Table of contents

A generalization of the Bloch equations for a spin ½ particle to those valid for any multipole is obtained by use of tensor operator techniques. As an application of the Bloch-type formalism, we discuss the interaction of coherent and incoherent light with a gas in a magnetic field as contrasted with the description of a Zeeman laser.

An iterative procedure, based on a steepest-descent criterion, is developed for the estimation of eigenvalues of self-adjoint differential operators. The procedure is applied to the vibrational Hamiltonian for He₂, using the potential energy functions of Matsen and Scott. The first six energy levels of the ¹Σ_u⁺ and ³Σ_u⁺ states are estimated.

The complete quantum-dynamical solutions for a free particle moving in one dimension and for a free particle moving on a ring are obtained in terms of classical path sums. The propagators and Green operator kernels, and, in particular, the kernel of the spectral operator I_E = δ(E - H) are expressed explicitly in this way. Relations between quantum and classical mechanics which hold for low quantum numbers and relatively small values of the classical action are discussed. They are the correspondence identities.

The Bohr-Sommerfeld identity between the hydrogen energy levels given by the old quantum theory and the actual energy levels is explained. The quantum-dynamical solution of the hydrogen-like problem is obtained for bound states in terms of sums over classical paths using the symmetric representation of Fock. The classical analogue of the Fock theory of the hydrogen atom is discussed. Schwinger's expression for the Green function is obtained in a new form.

The lifetime of free positrons being annihilated in various condensed substances displaying molecular or covalent bonds has been measured. It is found that the structural properties of the molecules and of the materials have only a negligible effect on the free-positron decay rate. Each atom contributes to the annihilation with an effective number small xi of electrons irrespective of the bond. This effective number is found to depend on the atomic size and on the electronic configuration of the atomic external shell. In particular, the ratio of the number small xi to the number v of electrons in the external shell is found to increase with increasing atomic size. As a by-product, a criterion has been obtained for the identification of the free-positron component in the time annihilation spectra of polymeric compounds which display a complex spectrum.

Data obtained from quantum defects can be extrapolated along isoelectronic sequences and used to obtain phases for electron scattering by neutral atoms. An analytical theory is developed for the case in which polarization potentials are neglected.

An alternative theory uses simple model potentials which include polarization, and which contain an adjustable parameter. Numerical methods are used to determine the parameter from quantum defect data and to make extrapolations. Phases are obtained for electron-helium scattering; the result obtained for the scattering length is λ = -1.12±0.05.

Energy distributions of protons produced in dissociative collisions of H₂⁺ with H₂, Ar and Xe have been measured in the forward direction at 10 keV. An experimental method giving the corresponding excess kinetic energy spectrum in the centre-of-mass system of H₂⁺ is briefly discussed. The measured spectra are explained in terms of electronic and vibrational transitions to dissociative states of the H₂⁺ molecular ion. The dissociation cross section by vibrational transitions in the 1sσ_g state is estimated to be, respectively, σ_v = 1.4 × 10^-18 cm², σ_v = 4.3 × 10^-18 cm² and σ_v = 5 × 10^-18 cm² for hydrogen, argon and xenon targets. The dependence upon the target density is interpreted as a double collision dissociation process of the type H₂⁺ -> H₂^0* -> H⁺. The influence of ion source parameters is also studied. Comparison is made with theory and the agreement is found to be good if one makes allowance for the dependence of dissociation upon the internuclear axis orientation.

The polarization of the core of an alkali atom by the residual interaction with the valence electron which Hartree-Fock theory ignores should lead to a slight positive hole near the valence electron. This should reduce the oscillator strengths of low-lying transitions of the valence electron, where this electron is sufficiently close to the core to polarize it strongly. The physical picture is verified, and the magnitude of the effect calculated by treating the residual interaction as a perturbation. The reduction found in the oscillator strength of the lowest ns -> np transition of the valence electron increases from less than 1% in lithium to 16% in caesium.

The properties of Rydberg series of autoionizing levels of He converging to the n = 2 level of He⁺ are discussed on the basis of the assumption that the Schrödinger equation is approximately separable in the coordinates R = (r₁² + r₂²)^1/2 and α = tan^-1r₂/r₁. For each Rydberg series a `potential' in R is obtained, and the properties of the autoionizing states follow from the shape of the `potentials'. Energies of some of the states are calculated and compared with energies from experiment and more conventional calculations.

It is shown that, for optical dipole transitions between states which are degenerate in the limit of the nuclear charge Z being infinite, the dipole length formula is likely to give greater accuracy than the dipole velocity and dipole acceleration formulae. Formulae which are exact to O(1/Z³) are given for the transition matrix elements for the 2 ¹S-2 ¹P and 2 ³S-2 ³P transitions in the two-electron systems.

