Table of contents

A study is made of the statistical thermodynamics of cross-linked polymers. It is argued that several physical and mathematical hypotheses or approximations are involved in making progress and these are enumerated. The analysis concerns that part of the behaviour of the system solely due to cross links and entanglements, and it is shown that under certain conditions a cross link will have a distribution (2/πLl)^-3/2 exp(-2r²/L_al) about its mean position, where L_a is the average length between links and l the monomer length. This value can be obtained in a calculation making the total entropy a bound. The elastic free energy has the form

where N is the number of polymers, L the length of a polymer, _i the strains, and small omega, macron satisfies

where M is the number of cross links, V the volume and C a constant. This expression offers a form where the effect of entanglement is added to the effect of the cross links.

The ferrous content of hafnium-doped yttrium iron garnet crystals has been determined by a number of methods and microwave measurements of the temperature dependence of the anisotropy constant found to give the most reliable value. The Fe²⁺ ions have a doublet energy level lying lowest with a maximum splitting of 75 cm^-1.

Thermal annealing of the F band in LiF irradiated with x-rays at room temperature has been studied. Four annealing stages were identified. These occur in the temperature ranges 25-60, 120-200, 200-300 and 300-400 °c. The last two stages are very prominent and reproducible. A detailed analysis of the annealing kinetics of the 200 to 300 °c stage was made. This stage is characterized by second-order kinetics with an activation energy of about 3·2 ev.

The irreducible corepresentations of the magnetic space group of the canted antiferromagnet NiF₂ have been derived and the symmetries of the magnons all over the Brillouin zone have been determined. The dispersion relations for magnons in NiF₂ have been calculated and the use of the theory of spin-space groups for NiF₂ is illustrated.

Vibrational series headed by zero-phonon lines have long been known in the absorption and emission spectra of bismuth-doped calcium oxide phosphors. Single crystals of undoped and bismuth-doped calcium oxide have been obtained, making possible Zeeman and uniaxial stress experiments on the zero-phonon lines. The absorption line in bismuth-doped calcium oxide at 4.2°K is at 367.02 nm (27239 cm^-1) and is shown to be an electric dipole transition occurring at a centre of cubic symmetry. Using a pulsed magnetic field of 120 kG, a Landé g factor of 0.40 has been found for this transition. In the single crystals of undoped calcium oxide at 4.2 °K there is a zero-phonon absorption line at 355.74 nm (28102 cm^-1). This is also observed in proton or neutron irradiated samples and has been found previously in additively coloured crystals. This absorption line has also been shown to be an electric dipole transition occurring at a centre of cubic symmetry, but here there is no detectable Zeeman splitting. These results are discussed with reference to the possible assignment of this line to the F centre.

Phonon dispersion relationships have been determined for the acoustic branches in MnO along the [100], [011] and [111] directions by neutron spectrometry. The measurements have been interpreted in terms of the rigid-ion and the shell model. The rigid-ion model does not fit the data adequately and the shell model, while producing a much better fit to the data, cannot be said to give a satisfactory description.

The electron-phonon interaction matrix element is examined in a modified tight-binding approximation. It is found that the matrix element is proportional to the bandwidth and therefore very small for an extremely narrow band. Paradoxically, in the usual Bloch tight-binding approximation the same matrix element which is proportional to the well-known Bloch constant C, is also found to be proportional to the bandwidth for intra-band transitions. The reasons for this equivalence are discussed. Inter-band transitions where the two matrix elements differ grossly in magnitude are also considered.

A method is given for the evaluation of two-dimensional small-angle folding integrals in the particular case of cylindrical symmetry. The associated problem of deconvolution of scattering patterns with cylindrical symmetry is also considered. The present results are particularly applied to problems in the small-angle scattering of electrons (incident energy range 20 kev-1 Mev). The good accuracy of the method of computation is illustrated by calculating the angular distribution of electrons after ten scattering events using a test first scattering distribution of Gaussian form. The general procedure also allows the correction of the electron scatter patterns for convolution effects due to the finite angular divergence of the incident electron beam and the finite size of the collector aperture.

The free energy expression of Wohlfarth for a very weak itinerant ferromagnet is amended by the addition of volume-dependent terms given by Belov. Hence the forced magnetostriction and pressure dependence of the Curie temperature are calculated. The effects depend on the volume dependence not only of the effective interaction energy but also of the fine details of the band structure. Magnetostriction measurements on ZrZn₂ by Ogawa and Waki are used to estimate that here partial differentialT_c/partial differentialP = -2·4 degK kbar^-1. A variety of results relating to the effect is deduced from elementary thermodynamics. For a very weak itinerant ferromagnet corrections to the result deduced from the Belov equation are obtained. Reference is made to recent measurements by Fawcett et al. on dilute platinum and palladium alloys.

