Table of contents

It is shown that, to be consistent with the formalism of statistical mechanics, the absolute temperature T of, and the reversible heat δQ for, a thermodynamical system moving with a uniform velocity v with respect to a given frame of reference K should be related to the values T₀ and δQ₀ appropriate to the rest frame K₀ by the old Planck-Einstein formulae T = T₀(1 - v²/c²)^1/2, δQ = δQ₀(1 - v²/c²)^1/2 and not by the alternative formulae T = T₀/(1 - v²/c²)^1/2, δQ = δQ₀/(1 - v²/c²)^1/2 which have been proposed by some authors in the recent past. In view of the requirements of covariance, the validity of the former formulae appears to be logically necessary. The Planck-Einstein formulae for other variables, such as the total energy E and the total momentum P, are also upheld.

Quantum defects provide a measure of the differences between energy levels of many-electron atoms and energy levels of hydrogenic systems. The theory for central potentials has been developed by Hartree in 1927 and 1929, Ham in 1955, and Seaton in 1955 and 1958.

The present paper is the first in a series concerned with a many-channel generalization of quantum defect theory. Wave functions ψ(N+1) for an atomic system containing N+1 electrons are expanded in terms of products ψ_α(N)F_α where the ψ_α(N) are functions for the corresponding N-electron system and the F_α are functions for the added electron. The F_α satisfy coupled differential equations with potentials V_αβ(r) = -1/r(Z-N)δ_αβ + ½U_αβ(r), Z being the nuclear charge. If ψ_α(N) or ψ_β(N) is a bound state, U_αβ goes to zero faster than 1/r as r → ∞. The present discussion is restricted to systems for which Z-N > 0 and the simplifying assumption is made that U_αβ = 0, for r > r₀ where r₀ is finite.

For all values of the total energy E the asymptotic forms of the functions F_α can be expressed in terms of a scattering matrix S. It is shown that S can be expressed in terms of known functions and of a matrix IJ^-1 where the matrices I and J are analytic functions of E. Once the matrix IJ^-1 is known one may calculate the bound-state energy eigenvalues, allowing for configuration interaction and series perturbations, the asymptotic forms of the normalized bound-state eigenfunctions, elastic and inelastic collision cross sections, and the positions and shapes of resonances in cross sections.

The theory is summarized and practical procedures of spectrum analysis are discussed. The He I spectrum is analysed using least-squares fitting procedures. An improved value of the series limit is obtained. Continuum phases obtained from extrapolation of quantum defect data are compared with phases calculated by Sloan in 1964 using polarized orbital theory. The Ca I ³D₁ series is analysed using the two-channel theory and allowing for admixture of the configurations 4sn₁d and 3dn₂s. Good agreement with experimental energies can be obtained using a Y matrix which is a linear function of the energy. The structure of perturbations and resonances is discussed, particular attention being paid to the normalization properties of the wave functions.

The theory of I is extended to the case of scattering by He⁺, taking explicit account of the long range of the potential coupling between the levels 2s and 2p. The asymptotic forms of the general solution, corresponding to the coupled equations which represent the problem, are expressed in terms of the matrix Y. It is then possible to express through this matrix all asymptotic quantities such as the phase shifts or the amplitudes of the autoionized levels. The theory is applied to the particular case of the system He⁺ +e^- in the states ¹P and ³P. The matrix Y is obtained from the R matrix calculated by Burke et al. in 1964 for electron scattering by He⁺. On the assumption of a quadratic approximation, the matrix Y is extrapolated below the excitation threshold of the second quantum level. The elastic phase shifts and the positions of the autoionized levels are obtained. Where comparison is possible, our results are in excellent accord with those obtained by direct solution of the corresponding coupled equations by Burke et al. in 1964.

Photoionization cross sections and oscillator strenghts for the absroption of radiation by neutral atomic calcium in its ground state are calculated with allowance for configuration interaction effects, using radial wave functions with asymptotic forms obtained on application of many-channel quantum defect theory. Calculations are first carried out assuming a two-channel system, an attempt subsequently being made to improve the results obtained by including a third channel.

