Table of contents

An exact statistical-mechanical treatment of a system of monoenergetic particles with mutual interaction is given and shown to lead to a second-order phase transition. For comparison a self-consistent BCS-type treatment is developed and shown to be in substantial agreement. Fluctuations are treated and shown to become large at temperatures approaching the transition from above.

It is shown that an assumed annual cyclical variation of the gravitational constant is sufficient to eliminate the systematic errors which have been noted in the last two redeterminations of the gravitational constant. Such a variation is consistent with a previously proposed theory, is not in contradiction with the latest experimental measurements of the acceleration due to gravity, or with the null Eotvos result, and is thus consistent with the weak equivalence principle.

It is shown that time-dependent Hartree-Fock theory can be used to calculate the interaction potential of atomic systems at large separations. Explicit calculations are performed for a pair of helium atoms with the result that at a separation Ra₀ the interaction potential, in atomic units, is given by

V(R) = -{1.376/R⁶ + 13.18/R⁸ + 169.8/R¹⁰ + O(1/R¹¹)}.

Comparison with other data suggests that the coefficients are too small by about 7% and the interaction potential is accordingly more accurately represented by

V(R) = -{1.47/R⁶ + 14.1/R⁸ + 182/R¹⁰ + O(1/R¹¹)}.

Calculations of the non-additive long-range interaction potential for three helium atoms are also reported.

Close-coupling calculations are carried out of the 0-2 rotational excitation of H₂ by the impact of H, He and H₂ and of the 0-2 rotational excitation of D₂ by H impact. The failure of the distorted-wave approximation at high impact energies is demonstrated. The close-coupling result for He impact at 0 15 eV differs by less than 10% from the result of a semi-classical calculation. Excitation cross sections and angular distributions and elastic cross sections are calculated. For H-H₂ and H₂-H₂, rate coefficients are also given.

The theory of Sandars and Beck has been tested accurately by calculating, using relativistic Hartree-Fock atomic wave functions, the relativistic quadrupole interaction constants B for the ground states of Mn and ¹⁵¹Eu and the relativistic dipole interaction constant A for the ground state of ¹⁵¹Eu. The results agree in sign and order of magnitude with the experimental values obtained by Sandars and Woodgate and Evans et al.

Perturbation theory is used to determine the approximate location of the energies of the 2s2p ^1,3P autoionizing levels of the two-electron systems.

Extensive calculations are being made of the cross sections for electron impact excitation of the metastable levels of the 3p^q (q = 2, 3, 4) configuration. These cross sections are important in the interpretation of gaseous nebulae spectra. Collision strengths are reported for ions in the 3p³ isoelectronic sequence. The calculations are made using an improved distorted wave method recently formulated by Seaton.

Collisions between electrons and oxygen molecules and ions have been studied using a mass spectrometer. The formation and subsequent ionization of O ⁺(⁴S), O ⁺(²D) and O ⁺(²P) to form O^{2 +} has been shown to take place. Appearance potentials for O^{2 +} and O^{3 +} formed by electron collision with oxygen molecules have been measured, as well as the ionization potential of doubly charged oxygen. The A ³Σ_u⁺ state of O₂^{2 +} has been detected. The Wigner spin rule has been used to explain the different charge-exchange rates between the A ³Σ_u⁺ and X ¹Σ_g⁺ states of O₂^{2 +} and oxygen, nitrogen and argon gas.

The modified close-coupling approximation for e-H scattering is proposed, which exactly accounts for dipole and quadrupole polarization without losing the extremal properties for phase.

The equilibrium fractions of He^- produced by He⁺ impact on hydrogen gas in the energy range 80-200 keV have been measured. The effect of hydrogen gas target thickness on the He^- fraction when formed from He⁰ and He⁺ has also been studied at 100 keV. The He^- fraction increases to a maximum value as the target thickness is increased and then falls to the equilibrium value. This effect is more pronounced when He⁰ is the incident particle, and can be explained if the negative ion is formed from the metastable neutral particle.

