Table of contents

The Schrodinger equation for a hydrogen atom can be solved in spherical polar, parabolic and spheroidal coordinates. Relations between these types of solution are obtained by an algebraic method depending on the existence of an extra constant of the motion for the two-centre Kepler problem. Limiting conditions and certain orthogonality relations are studied. It is shown that, in Fock's method of stereographic projection of momentum space onto the surface of a four-dimensional hypersphere, one cannot find an orthogonal coordinate system to separate the equations of motion, corresponding to the use of spheroidal coordmates in configuration space. This is in marked contrast with the situation with spherical polar and parabolic coordinates.

The method of polarized orbitals has been employed to obtain a representation of the singlet P-wave continuum functions of the negative hydrogen ion. These functions were combined with twenty- and seventy-parameter bound-state wave functions to calculate the bound-free absorption cross sections of the negative hydrogen ion using the length and velocity alternatives for the dipole operator. It is found that the results of Geltman, Doughty et al. and the present calculations in the length formulation are in close agreement, but that there is a considerable disparity at long wavelengths between these results and the experimental photodetachment results of Smith and Burch and of Branscomb and Smith.

Theoretical oscillator strengths of atomic helium are adjusted to reproduce accurate theoretical oscillator strength sums. The adjusted set of oscillator strengths is then used to calculate those properties of helium which may be written as oscillator strength sums. The results are in good agreement with experimental data and other theoretical calculations.

The classical theory of electron capture is adapted for use with the Hartree-Fock-Slater description of the atom (instead of with the Thomas-Fermi description). Calculations are carried out on electron capture by protons from neon and from argon. Good accord is obtained with the experimental data of Stier and Barnett, and Barnett and Reynolds.

A Feshbach resonance consists of a particle temporarily bound to an excited state of a complex target. For the case of a two-electron system, a perturbation theory is developed for the associated complex resonance energy, as defined by Siegert, by treating the coupling responsible for the emission of a particle as small. The resulting expressions for the width and energy shift agree with Feshbach's results.

For the class of potentials vanishing beyond a certain point we have derived a new dispersion formula in which the scattering matrix S(k) is written as the ratio of two conditionally convergent partial fraction sums. The numerator displays the poles of S(k) at the complex eigenvalues k_n and the denominator displays poles at -k_n, the zeros of S(k). The expression is manifestly unitary and remains so when an approximation depending only on a finite number of poles is made. Besides the S-wave theory, higher-wave scattering is treated, and among particular illustrations is a generalization to higher waves of the effective range approximation.

Differential and total cross section formulae in the proton-atom binary-encounter collision model have been derived in a very simple way by using the momentum transfer as a variable. By simultaneously using the velocity of the incident particle as a variable, we show clearly the analogies and differences of the proton-atom and electron-atom binary-encounter collision theory. Possible applications of the theory are discussed and comparisons with other theoretical models and experimental cross sections are made for the ionization of atomic hydrogen by protons and electrons.

The usual sum over intermediate states occurring in the expression for the second-order correction to the energy of a system of particles, such as a light nucleus, is reduced to solving a few inhomogeneous differential equations. A method of easily handling the Pauli principle which occurs as a projection operator in the inhomogeneous terms is introduced. If oscillator states are used as the unperturbed basis states, all differential equations become essentially one-body equations.

A Green function method has been devised to treat the effect, in a nucleus, of four nucleons coupling to zero momentum (quadrupling). The method permits the computation of the respective contributions by the quadruples and the pairs to the energy gap in nuclear spectra. A rough numerical estimate shows that the quadrupling gap is considerably larger than the pairing one, and is in good agreement with the experimental values for the energy gaps in light nuclei.

Both the Mayer theory of condensation and series expansion developments for the Ising and Heisenberg lattice models make significant use of the concept of multiply connected graphs or `stars'. However, the problem of suitably classifying and enumerating the different types of star which can be formed from p points and l lines is still far from solution. Recently Sykes et al. described how these graphs could be classified into homeomorphic types. In this paper we discuss the enumeration of these homeomorphic types and give short tables of such types containing up to five principal points. We then describe an allied method of counting the number of stars belonging to each homemorphic type, which enables us to produce a detailed list of all stars containing up to 12 points and 12 lines.

