Table of contents

A rigorous inequality is derived relating the specific heat of a system, the temperature derivative of the expectation value of an arbitrary operator and the mean-square fluctuation of the operator in an equilibrium ensemble. The class of constraints for which the theorem is shown to hold includes most of those of practical interest, in particular the constancy of the volume, the pressure, and (where applicable) the magnetization and the applied magnetic field.

The rigorous inequality derived in the preceding paper, together with certain plausible assumptions about the correlations in the neighbourhood of the critical point, is shown to imply the relations sv >or= 2 - α, sv' >or= 2 - α' for the exponents describing the asymptotic from of the singularities at a critical point, where -α and -ν are the exponents for the specific heat and the correlation length respectively above T_c, the primed quantities are the corresponding exponents below T_c and s is the number of dimensions of the system concerned.

Matrix Hartree-Fock wave functions are calculated for the ²Σ⁺ ground state and excited ²II levels of BeF. The calculations indicate that the first excited ²II state is derived largely from a configuration... 1π⁴2π but that a small contribution of the configuration... 1π³5δ² results in the spin-orbit coupling constant having a small negative value, the state thus appearing inverted.

Experimental data on optical absorption, electron scattering and optical refractivity are used to construct a model dipole spectrum of molecular nitrogen which is consistent with the oscillator strength sum rule. The spectrum is used to calculate various dipole properties. The spherically symmetric van der Waals coefficient for a pair of nitrogen molecules is 73.4 A.U., the mean excitation energy describing the slowing down of fast particles is 82.1 eV and the Rayleigh scattering cross section at Lyman α is 5.6 × 10^-24 cm². The vibrational Raman scattering at Lyman α is estimated to lie between 10^-27 and 10^-26 cm².

The hyperfine coupling constants of the 1s²2p ²P_3/2,1/2 states of lithium are calculated using the 45 configuration wave functions of Weiss. Excellent agreement with experiment is obtained for A_1/2. The calculated value of A_3/2, however, appears to agree poorly with the numbers derived from recent level-crossing experiments on ⁷L₁.

A theoretical discussion of the ionization of an atom by the simultaneous absorption of many photons is presented. This process may be relevant to the production of plasmas by concentrated beams of intense radiation from lasers. Approximate methods are developed for evaluating the quantum-mechanical expression for the ionization probability. It is shown that the r E form of the interaction Hamiltonian is greatly preferable to the p A form.

Comparison of the numerical results with experiment shows that at moderate pressures this process can at best only provide the initiating electron, and the growth of the plasma must then take place by other means, for example inverse bremsstrahlung. Even to sustain this argument it is necessary to assume that the radiation is normally incoherent, and the role of inhomogeneities in increasing the overall ionization probability is stressed. At low pressures, where the second process is inhibited, better agreement is found. Finally, the enhancement of the ionization probability which occurs when a natural atomic level lies close to an integral number of photon energies above the ground state is illustrated by a specific example (lithium). A fourfold drop in the threshold flux for ionization is predicted when a ruby laser is tuned through such a resonance.

In the general classical theory of collisions of electrons and protons of charge ±e and velocity v with hydrogen atoms at rest, classical impulse and Monte Carlo methods apply to ionization and charge transfer. At the other extreme, for very weak excitations and intermediate incident particle energies, classical adiabatic perturbation theory is preferable. In this limit the differential cross section with respect to energy transfer ΔE is given. The theory on its own may be applied only to fairly highly excited quantum states. Regions of validity of the approximation and corresponding quantum cross sections are obtained.

The results of a six-state close-coupling calculation of low-energy electron scattering by atomic hydrogen are presented. The first six hydrogen atom eigenstates are retained in the expansion of the total wave function and the resultant equations solved on a C.D.C. 6600 computer. Resonances are found below the n = 3 excitation threshold of atomic hydrogen and they are shown to play an important role in the excitation cross sections Q(1s-2s) and Q(1s-2p). For energies insufficient to excite these resonances it is found that the three-state close-coupling approximation gives reliable results. Finally, cross sections are given for the transitions between the n = 1 and 3 and the n = 2 and 3 levels of atomic hydrogen as well as for elastic scattering on the ground state.

