Table of contents

Little attention appears to have been paid to the effect of relativity on atomic wave functions since White studied the matter in 1931. We present graphs and tables to show the order of magnitude of corrections to the hydrogenic charge distributions for Z = 80, and study the systematic correction to various hydrogenic expectation values as a function of atomic number. We discuss the use of our results to predict relativistic expectation values in complex atoms in the light of recent work by Garstang and Mayers.

Simple rigorous upper bounds are obtained for the transition probability from the initial state of a two-state quantal system with a time-dependent Hamiltonian. The bounds are compared numerically with the exact transition probability for some special cases.

The total integrated angular change of state of the system is defined, and is used to obtain the inequalities.

The Born and closure approximations are used to obtain an expression (in the form of a series) for the dissociation cross section σ₁ for the process H₂⁺ + He → H + H⁺ + He. Taking the first term only of the series, and assuming the LCAO approximation for the H₂⁺ wave function, that overlap terms can be neglected, and that the hydrogen and helium atoms are in their ground state after the collision, the high-energy limit for the dissociation cross section is found to be 1 59 × 10^-14/E cm², where E is the energy of the incident H₂⁺ in keV, as compared with the experimental values 1 25 × 10^-14/E cm², or 1 10 × 10^-14/E cm². The effect of initial vibrational excitation of the H₂⁺ is also examined, and found to increase σ₁ by up to 20%, comparable with the experimental results of Barnett and Ray, and McClure for the dissociation of H₂⁺ in hydrogen gas.

A simple formula, based on the classical impulse approximation, is given for the cross section for electron detachment collisions between fast H^- ions and neutral atoms or molecules The utility of the formula is demonstrated by comparisons of calculated and measured cross sections.

The method of polarized orbitals is used to obtain a wave function for an electron in the field of a Li²⁺ ion. For low electron energies the polarizedorbital phase shifts are in reasonable agreement with extrapolated experimental quantum defects. The photoionization cross section for Li⁺ is calculated in the dipole length and velocity formulations using the polarized-orbital wave function for the continuum state and six-parameter and twenty-parameter Hylleraas-type wave functions for the ground state. Oscillator strength sums suggest that the calculated cross sections are accurate to about 3 5%. A comparison of the present calculations with experimental data on metallic lithium show that absorption coefficients of Li⁺ and metallic lithium are not equal.

Beams of H⁺, He⁺, Ne⁺, Ar⁺ and Kr⁺ ions at energies between 60 keV and 250 keV were passed through various thin gas targets where fast neutral atoms were formed by electron capture. The small-angle scattering associated with such collisions was measured by scanning a particle detector across a diameter of the emergent beam. Differential electron capture scattering cross sections were measured in H₂, He, Ne, Ar and Kr gases.

Studies of the charge neutralization of 60-450 keV Ne⁺, Ar⁺ and Kr⁺ ions during passage through both `thin' and `thick' targets of hydrogen and the rare gases have allowed the total charge transfer cross section Q and the equilibrium neutral fraction F_{0 ∞} to be determined In three cases, curves showing the variation of Q with impact energy show unexpected secondary maxima which are probably due to a process of charge transfer with simultaneous multiple ionization of the target.

The differential cross sections of most of the energetically allowed reactions of deuterons on ¹²C are calculated using the optical model and the Hauser-Feshbach and distorted-wave theories Comparison is made with the experimental data.

A method by Post, which has been used to calculate lower bounds on the energies of the lowest spatially symmetric or antisymmetric states of many-identical-particle systems, is shown to apply quite generally to the ground states of systems of bosons or fermions for any pair interactions whatever. In particular, the N-nucleon system is considered. No ordinary central pair potential was found to give the empirical binding energies of ²H, ³H and ⁴He. For the exponential and square-well potentials the calculated bounds on the energies are closer than 1% over wide ranges of the potential parameters. The method justifies and generalizes certain `equivalent two-body' calculations originated by Wigner.