The dependence of the mean square end-to-end distance <R_n²> on the number of links n of a polymer chain subject to excluded volume conditions has been studied, using Monte Carlo techniques on a digital computer. The individual molecules of the chain are considered as spheres, and the chain is not constrained by a lattice framework. The results, up to n = 90, have been fitted to the usual formula <R_n²> = an^γ using the method of least squares, for different values of the excluded volume ratio, i.e. the ratio of the minimum permitted centre separation of any two non-consecutively added spheres to the corresponding distance when the addition is consecutive. For excluded volume ratios of 0.1 and 0.2, γ has been evaluated as 1.050 ± 0.021 and 1.065 ± 0.017 respectively while, from two independent investigations of 20-link chains, with excluded volume ratios of 0.5, γ has been determined as 1.161 ± 0.017. These results suggest that the formula <R_n²> = an^γ, where a only is dependent on the magnitude of the excluded volume, may apply only for excluded volume ratios in excess of a certain minimum value. Assuming that the γ value of 6/5 arrived at by Edwards in 1965 is correct, this minimum value has been tentatively assessed as approximately 0.5.

The configurational statistics of a simple macromolecular chain is studied by applying the mathematical methods of Markov chains. The progress of the chain is governed by a transition probability matrix involving two probability parameters p and α. In terms of these parameters a few useful properties of the system such as the degree of helicity and expected square of the end-to-end distance are calculated. The same properties are also calculable when the formalism of the Ising model is applied to the macromolecular chain. This leads to the evaluation of p and α as a function of temperature in terms of the thermodynamic quantities such as heat of fusion and change of entropy. The self-consistency of p and α is checked from the value of the expected square of the end-to-end distance which agrees well with the data of Hoeve and others.

It is shown that the neutron paramagnetic scattering from an Ising system can, unlike the Heisenberg case, be solved exactly at limitingly high temperatures. The energy distribution of the scattering consists of a number of discrete levels, the envelope of which is close to Gaussian in form. The solution is given for a particular model comprising a simple cubic lattice with nearest-neighbour interactions and spin 5/2 ions. A comparison is made between the second and fourth moments of the distribution and the known moments for the Heisenberg system.

A method is described for the calculation of the time dependence of the component of the macroscopic nuclear magnetic moment of a solid in the direction of a steady magnetic field H₀ following the application of a strong, transverse radiofrequency field. Observation of the time variation of the magnetization corresponds to a free induction decay experiment performed in a frame of reference rotating at the frequency of the transverse field, and the decay function is proportional to the Fourier transform of the rotary saturation line shape. A marked lengthening of the decay is expected when the static field effective in the rotating frame is inclined at 54.7° to H₀ and the method is potentially useful as a line-narrowing technique. The theory is applied to the fluorine resonance in calcium fluoride and good agreement is found with the experimental results of Goldburg and Lee. The theory makes use of a unitary transformation method described in an earlier paper. A close correspondence is found with certain of the results already established for the related problem of motional narrowing by rapid rotation.

Ferromagnetic resonance has been observed in a series of NiCu, NiW, NiCo and NiCr alloys. The composition ranges included 0-62 at.% Cu, 0-9 at.% W, 0-50 at.% Co and 9-11 at.% Cr. The investigations have covered the temperature range 10 °K-290 °K at 9 44 Gc/s and 35.3 Gc/s. For alloys with Curie points below room temperature the measurements extend into the paramagnetic phase. The g value for the NiCu and NiW alloys is independent of composition in the ferromagnetic phase and equal to that for pure nickel. For the NiCo alloys the g value varies with composition falling to 2.14 ± 0 03 for the 50 at.% Co alloy. This is explained in terms of a model which assigns fixed moments and g values to the nickel and cobalt lattice sites. The linewidth in some alloys is strongly influenced by crystal anisotropy effects. When these are not important, the linewidth is found to vary with frequency closely in proportion to ν^1/2. Dilute alloys of nickel with copper and tungsten exhibit a maximum in the linewidth in the vicinity of the Curie point for each alloy.

Ferromagnetic resonance has been observed in a series of alloys of CoPt and CoPd together with one FePd alloy. The composition ranges were 74.7-0 5 at.% Co for the CoPd alloys, 3.5-1.11 at.% Co for the CoPt alloys, and 2.2 at.% Fe for the FePd alloy. The investigations covered the temperature range 4 °K-290 °K at 9.44 Gc/s and 35.3 Gc/s. The g values for the alloys vary with composition. An explanation of this behaviour is given using the sublattice magnetic moment data obtained from neutron scattering experiments. The linewidth in the CoPd alloys shows a marked contribution from crystal anisotropy effects. These do not appear to play such an important role in the CoPt and FePd alloys. When crystal anisotropy effects are not important, the linewidth varies with frequency closely in proportion to ν^1/2. The FePd alloy exhibits a maximum in linewidth in the vicinity of the Curie point.