The contribution of direct electric octupole excitation to the generalized oscillator strength for electron impact has been calculated, and numerical results obtained for the 4 ²S-4 ²F transition in potassium. The contribution is insufficient to account for the observed cross section for this transition.

Average values of differential cross sections for electron capture by protons, as calculated using the wave version of the first Born approximation, are defined for neutral atomic targets that are randomly oriented with respect to the direction of incidence. Only for transitions between S states of the atomic target and the residual ion are the constituent cross sections independent of the orientation of the atomic frame of reference. The averaging procedure, used to remove the dependence upon the relative orientation between the atomic frame of reference and the direction of incidence, is exemplified by a calculation of the averages for transitions from S states to P states, and P states to P states.

The perturbed stationary state method is applied to

He(1s²)+H⁺ -> He⁺(1s)+H(1s)

but with neglect of momentum transfer. The calculations of Green et al. and of Sin Fai Lam show that the approximation is certainly untenable at energies above 16 keV. The experimental results of Hasted and Stedeford and of Stier and Barnett for the sum total of all processes involving charge transfer from helium to protons are exceeded from 3 to 10 keV. The differential cross section at small angles of deflection compares reasonably well with the experiments of Helbig and Everhart for energies below 5 keV.

Collisions involving one-electron capture and loss during the passage of 60-450 kev helium beams through hydrogen have been investigated by a technique which allows the fast metastable atom population of the beam to be determined. One-electron loss cross sections σ₀*₁ and σ₀₁ for metastable and ground-state helium atoms have been separately determined. σ₀*₁ is found to be between 13 and 3 times greater than σ₀₁ in the present energy range. It is shown that at energies between 80 and 250 kev, where the formation of He²⁺ is relatively unlikely, a satisfactory description of the charge equilibration of fast helium beams can be obtained in terms of charge-changing collisions involving He⁺ ions and ground-state and metastable helium atoms. Cross sections σ₁₀ and σ₁₀* for one-electron capture into metastable and ground states of helium have been estimated in the range 75-250 kev where it is found that σ₁₀* is between 30% and 50% of σ₁₀.

Cross sections have been measured for the reaction Li⁺ +e -> Li²⁺ +2e for electron energies between threshold and 3 keV. The cross section reaches a maximum value 4.15 × 10^-18 cm² ± 6% at an electron energy of 325 eV. At higher energies the results are similar to those of Rapp and Englander-Golden for helium when the latter measurements are scaled classically. The reaction Ba⁺ +e -> Ba²⁺ +2e has been investigated for electron energies between threshold and 2 keV. At an electron energy of 18 eV the cross section increases abruptly from 1.6 × 10^-16 cm² to 4.25 × 10^-16 cm². This is interpreted as the onset of autoionization (inner-shell excitation).

Thermal and suprathermal energy gas reaction rates for exothermic processes involving molecules or molecular ions are treated generally in terms of the adiabatic criterion. An absolute cross section function is derived in terms of a near-resonance energy defect which allows the determination of the dependence of reaction rate on temperature under conditions of complete or partial thermodynamic equilibrium. The reaction rates for seventeen exothermic processes are evaluated at 300°K and in many cases over a temperature range 150°K-4800°K: (i) under complete temperature equilibrium, (ii) with vibrational and rotational temperatures 300°K and (iii) at various vibrational temperatures of the diatomic partner but at a fixed translational and rotational temperature of T = 300°K. Special attention is given to aeronomically important reactions.

It is apparent from these calculations that the exit channels of the triatomic quasi-molecule of collision are sufficiently well determined by the energy levels of the infinitely separated initial reactants to make the consideration of such levels useful in predicting reaction cross sections.

A general expression has been obtained for the angular distributions of electrons scattered resonantly by molecules. Specific expressions are given for particular types of molecules, and angular distribution coefficients are tabulated for these types.

The transitions between excited states in hydrogen have been studied by perturbing a glow discharge with a high-frequency electric field. The relative phases and depths of modulation of the light emitted in the first four Balmer lines have been measured at 4, 8 and 15 MHz. A model based on the work of Bates et al. is developed and the predictions give reasonable agreement with experiment. A comparison is made with the results of previous theories. It is shown that the self-absorption of line radiation by the plasma has a critical effect on the response to the perturbation. Suggestions are therefore made concerning the best kind of plasma for testing the model and determining its parameters.

The equilibrium constant for dimerization K for quadrupolar gases has been computed under two limiting approximations: (i) K is calculated on the fixed relative orientations with a 12-6-5 potential model and then these values are averaged by giving equal weight to all orientations; (ii) the potential is `pre-averaged' over all orientations and then K is calculated. The former method is more appropriate for slow rotating molecules, and the latter one for fast rotating molecules.

