Table of contents

The entropy of very long flexible molecules in the presence of topological constraints is studied, and a formula deduced which needs the probability that a random walk will have a particular topological specification. Examples are solved, including a plane random walk sweeping out a given angle around a point in the plane which is generalized to three dimensions including the passage of a random walk past many lines in space, and the probability that a random walk will penetrate through or become multiply entangled with a closed ring.

The calculation of the retarded fourth-order resonance interaction between identical molecules with one shared quantum of excitation energy is described. The interaction is calculated in the electric dipole approximation for molecular transitions with the help of time-ordered graphs and a reduced Hamiltonian which treats molecule-photon and molecule-molecule interactions on the same footing. Interactions involving spins, valence forces and permanent electric or magnetic multipoles are not considered. In the near and wave zones the leading terms of the retarded interaction are proportional to R^-6 and R^-2 respectively, where R is the intermolecular separation.

This paper provides a prescription for solving propagation and radiation problems in linear homogeneous dispersive anisotropic media in uniform motion. The prescription stems from the central idea that the influence of a material medium on the propagation of electromagnetic waves through it can be described by a single constitutive equation. This equation defines a conductivity tensor that includes both the electric and magnetic structure of the medium. By using the equations which relate plane-wave fields and currents in one inertial frame, denoted by S', to those in another, denoted by S, where S' moves with uniform velocity relative to S, a transformation law which establishes the connection between the conductivity tensors in S and S' is derived. As an illustration of the application of the method to anisotropic media the dispersion equation is derived for interstreaming plasmas in the presence of a uniform magnetic field.

This paper describes the solutions to some problems of propagation and radiation in moving media. For example, the propagation characteristics of plane waves in a dispersionless uniaxial medium in motion are analysed. The radiation emitted by an electric dipole in such a medium in motion is calculated when the dipole axis points either parallel or perpendicular to the axis of translation, which is taken to be parallel to the optic axis. It is shown that the effect of the motion is to enhance the radiation intensity in all directions. Similar problems for an isotropic warm plasma are investigated. The radiation intensity in the longitudinal wave is always increased by the motion of the plasma. On the other hand, the radiation intensity in the transverse wave can either be decreased or increased by the plasma's motion, depending on whether the dipole axis is parallel or perpendicular to the velocity of the plasma.

The characteristic energy loss spectra of 200 ev electrons reflected from freshly evaporated Al, Ge, Cu and Au have been obtained. The spectra are found to be dependent on the angle through which the primary electrons are scattered. Energy losses which are due to the excitation of volume plasma oscillations are found to decrease in intensity with decreasing scattering angle while those due to excitation of surface plasma oscillations are found to increase in intensity, thus by obtaining the energy loss spectrum of a given material at two scattering angles (90° and 20° in the present experiments) it is possible to identify those energy losses which result from excitation of plasma oscillations. The plasma losses of Al and Ge were identified in this manner. Copper and gold have more complex spectra than Al and Ge. Only the 3 6 ev energy loss of Cu showed a definite intensity variation with scattering angle and is interpreted as the surface plasma energy loss.

A modification due to Rudge of an approximation suggested by Ochkur is used to calculate the following excitation cross sections in helium. 1 ¹S -> 2 ³S, 2 ³P, 3 ³S, 3 ³P, 3 ³D and 2 ¹S -> 2 ³S. Various atomic wave functions are used and a comparison of the results is made with experimental data.

The general dependence of the energy levels belonging to the discrete spectrum of a multi-particle Hamiltonian on the particle mass is derived. The implications of this result in molecular systems are discussed.

The purpose of this paper is to calculate the projection operators for different helicity states of spin 3/2 particles.

The moment of inertia of the ²⁸Si nucleus is calculated using the formalism developed by Thouless and by Peierls and Thouless. The approximate Hartree-Fock single-particle wave functions are chosen to be the harmonic oscillator wave functions with only two parameters, which were determined by minimizing the ground-state expectation value of the Hamiltonian, and the two-body interaction is taken to be the Rosenfeld mixture with Gaussian form. A check on the self-consistency of the problem is made by calculating the mass of ²⁸Si. The values of the moment of inertia obtained are rather less than the rigid-body value.

The beta decays of ²²Mg, ³⁰S and ³⁴Ar have been investigated with the following results.

For ²²Mg, τ_1/2 = 3 99 ± 0 12 sec. The decay proceeds in (59 ± 2)% of the cases to the 656 kev 0⁺ level in ²²Na, and in (41 ± 2)% of the cases to the 583 kev 1⁺ level in ²²Na. The end point of the superallowed beta particles was determined using the ²⁰Ne(³He, n)²²Mg reaction to be 3 146 ± 0 035 MeV.

