Table of contents

Recent work on the asymptotic momentum expansion of Schrodinger two-body scattering amplitudes by Buslaev and Hunziker has made it possible to derive identities called `moment relations' by a method analogous to that used for partial-wave amplitudes (Jost functions). These identities involve the total cross section and the differential cross section in the forward direction.

The results are used to derive simple analytic approximations to the total cross section and are compared numerically using a Gauss potential with the cross sections obtained by summing over the first seven partial waves using exact numerical phase shifts and also approximate phase shifts due to Swan. The moment approximations are not so good as those obtained from Swan's phase shifts as far as was tested.

In recent work on the asymptotic form of correlation functions in `classical' fluids it has been proposed that the direct correlation function f(r) ~ -ϕ(r)/kT at large r, where ϕ(r) is the interatomic potential. It is pointed out here that for the Born-Green theory, and for ϕ(r) which has the van der Waals form -Ar^-6 at large r this relation is modified to read f(r) ~ - (ϕ(r)/kT)(1/K_T(2P-n₀kT)) where n₀ is the mean density, P the fluid pressure and K_T the isothermal compressibility. The multiplying factor can be significantly different from unity and negative. It is emphasized that small-angle scattering of X rays or neutrons from liquid argon, for example, could in principle decide between the two results.

A microcanonical distribution function depending on the total energy E and the total linear momentum P of an isolated system is examined. For P = 0 the generalized microcanonical ensemble is found to be thermodynamically equivalent to the usual microcanonical ensemble.

The conventional symmetrized definition of a noise current power spectrum is shown to be inappropriate at optical frequencies. A new definition is developed which takes account of the fact that optical detection is accomplished by photon annihilation. Using this definition, fluctuation-dissipation theorems are established for a medium in thermal equilibrium and for a pumped Raman-active medium. The theorems relate the power spectrum to the first- and third-order conductivity tensors in the thermal equilibrium and Raman-active cases respectively, the zero-point fluctuation noise is absent in both cases. The consequences of time reversal invariance and the emission of Stokes, anti-Stokes and spontaneous radiation are discussed.

A compact and at the same time general tensor formalism adapted to the theoretical prediction of a wide range of non-linear processes due in general to multipole interactions between atomic or molecular systems and electromagnetic fields is proposed. Recurring to a semi-classical method, the expectation values of the 2ⁿ-pole induced electric moment P_e⁽ⁿ⁾ and 2ⁿ-pole magnetic moment P_m⁽ⁿ⁾ for a transition k → l are computed explicitly to within the third order of quantum-mechanical perturbation theory approximation. The formalism is demonstrated with regard to its utility and simplicity on the example of electric and magnetic multipole light scattering with multiharmonic frequencies ω_kl + ω₁ + ω₂ +... + ω_r.

The minimal electromagnetic interaction of a charged, massless vector particle is shown to be consistent.

The electric polarizabilities of the fourteen bound vibrational states of molecular hydrogen are calculated and found to range from 5.315 a₀³ for the ground state to 10 065 a₀³ for the thirteenth excited state. These values are subsequently used to compute the mean polarizability for temperatures ranging from 2000 to 5000 °K. The mean polarizability is found to vary over this temperature range from 5.350 a₀³ to 5 630 a₀³. The results are discussed in connection with recent measurements of the polarizability of atomic hydrogen in which the vibrational states of the molecular hydrogen present in the experimental gas were highly excited.

A variational principle for the phase shift which is analogous to that originated by Tamm is derived, and its connection with certain approximations for the phase shift is discussed. Further, the Tamm variational principle is used to establish an integral equation involving the Kato functional. The Tamm variational principle is then generalized to give a variational principle for the scattering amplitude. Finally, application is made to the scattering of electrons by the static field of a hydrogen atom.

Details are given of a crossed-beam technique for the examination of molecular spectra. Using this technique cross sections for the excitation of the B ²σ state of N₂O⁺ by electron impact on N₂O have been measured. The maximum cross section of the v = 0 level is 26.1 × 10^-18 cm² and the ratio of the effective cross sections of the (0, 0), (0, 1) and (0, 2) bands of the B ²σ-x ²II system is 1: 0.52: 0.13.

A method which takes account of electron-electron repulsion is proposed for calculating cross sections for the excitation of neutral atoms by electron impact. The method owes much to the recent work of Vainshtein, Presnyakov and Sobel'man, but differs from this latter in several important respects. At high energies, the new approximation converges to the Born approximation and, with the possible exception of the threshold region, the cross sections are smaller than the corresponding Born cross sections. The maximum is displaced considerably to higher energies, as compared with the Born maximum.

Calculations for the 1s-2s and 1s-2p transitions in atomic hydrogen are in excellent agreement at intermediate and high energies with the experimental results of Fite et al. An unexpected characteristic of the theory is the prediction of a sharp maximum in the cross section just above the excitation threshold. In particular, the calculations in the threshold region for the 1s-2p transition in atomic hydrogen are in good accord with the recent observations of Chamberlain, Smith and Heddle. Results are also presented for the 3s-3p transition in sodium.