In the case of the two-channel approximation, for which the fitting to experimental data is fairly accurate, it would appear that where there is discrepancy between theory and experiment this is due to neglect of the third channel. For the three-channel case, the discrepancies are probably due to inaccuracies in the fitting procedure. A further source of inaccuracy occurs when there is cancellation between positive and negative contributions to the total line strength matrix element.

The theory is presented for obtaining, by a Monte Carlo method, the classical cross sections for charge transfer and ionization of hydrogen atoms by protons, in which the atoms are chosen initially from a microcanonical ensemble. With minor modifications the theory applies to any collisions between three charged particles which do not all have charges of the same sign. Scaling laws and an iterative solution of Kepler's equation are discussed.

The Monte Carlo method is used to obtain total classical cross sections for ionization and charge transfer of hydrogen atoms in any level n by protons in the energy range 38/n² kev to 218/n² in the laboratory frame. The cross sections have normal statistical errors of about ±10% for ionization and ±9% or more for charge transfer. For n = 1 the classical ionization cross section agrees with experiment to within about 10 or 15 per cent over the entire range of energies, showing that the agreement between earlier classical theories of ionization and experiment is not simply the result of further dynamical approximations. The classical charge transfer cross section is somewhat higher than experiment. A generalized correspondence principle is derived which states that, for a non-relativistic quantal system of charged particles, the classically scale-invariant quantities tend to their classical values when the linear dimensions tend to infinity and other quantities are also scaled. The above cross sections are thus correct in the limit n -> [infinity]. Validity and application of the results are discussed.

Cross sections for ionization of hydrogen atoms by protons are obtained in the classical impulse (binary encounter) approximation. Cross sections for two velocity distributions of the electron are compared with the Monte Carlo calculations of Abrines and Percival for incident proton energies of 38 keV to 217 keV. They agree to within 10% at 217 keV, but even with the correct velocity distribution the maximum classical impulse cross section at about 38 keV is too large by a factor of 2. A reason is given for the better agreement of the impulse approximation with the total loss (ionization plus charge transfer) cross section.

Calculations are performed of the lifetimes of the 1s2s2p ⁴P_5/2 states of He^- and lithium which decay to the 1s ²f ²F_5/2 continuum through the spin-spin interaction. A variety of wave functions are used, including the correlated wave functions of Holøien. Without correlation the lifetimes lie close to 4 × 10^-4 sec and 6 × 10^-5 sec for He^- and lithium, respectively, but are increased by an order of magnitude by the inclusion of correlation.

Configuration interaction wave functions are used to determine the dipole polarizabilities of normal Li, Be⁺,..., O⁵⁺. A value of α = 23.97 × 10^-24 cm³ is obtained in the case of lithium, which is within the error range of the most recent experiments.

The van der Waals interaction of neutral atoms may be characterized by two-body constants C^AB and three-body constants C^ABC. A recent analysis of spectroscopic, refractive index and Verdet constant measurements has been combined with a simple technique for evaluating oscillator strength summations to give C^AB for all pairs and C^ABC for all triplets of the atoms H, He, Ne, Ar, Kr and Xe. The results are believed to be accurate to within 10%.

A middle-point matching procedure is described for obtaining the zero energy scattering solution of the two-nucleon system when tensor forces are present. This yields an accurate and rapid calculation of the effective-range parameters, and, as an example, some numerical results so obtained are presented.

Differential cross sections have been determined for the elastic neutral pion photoproduction from deuterons at recoil deuteron angles of 25°, 30° and 37° (lab.) within the photon energy range 220 < Eγ < 280 MeV. The experiment was carried out by detecting the recoil deuteron with a counter telescope capable of distinguishing deuterons from protons and measuring their energy. These and previous results have been compared with an impulse approximation calculation.

The normal modes of magnesium have been investigated by means of the inelastic scattering of slow neutrons. The frequencies of about 1000 phonons have been measured with a time-of-flight apparatus, and the values along symmetry directions obtained by an interpolation method. Previous workers have found that their results could be fitted by force models extending to fourth-nearest neighbours or less. The present measurements cannot be fitted by any tensor force model extending only to fourth-nearest neighbours.

The motions of the atoms for a simple force model are discussed.