Recent measurements of the intensity of muons below 3000 m.w.e. (metres water equivalent) underground have been combined to give an estimate of the depth-intensity relation and the angular distribution for atmospheric muons down to a maximum depth of approximately 9000 m.w.e.

It is found that the vertical intensity can be represented by the expression

I(0, h) = 9.8 × 10^-7 exp(-h/λ)cm^-2sec^-1sterad^-1

where h is the depth in m.w.e. and λ = 810 ± 50 m.w.e.

The corresponding angular distribution at vertical depth h is given by

I(θ, h) = I(0, h) secθ exp{-(h/λ)(secθ - 1)}

where θ is the zenith angle. The implication of the above angular distribution for the estimate of neutrino fluxes in deep underground cosmic-ray experiments is discussed.

Inelastic neutron scattering from magnesium oxide at room temperature has been measured and the frequencies of 656 phonons with wave vectors in mirror planes deduced. The results are shown to be in good agreement with dispersion relations calculated on the basis of a shell model, the parameters of which were obtained from independent measurements of the elastic constants, dielectric constants and infra-red dispersion frequency.

The inelastic spectrum of cold neutrons scattered incoherently from condensed hydrogen shows a quasi-elastic peak and an ortho-para conversion peak. A theory is developed to calculate the broadening of these peaks due to the hindering of the rotation of ortho-hydrogen molecules by anisotropic inter-molecular interactions. The quadrupole-quadrupole, van der Waals and valence interactions are taken between pairs of molecules and contributions to the broadening up to the first power of the ortho concentration c are discussed. The quadrupole-quadrupole term is seen to be dominant, broadening the ortho-para peak in agreement with experiment but contributing a width to the quasi-elastic peak which is less than the resolution broadening. The theory is extended to the analogous incoherent scattering peaks in deuterium with similar results.

Molecular motions in liquid and solid hydrogen and deuterium have been investigated by the inelastic scattering of cold neutrons. In all cases discrete peaks are observed corresponding to δJ = 0 and J = 1 → 0 rotational transitions. In solid hydrogen there is no broadening of the elastic peak (J = 1 → 1), but the inelastic peak (J = 1 → 0) is appreciably broader than the experimental resolution. The peak widths are in fair agreement with the theoretical predictions of Elliott and Hartmann, confirming that quadrupole-quadrupole interactions are dominant. In liquid hydrogen both elastic and inelastic peaks are broadened by diffusive motion of the molecules, consistent with macroscopic measurements of the diffusion coefficient. In both solid and liquid hydrogen, scattering by intermolecular vibrations is observed; the liquid hydrogen results are fitted approximately by a simple model containing one arbitrary parameter. Measurements on solid and liquid deuterium show the same qualitative features of rotational and diffusive motions, but are complicated by the appearance of peaks due to coherent scattering.

The specific heats of the alloys MnNi and MnNi₃ have been measured between 0 3 and 4 °K. Large nuclear specific heat contributions were observed, corresponding to magnetic hyperfine interactions at the ⁵⁵Mn nuclei. The effective hyperfine field for Mn in MnNi was found to be 235 kOe. For MnNi₃ three values 259, 318 and 335 kOe were found, corresponding to increasing atomic and magnetic order in the alloy. A correlation for the Mn-N1 alloy system between the hyperfine field and the magnitudes and directions of the magnetic moments on the nearest-neighbour atoms is discussed. Values for the electronic specific heat coefficient γ were also obtained. For MnNi, γ = 1 24 mJ degK^-2 mole^-1 while, for MnNi₃, γ was found to decrease upon ordering from 8 85 to 3 95 mJ degK^-2 mole^-1. This is in agreement with the results of other workers.

Electron spin resonance at X band has been observed in Cu 4 at.% Mn and Cu 15 at.% Mn alloys throughout the range 2-295 °K. For specimens cooled in zero magnetic field an essentially antiferromagnetic-like behaviour is observed but this is greatly modified if the samples are cooled in a field of 5 kOe. The general features of the results are in agreement with the ferromagnetic/antiferromagnetic model proposed by Kouvel to explain the observations on the static magnetic susceptibility, although the present work points to the Néel temperatures being very much lower than those originally assessed.