Some calculations have been made of the acoustic impedance of liquid ³He, in order to analyse the experimental results of Keen, Matthews and Wilks and of Betts, Keen and Wilks According to the theory of a Fermi liquid, due to Landau, the propagation of sound in liquid ³He should show an anomaly when the period of oscillation is made less than the relaxation time of the liquid. Landau predicts that a new type of wave motion, called zero sound, should propagate in this case The calculations reported here show that there is good agreement between the measured value of the acoustic impedance and that expected for a Fermi liquid supporting zero sound It is concluded that the experimental results provide strong evidence for the existence of zero sound. The agreement between theory and experiment, while close, is not perfect, so we have examined the approximations made in calculating the acoustic impedance in the hope of obtaining further information. It seems that the sequence of parameters describing the interaction of quasi-particles, F₀, F₁, F₂,., which is usually carried only as far as F₁, should be extended to include F₂ as well Rough estimates of F₂ are made by two different methods, the agreement is not very close, but is considered satisfactory

Natural and magneto-optical rotations in an ideal diamagnetic gas have been related by means of statistical mechanics to various electric and magnetic polarizability parameters of an individual molecule By admitting electric and magnetic dipoles of second and third order in H, the external magnetic field, and by taking into account the orientation of molecules by the external field, higher-order field effects have been calculated The composite natural and induced rotation θ_T per unit path length may be expanded as a power series in H

θ_T = (ω/2c)(2πN/v)(A + BH + CH² + DH³ + ...)

where A, B, etc., are functions of molecular polarizabilities Calculation of A and B yields the familiar expressions for natural rotation and the Verdet constant. C, like A, exists only for optically active molecules, so that the term in D is required to describe departure from the Verdet constant of other molecules at high fields C and D each contain a temperature-dependent term which describes orientation effects due to the field, as well as a temperature-independent term which arises from the modification of molecular parameters by the external field. For spherically symmetric molecules all temperature-dependent terms are zero. The temperature-independent terms cannot be evaluated with the data at present available, and tentative order-of-magnitude calculations indicate that for non-spherical molecules the modification of the Verdet constant due to orientation effects is of the order of 1 part in 10⁶ at fields of 10⁵ gauss

Microwave conduction in n-type germanium in the presence of a high steady electric field is studied The carrier concentration is assumed to be high enough to ensure that intercarrier collisions enforce a Maxwellian distribution on the carriers and the momentum loss by the carriers occurs mainly through scattering due to lattice vibrations. The effect of anisotropic effective mass on the microwave conduction characteristics is discussed, and the anisotropy is found to be small at room temperature Numerical values of microwave conductivity and the change in dielectric constant are found to agree with the experimental results of a 4 68 Ω cm n-type germanium sample, and the best fit with the experimental characteristics is obtained for a value of the optical deformation potential constant D₀ between 0 7 to 0 8 × 10⁹ ev cm^-1.

In this paper the authors have investigated the generation of the third harmonic by microwaves, in a homogeneous semiconductor at low temperatures. The effect of ionized impurity scattering has been taken into account and it has been shown that, at low temperatures, the effect of the optical mode of scattering on the amplitude of the third harmonic is small. An approximate solution for the amplitude of the third harmonic part in the reflected and transmitted components of a large amplitude electromagnetic wave, incident normally on a slab of semiconducting material, is obtained, and numerical results have been presented for a few typical parameters It has been shown that a third harmonic of about 12% may be generated in the reflected wave when an incident electromagnetic wave of amplitude 1 e s u. cm^-1 is allowed to fall on a typical germanium sample at 77 °K

The possible existence of a resistance minimum in dilute alloys with transition element impurities which do not exhibit localized magnetic moments is examined. The interaction mechanism consists of a combination of potential scattering of conduction electrons by impurity ion cores and exchange interactions of the conduction electrons with the electrons of the `impurity atoms' In the formulation the role of both empty and occupied impurity states is taken into account.

The resulting third-order contribution to the resistivity from the present interaction, π_s, has the form - cπ_s⁽⁰⁾T, where c is the impurity concentration, π_s⁽⁰⁾ a positive constant and T the temperature. The magnitude of the coefficient π_s⁽⁰⁾ is found to be of the right order.

