Table of contents

The nuclear spins in solid ³He are coupled to one another as a result of the Pauli exclusion principle. The exchange energy is so small that the Pauli principle can be treated rigourously as a boundary perturbation, and the coupling of the spins is given correctly by the Heisenberg Hamiltonian. The coupling is antiferromagnetic in sign if, as is to be expected, two-particle exchange predominates, but may be ferromagnetic if three-particle exchange is more important. Only if four or more neighbouring particles can change places simultaneously must a more complicated form of the Heisenberg Hamiltonian be used. Herring's formula for the exchange energy is generalized to include exchange induced transitions between nearly degenerate levels of the system (spin-lattice coupling). It is demonstrated that coupling between the spin system and the lattice can be calculated if the exchange frequency is known as a function of strain, and a rough argument is given to show how different components of the strain tensor may affect this frequency. Temperature dependence of the exchange frequency is discussed. It is shown that at moderate temperatures thermally excited vacancies produce a new mechanism for spin exchange, which may work towards parallel alignment of the spins in the body-centred phase, but towards anti-parallel alignment in the hexagonal phase.

A problem suggested by the study of exchange in solid helium is set up; this is the problem of calculating the exchange energy of two identical spherical particles enclosed in a box. The eigenfunction of a six-dimensional wave equation must be calculated in a region in which it is very small. The calculation is carried out by a random walk method.

A discussion is given of the cellular method and the various sources of error in it. Numerical examples show that the method is practicable and that it provides accurate and unambiguous energy eigenvalues, subject, like in other methods, to the validity of the spherical approximation to the potential, which is discussed. An interpolation scheme for the bands is given. Methods to obtain the density-of-states curves, Fermi energies and Fermi surfaces are given and illustrated with some results for Be metal.

A comparison is undertaken of graphs which contribute to the partition functions of an infinite assembly and a finite assembly wrapped on a torus. This leads to an order of magnitude estimate of the maximum in the specific heat of finite two- and three-dimensional Ising models. It is found that for a normal size assembly and for a single crystal the specific heat deviates from its limiting value for an infinite assembly when the temperature is within a factor of about 10^-7 of the Curie temperature. This figure would be modified to perhaps 10^-6 for the usual multi-grain structure. A similar discussion is undertaken for the magnetic susceptibility above the Curie temperature.

This paper is a sequel to a previous one (containing an error which is corrected) in which the general hydrostatics of an electrically conducting liquid under an electrostatic field was treated, taking into account the electric double layer, the theory being considered as rigorous within the framework of macroscopic physics. In that paper a set of equations was obtained governing the shape of a liquid boundary. Two problems were solved in the first approximation, but no general method of solution was given and a procedure of guess-work was used at an important stage. In this paper a general method is given with forms pertaining to axisymmetric two- and three-dimensional situations, respectively, and it is applied to find the second approximation to the solution of one of the previous problems, that of the two-dimensional sessile drop. The treatment of this problem involves the solution of an interesting electrostatic problem and the tabulation of a function.

A stochastic model for the evaluation of the classical velocity autocorrelation function of an atom in a simple liquid is derived. It is shown that in the harmonic approximation the mean frequency of the thermal vibration ω₀ can be expressed in terms of the pair potential and the pair correlation function. The calculated value of ω₀ for liquid argon is in excellent agreement with the value found from a comparison of the theoretical velocity spectrum with one obtained recently from a Monte Carlo calculation. The velocity spectra of a methane molecule and of a hydrogen molecule in liquid argon are calculated to aid in the interpretation of recent neutron scattering experiments. The calculation of the scattering cross sections will be given in a separate paper.

The velocity spectrum obtained in an earlier paper is used in a calculation of the cold neutron scattering cross section for a dilute concentration of hydrogen in liquid argon. Since the theoretical cross section contains no unknown parameters, a comparison with recent observed cross sections provides a good test of the validity of the itinerant oscillator model of liquids.