The scattering of an electron by a hydrogen atom is considered with particular reference to the errors implicit in the Born and Born-Oppenheimer approximations. A new approximation to the scattering amplitudes is proposed with a view to minimizing these errors. Results of calculations using this method are shown to be encouraging, the cases considered being the elastic scattering of an electron by a hydrogen atom in its ground state and the excitation of the 2s state from the ground state.

A new calculation of the ionization cross section for the process e^- +He⁺(ls) → He²⁺ +e^- +e^- in the Born-exchange approximation is reported and the results shown to be in close accord with experimental values.

Analytical representations of the dipole spectrum of the ground state of helium are derived. From them are calculated the Verdet constant, the mean excitation energies I and I₁ characterizing stopping and straggling of a beam of fast particles absorbed in helium and the leading coefficient C(a, b)/R⁶ of the long-range interaction between atoms a and b a distance R apart. I is 42.0 ev, I₁ is 73 ev, C(He, He) is 1.471 A.U. and C(He, H) is 2.830 A.U.

A calculation of the 1 ¹S-2 ³P excitation of helium by electron impact is described. The cross section is calculated by means of the distorted-wave method with full allowance for both direct and exchange potential distortion of the colliding electron wave functions. The calculation is performed with a more accurate ground state wave function than the one used by Massey and Moiseiwitsch. Finally, a comparison is made between the calculated cross sections and experiment.

Cross sections for excitation of ¹S states are calculated in the Born approximation. For the 1 ¹S-2 ¹S differential cross section a variety of wave functions are used. For 1 ¹S-n¹S total cross sections, n = 2... 10, a self-consistent field function is used for the ground state and a truncated Whittaker function, orthogonalized to the ground state function, is used for the excited states. The theoretical results are compared with those obtained experimentally.

A proposal is made that the atom-atom interchange process A ⁺B_I + B_II → A⁺B_II + B_I contributes sufficiently to account for the anomalous low mobilities, at high field strength-pressure ratio, of molecular ions N₄⁺ in N₂, Ar₂⁺ in Ar, H₃⁺ in H₂, N₃⁺ in N₂, C₂O₂⁺ in CO. Limits for the cross sections are calculated from the mobility data and fall in the region 0 5 to 9.5 × 10^-15 cm² at impact energies of the order of 1 ev.

A method for obtaining beams of polarized electrons, by transfer of polarization from an atomic beam to a cross-fired electron beam, is investigated using the formalism of the Stokes vector and the interaction matrix. The transfer takes place through the mechanism of spin exchange in elastic collisions, and this effect is discussed together with problems arising from the statistical nature of the transfer process. The theoretical requirements for a workable system are translated into practical terms and details are given of a proposed experiment. This incorporates a device capable of trapping electrons of low energy for periods of a few milliseconds. The system is estimated to yield a pulsed beam of electrons of a few ev energy, with an intensity of about 10^-8 A and a polarization approximately equal to that of the atomic beam.

⁹Be(d,n)¹⁰B neutron time-of-flight spectra are presented. These spectra provide no evidence for the formation of any level near 2.86 MeV in ¹⁰B for 600 keV deuterons.

The form of the localized orbitals in a one-dimensional lattice of like atoms with half-filled energy bands is considered. It is shown how it is possible to define two sets of equivalent orbitals, the members in each set being centred on alternate atoms. Either set of orbitals just spans the same space as the Bloch waves belonging to the lower half of the energy band. The relationship between the band Wannier function and these half-zone equivalent orbitals is discussed.

It is shown that the augmented plane wave, orthogonalized plane wave, and Green's function treatments of electronic band structure can all be derived from the general principle that the band structure depends only on the scattering properties of the core potential. In each case a model potential in real space is constructed with the same scattering properties as the original potential and when this is converted into a momentum representation the usual results are obtained.

A general method is described for the calculation of the relaxation function G(t) for a macroscopic nuclear magnetization perpendicular to a strong steady magnetic field, under the influence of the nuclear dipole-dipole interaction. This function is the Fourier transform of the nuclear magnetic resonance absorption line shape. Approximations are introduced, the form of which depends upon whether the short-term or the long-term development of G(t) is required. A third type of approximation leads to the equation of Lowe and Norberg. The theory is applied to the case of the fluorine resonance in calcium fluoride with the steady field in the [100] direction. The first approximation gives a curve that differs only slightly from that of Lowe and Norberg over the range of t for which the latter is in good agreement with experiment. For larger t the disagreement with experiment is less severe, the characteristic oscillation of G(t) being present, though with too large an amplitude. In contrast the second approximation gives a curve which though overdamped provides a better description of the approach of the spin system to equilibrium. The theory may be extended to problems in which the dipolar Hamiltonian possesses an explicit time dependence. In this form it is applied to the case of motional narrowing due to rapid rotation of the sample about an axis inclined at 54.7° to the magnetic field, for which the absorption spectrum consists of a narrow central line flanked by an infinite series of sidebands spaced at intervals corresponding to the rotation frequency. It leads to a separation of the two components of the second moment of the spectrum due respectively to the generation of sidebands and to the finite width of the lines. Though the sum of these contributions is invariant to rotation they themselves are not. An assumed form for the shape of the lines leads to a linewidth which for rapid rotation depends inversely on the cube of the rotation frequency and is in substantial agreement with an earlier statistical theory.