For ³⁰S, τ_1/2 = 1 18 ± 0 04 sec. The decay proceeds in (20 ± 1)% of the cases to the ground state of ³⁰P, and in (80 ± 1)% of the cases to the 676 keV 0⁺ level in ³⁰P. Less than 1 6% of the decays proceed to the 707 keV 1⁺ level in ³⁰P.

For ³⁴Ar, τ_1/2 = 0 85 ± 0 10 sec. There is no decay that proceeds with more than 20% of the main decay which is to the ground state of ³⁴Cl.

fτ values are deduced and discussed.

The cascade relations in the level scheme of ¹³¹Cs were studied with a sum coincidence spectrometer. A number of gamma rays as well as an additional 967 kev transition were observed. The latter suggests a spin of 5/2⁺ to the 1046 kev level. The new levels at 697 and 585 kev are well established in the present work.

Measurements have been made of proton polarization, proton relaxation, and the shape, intensity and spin-lattice relaxation rate of an electron spin resonance line from damage centres, in a lanthanum magnesium nitrate target during a bombardment with 146 Mev protons. The damage centres exhibit an inhomogeneously broadened line centred on a g value of 2 002 ± 0.002, and their energy of formation is found to be 1 0 ± 0 4 ev. At 9 5 Gc/s spin-lattice relaxation rates of (7 ± 3) × 10² sec^-1 and (5 ± 2) × 10² sec^-1 were measured at temperatures of 4 2 °K and 1 4 °K respectively. On the assumption that these centres are the sole cause of the observed decay in polarization, it is deduced that about half of the decay may be attributed to relaxation of the protons. The remainder of the decay may be attributed to detuning of protons, providing each damage centre occludes an average of about 700 protons. From the relaxation data, an order-of-magnitude value for the damage-centre spin fluctuation rate of 10⁵ sec^-1 at 35 2 Gc/s is derived. This value is used in a comparison of the observed damage-centre contribution to proton relaxation with that predicted by different theoretical models. The hindered-diffusion model of Khutsishvili, using his definition of a diffusion barrier, fits the observations to within a factor of 2. His value for the diffusion barrier radius is found to be the same as the radius of occlusion derived from the decay in polarization, namely 16 Å. The shell-of-influence model of Schmugge and Jeffries leads to a relaxation rate that is faster than the observed one by a factor of 50. Unhindered diffusion, as analysed by de Gennes, is too efficient by a factor of about 400 to fit the observations.

Nuclear hyperfine interactions in Dy metal have been investigated in a temperature range spanning the ferromagnetic, antiferromagnetic and paramagnetic regions (4 2-300 °K), using the Mossbauer effect in ¹⁶¹Dy. The effect was observed in both the 25 6 kev and 74 5 kev γ-ray transitions. No detectable changes in the hyperfine spectra were observed in traversing the ferromagnetic-antiferromagnetic transition. The temperature variations of the effective magnetic field and the electric field gradient at the Dy nucleus are shown to be in fair agreement with a free-ion theory. The ratios of excited-state moments to ground-state moments for both transitions were also measured. For the 25 6 kev transition, values of -1 21±0 02 and +1 01±0 03 were obtained for the magnetic dipole and electric quadrupole moments respectively, and for the 74 5 kev transition similar ratios of +0 84±0 05 and +0 48±0 04 were obtained. Using independently measured values for the ground-state moments we have thus estimated the excited-state moments and compared them with values predicted by nuclear theory.

The g values and hyperfine structure constants of [Cl₅OMo]^2- have been studied theoretically, and it has been found necessary to include the contribution of excited states that are not usually taken into account The values 2 Å and 2 3 Å have been used for the Mo-O and Mo-Cl distances respectively, and it was possible to fit the measured g values into the formulae derived, on the assumption of reasonable values for the unknown parameters. It was not possible to fit the measured g values into the formulae of previous theories for the vanadyl ion if we assume the same interatomic distances.

From the hyperfine structure constants A and B, two sets of values of <r^-3> have been obtained as a function of the unknown parameters, one set for each sign of B. The set that gives values smaller than those predicted by the formulae of Elliot and Stevens was chosen in that set the value of <r^-3> is between 3a₀^-3 and 4a₀^-3 when the unknown parameters are within the expected range of variation

The fractional magnetic field shift of the nuclear magnetic resonance of ¹⁹F in K₂OsF₆ is calculated by means of the strong crystal field theory, intermediate coupling of spin and orbital moments, and the LCAO method of Stevens. An order-of-magnitude agreement with the experimental results of Elwell is found. The value of the nuclear magnetic resonance result in studying the bonding parameters in transition metal complexes having non-magnetic ground states is emphasized. Limitations in the usual estimation of the extent of sσ bonding are pointed out.

