, Δ = 1fraction nine-sixteens. In two dimensions γ is exact and Δ is correct to 0.6%; the three-dimensional indices are correct to 0.2%. Just below the critical temperature

where in three dimensions 0.303 <or= β <or= 0.318 and 1·55 <or= Δprime <or= 1.65. If the even derivatives are to have the same index above and below T_c, then β = fraction five-sixteens and Δprime = 1fraction nine-sixteens. The latter index lies very close to the lower limit so that, although we cannot rule out Δprime = Δ, Δprime = 1fraction five-eighths is not unlikely. In two dimensions our results are consistent with linearity of the index and, if this is assumed, then β = fraction one-eighths and Δprime = 1fraction seven-eighths exactly and there is symmetry about T_c.

The amplitudes C_2n⁺ and C_n^- are estimated for n = 1 to 6.

A calculation is given of the additional zero-point sublattice deviation created by a single non-magnetic impurity ion in a Heisenberg antiferromagnet. A knowledge of this quantity is shown to permit a measure of the sublattice magnetization in the pure host to be made from an observation of the cross section for the diffuse elastic scattering of thermal neutrons from the mixed system. The form of the cross section is discussed in detail. It is shown that the pure host sublattice magnetization can be obtained directly from a relative measurement of the intensity of the first Bragg peak and the diffuse elastic scattering intensity at this position. The calculation of the magnetic defect created on the host is based on the linear spin-wave approximation and is accomplished with the use of thermal two-time Green functions. We also discuss how nuclear magnetic resonance and Mössbauer experiments can be used with the mixed system to give an estimate of the sublattice magnetization in the pure host.

Second moments, linewidths and spin-lattice relaxation times have been measured in the three isomeric solid trimethylbenzenes between 4 °k and their melting points. The observed spectra at 4 °k are characteristic of solids in which reorientation of the methyl groups modifies the spectra. This motion, presumably describable as a quantum-mechanical tunnelling, does not provide a relaxation mechanism and the spin-lattice relaxation times are very long.

Above 85 °k the high-field relaxation times are compared with observed rotating frame (H₁ = 3 G) relaxation times. These indicate the occurrence of additional molecular motions too slow to promote spin-lattice relaxation in a magnetic field of 6000 G, and sometimes too slow even to modify the observed spectrum.

The following measurements have been made: the proton linewidth for HBr and 50% HBr/DBr in all solid phases by both steady state and Bloch decay methods, the deuteron spin-lattice relaxation time for DBr in the liquid and upper solid phase, the proton spin-lattice relaxation time for HBr and 50% HBr/DBr in the solid state at resonance frequencies of 21·5, 20·8, 14·3 and 9·0 MHz and the proton spin-lattice relaxation time in the rotating frame for radio-frequency fields of 14·5, 12·0 and 5·3 G.

The measurements have been interpreted in terms of the various nuclear spin interactions and important information has been obtained about the molecular motion. The deuteron quadrupole coupling constant in DBr is estimated to be 250±50 kHz.

It is found that for HBr in the upper temperature phase below the melting point the molecules are reorienting rapidly and diffusing slowly and the corresponding correlation times and diffusion constants are derived. On melting the reorientation rate hardly changes but the molecules become free to translate much more rapidly. It is possible that there is an observable spin-rotation contribution to spin-lattice relaxation in the upper solid phase in HBr. The relaxation in the lower temperature region of the upper phase has a strong contribution from proton-bromine dipolar relaxation of the second kind.

A hysteresis effect similar to that observed in dielectric constant measurements is found around the lower of the upper pair of phase transitions in HBr. The relaxation in the middle phase is also controlled by proton-bromine interactions giving dipolar relaxation of the second kind but with the quantization of the bromine nucleus controlled by electric field gradients. The necessary theoretical analysis is given for relaxation with or without electric field gradient quantization for T₁ and T_1ρ. The linewidth in this phase shows motional narrowing of the second kind. The values of the molecular reorientational correlation time and the bromine nucleus spin-lattice relaxation times are predicted. The proton spin relaxation in the lowest temperature phase in HBr/DBr is closely related to the dielectric loss measurements. The interpretation is complicated by the strong bromine quadrupole interaction.

`Vapour snakes' were observed during the solidification of HBr and DBr.

The following measurements have been made: the proton linewidth for HCl, in both solid phases by both steady state and Bloch decay methods, the proton spin-lattice relaxation time for HCl and for 50% HCl/DCl in the solid state, at resonance frequencies of 21·5, 20·8, 14·3 and 9·0 MHz, and the proton spin-lattice relaxation time in the rotating frame for radio-frequency fields of 14·5, 12·0 and 5·3 G.