A coupled Ising chain model of the spin interactions in certain rare-earth chlorides and hydroxides is developed. The specific heat and susceptibility are derived analytically as functions of temperature in the two-chain case by the transfer-matrix method. These results are then compared with the available experimental data on the rare-earth chlorides and interaction parameters are derived for PrCl₃ and NdCl₃.

The transverse dynamical spin susceptibility of a metal in a static magnetic field is calculated within the random phase approximation. The new feature of the work is that the existence of Landau levels is taken into account. The dynamical susceptibility is used to discuss the spin-flip excitations of the system. These include Stoner single-particle excitations, spin waves and some other collective modes. The spin-wave spectrum obtained is directly related, in the long-wavelength limit, to that obtained by Platzman and Wolff using Fermi liquid theory. The pronounced anisotropy of the spin-wave spectrum, as observed by Schultz and Dunifer in the alkali metals, is obtained in the present theory through the introduction of the Landau levels. A discussion is given of q_max, the wave vector at which the spin waves merge with the Stoner continuum. Other collective modes obtained include some in the nature of cyclotron resonances with spin-flip, and some which lead to an instability of the normal Hartree-Fock state at very low temperatures, as discussed previously by Celli and Mermin.

The optical constants of gadolinium have been measured by internal reflection from evaporated films on silica glass prisms, at 293 °K and 105 °K, in the spectral region 0·50 to 5·6 eV. Absorption bands were observed at 0·70 eV and 1·1 eV in the ferromagnetic state. No evidence of absorption bands owing to transitions of f electrons was found. An extrapolation of the room temperature data to low energies suggests a value of 0·5 for the effective number of free electrons per atom.

The optical interband absorption of aluminium is calculated using a pseudopotential model with Ashcroft's empirical parameters. The calculation predicts an absorption peak at 1·55 eV in agreement with published experimental data. The predicted absorption is considerably sharper than that observed experimentally, but the areas under the absorption peaks agree within 10%. There is at present little experimental evidence for a second predicted absorption process centred at 0·55 eV.

We have investigated the L_III absorption edges of ⁸⁰Hg and ⁸¹Tl in the metals and their respective compounds. It is found that owing to chemical combination the L_III absorption edges of ⁸⁰Hg and ⁸¹Tl are, in general, shifted towards the high-energy side. A white line at the absorption edge has been detected for HgO, TlBr and Tl(NO₃)₃.

If ψ_k(r) = R_k(r) exp{iϕ_k(r)}, with R and ϕ real, and R positive, it is suggested that there may be some advantages in using the equations determining R and ϕ for deriving dispersion relations, constant-energy surfaces and charge distributions in perfect crystals. The idea is illustrated in the one-dimensional and two-dimensional separable case.

A two-band superconductor model with one normal d band and one superconducting s band in the presence of nonmagnetic impurities is investigated. The nonmagnetic impurities cause interband as well as intraband scattering. The phase transition temperature of the superconducting s band is obtained in the limit of low impurity density, and is found to be lower than that of the corresponding pure two-band superconductor.

Results of measurements of the linear thermal expansion of single-crystal rods of pure antimony and bismuth over the temperature range 200 °K to 8 °K, and 200 °K to 4·5 °K respectively, made using an optical lever dilatometer, are discussed. The principal linear expansion coefficients α_{perpendicular} and α_parallel in the directions perpendicular and parallel to the trigonal axis are tabulated and shown graphically as functions of temperature. For both materials α_{perpensicular} is considerably less than α_parallel, with antimony showing the greater anisotropy. α_{perpendicular} for antimony is negative below 20 °K, passing through a minimum at 14·8 °K. Calculated values of the Grüneisen parameters, γ_{perpendicular} and γ_parallel, corresponding to α_{perpendicular} and α_parallel are tabulated together with estimates of the high-temperature limiting values of γ_{perpendicular} and γ_parallel.

An analysis of the dynamical spin states of alloys typified by palladium-iron is given. It is shown that the spin-wave spectrum of Doniach and Wohlfarth is valid only over a very small momentum range. In general, the dynamical states are spin waves but with finite damping. The enhancement of the electronic specific heat is calculated. The results are in reasonable agreement with experiment, where the comparison is possible.

