Table of contents

From the solution of Liouville's equation for a particle in a spherical potential the vector invariants, other than the angular momentum, are found. For purely repulsive potentials the symmetry group of the system is found to be the Lorentz group, and the canonical transformations generated by the two vector invariants give functional representations of the Lorentz group.

For the ground state of the H₂⁺ molecule the function ψ = (1 +pRξ) exp (-αRξ) cosh (βRη) which is separable into elliptic coordinates, has been studied. The calculated energy is lower than that of any of the existing variational functions. The integrals <X²> and <Z²>, as well as the energy values for different nuclear separations, have also been calculated.

A one-parameter iteration-variation method is developed and used to calculate phase shifts for scattering by model potentials. Some reasons are suggested for the failure to obtain convergence in certain energy ranges. Alternative iteration-variation methods are proposed and some typical numerical results are given.

A matrix method is used to calculate the energy eigenvalues and the corresponding orbital wave functions for a number of ³S states of the helium isoelectronic sequence within the framework of the `fixed core' approximation. The wave functions are employed to calculate the expectation values of a number of one-electron operators.

Results are presented for the 2 ³S, 3 ³S and 4 ³S states of He, Li⁺ and Be²⁺.

Hartree-Fock integro-differential equations for the excited singlet S states and in particular for the 1s2s ¹S state of helium-like ions are studied by the examination of the first-order equations in a perturbation expansion in powers of 1/Z; the complete set of hydrogenic eigenfunctions for s orbitals is used as the basis for the solution of these equations. The statement of a theorem due to Froese is shown to be in error and is corrected. Considerable insight into the roles of various zero-order states in the first-order function and the effect of various constraints on these is gained. It is shown that, for the nsms¹S state, the zero-order ns²¹S and ms²¹S states play a very important part in determining the effect of the constraints on the first-order Hartree-Fock functions.

The inaccuracy inherent in computing classical differential scattering cross sections by the use of numerical differentiation is removed. The cross section is represented by an analytic form of the derivative capable of being evaluated to any desired degree of accuracy. The general method is demonstrated for scattering by a Langevin field and the results are compared with an incremental procedure adopted by previous authors.

A simple approximation is proposed for investigating the failure of an atomic closed-shell core to follow adiabatically the motion of an outer electron in an almost non-penetrating orbital, on the condition that the outer electron has insufficient energy to excite the core. The method is variational and provides approximate upper limits to the non-adiabatic corrections to the energy levels. Calculations are done on the ²F levels of Na and K and on the ²F and ²G levels of Mg⁺ and Ca⁺. The correction is found to decrease the magnitude of the energy change due to dipole polarization by 6% in the ²F levels of K and somewhat more in the ²F levels of Ca⁺; for lighter ions, and also for ²G levels, the percentage changes in the dipole polarization energies are smaller. The effect of the correction on quadrupole polarization is greater but the calculations are much less accurate.

The dipole polarizabilities of four closed-shell ions, Na⁺, Mg²⁺, K⁺ and Ca²⁺, are evaluated from observed spectra, using theoretical values of the quadrupole polarizabilities taken from the literature and including a number of corrections, in particular a non-adiabatic correction and perturbation corrections up to the fourth order. These additional, previously neglected, corrections are partly positive and partly negative, and nearly cancel each other: consequently the results, in which the probable error is about 1%, are close to results previously obtained from observed spectra.

A new formula is derived for the dipole-dipole term in the van der Waals interaction between two spherically symmetrical atoms. The approximation consists in taking the distortion of each atom in the instantaneous dipole field of the other atom to be proportional to the corresponding adiabatic distortion, and the co-efficients of proportionality are variationally determined. Good agreement with the results of other calculations is obtained in a number of cases involving atomic hydrogen and the inert gases.

A crossed beam method has been used to measure the cross section for proton production in single collisions between H₂⁺ ions and electrons at incident electron energies from 5.0 eV to 1000 eV. The measured cross section is the sum of the dissociative excitation cross section plus twice the dissociative ionization cross section. Its magnitude is in good agreement with other experimental data and with theory, but there is some disagreement over the rate of fall-off of the cross section at high energies. The present data at electron energies above 40 eV can be represented approximately by Q = (255/E)log(E/13.8) where Q is in units of πa₀² and E in eV. The measured cross section was found to be independent of ion source pressure over a wide pressure range, and the implications of this with regard to the vibrational-state population distribution of the H₂⁺ ions are discussed.