Previous work for S-wave scattering of electrons by atomic hydrogen calculated with a correlation method is extended to higher partial waves. The wave function used is more flexible than that with which the earlier results were obtained. With the use of the bound principle, which is satisfied by this method, cross sections are found for the 1s-2s and 1s-2p transitions in the electron-hydrogen-atom system in the energy region between the n = 2 and n = 3 thresholds which appear to have converged. These cross sections can serve as normalization values for experimental results. Comparison with close-coupling results obtained with expansions including three- and six-states shows that, in the energy region concerned, both these methods are acceptable. A new shape resonance in the ¹P wave just above the n = 2 threshold is reported.

The results for transitions between the n = 1 and n = 2 states of atomic hydrogen obtained in the previous two papers are folded with the appropriate electron energy distributions and compared with experiment. Good agreement is obtained with the shape of all the cross sections, with the normalization of the 1s-2p cross section and with the ratio of spin exchange to total 1s-2s cross section, but the theoretical result for the 1s-2s cross section is a factor of 2 greater than that obtained by the normalization of the experiments to the Born approximation above 300 ev. It is suggested that the use of the theoretical results given in the previous papers may be a preferable way of normalizing the experimental data. Finally, results are given for the polarization of the Lyman α radiation induced by electron impact close to threshold.

The ¹S, ³S, ¹P, ³P and ¹D partial cross sections for e^--H scattering given in two of the preceding papers are analysed using many-channel resonance theory. Partial widths, resonance positions and the background scattering matrix are obtained. The peak at 11.5 ev in the 1s-2s excitation cross section is shown to follow from interference of resonant and non-resonant processes. This interference is examined in detail and is shown to imply that the scattering matrix can be simplified by transforming it to a representation which diagonalizes the long-range 1/r² potentials coupling the degenerate n = 2 states. The scattering matrix in this representation is obtained, and it is found that the resonance states decay into the same linear combination of states at the n = 2 threshold that are excited from the ground state. Some general results in multi-channel resonance theory, applicable to such situations, are then proved. Finally the origin of the ¹P H^- autoionizing state at 10.24 ev is discussed, and its position and width are obtained.

The photoabsorption coefficient of H^- in the region from 0.65 to 3.5 ryd is calculated using a three-state close-coupling approximation for the final state. Particular attention is given to the region near 0.751 ryd where a narrow shape resonance exists. Its oscillator strength is estimated to be 0.044. The contribution of processes leaving the hydrogen atom in an n = 2 excited state are studied. They are found to contribute 17% of the total. The oscillator strength sum rule is checked within 3%.

The Born approximation is used to calculate the cross sections for capture of electrons from hydrogen and helium atoms by positively charged mu mesons. It is found that for low energy the cross section exceeds those for the corresponding capture processes by protons. This shows that chemical reactions involving mu mesons are of considerable importance, and supports the conclusion by Nosov and Yakovleva in their treatment of depolarization of μ⁺ mesons in matter.

An attempt has been made to study the detailed characteristics of the capture of π^- mesons in light elements, in particular oxygen. This study was made as an initial investigation into the feasibility of the use of π^- mesons in radiation therapy for the treatment of carcinomas. There is a yield of 1.77 ± 0.08 multiply charged particles per π^- capture in oxygen with a mean kinetic energy of 6.9 MeV. These particles have a high relative biological efficiency and low oxygen enhancement ratio so that the biological effect of an irradiation with stopping π^- mesons would be characterized by these heavy particles.

The frequency distribution of the various modes of interaction are given together with energy spectra for protons, alpha and heavier particles.

Neutron paramagnetic scattering is discussed generally in terms of the relaxation function of the magnetic system. The expansion of the scattering function F(κ, ω) in terms of moments of the energy transfer is evaluated to the fourth moment for the Heisenberg Hamiltonian and for limitingly high temperatures. Consideration of these moments leads to a better approximate form for the cross section at large scattering vector than the usual Gaussian form. The interpretation of experimental data is discussed and it is shown that important inaccuracies may arise from the use of the Gaussian approximation. A procedure for making good use of experimental information is given for data from both single crystals and polycrystals. In the single-crystal case it is possible to determine exchange constants to a large number of different atoms, whilst for the polycrystal it may be difficult experimentally to separate more than two different exchange constants.