The measurements have been interpreted in terms of the various nuclear spin interactions and important information has been obtained about the molecular motion.

It is found that for HCl in the upper temperature phase below the melting point the molecules are reorienting rapidly and diffusing slowly, and the corresponding correlation times and diffusion constants are derived. On melting the reorientation rate hardly changes but the molecules become free to translate much more rapidly. It is possible that there is a small spin-rotation contribution to spin-lattice relaxation in the upper solid phase in HCl. In the lower temperature part of the upper phase the proton relaxation, both conventional and rotating frame, contains a contribution due to dipolar relaxation of the second kind. In the lower temperature phase we have difficulty in interpreting the spin-lattice relaxation times but they and the linewidths are consistent with very little, if any, large-scale molecular motion in this phase. It seems likely that the proton relaxation is due to dipolar relaxation of the second kind with chlorine nuclei controlled by quadrupole interaction.

The disturbances in magnetization around a wide variety of impurities in nickel (Zn, Al, Ga, Si, Ge, Sn, Sb, non-transition metals; Nb, Mo, Ru, Rh, 4d elements; W, Re, 5d elements) have been measured by the neutron diffuse scattering method. This work is an extension of an earlier study (by Collins and Low) of the 3d transition elements V, Cr, Mn, Fe dissolved in nickel. Two distinct kinds of magnetization distribution have been observed. In one category (Mn, Fe, Rh) the magnetic disturbance is positive and confined to the impurity site. The other solutes lead to a widespread negative magnetic disturbance extending some five ångströms into the nickel lattice. With the exception of V, the individual scattering curves of the second class approximately superpose, when normalized at a single point. This leads to the conclusion that the spatial form of the moment disturbance in the nickel matrix is similar for Zn, Al, Ga, Si, Ge, Sn, Sb, Nb, Cr, Mo, Ru, W and Re and suggests that this wide variety of impurities affects the nickel matrix through the same mechanism. Differences between the impurities show up in the strength of the disturbances. In discussing the nature of these magnetic defects, a molecular field theory model is used to illustrate a possible mechanism for linking the spatially widespread losses of moment with a more localized driving force. A transition element solute gives rise to a widespread negative defect or to a positive disturbance confined to the solute site depending on whether or not formation takes place of the bound impurity states proposed by Friedel.

The Mössbauer spectrum of FeAl₂O₄ consists of three absorption peaks at 0, 1.46 and 2.56 mm s^-1 relative to a ⁵⁷Co/Cu source. The spectrum can be resolved into two quadrupole-split spectra. The outer pair of lines corresponds to a quadrupole splitting of 2.76±0.10 mm s^-1 and an isomer shift of 1.52±0.10 mm s^-1 and may be attributed to the Fe²⁺ ions at the B sites. The inner doublet with a quadrupole splitting of 1.39±0.10 mm s^-1 and an isomer shift of 1.10±0.10 mm s^-1 can be attributed to the Fe²⁺ ions at the A sites. The degree of inversion x was determined from the relative areas under the two spectra to be 23±2%. This is in agreement with the value of 25% determined from the intensities of the X-ray diffraction powder pattern.

The hyperfine fields at ¹¹⁹Sn nuclei in ordered Co₂MnSn have been measured using the Mössbauer effect. The field values obtained are compared (i) with other measurements on Co₂MnSn which are in disagreement, and (ii) with hyperfine fields reported for other similar Heusler alloys. In addition a value of +0·67±0·01 nuclear magnetons is obtained for the nuclear magnetic moment of the first excited state of ¹¹⁹Sn.

The three second-order shear constants of hexagonal materials are joined by five third-order ones when volume-conserving deformations are considered. Further, when the total energy of the material is decomposed there is a first-order shear constant for each component. These nine shear constants, denoted by g_i, are listed.

Electrostatic contributions are calculated at nine values of c/a: g₄, g₅ and g₉ show considerable variation. Repulsive ion-ion contributions are calculated for three neighbourhoods, the coefficients involved being given as polynomials in c/a: g₂, g₅ and g₆ are the most sensitive to c/a.

The minimum in the electrostatic energy occurs at c/a = 1·6360±0·0002 and not at the ideal ratio.

A full description of interionic forces is seen to require the use of a complex electron-ion scattering amplitude. A prima facie case is established for the validity, in this connection, of phase shift data which have already been successfully and extensively used in electron transport theory.