Spin-lattice relaxation times, second moments and linewidths have been measured in 2,4-dimethylpentane, 2,2,4- and 2,3,4-trimethylpentane from 10 °K to their melting points. Comparison of experimental results with theoretical second-moment values indicates the presence of some reorientation of methyl groups even at the lowest temperatures. At low temperatures the spin-lattice relaxation times are much longer than is usual in the presence of molecular motion.

The 2,3,4-isomer can exist in two forms below 130 °K. one form has glass-like properties and also shows a time dependence of T₁ at 20 °K.

The fluorescence of Cr³⁺ in GdAlO₃ has been measured as a function of temperature. It is found that the single-ion spectra may be interpreted by including Heisenberg-type magnetic interactions of the Cr³⁺ ion with its neighbouring Gd³⁺ ions and also the magnetic interactions between the Gd³⁺ ions themselves. The effect of these interactions it to split the sharp Cr³⁺ fluorescent lines into four bands, the separations of which change little between 77 °K and 4 2 °K but decrease below the Néel temperature of GdAlO₃. The values of the exchange constants J_{Cr³⁺ - Gd³⁺} and J_{Gd³⁺ - Gd³⁺} are found to be 2 3 cm^-1 and -0 06 cm^-1, respectively.

The functional integral formalism which has been applied to the Ising model of ferromagnetism is used here to treat the interaction of the annealing process and magnetization in a dilute magnetic alloy AB (magnetic component A). Only the possibility of 50 50 component ordering is considered. A variational principle is derived which leads to an upper limit for the free energy of the model. The approximation obtained by minimizing this upper limit would appear to be an extension of the Horwitz-Callen theory.

Order-disorder and magnetic transition curves are presented for three values of the ratio R of the spin-exchange energy to the mixing energy. Component ordering is found to inhibit the magnetic transition in alloys which are rich in component B.

The inelastic neutron cross section is calculated for a Heisenberg ferromagnet containing either a ferromagnetically or an antiferromagnetically coupled impurity. This is achieved by constructing the appropriate total spin operator Green functions from their equations of motion and using a decoupling scheme that renders the analysis equivalent to employing the linear spin-wave approximation It is shown that the cross section as a function of neutron energy change exhibits peaks when the condition for an impurity mode is satisfied. The cross sections contain both diffuse and coherent terms. The former are of direct experimental interest and are discussed in detail for both types of impurity.

The high-temperature expansion for the specific heat of the Ising model of a ferromagnet (face-centred cubic lattice) is examined. It is concluded that the specific heat at constant field, C_H/R, diverges at the Curie point as 1 09 (1 - T_c/T)^-1/8 - 1 24 approximately. The critical energy of the face-centred cubic is estimated to be U_c/U_o = 0 2477 ± 0 0010 and the critical entropy S_c/R = 0 5901 ± 0 0010.

The exact determinantal equation satisfied by the coherent wave vector in a statistically defined medium of spherical scatterers, which was obtained by Lloyd in III of this series of papers, is used to show two things.

First, the determinant can be written as an infinite series similar to that obtained by ordinary resummation methods, except that it does not involve elements of the T matrix off the energy shell.

Secondly, the first two terms of the series are specialized to the case of point scatterers which are uncorrelated except for an infinitesimal sphere of exclusion about each of them, and the results found to disagree with those of other calculations; this discrepancy is clarified.

The formalism is generalized to deal with vector waves, and it is shown that, for electric dipole scattering, the simplest approximations to the functions appearing in the determinant lead to the Lorentz-Lorenz law.

The properties of a `loaded continuum' - an assembly of rigid spheres embedded in an elastic medium of zero density - are investigated. When the spheres are arranged in a perfect lattice the model shows the characteristic dynamical properties of a harmonic solid. Some of the qualitative effects of structural disorder have been investigated by means of this model.

This paper is a qualitative discussion of some features of the electronic structure of metals that are almost independent of the arrangement of the atoms. In any densely packed assembly, solid or liquid, the atomic volume fixes the overlap between neighbouring atomic potentials, and hence determines the approximate position of the muffin-tin zero, _MTZ, the energy at which a classical electron would be free to move from cell to cell. The general electronic structure of the metal depends upon _MTZ and upon the way in which each atomic potential is modified into a muffintin potential.

For example, recent theory has confirmed the argument of Wigner and Seitz that the bottom of the nearly free-electron bands of the pseudopotential formulation must lie a little below _MTZ. This is, therefore, the main parameter determining the distance from the Fermi level to the core states, which cannot be calculated accurately by the pseudopotential method. The transition from tight-bound to nearly free bands as the atoms are brought together, and the position of d bands, conceived as resonances of the muffin-tin potentials, also depend critically on _MTZ.