Measurements of specific heat have been made in the temperature range 1·4 to 4·2 °K on a series of Pd-Co and Pt-Co alloys. The cobalt content varied between 0·5 and 94·8 at.% in the Pd alloys and was less than 4 at.% in the Pt alloys. In the Pd-Co alloys the temperature dependence of the specific heats was analysed in terms of lattice, electronic, nuclear and magnetic spin wave contributions. From the nuclear contribution, the effective average internal magnetic field H_eff at the cobalt nuclei was deduced and found to be approximately proportional to the cobalt concentration. Addition of 0·5% Co to Pd produces an enhancement of about 40% in the total specific heat. This effect is thought to be mainly of magnetic origin. The specific heat values for the Pt-Co alloys were compared with that of platinum and the entropy change ΔS associated with the magnetic disordering was estimated. The results could be represented by ΔS = cR ln (2S_T+1) where c is the cobalt concentration and S_T has the value 0·8. The Curie temperatures T_c deduced from the specific heat results were found to be approximately half the values reported from magnetization measurements. This is shown to be due mainly to incorrect anlysis of the magnetization data which gives erroneously high values to T_c and to a lesser extent to random variations in solute concentrations in the specimens.

The thermal conductivity of sintered semiconductor alloys is calculated using the Klemens-Callaway model assuming that the boundary scattering can be adequately described by a mean free path. It is shown that the relative increase in thermal resistance increases with the amount of alloy disorder scattering and should be substantial in sintered germanium silicon alloys with particle sizes of the order of one micron.

New measurements are presented of conduction induced in arsenic trisulphide films by beta and cathode rays as a function of film thickness and incident beam current. Films between 2 and 20 μm thick have been examined, the beta and cathode rays being sufficiently energetic to penetrate the films completely.

It is concluded that the majority of the carriers are generated along the paths of the primary electrons and only those that escape recombination in these columns are available for conduction. The low probability of escape is responsible for the absence of saturation of the induced conduction with bias voltage. Under steady-state conditions the free path of carriers which escape recombination in the columns is normally long enough not to reduce the efficiency of their collection appreciably.

Cathode rays induce steady-state conduction in amorphous dielectric films only if their energy exceeds a threshold. A comparison between values for the threshold and theoretical data for the differential dissipation of energy of fast electrons in solids shows that at the threshold the electrons penetrate only half the thickness of the films. The space charge which develops under these conditions and which is made responsible for the occurrence of the threshold is studied by experiments in which both sides of self-supporting films of arsenic trisulphide are exposed to cathode rays.

The interpretation of the results is discussed.

Ionic conductivity of pure and doped KI is measured in the temperature range 200 to 700°c. A large contribution to the conductivity by the anion vacancies is observed in pure KI at high temperatures. The solubility of Ba²⁺ in KI is much higher than that of Cd²⁺, Pb²⁺ and Ca²⁺. The behaviour of KI doped with Ba²⁺ is similar to the behaviour of KCl and NaCl doped with divalent cation impurities. The solubility of CO₃^2- in KI is also good. Crystals doped with CO₃^2- show an increase in conductivity at high temperatures owing to the increase in anion vacancy concentration. At lower temperatures the conductivity of KI is suppressed by the CO₃^2- impurity. The values obtained for various energies of activation in KI crystals are E_c = (0·72±0·02) ev for the migration of a cation vacancy, E_A = (1·50±0·05) ev for the migration of an anion vacancy, W_s = (1·6±0·1) ev for the formation of a separated pair of Schottky vacancies and W_c similar 0·26 ev for the association of a cation vacancy with Ba²⁺ impurity.

Experimental observations are reported of high-frequency currents associated with thin photoconducting cadmium sulphide platelets having flat and parallel faces. They appear under conditions when `round-trip gain' due to acousto-electric amplification is achieved. They are correlated with departures from Ohm's law. The results are confined to conditions under which a linear theory is believed applicable. The linear theory is outlined and comparison is made with a variety of experimental observations. Semiquantitative agreement is obtained.

Anharmonic parameters of the Buckingham type potential function have been computed at 0 °K for solid argon, krypton and xenon. The contribution of cubic and quartic terms of potential energy expansion to Helmholtz free energy has been taken into account by an iteration method. The results are compared with quasiharmonic parameters of the same potential. The effect of anharmonicity is found to be negligible in the case of krypton and xenon.