Earlier studies by neutron diffraction of the occurrence of two antiferromagnetic structures in platinum-iron alloys in the neighbourhood of the composition Pt₃Fe have been extended to the ternary system in which some of the iron is replaced by manganese. The characteristic (½00) structure only appears when there are more than 25% of iron atoms. Manganese atoms function differently from iron, in agreement with the fact that Pt₃Mn is ferromagnetic.

Single crystals of chromium-manganese alloys containing 0.12, 0.44 and 1.03 wt% Mn have been grown and their magnetic structures determined from 78 °K to their respective Néel temperatures. Measurements of the susceptibility of these crystals and of Young's modulus of the polycrystalline material have also been made over the same temperature ranges. The data confirm earlier magnetic structure work on polycrystalline material, and serve to illustrate a theory due to Tachiki and Nagamiya originally proposed to explain the antiferromagnetism of chromium.

The Mössbauer absorption spectrum of ⁵⁷Fe in single crystals of FeCl₂.2H₂O has been studied. In the paramagnetic state the b axis is found to be the symmetry axis z of the crystalline electric field, and the quadrupole splitting is +2.63 mm sec^-1 (½eQV _zz/h = +30.6 Mc/s) at 77 °K. The orbital ground state of the Fe²⁺ ion is |xy⟩, referred to the directions of the ligands as axes, and there is an excited orbital state |xz⟩ at 520 cm^-1. In the antiferromagnetic state the hyperfine field is 255 kG and the axis of spin alignment is shown to lie in the ac plane close to the direction of the short Fe-Cl bond (the x axis), in agreement with susceptibility data. The asymmetry of the electric field gradient is given by η = (V_xx - V_yy)/V_zz = 0 2. By considering the effect of spin-orbit coupling on the orbital ground state the value of the quadrupole splitting and the g values may be satisfactorily understood.

The absorption of light by optically pumped atoms undergoing magnetic resonance is calculated from expressions based on a simplification of Barrat and Cohen-Tannoudji's theory of optical pumping. The density matrix for the system is first derived in orders of approximation which correspond to successive cycles of pumping. The off-diagonal elements of the matrix, and consequently the absorption of light, are modulated. The amplitudes of modulation are resonance functions which are, in general, more complicated than solutions of the Bloch equations. The selection rules for electric dipole radiation limit the possible frequencies of modulation to ω₀ and 2ω₀ for primary interactions (ω₀ is the applied frequency), but modulation at higher frequencies can arise through the circulation of coherence in optical pumping cycles. The functions which represent the amplitude of modulation at 4ω₀ are derived in first and second order, for comparison with experiment.

Optical pumping experiments have been performed on helium 4 atoms in the metastable level 2 ³S₁ with the object of studying the modulation of light in absorption in the longitudinal and transverse beams. Magnetic resonance curves were obtained similar to those found in emission in double-resonance experiments. With a particular geometrical arrangement employing circular polarizers, the Bloch magnetic resonance curves were obtained.

Modulation at four times the applied frequency was obtained with a linear polarizer in the pumping beam. This modulation is ascribed to the circulation of coherence in the optical pumping cycle. From the behaviour of the resonance signals as a function of the frequency and amplitude of the stimulating field, it is inferred that a substantial part of the signal arises from coherence which has survived two complete cycles.

The observations are satisfactorily explained by the theory developed in the previous paper.

Magnetic resonance in the ground states of ³He has been studied through the interaction with metastable atoms in a gas discharge. Modulation showing the characteristics of the ground-state resonances is observed in a transverse beam of light absorbed by the metastable atoms. This is evidence of the transfer of transverse magnetization (coherence between eigenstates) by collision. A theory is developed which explains the observations in detail.

The fact that coherence can be transferred by spin exchange in collision offers the possibility of exploitation in level-crossing or modulation experiments on spectroscopically inaccessible systems.

Existing theories of optical pumping do not take into account the relative phase of the optical field which monitors the resonances. In experiments on excited atoms the optical phase relations have a profound effect on magnetic resonance curves. Signals in the forward-scattered light show Doppler broadening, modified by strong coherence narrowing; a result quite different from that for the laterally scattered light. The question is raised as to whether similar effects are to be expected for resonances in the ground states.