Paramagnetic neutron scattering has been observed from an alloy of 65% Fe-35% Ni at temperatures up to about twice the Curie temperature and for energy transfers up to about 0.1 eV. The apparent iron atom moment is 1.4 ± 0.3 μ_B and is not sharply temperature dependent whilst the moment in the ferromagnetic state is 2.3 ± 0.25 μ_B. The existence of so much paramagnetic scattering at such small energy transfers indicates that atomic moments in the paramagnetic phase are defined for a relatively long period of time. This is presumably due to correlation effects between electrons.

Exact formulae for transport coefficients are available in the literature by calculating the flux caused by certain external forces. These expressions cannot usually be evaluated in perturbation theory since the transport coefficient involved is usually infinite in the absence of scattering; for example, electrical conductivity is infinite for a system without scattering. In this paper exact formulae are derived relating forces to fluxes, for example for the electrical resistance of a system. In certain special cases these can be evaluated in perturbation theory and manipulations of the perturbation series are no longer required, but for the general case they suffer from the same difficulties as the direct expressions.

The S- and P-wave scattering of electrons by singly ionized helium for energies which are insufficient to excite the second quantum level of the ion is calculated in the 1s-2s-2p close-coupling approximation. All the expected Rydberg series of resonances, converging on the second quantum threshold of the ion, are found for each total spin and total angular momentum state considered. These are interpreted as compound atomic states which autoionize with a characteristic lifetime for each series. The photoionization cross section of He from the ground state to the resonant continuum using both the dipole length and the dipole velocity operators is evaluated. Good agreement is obtained with recent experimental results of Madden and Codling.

The differential and total cross sections for the resonance charge transfer reaction H⁺ + H (1s) → H (1s) + H⁺ are calculated in a distorted wave approximation retaining terms of third order in the interaction potentials. It is found that, for energies above 10 kev, the only significant third-order term arises from the incident proton-electron interaction, the proton-proton term being smaller by a factor equal to the electron-proton mass ratio. In the infinite energy limit the third-order contributions vanish as E^-1/2 relative to the first Born terms, while at moderate energies (10 kev to 500 kev) they result in an increase of about 40% in the calculated cross sections.

The probability of capture into excited states of the hydrogen atom is investigated. The analysis yields a correction factor which, combined with the cross sections for 1s-1s transitions, yields an estimate of the cross sections for capture into all final states. These results are compared with the existing experimental data and a generally good agreement is obtained.

It is shown that the impact parameter method is consistent with detailed balancing even though only a finite number of states is included in the expansion set. Both atomic and molecular eigenfunction expansions are discussed. Conditions on the coupling coefficients are given which are sufficient for probability conservation and detailed balancing. It is emphasized that in the case of nonorthonormal expansions the matrix of coupling coefficients should not be Hermitian symmetric. The averaging processes used by Rapp and Francis and by Bates and Williams to symmetrize this matrix are shown not to be necessary for detailed balancing and to be inconsistent with the proper expression of probability conservation. Alternatives to the averaging process are suggested.

Using the conservation of the number of electrons in single atom scattering and a multi-ray phase grating theory with an exact complex atomic scattering factor, it is shown that the absorption coefficient would be expected to vanish. This is inconsistent with the recent results of Boersch et al. who used a two-ray grating theory to deduce a large absorption coefficient from the imaginary part of the atomic scattering factor.

The usual result for the Compton profile intensity, relating it to the momentum distribution of the electrons in the scatterer, is derived by a simple photon theory, based on energy and momentum conservation laws.

The result is shown here to follow from the wave-mechanical scattering theory of Waller and Hartree provided: (i) a one-electron approximation is adopted, (ii) electronic binding energies are assumed small compared with the energy transfer from the photon, (iii) the electron wave functions for the relevant continuum states are approximated by plane waves.

A numerical method previously employed for the calculation of vibrational excitation cross sections and vibrational relaxation in carbon dioxide has been applied to the case of vibrational excitation of carbon dioxide by collision with molecular hydrogen.