The effect of the Jahn-Teller coupling on the shape of the d ↔ d transition of a 3d⁶ ion in a cubic crystal is calculated on a simplified model in which the kinetic energy of the nuclei is neglected in the excited state. The result is applied to Fe²⁺ in MgO by the use of a modified Debye model, using data from compressional experiments. The predicted splitting of the d ↔ d transition is about half that actually observed. The reduction of the ground state g value by the Jahn-Teller coupling is small.

Theoretical calculations of the rates of the possible vibrational relaxation processes for binary mixtures of O₂ with CH₄, CD₄, C₂H₄, C₂H₂, H₂O, D₂O and HDO are presented, based on the quantum mechanical theory of Schwartz, Slawsky and Herzfeld. These predict mechanisms involving vibration-vibration and vibration-translation energy transfer which are in qualitative agreement with the available experimental data.

Spin-lattice relaxation time measurements were carried out on two orange ruby samples containing both Cr³⁺ and Cr⁴⁺ ions. The frequency used was 9 27 Gc/s and the temperature range 1.6-4 2 °K. By comparison with the behaviour of vapour phase grown pink rubies of similar Cr³⁺ concentration, it was shown that the Cr³⁺ ion relaxation was not seriously affected by the Cr⁴⁺ ions, even in the region where their adsorption lines overlapped. There were about 10¹⁶ spins of Cr⁴⁺ present, and approximately 10¹⁸ of Cr³⁺. It is thus improbable that Cr⁴⁺ is the fast relaxing ion thought to be the cause of the concentration-dependent relaxation times observed in some Verneuil-grown pink rubies.

Θ₀^c(cph) and the ratio K ≡ 100{Θ₀^c(cph) - Θ₀^c(fcc)}/Θ₀^c(fcc) for the ideal inert gas solids was calculated, using the quasi-harmonic approximation and an (m - 6) Lennard-Jones all-neighbour force model; K was found to be about 2%. The neglect of explicit anharmonic contributions to K is discussed. A table of the relevant all-neighbour sums is given.

It was found that the use of the ideal axial ratio γ_I = Θ(8/3) to characterize the close-packed hexagonal lattice limits the accuracy to which K can be calculated to about one decimal place, and Θ₀^c(cph) to about two decimal places.

King's, Phillips' and Hagan's intensity measurements on the strong bands of the C₂ Swan system have been re-interpreted. It was found that variation of electronic transition moment with internuclear separation is given by R_e(r) = -0 150 (1-5.07r) A.U.; 1.10 Å < r < 1.50 Å. A table of smoothed absolute band strength, oscillator strengths and Einstein A coefficients is given.

The observation of a second pair of satellites, outside the usual electron paramagnetic resonance spectrum of one line and two satellites from Nd³⁺ ions in La₂Mg₃(NO₃)₁₂. 24H₂O, is reported. The formation of such satellites is analysed theoretically using a simple model containing three spins coupled to each other by dipolar interactions of differing strengths. The analysis shows that, while the existence of any two of the interactions is sufficient for the formation of doubly forbidden satellites, all three contribute to satellite intensity.

The rate of inter-carrier energy exchange in a semiconductor is calculated. The energy exchange is considered to arise from simple intra-band collisions within a Maxwellian distribution of carriers. The calculation uses first-order time-dependent perturbation theory based on Bloch waves and a screened Coulomb interaction. The carriers are assumed to have a scalar effective mass. Previous estimates of the critical concentration of hot carriers have been based on a classical expression for the rate of inter-carrier energy exchange. The present theory shows, however, that this approach is not really justifiable. Revised calculations are therefore given for the critical concentration; the case of electrons in InSb is treated as an example. The present values of critical concentration are of the same order of magnitude as earlier estimates, but differ from them by a factor which is dependent on the range of the inter-electronic interaction. Expressions are also given for the mean energy exchange occurring in a single inter-carrier collision.

Effect of overlap between the magnetic electron wave function of a central metal ion and the electronic wave function of surrounding ligands on the crystal field parameters of rare-earth ions in rare-earth crystal has been investigated. For the Pr³⁺ ion in PrCl₃ crystal overlap effect is shown to affect the A₆⁰ parameter in an important way, while it is expected to be small for the A₂⁰ and A₄⁰ parameters. Results are discussed in the light of previous estimations of crystal field parameters for LaCl₃ structure by Hutchings and Ray based on an ionic approximation.