The problem is analysed in terms of the classical theory of the propagation of light in polarizable media. The (complex) polarizability of the sample under conditions of optical pumping and magnetic resonance is calculated to first order in perturbation theory. An expression is obtained similar to that found for resonance in the excited states, but different in so far as the factor which describes the magnetic resonance (the density matrix for the ground states) is independent of the factor which describes the Doppler effect (the plasma dispersion function). Since the intensity of the transmitted light is a function of the imaginary part of the polarizability, the conclusion is that the ground-state resonance curves should not be Doppler-broadened, which is in agreement with the known facts.

We have measured the angular correlation of radiation from positrons annihilating in sodium as a function of temperature. As the temperature is increased the angular correlation data change as though the angular resolution had worsened. The electron motion in sodium is known and is unlikely to cause this phenomenon. We thus attribute this effect to positron motion. The results can be given a one-parameter characterization by assigning to the positron an effective temperature (1.9 ± 0.4)T or an effective mass (1.9 ± 0.4)m. Possible causes of this effect are discussed.

A method of evaluation of lattice Green functions using a Fourier series expansion of their imaginary parts is developed. It is shown, through an example, that numerical evaluation of the real and imaginary parts of the Green functions can be done by this method with much greater ease than by direct integration of the Green function integrals, with the added advantage that the real and imaginary parts of the Green functions can be represented by simple analytic expressions with good approximation.

A molecular model has been developed to study the vibrations of U centres in caesium iodide. Employing the rigid ion model with nearest-neighbour short-range forces, the dynamical matrix of order 27 × 27 was solved to obtain the frequencies of the localized modes and the perturbed lattice modes. The results are compared with those obtained from the Green function method.

An experimental investigation has been made to find out how a conductor in a plasma behaves when an alternating potential is applied to it with a frequency in the neighbourhood of the plasma frequency. Attention was paid to the way in which the mean current to the probe (the `rectified' current) depended upon the frequency, particularly when it approached the plasma frequency and when the current became abnormally large (`resonance rectification'). The probe-to-plasma admittance was also investigated as a function of frequency.

The experimental results on rectified current and admittance are compared with values derived from the theory of Buckley for a spherical plasma probe. This theory gives a detailed description of the way in which these quantities depend on the electron-neutral collision frequency and on the frequency of the applied alternating potential. In many respects previous theories are unrealistic because separate sheath, radio-frequency penetration and unperturbed plasma regions, each having different physical characteristics, have been considered. Buckley, using values of sheath potential computed by Bernstein and Rabinowitz from Poisson's equation for spherical symmetry, has shown that it is not necessary to separate the various regions. The effect of collisions between electrons and neutrals is also taken into account in a realistic way by including a collision term in the Boltzmann equation. The experimental data on rectified current and probe-to-plasma admittance are found to agree with the theoretical data which refer to particular values of the ratio of the electron-neutral collision frequency to angular plasma frequency.

The two different mechanisms which have been used to describe the orientation of anisometric molecules in a simple liquid subject to laminar flow lead to similar expressions for the angular-dependent distribution functions. The application of a static electric field to the flowing liquid is not capable of providing a test as to the validity of either mechanism, as was originally suggested by Tolstoi. A general optical treatment using the distribution functions relates the directions of the principal refractive indices and the birefringence with the magnitude of the applied hydrodynamic and/or electric field.

A more extensive analysis of negative pion-proton elastic scattering reported in a previous paper is presented. It is concluded that the elastic-scattering data are consistent with the analysis of Hamilton and Woolcock. Some further analysis of charge-exchange scattering at low energies is also presented.

The crystal field parameters V₂⁰ and V₂² have been derived for Hg-Ar, Hg-Kr and Hg-Xe at 4 2 °K.

Plasmas produced by focusing a 400 MW ruby laser beam onto surfaces of C, Ti, Mn, Fe, Ni, Cu and Zn, and into argon gas, have been observed in the extreme ultra-violet region. Line and continuum radiation have been recorded down to 25 Å, including new spectra of Fe XV and Fe XVI and Ni XVII and Ni XVIII. The value of such plasmas for line classification work is demonstrated.

It is shown that the cluster series for the free energy and susceptibility of the Heisenberg model in zero field are considerably simplified by allowing the magnitude of the spin to become infinite.