Cross sections have been calculated for collision energies up to 2.5 eV and coupling between five molecular states, distributed between the three vibrational modes, has been retained. For the inter-molecular potential the Lennard-Jones parameters tabulated by Hirschfelder et al. have been used.

The dependence of vibrational relaxation in carbon dioxide on hydrogen concentration has been determined in terms of these cross sections for temperatures from 300 to 600 °K. The results are in qualitative agreement with the sound absorption measurements of Winter and reproduce the experimental observation that hydrogen appears to be less efficient at 400 °K at inducing vibrational transitions in carbon dioxide than at 300 °K. It is concluded that such anomalous behaviour can be accounted for when proper allowance is made for the coupling between the molecular vibrational states.

Fraser suggested that, in the expression for the transition probability, the electronic transition moment and the Franck-Condon overlap integral are separable, both depending on the initial and final quantum numbers of the band. A good approximation in many cases, Fraser's formula is inaccurate for certain molecular bands. An improved formula is found including, in addition to the electronic transition moment and the Franck-Condon overlap integral, a third factor which also depends on the vibrational quantum numbers. Using this formula, the degree of accuracy of Fraser's approximation is calculated. An example shows that it may be of the order of 20%, thus justifying the use of the `correction' factor.

The anisotropy of ytterbium iron garnet and ytterbium-doped yttrium iron garnet has been measured by the torque method in applied fields of 15 000 Oe at temperatures between 1.5 °K and 300 °K. Theoretical values of the anisotropy were calculated from optical energy level data derived by other workers and these were compared with the experimental results. At temperatures below 4.2 °K, the experimental measurements of anisotropy energy showed anomalies along certain crystallographic directions and the magnitude of the effects has been measured as a function of temperature and concentration of ytterbium. Possible mechanisms to explain the occurrence of these anomalies are discussed.

The absorption spectrum of Ca 1 in the Schumann ultra-violet has been photographed by means of a King furnace and a 3 m vacuum spectrograph. Of the 82 lines found, 81 are classified reasonably in six series of two-electron transitions, viz. 4 ¹S₀-3dnp,nf^1,3P₁⁰, ³D₁⁰, the remaining line being probably 4 ¹S₀-4p5s ¹P₁⁰, which mixes strongly with the former series.

Pronounced effects of configuration mixing of the 3d np, nf levels amongst themselves and with the 4s E(1)⁰ continua (`autoionization') are present.

With the same apparatus the singlet principal series, 4 ¹S₀-4snp ¹P₁⁰, has been extended from n = 13 to n = 33, leading to an improved value for the ionization potential. The level previously ascribed to 4s6p ¹P₁⁰ is now identified as 3d4p ¹P₁⁰.

The 6 ²P_1/2⁰-n²S_1/2, n²D_3/2 series of Tl I have been obtained in absorption, at good dispersion, and measurements made from n = 7 to 43 and n = 9 to 34 respectively. The series 6 ²P_3/2⁰-n²D_3/2,5/2 have been resolved for n = 10 to 12, and the ²D_5/2 series extended to n = 24. An amendment to the Tl I ionization potential of 2 5 cm^-1 is suggested. The level 6s6p²⁴P_1/2, which produces a perturbation in the n²S_1/2 series levels, has been resolved into a close doublet with an interval of 2.8 cm^-1 presumably due to partially resolved hyperfine structure.

Several zero-phonon lines are seen at 4.2 °K in the absorption spectra of single crystals of magnesium oxide after irradiation with high energy neutrons. The polarization spectra of two of these lines at 27 650 and 24 200 cm^-1 have been examined under uniaxial stresses of up to 30 kg mm^-2 applied along the crystal directions [100], [111] and [110]. Both lines have the same splitting patterns and are associated with centres having [110] orientation and orthorhombic symmetry.