Excitation of helium, nitrogen and nitric oxide by 60-400 kev protons and fast hydrogen atoms has been studied by the spectroscopic analysis of radiation emitted in the spectral range 3900-6000 Å during the spontaneous decay of excited products of single collisions. Absolute cross sections for the emission of some of the more intense lines have been determined and, in the case of helium, excitation functions for the 4 ¹S, 5 ¹S, 4 ¹D and 5 ¹D levels have been obtained by the use of known transition probabilities.

The differential cross section for the scattering of 97.1 MeV positive pions by protons has been measured at fourteen angles between 36.3° and 164.1° in the centre-of-mass system. After the subtraction of first-order Coulomb effects the cross section is given (in mbn sterad^-1) by dσ/dΩ=(2.873 ± 0.039) - (2.872 ± 0.057) cosθ + (5.08 ± 0.11) cos²θ

The total cross section was measured by attenuation to be 56.1 ± 1.7 mbn.

The values of the phase shifts uncorrected for D-wave or higher-order Coulomb effects were found to be α₃₃ = 20.95 ± 0.32°, α₃₁ = -2.24 ± 0.18°, α₃ = -8.59 ± 0.23°.

The magnitudes of the above corrections are discussed.

These phase shifts were used, together with previous measurements of negative pion elastic scattering, to derive the following values for the T = ½ phase shifts at the same energy: α₁ = 7.93 ± 0.42°, α₁₁ = -1.90 ± 0.30°, α₁₃ = -0.60 ± 0.30°.

The angular distribution of γ rays from the charge-exchange scattering of 96.6 MeV negative pions by protons has been measured at 14 angles between 20° and 151.2°. The γ-ray detector utilized a lead converter and counter telescope. The correction for loss of electrons in the converter was determined experimentally and the calculated efficiency of the detector checked by direct measurement with a tagged beam of γ rays.

Analysis of the distribution of γ rays yielded the following differential cross section for the charge-exchange scattering (mbn sterad^-1 c.m.): dσ/dΩ=(0.694 ± 0.047) - (1.56 ± 0.05) cosθ + (1.89 ± 0.14) cos²θ

Using previously determined values for the T = 3/2 phase shifts, those for the T = ½ state were found to be α₁ = 8.28 ± 0.56°; α₁₁ = -5.0 ± 1.1°; α₁₃ = -0.23 ± 0.47°.

Measurements of the attenuation cross section of uranium have been made for photons of eleven discrete energies in the region 10 to 300 keV in order to resolve large discrepancies between two previously published estimates of this cross section. The present results he between these estimates and agree with some other recent experimental work. They are also in good agreement with the sum of the photoelectric, coherent scattering and incoherent scattering partial cross sections obtained from theoretical calculations.

The experimental facilities offered by the Haverah Park extensive air shower array are described. This array contains four deep Cerenkov detectors, each of area 34 m², spaced 500 m apart. The response of the detectors to the different components of the cosmic radiation are described and the performance of the array as a whole is considered. The array is efficient in detecting showers from primary particles having energies in the range 10¹⁷-10¹⁸ ev and responds efficiently to showers arriving at large zenith angles.

The present paper attempts to evaluate the second virial coefficient of a polymer solution and to assess its dependence on chain length. A lattice model is used, and the `lattice constants' c_nm of two separated chains of lengths n and m are calculated for values from 2 to 12 for a number of two- and three-dimensional lattices. The use of two parameters m, n allows polydispersity to be taken into account. For a polydisperse solution the second virial coefficient is determined by b_nm = -c_nm/c_nc_m, where c_n is the lattice constant of a single chain of length n. The b_nm are fitted to an asymptotic formula A(nm)^γ, and it is suggested that γ = 7/8 in three dimensions and 3/4 in two dimensions. Some experimental tests are suggested.

A diagram technique is developed for calculating high-temperature expansion coefficients for magnetic systems with arbitrary symmetric tensor interactions between all pairs of spins. General results are obtained for the first three coefficients in the susceptibility and magnetic specific heat series, in terms of various lattice sums involving the long-range parts of the interactions. The case where both Heisenberg exchange and dipole interactions are present is considered in detail, and the results are applied to GdCl₃, a ferromagnetic insulator ordering at 2.2 °K. Two possible sets of values for the two main exchange constants J₁ and J₂ are found by comparison with experiment: either J₁ = 0.073 ± 0.004 °K, J₂ = 0.007 ± 0 003 °K or J₁ = -0.025 ± 0.006 °K, J₂ = 0.040 ± 0.002 °K, where a positive sign denotes a ferromagnetic interaction. It is not possible to decide between these on the basis of the limited bulk measurements so far made, though resonance measurements on dilute salts favour the second possibility.