Experiments on an absorber vibrated at ultrasonic frequency are described. The spectrum is used to calibrate the spectrometer for a measurement of the ⁵⁷Fe hyperfine spectrum, and it is found that the hyperfine fields in the bulk iron and in the domain walls are equal to within 0.1%.

The Mossbauer spectra of ⁵⁷Fe, embedded as a dilute impurity in 32 different metallic host lattices, have been studied. Most of the absorption spectra consist of a single line, but in a few cases hyperfine splittings have also been observed. The shifts in the energy of the 14 4 keV gamma ray, when one host element is substituted by another, have been measured and appear to be related to the electronic structure of the host. A comprehensive list of earlier data is also given.

Nuclei of the isotope ⁵⁴Mn have been aligned in iron, cobalt and nickel hosts at temperatures between 0 011 and 0.03 °K. From measurements of the γ-radiation anisotropy, values for the hyperfine magnetic fields acting on manganese nuclei in dilute FeMn, CoMn, and NiMn alloys were obtained. It was shown that the hyperfine field on manganese nuclei in nickel is negative. A discussion of the effects of incomplete solution of the active nuclei in the hosts and of incomplete sample magnetization upon the deduced field values in nuclear orientation experiments is given. The hyperfine field values obtained are compared with those from nuclear magnetic resonance experiments. It is shown that the hyperfine fields on the nuclei of copper, nickel and cobalt atoms in dilute alloys of iron, cobalt and nickel vary systematically with the host and localized magnetic moments whereas there is no such simple behaviour for the fields on iron and manganese nuclei.

Measurements were made of the anisotropy and plane polarization of the 768 kev γ radiation from ⁹⁵Nb nuclei oriented in iron at temperatures from 0.013 to 0.05 °K. A value δ = 0.05 ± 0.01 was obtained for the E2/M1 γ radiation mixing ratio. The magnetic moment of ⁹⁵Nb was found to be 6.3 ± 0.6 n.m. From low-temperature measurements of the anisotropy and circular polarization of the γ radiation from ⁹⁵Nb nuclei in cobalt, the hyperfine field on niobium nuclei in a dilute alloy of niobium in cobalt was found to be H_e = -189 ± 8 kOe.

A theory for the contribution of Sm³⁺ ions to the anisotropy of the iron garnet is developed through several stages of approximation on the assumption that the crystalline Stark and exchange fields at each of the rare-earth sites in the garnet have predominantly, though not perfectly, cubic symmetry. The ground state of each Sm³⁺ ion is first assumed to have had its angular momentum partially and anisotropically quenched by the cubic component of the Stark field. Still in the cubic approximation, the theory is refined to include (i) a partial decoupling of the spin from the orbital angular momentum by the Stark field and (ii) an enhancement dependent on temperature of the average spin angular momentum by the exchange field. The assumption of perfect cubic symmetry is then dropped, and local rhombic distortions of the exchange and Stark fields are introduced and treated perturbatively.

It is found that as long as the rhombic distortions are not too large there is a region in temperature in which the anisotropy varies as the inverse third power of the temperature. It is suggested that this behaviour is realized experimentally to a good approximation in the magnetic torque data obtained between 80 and 300 °K by Pearson.

From Pearson's data, a value βH_ex(T = 0 °K)/k approximately equal to 29 °K is extracted for the isotropic component of the exchange field. This value for the exchange field, when inserted in the perfectly cubic theory, gives approximate agreement with the anisotropy measured at low temperatures in garnets lightly doped with samarium. For the concentrated samarium garnet, however, the perfectly cubic theory gives the wrong direction of easy magnetization and, with this value for the exchange field, a Schottky anomaly at too low a temperature. The discrepancies are removed by inclusion of rhombic distortions. In particular, one choice for these distortions, appropriate for the concentrated garnet, leads correctly to the prediction that the (110) is the easy direction of magnetization, explains satisfactorily the ionic contribution to the specific heat between 5 and 20 °K, and locates correctly a prominent ionic absorption level which has been observed in the far infra-red.