In combination with the ideal and residual resistances, the above expression gives a resistance minimum at a certain value of temperature. A plot of the theoretical dependence of resistance on temperature shows a minimum around 20 °K in T₁ containing transition element impurities This is in agreement with available experimental results

The surface stresses used by Gersdorf to calculate the non-uniform form effect in magnetostriction are shown to be present by a method that does not rely on a specific model of electromagnetic conditions at the surface of the body. It is argued that it is probable that this is the only contribution that need be taken into account

The system described by the Hamiltonian shown is investigated on the basis of exact high-temperature series expansions and the Green function technique. Expansions for the zero-field (B = 0) free energy and susceptibility (valid for arbitrary lattice structure) are developed to fifth order in reciprocal temperature for the S = ½ system with nearest-neighbour forces. The critical behaviour is discussed in two and three dimensions using the random phase approximation and Padé approximants. In particular the variation of the critical temperature with η is accurately determined for common two- and three-dimensional lattices. It is conjectured that the susceptibility index γ remains constant for 0 ⩽ η < 1 but changes discontinuously at η = 1.

It will be shown that the decoupling procedure commonly used in the theory of Green's functions is equivalent to a simple approximation to the self-energy whenever a diagrammatic analysis is possible. This equivalence will be shown, using as an example the theory of spin waves in ferromagnetism.

Expressions are derived from which the local mode frequencies of a cluster (of any length) of isotopic impurities embedded in a monatomic chain may be calculated. A simple general formula is presented for the mass ratios at which local mode frequencies emerge from the continuum.

Studies of the energy transfer process at room temperature in the doubly doped system Cr, Nd Yt₃Al₅O₁₂ are reported The efficiency of the transfer process was measured by comparing the fluorescent intensities of Cr³⁺ and Nd³⁺ ions in crystals with a variety of doping levels and also by studying the time decay of the Cr³⁺ emission. From a comparison of the results it appears that transfer to Nd³⁺ ions occurs predominantly from the ²E Cr³⁺ state. The overall enhancement of Nd³⁺ emission due to transfer from Cr³⁺ ions was also measured when exciting the ions with unfiltered light from a tungsten-iodine lamp.

The nuclear magnetic resonance spectrum of ¹¹⁵In was recorded at room temperature in an InBi intermetallic compound From the observed spectrum the isotropic Knight shift (+0 18%), the axial component of the Knight shift (+0 02%) and the quadrupole coupling constant (6 6 Mc/s) at the site of the ¹¹⁵In nucleus have been measured. With the assumption of an ionic model, the quadrupole coupling constant at the site of the ¹¹⁵In nucleus in the compound has been calculated and compared with the observed one.

Both the change in Knight shift with concentration and the quadrupole broadening of the magnetic resonance of the ⁷L₁ nucleus in a series of dilute lithium alloys has been investigated. An attempt is made to correlate both sets of experimental data in the terms of Friedel's model of the scattering of electrons at the Fermi surface. It is shown that a set of phase shifts chosen empirically to give the experimental Knight shift also accounts satisfactorily for the residual resistivity and the quadrupolar broadened line shapes.

The electron spin resonance spectra of magnesium oxide show that Cr³⁺ and Mn⁴⁺ ions occupy substitutional sites in the lattice. Their symmetry will be either octahedral or lower depending on whether the necessary chargecompensating defect is remote from or near to these ions. A new spectrum due to Cr³⁺ in sites with tetragonal symmetry has the following spin-Hamiltonian parameters g = 1 9782 ± 0 0003 and D = 798 ± 3 × 10^-4 cm^-1. This is attributed to a charge neutral configuration in which the nearest- and next-nearest-neighbour cation sites along a [100] axis are respectively vacant and occupied by an Al³⁺ ion. In powder samples Mn⁴⁺ may be found in both octahedral and axially symmetric sites The spin-Hamiltonian parameters are respectively g = 1 9938 ± 0 0003 and g = 1 995 ± 0 001, D = 280 ± 20 × 10^-4 cm^-1. In both cases A⁵⁵ = 70 8 ± 0 5 × 10^-4 cm^-1. A sharp fluorescence line at 15 279 cm^-1 with associated vibronic side bands has been identified with the Mn⁴⁺ ion in sites with octahedral symmetry.