The magnon spectrum of GdCl₃, a two-sublattice ferromagnet in which the dipolar interactions are comparable with the two dominant exchange interactions, is investigated analytically and numerically. The spin-wave energies are found by deriving a generalization of the Holstein-Primakoff result, valid for a system of two equivalent sublattices with arbitrary symmetric interactions between all pairs of spins. The dipole-dipole interactions are treated exactly, by using the Ewald technique to evaluate certain k-dependent lattice sums and their derivatives at k = 0. In the long-wavelength region a formula is obtained for the quadratic approximation to the spectrum, which shows a complicated angular dependence, and this is used to calculate the form of the exponential magnetization curve at very low temperatures. The full dispersion relations are then obtained numerically, without renormalization, using the two possible sets of values for the exchange constants J₁ and J₂ derived from high-temperature experiments. For set A (J₁ = 0.073 °K, J₂ = 0.007 °K) the curves for different values of θ_k cross one another, and the `optical' mode energies are unusually low; for the more likely set B (J₁ = -0.025 °K, J₂ = 0.040 °K) the spinwave band becomes wider as k increases, and its lower edge is almost flat for an appreciable range of wave numbers. By integrating numerically over the Brillouin zone the zero-point spin defect is shown to be small, and the magnetization and specific heat are found as functions of temperature.

The influence of spin-phonon coupling on the specific heat of a system of paramagnetic ions in a lattice, the so-called Schottky anomaly, is calculated using the resolvent operator version of perturbation theory. It is found that this coupling does not cause the sharpening of the Schottky peak observed in cerium ethyl sulphate by Meyer and Smith. This is in conflict with earlier results obtained by Sears and by Tucker using Green function theory. The disagreement is traced to their use of an incorrect decoupling approximation. Finally the lifetimes of coupled spin-phonon modes are calculated.

The longitudinal relaxation model, which explains the relaxation effects observed in ferromagnetic resonance for ferrites doped with anisotropic ions, is adapted to investigate the effect of bottleneck in the energy flow to the lattice. It is deduced that this could occur if the heat capacity of the phonon (or magnon) modes to which the doping ions relax were too limited and these modes did not relax rapidly to the bath. Comparison of the expressions which are developed with experiment indicates that bottleneck effects are mild, if present at all, in ytterbium-doped yttrium iron garnet, and that this unwelcome complication is probably not important in general.

Previous work on crystal-field theory for UO₂ has not made use of the fact that in its ordered antiferromagnetic state its structure is properly described by one of the magnetic (Shubnikov) point groups. Here the theory of the splitting of atomic energy levels in a crystalline field with the symmetry of one of the magnetic point groups is applied to UO₂. This shows that selection rules do not necessarily invalidate the theory given by Blume of the first-order antiferromagnetic phase transition in UO₂.

Neutron diffraction studies of the alloy system Au₂Mn₂ - _lAl_l have established the coexistence of antiferromagnetic and ferromagnetic structures in between the extreme compositions of Au₂Mn₂ and Au₂MnAl. The magnetic moment of the manganese atoms remains throughout close to 4 2 μ_B. Both slow-cooled and quenched samples have been examined and the neutron conclusions have been largely correlated with concurrent measurements of bulk magnetization, explaining in particular the low values of μ_B, to which the latter measurements invariably lead.

It is shown further that the attainment of perfect Heusler-type ordering, near the composition Au₂MnAl, causes the appearance of a spiral-spin structure for which the spiral axis is parallel to a cube edge. Roughly a half of the material may appear with this new structure, which has a Néel temperature of 64 °K and in which the manganese moment is again close to 4.2 μ_B.

The spatial distribution of the magnetic moment disturbance associated with six non-transition element impurities (Al, Ga, Si, Ge, Sn, Sb) in ferromagnetic iron has been measured using the neutron diffuse-scattering technique. Of these solutes Al produces the least perturbation at neighbouring iron atoms and creates a defect which comes close to being a simple magnetic vacancy. The other five impurities also give rise to effects which approach this description. However, small disturbances extending some ångströms into the iron host are present in all cases. The magnetic moments on iron atoms situated about 4 or 5 Å from the impurity always exceed the moment on pure iron by amounts of the order of a per cent. The magnetic disturbances are observable only by virtue of the large number of iron atoms affected.