Measurements of the magnetic susceptibilities of gadolinium, terbium, dysprosium, holmium and erbium gallium garnet and of terbium and holmium aluminium garnet have been made between 0.6 °K and 4 2 °K. Antiferromagnetic transitions have been observed at 1.35 °K for terbium aluminium garnet, at 0.8 °K for holmium aluminium garnet and at 0.8 °K for erbium gallium garnet. The results are discussed in relation to the theoretical work of Capel.

Recent measurements of spin-wave dispersion relations in ferromagnetic terbium by Møller and Houmann are analysed in terms of `interlinear' exchange parameters giving the coupling between a central atom and lines of atoms parallel to the c axis. The fitted parameters show the markedly oscillatory behaviour that would be expected from a Rudermann-Kittel-Yosida interaction.

A phenomenological description is given of the relaxation effects which occur in ferromagnetic resonance and in magnetic anneal. The expressions which result will allow the correlation of the experimentally observed effects without detailed assumptions about the electronic energy levels of a crystal. The effects concerned are (a) the ferromagnetic resonance linewidth, (b) the departure of the apparent g value of a crystal from its true value, (c) the frequency dependence of the apparent anisotropy constant and (d) the dependence of the magnetic anisotropy field upon the orientation of the intrinsic magnetization in past time. The formulation was developed to aid in the study of ferrite crystals which contain both ferrous and ferric iron, since such crystals show all of the above effects in different temperature ranges. The relationship to theories based on microscopic energy levels is indicated.

The spin-lattice relaxation time for the conduction electrons has been measured for several dilute alloys of Li in both the solid and liquid states. The relaxation time was found to increase by a few per cent at the solid-liquid transition, suggesting a rather small change in the density of states of the Fermi level. In addition, results are presented for the g shift of the conduction electrons with respect to the free-electron value.

The occurrence of giant magnetic moments (similar 12 to 14 Bohr magnetons) in some dilute alloys of Fe or Co dissolved in Pd, Pt or mixed 4d-5d series systems is theoretically analysed by taking into account the exchange interaction between the impurity and the spins of neighbouring host atoms, lying within a radius of the order of 8 Å, embracing over 100 atoms. The model Hamiltonian considered includes, in addition to the usual s-d mixing, the mixing between the host localized states (unfilled inner shells) and conduction s-electron states, as well as the exchange interaction mentioned above. Making use of the double-time Green function technique, the net moment associated with the impurity centre and the host neighbours is obtained and found to be a monotonically increasing single-valued function of the matrix susceptibility χ. The matrix susceptibility is attributed to the incompletely filled inner shells (as amply evidenced by the observed paramagnetic temperature dependence of χ for Pt, Pd). A plot of the theoretical curve for the net moment against the host susceptibility is given and is in good agreement with available experimental data.

The Madelung method of plane-wise and line-wise evaluation of lattice sums is used to calculate the electrostatic energies of metals consisting of point charges at the lattice sites and a uniform negative compensating background charge. The resulting expression is rapidly convergent and provides a simpler and more natural approach than the Ewald method in some problems.

It has been conjectured that the electrostatic energy of the hexagonal closepacked structure has a minimum at the ideal axial ratio (8/3)^1/2. It is shown that this is not true and that the minimum occurs at an axial ratio of 1.6356 ((8/3)^1/2 = 1.6330). The difference in energy, however, is very small.

Plane-wise summation is used to calculate the electrostatic energy of growth and deformation stacking faults in hexagonal metals. This approach is simpler and yields more physical insight than the Ewald method, and can be readily generalized to any form of ion-ion potential and to more than one stacking fault.