The oscillations of a magnetized plasma are studied for arbitrary initial distributions of the electrons and ions. A dispersion equation is derived and it is found that the effect of an anisotropy is to add first-, third- and fifth-order terms to the usual fourth-order equation for an isotropic distribution. The dispersion equation for longitudinal oscillations is found to have the same form for all distributions, but the equation for transverse oscillations contains an additional term due to the anisotropic part

Longitudinal electron oscillations are studied for an arbitrary distribution along the magnetic field and a Maxwellian distribution in the transverse plane. The complex frequencies are found in the long- and short-wavelength regions. To first order the real part of the frequency γ for long wavelengths is independent of the magnetic field but is a functional of the distribution function. For short wavelengths γ is independent of the distribution but exhibits logarithmic oscillations characteristic of the field.

The dielectric constant of polar liquids is calculated by assuming a model of the molecule which describes its interaction with the electric field and by introducing means of calculating certain averages demanded by the statistical mechanical theory.

In this paper we make a calculation for the time-dependent case which parallels the approach of Onsager. The model is introduced at the beginning, and the average torque on a molecule due to the local field is found by solving quasistatically the problem of a polarizable dipole performing small librations in a cavity in a continuum having a complex dielectric constant. The result of the calculation predicts a non-exponential decay of the macroscopic polarization.

The results are compared with the treatment of similar type by Scaife, and with treatments by means of the Kubo theory of non-equilibrium statistical mechanics. These latter allow the introduction of the model at a late stage in the calculation, as in the theories of Kirkwood, Frohlich and Buckingham on the static case.

Such a treatment, as carried out by Glarum, and a modification of it by the authors are considered. It is found that the results of the present paper show a certain agreement with the results of Scaife and with the authors' modification of the treatment of Glarum.

Two types of methods have been previously used to relate the macroscopic decay of polarization in a dielectric specimen to a microscopic decay function defined in terms of molecular motions. The first is based on Kubo's theory of non-equilibrium statistical mechanics combined with Kirkwood's theory. The second is based on the use of Debye's rotational diffusion equation.

Glarum has made a calculation of the first type, which predicts an exponential macroscopic decay in the case of a liquid assumed to have particularly simple properties. Scaife and the authors have made a calculation of the second type, which for the same liquid agree with each other in predicting a non-exponential macroscopic decay, but disagree with Glarum's calculations.

In this paper it is shown that, if Glarum's calculation is modified at one point, the results then agree with the calculations of the second type made by Scaife and the authors. The modification consists in the use, at a certain point, of a macroscopic decay function which we feel to be more justified physically than that used by Glarum.

If the chemisorption of an atom (or molecule) on a metal involves a sharing of the conduction electrons with the adsorbed atom it follows that, because two chemisorbed atoms share the same electrons, there will be an interaction between them. This interaction, which is an indirect one, can arise because a discrete atomic level which lies in the conduction band of the metal is broadened and becomes a virtual level when the atom is chemisorbed. The two virtual levels going with two chemisorbed atoms only coincide when the atoms are infinitely far apart, otherwise the virtual levels are split. If the Fermi level falls near the original virtual level, the occupancy of the two virtual levels will be sensitive to this splitting, and because of this there will be a significant interaction between the chemisorbed atoms. This is likely to occur with the alkali metal atoms on tungsten and nickel, and with hydrogen, but not for many molecules. An investigation using a simple model shows that the indirect interaction is long range (inverse square law) and oscillatory.

An effect is described, to be called the band shift effect, which arises from the interplay of strong correlation theory and the first-order energy shift of a band calculated by means of tight-binding theory. This leads to a spin dependence of the energy which, exactly in the half-filled band case and approximately otherwise, leads to a Heisenberg exchange Hamiltonian. The coupling constant from this effect is calculated for nickel and found to be of the order of magnitude observed empirically. In the presence of this effect, it is found necessary to use a condition for ferromagnetism based on the free energy of the system rather than on the occupation numbers for different electron spins The half-filled band case usually leads (in the limit of infinitely strong correlation) to ferromagnetism, without regard to the shape of the band.