Measurements have been made of the susceptibilities of copper-zinc alloy phases containing small amounts of chromium, manganese, iron and cobalt. A Curie-Weiss analysis is possible for some of the alloys and effective moments derived from it are discussed in terms of current theories of such materials; in other alloys unmagnetized configurations are found. In solution in the copper-zinc phase a moment appears on solute iron atoms as the copper content increases from 15 to 19 at. %.

A pronounced low-temperature resistance minimum has been found in dilute solutions of iron in copper-rich alloys of the copper-zinc phase, but in zinc-rich alloys of the same phase the minimum is very much weaker; this quenching of the resistance minimum correlates well with the trend of the iron magnetic moment found by Waszink and Coles. The temperature-independent resistivity contributed by the iron is very large, and almost independent of solvent composition. These results suggest that the iron impurity forms a half-filled virtual bound state which is partly magnetized at the copper-rich end of the phase, but very nearly demagnetized at the zinc-rich end. The variation of resistance at low temperatures is not well described by current theories of magnetic impurities, and in particular no indication can be seen of a critical temperature or of condensation into a non-magnetic ground state.

The Mössbauer effect spectrum of ferrous fluosilicate has been measured between 1.7 and 4.2 °K in external magnetic fields of up to 30 kG for both single-crystal and polycrystalline specimens. With the field H parallel to the trigonal axis z of the crystal (for which the susceptibility is small) only a small hyperfine splitting was observed, but for H along any axis x in the plane perpendicular to z a large effective field was found. The saturation values deduced for the components of the hyperfine field tensor were H_nx⁽⁰⁾ = -248 kG and H_nz⁽⁰⁾ = -550 kG. The spectrum for polycrystalline specimens was almost identical with that for the single crystal with the field perpendicular to z. The data are discussed in terms of the theory of hyperfine fields, and an estimate of the effective value of <r^-3> for the 3d-electron spin is made. The value of the electric quadrupole moment of the excited state of ⁵⁷Fe deduced from the quadrupole splitting is re-assessed at +0.18 bn.

Effective mass theory is reformulated in terms of one-electron Green's functions. This enables a generalization of effective mass theory to be made and leads to an approximate Green function for the periodic lattice. The Green function is evaluated for a two-band model and used to calculate the energy levels of a hydrogenic impurity.

The band structures, density-of-states curves and Fermi surfaces are presented for the hexagonal close-packed configurations of Sc, Ti, Y and Zr, as calculated by the cellular method. A study is made of the effect of the potential field on the band structure and Fermi surfaces of Ti. Heat capacities and bandwidths are discussed.

The electron density in a metal will be disturbed when an atom or molecule is chemisorbed. This effect has been investigated using a simple model which describes chemisorption as a sharing of the electrons of the system between the adsorbate and the metal. The disturbance proves to be long range, oscillatory and generally non-isotropic. Numerical estimates are given for alkali metal atoms on tungsten.

It has recently been predicted that in a suitable semiconductor the electron temperature may be reduced below the lattice temperature by an applied electric field. These calculations have either employed a three-term expansion of the electronic distribution function, or neglected all scattering other than that arising from polar optical modes. In the present paper the electron temperature is calculated as a function of electric field for both polar and non-polar semiconductors, including polar optical, deformation potential optical and acoustic, and intervalley scattering without expansion of the distribution function. Results are presented for n-type GaAs, InSb, Si and GaP. Finally, in discussing the approximations which remain in this theory, particular reference is made to p-type Ge, for which experimental data are available.

The effects of the electron - lattice interaction on the free-carrier magneto-optical properties of semiconductors are treated by developing a model due to Johnson and Larsen based on one-phonon excitation of magnetic Landau states. The main result is to explain a large discontinuous broadening observed by Summers, Harper and Smith in InSb at magnetic fields H given by eH/mc = ω_o, the lattice long-wave longitudinal optic frequency.

The thermal conductivity of the dissociating system N₂O₄ [A: r harp over l] 2NO₂ has been measured at 35 °c and 50 °c over a range of pressures by a precision hot-wire method. The present theory of relaxation effects in the heat conductivity of a chemically reacting gas mixture has been critically tested. It has been found, on varying the gas pressure, that the theoretical estimate of this effect holds only up to a certain upper limit of pressure, beyond which the observed effect becomes insignificant compared with the theoretical predictions. Moreover, the authors' recent suggestion of a probable effect of chemical reaction upon diffusion between the various chemical species has been well confirmed by the experiments.