The infra-red Faraday effect has been measured in p-type Si, GaSb and GaAs. These and previous measurements on p-type germanium are interpreted in terms of contributions due to light and heavy holes and intervalence band transitions. The contribution of the free holes to the total rotation is extracted for each material and combined with Hall effect data. These combined data are used to deduce the light and heavy hole masses in GaAs and GaSb. For GaAs the masses are found to be 0 089m₀, 0 51m₀ and for GaSb they are found to be 0 056m₀, 0 33m₀. In Ge and Si, where the masses are already known from cyclotron resonance experiments, the data are used to find the relative populations of the light and heavy hole states, giving 5 8% and about 7% respectively. The observed frequency dependences of the rotation due to intervalence band transitions in GaAs and Ge are compared with those deduced theoretically by summing the effects of the direct Landau transitions.

The response of an anharmonic crystal to low-frequency and long wavelength probes is discussed. The difference between the velocities of sound in the zero sound and first sound regions is emphasized. In the zero sound region, it is shown that measured elastic constants may, unlike the macroscopic elastic constants, not have the symmetry of a second-rank tensor in addition to the point group symmetry of the crystal. The size of the deviations from the symmetry of a second-rank tensor is calculated for alkali halides and shown to be about a per cent at 300 °K. In the first sound region the elastic constants and ultrasonic attenuation are calculated. The difference between these elastic constants and those of the zero sound region is shown to be about 10% for some elastic constants of alkali halides at 300 °K. The extent to which the propagation of first sound is adiabatic is also discussed. The velocity of the second sound mode is calculated as a function of temperature, but calculations suggest it is very heavily damped in alkali halides. Finally, the thermal conductivity of KBr is computed and compared with experiment. The extension of these results to the dielectric response function is also described.

The present paper generalizes a previous result of the author about the impossibility of constructing a physically acceptable solution of the Einstein-Maxwell equations corresponding to an infinite charged cylinder.

The optical properties of evaporated opaque films of indium, gallium and thallium have been studied in vacuum throughout the wavelength range of the visible spectrum, 0.4-0.65 μm, by a photographic method. This permitted the simultaneous measurement of the optical constants to be made immediately after evaporation.

The application of a free-electron model of the metals to the results is discussed, deviations from the predicted relationships being taken to indicate the photon energies at which electron interband transitions take place.

The absorption spectrum of the TeSe molecule has been investigated and bands, all degraded to the violet, have been obtained in the region λλ2520-2140. Most of these bands can be classified into two systems having the following vibrational constants (in cm^-1).

	Te	ωe	ωexe
II	42165	353.0	1.0
I	40680	353.0	1.0
x	0	318.0	1.0

Bands lying in the region λλ2300-2140 are possibly fragments of a third system which has the same lower-state vibrational frequency as the other two systems. The vibrational levels in state II show indications of a probable perturbation and predissociation. All the three systems seem to be due to the transition D ³II <- x ³Σ.

The electron mean free path in metal films and the bulk resistivity of the films are determined from two measurements of resistance and thickness without assuming the resistivity of the pure material of which the films are made. The value of the specular reflection coefficient at the film boundaries is found from measurements on a third film, but is very sensitive to small errors in resistance measurement.

The squares of the overlap integrals |∫ψ_v'ψ_v" dr|² and r centroids r_v',v" of the D ¹Δ → A ¹π system of MgO, which is of astrophysical interest for disk and spot spectra of the Sun, have been evaluated.

It is pointed out that a recent observation by Gough on the polarization of sensitized fluorescence constitutes an experimental demonstration of the transfer of coherence from mercury to cadmium atoms in collisions of the second kind. The implications of this are discussed.

The quartic anharmonic zero-point vibrational energies of the three cubic lattices are calculated using an approximation first introduced by Bethe. The results are in good agreement with those obtained by standard lattice dynamical methods.

Recent calculations of the anharmonic contribution to the Helmholtz energy of a face-centred cubic lattice with arbitrary nearest-neighbour central forces are used to estimate possible anharmonic contributions to the thermodynamic Grüneisen parameters of solid argon, krypton and xenon. For a Lennard-Jones 12-6 potential the anharmonic contribution to the Grüneisen parameter γ is found to be large and such that γ(Xe) > γ(Kr) > γ(Ar). The calculations also suggest that for these solids γ(T) should be quite rapidly decreasing well before the melting point.