The band shift effect is discussed from the point of view of alloy theory. The case of an alloy in which the bandwidth is small compared with differences in binding energy is closely similar to the strong correlation limit. The alloy is treated by t-matrix theory and it is shown how previous work may be improved to give the correct behaviour for the band shift effect This requires first a description of tight-binding theory as derived from the t-matrix formalism and then the extension of the treatment to include correctly all double-hopping terms in the summation.

The correlation function expression for the shear viscosity in an insulator is evaluated by double-time Green functions. It is shown that the relaxation-time expression recently obtained by Rice can be recovered by using the Hartree-Fock approximation for the correlation function.

The bulk viscosity and sound absorption in critical mixtures are studied theoretically using the general theory of anomalous transport coefficients developed in an earlier paper. The anomalous increase in the bulk viscosity near the critical point is found to be much greater than that of shear viscosity. As a natural extension of this result the anomalously large sound absorption in critical mixtures has been obtained by considering the frequency-dependent bulk viscosity. The result is found to be practically identical with the one which has been obtained by Fixman in an entirely different way and has been thought to be additional to that due to viscosities. An intuitive way of looking at our general theory is also discussed.

A two-band model of the temperature variation of the Hall effect in palladium and its alloys is proposed. The model may also be applied to metals and alloys in which the conduction does not necessarily take place in both an s band and a d band. It predicts a strong dependence of the Hall effect on impurities, the addition of impurity to a pure metal causing in general an extra Hall effect ΔR₀ (=R₀ (alloy) -R₀ (pure metal)) which varies with temperature T as ΔR₀ = D/(T + θ₁) over a wide temperature range. Experimental results on palladium and dilute iron-palladium alloys are reported, and are interpreted semi-quantitatively on the basis of the model.

The density of states, the one-electron propagator and the Fermi level were computed for an electron in a random array of impurities. The Monte Carlo method was used together with the tight-binding approximation. The density of states is very narrow at low concentrations, and it widens as the concentration increases. The Fermi level reaches the edge of the conduction band for c = 6 × 10¹⁷ cm^-3. The propagator is strongly damped at low concentrations, but it becomes oscillatory and much less localized as concentration increases. The results are essentially size independent and they show that the method is practically equivalent to the perturbative approach at high concentrations, but its validity extends beyond that, and it gives the correct result at low concentration as well.

The effect of conduction electron screening on the relevant electron-phonon interactions in InSb is considered, and compared with hot-electron results obtained in the temperature region below 20 °K in n-type InSb with electron concentrations of 5 × 10¹³-5 × 10¹⁴ cm^-3. From this comparison it appears that piezoelectric scattering is not important and that a consideration of optical and acoustical deformation potential scattering is sufficient to yield a qualitative agreement. The value assumed for the deformation potential constant is 30 eV.

A calculation is presented which describes the splitting of doubly degenerate electronic states of tetragonal and trigonal centres in cubic crystals by an applied uniaxial stress. It is shown that the shift and splitting of the states may be expressed in both cases in terms of four independent parameters. The approach follows that originally used by Kaplyanskii in 1964 for anisotropic centres having non-degenerate states and cubic centres having degenerate states. The calculation is used to tabulate the energy shifts of lines in the uniaxial stress spectra of A → E and E → E transitions within centres of tetragonal and trigonal symmetry, and the results are illustrated by application to the 3932 Å and 4874 Å colour centre zero-phonon absorption lines in lithium fluoride.

The effect of the overlap between the magnetic electron wave function of the central Pr³⁺ ion and the electronic wave function of the nearest Cl^- ions on the crystal-field parameters of the Pr³⁺ ion in a PrCl₃ crystal has been estimated. The Coulomb part of the overlap contribution, which always opposes the point-charge contribution, to A₆⁰ <r⁶> and A₆⁶ <r⁶> is nearly double the point-charge contribution, while for A₄⁰ <r⁴> and A₂⁰ <r²> it is 89% and 23%, respectively, of the point-charge contributions. The effect of the overlap correction to the parameters calculated by Hutchings and Ray on the basis of the point-electric-moment model of the ligands has been estimated. The calculation emphasizes the need for a detailed molecular-orbital treatment of crystal-field splitting of the energy levels of rare-earth ions, possibly allowing for the covalency.