Electron correlation and exchange make the construction of a self-consistent muffin-tin potential much more difficult. The Friedel sum rule is not necessarily valid for the muffin-tin wells, and the screened pseudopotential formula cannot be justified rigorously. In the spirit of the `neutral pseudo-atom' concept, where each ion is assigned a share of the conduction electron density, various elementary screening models are investigated, but it turns out that pseudopotential `core shifts', etc., are quite sensitive to the choice of approximation. Recent results on the electron density in metals and semiconductors suggest that the starting point for self-consistency calculation might even be to superpose the fields and charge clouds of neutral Hartree atoms.

The electronic energy bands of a two-dimensional liquid in which there are angular correlations between the positions of nearest-neighbour atoms are investigated, using an extension of the Greenian formalism of Phariseau and Ziman and a generalized coherent-wave approximation Near k = 0, and when the number of nearest neighbours is odd, it is found that, in addition to the normal bands in which the wave function has the same sign on each atom, bands occur in which the wave function is essentially of opposite sign on nearest-neighbour atoms. The energy bands in such liquids therefore closely resemble those in the corresponding crystal having two atoms per unit cell In particular, different band edges can be identified with wave functions possessing bonding or antibonding properties.

As the degree of angular correlation is decreased, the imaginary part of the wave vector or energy in these new bands increases and the bands ultimately disappear as the nearest-neighbour environment approaches spherical symmetry The energy states associated with these bands presumably become added to the symmetric-sign bands as states of large wave vector

The extension of these findings to three dimensions is conjectured and the results are applied briefly to a discussion of the energy band structure of amorphous germanium, which is a semiconductor, and of liquid germanium, which is a metal.

The density of electron eigenstates at a given place in a liquid chain is approximated by the density of states in a periodic chain, in which the spacing between potential wells is equal to the local average spacing of the liquid chain This approximation gives good results, at least for δ-function wells and positive electron energies The approximation can be extended to three dimensions.

The problem of continuous bands of Bloch states is rewritten as one of discrete roots of Dyson's equation for a given k vector Equivalent potentials which conserve chosen energy levels are shown to obey a general theorem of which the orthogonalized plane wave pseudopotential theorem is a particular case. Another interesting family is obtained by studying the kernel of the energy shift expansion. Combining both families yields equivalent potentials, from which a criterion is suggested for (1) giving a more technical meaning to the arbitrariness of the orthogonalized plane wave pseudopotentials, and (11) making the best choice The criterion is related to the radius of convergence of the perturbation expansion in terms of the strength of the coupling The conclusion favours the Austin pseudopotential

The band energy of silver has been calculated by the composite wave variational method at the symmetry points Γ X and L of the Brilloun zone, using the quantum defect method to find out the logarithmic derivatives of the radial wave functions inside the atomic cell. The results show that the p states at the zone faces are below the s states, and the band gap obtained at the zone face L is not sufficient for the Fermi surface to make a contact with the zone face L.

The Born-Mayer model with four repulsive parameters has been employed for the study of the pressure transitions in alkali halides. Satisfactory solutions can be obtained only by multiplying the van der Waals term by K, where K > 1 for most alkali halides It has been shown that the ratio of ρ₂/ρ₁ is equal to the structural constant, a = r_ii/r_ij The three- and two-parameter equations are thus special cases of the four-parameter equation and b₂ obtained is proportional to (1 25 r_l² + 0 75 r_j²)/2.

It is shown that the Boltzmann equation treatment of acoustic absorption is valid for arbitrarily large thermal phonon mean free paths, as long as the acoustic wavelength is much longer than the thermal phonon wavelength.

Some force constants for octahedral halide ions of symmetry O_h have been determined from the Raman spectral data. The trends in the force constants and vibrational frequencies have been discussed with reference to the nature of the bonding and the hybridization The X-Y and Y.. Y mean amplitudes have also been calculated.

Kirchhoff's integrals for a slit aperture formed in a non-planar screen are evaluated over the plane aperture and over the extended aperture. These are compared with an interative asymptotic solution of the integral equations, and with experimental measurements for slit widths of 3, 5, 6 and 10 wavelengths. The results indicate that Kirchhoff's approximation for non-planar problems, used over the extended aperture, is poorer than for planar problems. However, the approximation used over the plane aperture gives fairly good agreement with experiment, except for certain screen interaction effects.

A relationship between the axial magnetic field necessary to extinguish oscillation from a Brewster-angled laser and spontaneous intensity is derived. The results are used to measure the Doppler width of the spontaneous light at 6328 Å and, after an adiabatic collision process is allowed for, to find the gas temperature Using the same measurements, the gain of the system is evaluated as a function of input power, as is the ratio of the population of the upper and lower laser levels.

A direct comparison is made between the lower bound formulae of Stevenson and Temple, for the energy eigenvalues of a quantum-mechanical system. It is found that Stevenson's formula is better than that of Temple, which is in contradiction to a statement of Kato.