Lines of scandium, titanium and vanadium have been observed at wavelengths below 31 Å. These and some argon lines are classified as transitions in the Ne 1 and F 1 isolectronic sequences involving a principal quantum number change of 2-3 or 2-4.

Rotational analyses have been made of emission bands of the intravalency transitions A ²Σ⁺-X ²II_r (a), D ²Σ^--X ²II_r (a) and the Rydberg transition B ²Σ⁺-X ²II_r (a). Eleven perturbations in v = 0-2 of A ²Σ⁺ have been identified; six are due to one component of a new state G ²II (a), which could be well characterized by analysis of the perturbations. A sharp but weak predissociation in B ²Σ⁺ is ascribed to a state ⁴Σ⁺ leading to ground state atoms, whence D₀⁰(AsO) ⩽ 4.977 eV, 114.8 kcal mole^-1. In the ground state r_e = 1.6233 Å, k_e = 7.26 × 10⁵ dyn cm^-1.

The best available values for parameters characterizing the known electronic states of AsO are summarized in table 4.

Measurements of the dielectric properties of argon, helium, krypton, and xenon afterglows following a long pulse, high-frequency, electrodeless discharge are presented, over a signal frequency range 0 5 to 600 Mc/s. The pressure range investigated is 0 1 to 10 torr.

Computations of the detuning of a plasma condenser are compared with experimental values, the calculations being based on one of three models for the plasma afterglow in turn. In this way, the free electron gas is compared with a loosely bound electron gas, and with a combined population of free electrons and more numerous loosely bound electrons. Calculations based on the combined population model are shown to reproduce the experimental results closely.

A quantitative analysis is used to evaluate the plasma parameters, and the values for plasma frequency obtained are shown to agree with values obtained using the well-known cavity detuning method of measurement. The values obtained for ω₀, the natural frequency of oscillation of the loosely bound electrons, are in the range 2 × 10⁹ rad sec^-1 to 5 × 10⁷ rad sec^-1, decreasing from higher to lower values during the afterglow. ω₀ is generally found to be higher for higher gas pressures and for higher atomic weights of the neutral gas atoms. Little change in ω₀ is measured for an increase of three times the plasma rod size.

It is shown that the absolute temperature transforms as T = T₀/(1-v²/c²)^1/2; where T₀ is the temperature of a system defined in the rest frame of the system and T the temperature defined in a frame of reference moving with respect to the rest frame with velocity v. The result is in agreement with the results of Ott and Arzelliès.

Neutron diffraction measurements have been made at 294 °K and 78 °K on single-crystal samples of high purity chromium which have been allowed to cool through the Néel temperature in a field of 55 kG acting along a cube edge. It is shown that field cooling favours only domains containing spin density waves with a wave vector parallel to the field applied during cooling. The complementary effects produced by stress cooling are also indicated.

The technique of atomic beam magnetic resonance has been used to measure hyperfine structure separations and g_J values for the states..3d²4s²³F, J = 2, 3 and 4 of ⁴⁷Ti and ⁴⁹T1, for the states..3d⁷4s²⁴F, J = 7/2, 5/2 and 3/2 of ⁵⁹Co, and for the states.. 4d⁹5s ³D, J = 3, 2 and 1 and.. 4d⁹5s ¹D₂ of ¹⁰⁵Pd. The results yield Q (⁴⁹Ti)/Q (⁴⁷Ti) = 0 819(1), Q (⁴⁷T1) = +0 29(1)b, Q (⁴⁹T1) = +0 24(1)b and Q (¹⁰⁵Pd) = +0.8(1)b without correction for shielding effects.

Using high-temperature series expansions, an attempt is made to characterize the behaviour of the Ising model of a ferromagnet in a non-zero magnetic field. The functional form derived can be used to pass to the low-temperature side of the Curie point, and hence to determine the low-temperature critical behaviour. As a result, it is suggested that there are two basic parameters, in terms of which critical indices for all thermodynamic properties can be expressed.

Corrections to 1965 Proc. Phys. Soc.86 443.