By a semi-macroscopic method the following equation is derived

n²_στ - n²₀δ_στ = ((n²₀ + 2)/3){AIδ_στ + B(3I_στ - Iδ_στ

defining the change in optical permittivity tensor n²_στ due to strong light of intensity I. The quantity A describes isotropic changes in n_στ related with linear and non-linear polarization and opticostriction, whereas B accounts for the optical anisotropy induced in the medium by an intense (e g. laser) beam. The optical birefringence constant B is discussed in detail statistically for the case of multicomponent systems of interacting molecules; it is evaluated numerically for several liquids and compared with the experimental data - the third-order polarizabilities of their molecules are found to be of the order of 10^-36 e s.u.

The L emission spectrum of praseodymium has been investigated using a 40 cm curved mica crystal spectrograph. Three new diagram lines at 1841 5 X.U., 1922 2 X.U. and 2076 1 X.U., corresponding to the quadrupole transitions L_IN_IV,V, L_IIN_VI,VII(γ₈') and L_IIIN_VI,VII(β₇') respectively, have been observed for the first time. The data on the measurement of other L series lines of praseodymium have also been given with a view to removing the discrepancies in the earlier measurements.

The use of vacuum sparks and laser-produced plasmas has enabled many of the stronger transitions from these ions to be classified. New lines of Fe XII and XIII complete the classification of the intense members of the solar emission line group at 167 Å to 220 Å.

Some regions of the absorption spectrum of CeO, obtained in a King furnace at about 2000 °K, have been photographed in high resolution, and rotational analyses of four bands have been completed. The bands have two common lower states and it is suggested that these are the ³Φ₂, ³Φ₃ components of a...σϕ ³Φ lower state, which may be the ground state. The internuclear distance r₀ in ³Φ₃ is 1 814 Å, and the lower state vibration frequency is about 830 cm^-1.

Nuclear magnetic resonance saturation in liquids is described in terms of two competing, but analogous, processes which tend to bring the spin system to the lattice temperature in the stationary and rotating frames respectively

It is pointed out that the minimum magnetic field for the quasi-superconductive transition predicted by Frohlich can be reduced by taking advantage of Fermi surface anisotropy in uniaxial crystals

Somerville has pointed out that from the adiabatic calculation of Taylor and Harris one cannot, from a mathematical point of view, rule out the existence of a bound state of H₂^- with very small binding energy. From both theoretical and experimental work done on H₂^- since 1964, additional physical arguments are available to show that a non-adiabatic study is very unlikely to change the conclusion of the adiabatic work that a bound state of H₂^- does not exist.

Studies have been made of the absorption of photons (λ between 1000 and 800 Å) by the gas flowing from a hydrogen discharge. Metastable species have been detected in the gas in addition to ground-state molecular and atomic hydrogen The effects of these metastable particles on previously reported measurements of the photoionization cross section of atomic hydrogen are discussed.

The calculation of the Franck-Condon factors of the bands of the yttrium oxide (B ²Σ-X ²Σ) system of astrophysical interest has been made by the method of numerical integration using Morse wave functions. The r centroids, previously unobtained for this band system, are evaluated by the same procedure and also by the graphical method.

Owing to a mistake in algebra made by Holøien and Midtdal in 1955 when evaluating the matrix elements of the kinetic energy part of the Hamiltonian operator, the numerical values obtained for the energy and the coefficients of the corresponding eigenvector are in error The error is associated only with correlated basis functions because the correction term vanishes for those which are non-correlated. The corrected values obtained by the re-examination for lithium are more consistent with recent results from electron-impact experiments. New results for the energy and the corresponding eigenvector are given for a 15th-order subset both for He^- and Li