The electromagnetic emission from plasmas produced by irradiating solid targets with focused light from Q-spoiled ruby lasers is recorded with grazing- and normal-incidence spectrographs as a function of plasma radius and of distance from the target The plasma density and temperature are thereby evaluated. Observed linewidths of the order of 10³ cm^-1, in the grazing-incidence region are ascribed to pressure broadening with corresponding electron densities in excess of 10²⁰ cm^-3. The thermodynamic state of the plasma is considered

A brief discussion is given of the effects of varying laser power and target preparation on the degree of ionization of the laser-produced plasma, and hence on the extension of its use as a spectroscopic source for laboratory astrophysics.

A treatment of pressure broadening and shifts caused by a combination of resonance and higher-order interactions is developed on the basis of the phase-shift theory of impact broadening and applied to the recent measurements in the rare gases. Oscillator strengths of the resonance transitions from the levels ¹P₁ and ³P₁ are deduced for neon, argon and krypton. In each gas the ratio of these f values is in agreement with that calculated from the coupling parameters derived from the term and g values. It is predicted that the shift associated with the higher-order interactions is partially suppressed in the presence of resonance interactions and some evidence of this effect is found in the published results for helium.

The L emission spectrum of the element lutetium 71 has been investigated using a 40 cm bent-crystal spectrograph of the transmission type. The existence of the lines λ1193 95 X U. and λ1143 92 X U., corresponding to the non-quadrupole transitions L_IN_I and L_IO_I respectively, and a line λ1138 88 X U., corresponding to the quadrupole transition L_IO_IV.V, has been established for the first time The transtions L_IIIN_IV, L_IIIO_{IV V}, L_IN_IV, L_IN_V, L_IIO_II,III and L_IIIO_II,III reported earlier only by Allais have been confirmed. The lines λ1772.3 X.U and λ1530.2 X U., corresponding to the transitions L_IIIM_II and L_IIM_II respectively, could not be confirmed

Absorption spectra of solidified films of In-doped Ar, Kr and Xe have been obtained in the quartz ultra-violet and Schumann regions. The purpose of the experiments was to determine the effect of a matrix environment on the 5s5p²²S_1/2 and 5s5p²²P_1/2,3/2 autoionizing levels of In. Apart from a shift in energy and crystal-field splitting of these levels, the spectrum of In in the three matrices correlates well with that of In in the gas phase, particularly for In atoms trapped in Ar and Kr.

Infra-red spectra have been obtained for several related molybdenum co-ordination compounds [(C₆H₅)₄As]⁺[MoOBr₄]^-, [(C₆H₅)₄As]⁺[MoOCl₄]^-, [(C₂H₅)₄N]⁺[MoOCl₄]^- and [(C₄H₉)₄N]⁺[MoOCl₄]^- A band observed at about 1000 cm^-1 in all these compounds is assigned to the molybdenum-oxygen stretching frequency The precise frequency of this band depends primarily upon whether water is present in the anion and to a lesser extent on the associated cation

The problem of neutron scattering by an Ising paramagnet at infinite temperature is shown to give rise to a solvable one-dimensional random walk problem.

Calculations have been performed to examine the effects of back-coupling and distortion in the asymmetric reaction He²⁺ + H(1s) -> He⁺ (1s) + H⁺. It is shown that back-coupling is not important at α-particle energies above 100 kev and that the approximation to the distortion used by McCarroll and McElroy leads to incorrect cross sections.

The values of the electric field gradient have been calculated for the deuteron sites of the molecular systems BD₄^-, CD₄, ND₄⁺, AlD₄^-, SiD₄, PD₄⁺, DF, DCl, D₂O, D₂S, ND₃ and PD₃, using the one-centre expansion wave functions of Albasiny and Cooper and of Moccia. Reasonably good agreement with experiment is obtained.

A recurrence formula is derived which gives the 2^l-pole oscillator strength sums for atomic hydrogen for negative integral powers of the energy This is accomplished by solving a differential equation formulated by Dalgarno and Lewis.

In the letter it is pointed out that the diffuse part of the inelastic onemagnon cross section for scattering from a Heisenberg ferromagnet with a magnetic impurity has an identical form for all three cubic hosts. In a previous paper the author calculated the diffuse part of the cross section with explicit reference to a simple cubic host because the location of the impurity energy modes had only been calculated for this system. Recently these calculations have been extended to body-centred and face-centred cubic structures which most of the known ferro-magnets possess.