Fundamental data relating to Ziman pseudo-atoms (i.e. ions plus associated screening charges) are obtained for the alkali and noble metal ions, phase shifts being evaluated as functions of angular momentum and energy. In particular, the especially important parameters η_ι(k_F) and (k[partial differential]η_ι/[partial differential]k)_F are tabulated. These provide the basis for a comprehensive and systematic treatment of transport properties and interatomic forces in pure metals and alloys at arbitrary pressures. From the evidence of a number of applications already made, as well as by comparison, where overlap occurs, with the best calculations hitherto performed, it would appear that our results are reliable.

Calculations of the pressure dependence of the resistivities and thermopowers of the alkalis, both in the liquid and solid states, are made covering volume changes approaching 50%. Good qualitative agreement with experiment is obtained. The method depends basically on the Ziman pseudo-atom concept, the scattering from such pseudo-atoms being calculated as a function of Fermi level. A spherical Fermi surface is assumed throughout.

The thermal conductivity of a simple liquid with a spherically symmetric intermolecular potential is calculated using the correlation function theory of Green. The dynamical equations of motion are transformed into a collective variable description and a wave number cut-off is used to give the right number of degrees of freedom. Using Peierls-type second-order perturbation theory, the terms in the heat flux correlation are represented by a diagram expansion. The calculation is applied to liquid argon, the interparticle potential being a semiphenomenological parameter consistent with specific heat data. Order-of-magnitude agreement with experiment is obtained for the thermal conductivity.

Equations previously derived for the volume and internal energy of a simple monatomic liquid, using approximate geometric theory, are used to calculate the specific heats C_v, C_p and volume V of liquid argon as a function of pressure and temperature. A Lennard-Jones 12-6 interatomic pair potential is used. The results are compared with the corresponding experimental curves. The qualitative features of the (P, V) isotherms are reasonably reproduced, and at high pressures the densities are within 25% of the experimental values. The agreement of the specific heats is poorer, although the correct order of magnitude is predicted by the theory.

Results in diffraction obtained by Cornu's spiral can be interpreted as supporting Young's theory of diffraction and a suggestion is made about the nature of diffraction.

The decay of fluorescence following a brief pulse of optical excitation has been studied for the triplet resonance line in cadmium. The coherence existing between the m = ±1 Zeeman sublevels, as shown by the modulation of the fluorescence, was observed. The techniques used show the possibility of separate determination of lifetimes associated with the relaxation of coherence and with the decay of population.

A systematic means is suggested for identifying the molecular features associated with a given atomic transition in the spectra of impurity atoms trapped in solids. It is shown that free-radical spectroscopy can be used to obtain potential energy curves for excited states of van der Waals molecules. The ultra-violet spectra of Hg₂ and Cd₂ in inert gas matrices are discussed as illustrations.

By applying the Heisenberg model and Bogoliubov's variational method in the molecular field approximation to uniaxial ferromagnets with domain structures, the temperature is determined at which the local mean magnetization vanishes. It is shown that this temperature transition varies within the domain structure, being largest in domains and smallest in interdomain walls.

The rotational analysis of one band of each of five systems of BaF has been completed. The lowest known excited state A ²II is found to be a regular state, and B ²Σ⁺ and A ²II are nearly in the relation of pure precession predicted for σ6p and π6p. The internuclear distance r₀ in the ground state is 2.163 Å.

Proton spin-lattice relaxation in lanthanum magnesium nitrate containing different concentrations of neodymium has been studied using dynamic nuclear polarization at helium temperatures. The results agree with theory only at low concentrations and higher temperatures, suggesting that neodymium spin-lattice relaxation is probably not concentration independent at higher concentrations and lower temperatures.

The Lα_1,2 lines of praseodymium have been remeasured using a 40 cm curved mica crystal spectrograph of the transmission (Cauchois) type. In common with the trend observed by earlier workers in the case of other rare-earth elements, the values of the wavelengths (Pr Lα₁ = 2457.99 X.U., Pr Lα₂ = 2468.10 X.U.) obtained in the present investigation are a few tenths of one x-ray unit greater than those reported by Hjalmar.

Franck-Condon factors and r centroids have been computed for eight bands of the C ²Σ-X ²Σ system of RhC which is rather important as it is one of the only two diatomic metal carbides to be studied so far spectroscopically.