A model of a body-centred cubic lattice is considered, in which the polarization of phonons is taken into account in the description of scattering of phonons from a substitutional impurity involving changes in mass and force constants. In the limit of long waves, the total cross section is found to correspond to a Rayleigh type of scattering with resonance possibilities. It is shown that the cross section depends upon the direction of incidence and the state of polarization of incident phonons. This behaviour persists even in the limit of long waves and is manifested through the specific partial wave contributions to the total scattering cross section. The dispersion law for this model, which does not go over to the Debye model in the long wavelength limit, is held responsible for this behaviour. The case of an isotopic impurity is treated in detail and a directional asymmetry is predicted for the scattering process. The scalar model studied earlier is shown to be a special case of the model under consideration, with a specific choice for the force constants. The relevant Green functions are discussed in an appendix.

A study of high-dispersion emission spectra of NH and ND in the region 2460-2560 Å yields the following rotational constants for the d ¹Σ⁺-c¹Π band system B_e' = 14 409(NH), 7 693(ND), α_e' = 0 634(NH), 0 263(ND), D₀' = 1 666 × 10^-3(NH), 4 75 × 10^-4(ND), B_e" = 14 697(NH), 7 841(ND), α_e" = 1 015(NH), 0 393(ND), D₀" = 2 19 × 10^-3(NH), 6 18 × 10^-4(ND), all in cm^-1. These constants are consistent with the isotope effect. Approximate vibrational constants are also calculated.

The Franck-Condon factors q_{v' v"} and r centroids r_{v' v"} have been reported for the C ³Δ → X ³Δ band system of the astrophysically important zirconium oxide molecule. The r centroids have been found to increase with an increase in wavelength The values of Δr ≡ r_{v' + 1.v" + 1} - r_{v' v"} are found to be constant for a given sequence

Franck-Condon factors for v" = 0, 1, 2 and v' ⩽ 11 in the O₂ Herzberg I system have been calculated for realistic Klein-Dunham potentials and accurate wave functions. Corresponding Franck-Condon densities have also been determined in the photodissociation continua associated with the three progressions of absorption bands Absorption cross sections are adequately known only for part of the continuum adjoining the v" = 0 progression. Combination of such measurements with the densities indicates that the electronic transition moment increases slowly with ν. The total oscillator strength in this continuum is found to be near 2 × 10^-7.

Bands observed in absorption at about 1000 °C in the region 5500 to 8000 Å are ascribed to transitions A ²II_1/2-X ²II_1/2 in the molecules BiS and BiSe. A rotational analysis has been made of four bands of BiS. Constants derived for the ground states ²II_1/2 (or Ω = ½ in case c) are as follows BiS: ω_e = 408.7 cm^-1, χ_eω_e = 1.46 cm^-1, B_e = 0.1130 cm^-1, r_c = 2.319 Å. BiSe: ΔG_1/2 = 264.8 cm^-1. As in the corresponding system of BiO, the rotational lines appear to be unresolved blends of nuclear magnetic hyperfine structure components with an overall width in the range 30½ ⩽ J ⩽ 110½ of about 0.45 cm^-1.

The angular distribution for the ¹⁰B(³He, ³He)¹⁰B reaction has been measured at 9 8 MeV and optical model parameters have been determined.

Matrix Hartree-Fock wave functions are calculated for the low-lying configurations of the BeS molecule The energy of each state is computed variationally and estimates made of differences in correlation energy between the various levels The results indicate a ¹Σ⁺ ground state

The process of one-electron loss during the passage of fast helium atoms through gaseous targets has been investigated. Apparent cross sections σ₀₁ are shown to be dependent on the conditions of formation of the fast helium atoms.

The procedure of Edwards and Sherrington for expanding the zero-temperature Green functions of quantum many-body theory in terms of correlations about maximal quantum randomness is extended to cover themodynamic Green functions

Correction to Proc. Phys. Soc.88, 65