A new method for calculating the energy bands of transition metals has been developed based on the Korringa-Kohn-Rostoker method. This method promises to be more convenient and simpler than any method in use at present. By introducing an arbitrary energy shift the method splits the KKR coefficients into two rapidly convergent sums, one over direct and the other over reciprocal lattice vectors, and shows that the d bands are of tight-binding character. It bears a close relationship to the empirical `model Hamiltonian' or `combined interpolation scheme' of Mueller and Hodges et al. which may be derived as an approximation to it. In particular the hybridization terms derived from the new method are investigated and they are in good agreement with those that appear in the model Hamiltonian. These terms have not previously been explained.

Co-operative modes of motion in liquid lead have been observed using the acceleration of slow neutrons. The frequency-wave number relationship for the modes has been derived and is similar to that found in solid lead. The lifetime or interaction of the modes has been studied by fitting a number of models to the data. It is concluded that for wave numbers less than 2 Å^-1 the modes can be described by an extension of hydrodynamic theory to high frequencies, by admitting a sufficiently complicated damping function. A damping function which involves both solid-like and fluid-like terms is suggested.

The temperature dependence of co-operative modes of motion in liquid lead has been observed in the range 336-725 °c, using the inelastic scattering of cold neutrons. The positions and widths of modes near a frequency of 12·5 × 10¹² rad s^-1 do not change significantly with temperature. However, the width of the non-propagating mode at Q = 2·2 Å^-1 increases with temperature in a manner suggestive of single-particle motion. Simple models are employed to discuss these processes, although further theoretical work is needed to explain the data fully.

The elastic magnetic diffuse scattering of neutrons from a single crystal of PdFe (Fe similar ¼ at.%) has been examined at 4·2 °K. Differential cross sections were measured for scattering vectors along <100>, <110> and <111> directions in the crystal. The three sets of data superpose within experimental error, implying a corresponding degree of isotropy in the magnetization of the palladium host around each Fe atom. This result is discussed with the aid of a simple model to represent the palladium matrix.

Measurements of reflectivity due to free carrier dispersion in p-type GaAs and GaSb are reported. Differently doped specimens are compared to observe the behaviour well below the reflectivity minimum. The results obtained are combined with the results of earlier Faraday effect observations in these materials to deduce the light and heavy hole masses. These are found to be 0·068±0·015 and 0·50±0·02 in GaAs, and 0·044±0·007 and 0·33±0·013 in GaSb.

Specific heat measurements have been made on gadolinium, terbium and dysprosium aluminates in the neighbourhood of their Néel points. The entropy and energy changes associated with their transitions are compared with those for the Heisenberg and Ising theoretical models.

The previous theory by the author for the donor electron absorption in semiconductors with a spherical conduction band is extended to the case of a spheroidal conduction band. It is shown that the calculated absorption for Ge and Si is nearly equal to that evaluated by making the approximation of a spherical conduction band.

Through screening by outer (s) electrons the plasma oscillations of inner (d) bands are transformed into an acoustic phonon branch if certain conditions are fulfilled. Similar to the lattice-phonon branch this provides an interaction with free electrons which leads to superconductivity. Dependence of the transition temperature T_c on the number of d electrons follows Matthias' rule. Suggestions for increasing T_c are discussed.

It is shown that the electron-phonon coupling constant in narrow bands is much smaller than in wide bands. The contribution of incomplete d bands of transition metals to the total phonon-induced electron interaction, relevant for superconductivity, is much smaller, therefore, than generally believed.

The range of electron states near a band edge in liquid chains has been found to depend, not on an exponential decay factor as previously believed, but on the `beat period' of a related Bloch function. This finding affects extension of the work to real liquids, and the use of such approximative methods as the local density model.

X-ray and electron spin resonance studies are reported of the paramagnetic material [(CH₃)₄N]₂Cu[S₂C₂(CN)₂]₂ in single-crystal form, and also of this compound diluted in its diamagnetic nickel analogue. This mixed crystal is triclinic, and the copper ion has axial magnetic symmetry, with the Hamiltonian parameters g_parallel = 2·070, g_{perpendicular} = 2·017, A_parallel = ±162, A_{perpendicular} = ±42, P = ±8, Pprime = 0, P_z = ±5·3 and P_x = P_y = −or+3. (Except for g, the units are 10^-4 cm^-1.) The undiluted copper material is monoclinic and unit cell dimensions of a = 13·22 Å, b = 11·56 Å, c = 16·29 Å, β = 95·2° are reported with space group P2₁/c. A single electron spin resonance line was observed in all orientations, apparently exchange narrowed, at 9·27 GHz.