The percentage of dimers has been calculated from the contribution of dimers to the second virial coefficient for N₂, CO₂ and C₆H₆ on the 12-6-5 and Lennard-Jones 12-6 potentials. It is found that the presence of an appreciable quadrupole moment in the molecules greatly increases the tendency of molecules to form dimers.

The effect of quadrupole-quadrupole interaction on the initial pressure dependence of viscosity and thermal conductivity is evaluated by assuming equal probability for all the relative orientations of the interacting quadrupoles. The quantitative agreement between theory and experiment is quite satisfactory. It is concluded that the method of approximating the three-body encounters by two-body collisions between a monomer and a dimer may serve as an important tool for the study of transport coefficients of gases at elevated pressures provided the dimer concentration and the force parameters for monomer-dimer interaction are evaluated accurately. A new approach is suggested for calculating approximately the force parameters of the dimer.

This paper discusses the conditions under which a system which has an energy-level separation of h(ω₁ - ω₂) may be expected to radiate at ω₁ - ω₂ if excited by two coherent waves at ω₁ and ω₂.

The fluorescence excitation spectra of liquid benzene, toluene, p-xylene, mesitylene, 2-methylnaphthalene, 1,6-dimethylnaphthalene, naphthalene, fluorobenzene and fluoronaphthalene, and of cyclohexane solutions of the first six compounds, were observed in deoxygenated systems at wavelengths not less than 195 nm. The observed dependence of the relative fluorescence yield on excitation wavelength and concentration is analysed in terms of a model kinetic scheme involving the formation of higher excimer states D** from the higher excited molecular states X**. Data are obtained on the relative efficiencies of internal conversion from X** and D** to the corresponding fluorescent states. The nature of the competing processes is discussed.

Expressions are derived for the fluorescence response functions of organic solution systems in which non-radiative energy transfer occurs and obeys either Stern-Volmer or Förster kinetics. Scintillation pulse shapes are evaluated for typical crystals, binary and ternary liquid solutions, and binary plastic solutions. The scintillation decay time is inadequate to describe the time characteristics of a solution system, and alternative parameters are introduced. Criteria for a fast solution scintillator are established, and binary liquid and plastic scintillators with improved time characteristics are proposed.

The effect of non-radiative energy transfer and diffusion on the fluorescence decay of phenanthrene in the presence of acridine was observed in six solvents of viscosity η = 0.4 cp to 64.6 cp. The system obeys Stern-Volmer kinetics at low η, Förster kinetics at high η and intermediate kinetics at medium η. At η = 64.6 cp the decay obeys the Förster relation. At lower η it agrees satisfactorily with the relation of Yokota and Tanimoto, but it is less rapid than that predicted by Voltz et al.

An analytic solution to the time-dependent diffusion equation for the F layer have been constructed through perturbation theory. The solution is expressed in terms of arbitrary scale-height factor. A closed-form expression for the decay constant in terms of the recombination rate coefficient and the scale-height factor has also been derived. The results for a scale height of 80 km and scale-height factor 1.4 have been compared with the experimental results of Wright and Fine at different hours after sunset when production processes are negligible.

We report measurements of the shift-to-width ratio and shift of an Ar II line emitted by a collision-dominated plasma as functions of electron density and temperature. The object of the experiment was to check the high-temperature régime of recently proposed formulae for the shift-to-width ratio.

The plasma conditions were varied by employing a useful property of the Z-pinch source. The electron density of the plasma was obtained from the Stark width of the line studied, and the electron temperature from the ratio of the total intensity of this line to that of a nearby Ar III line.

The results are in good agreement with theory.

High-dispersion emission spectrograms of the λ 4502 band of NH and the λ 4484 band of ND have been obtained from a positive column discharge through a mixture of helium, nitrogen and hydrogen (or deuterium). New bands at λ 5254 (NH) and λ 5015 (ND) belong to the same system, which is assigned to the c ¹π-b ¹Σ⁺ transition. Rotational analysis gives the following constants for b ¹Σ⁺ (in cm^-1):

The capital Lambda doubling in c ¹π is compared with calculations which assume `pure precession' with b ¹Σ⁺ and d ¹Σ⁺.

The afterglow from a microwave discharge through argon containing about 4% oxygen at a pressure of 40 torr is a stable relatively intense source of the O₂ (A ³Σ⁺_u -x ³Σ^-_g) Herzberg I band system. Using it, the vibrational assignment of Broida and Gaydon has been confirmed.

Absorption spectra of BaH and BaD have been obtained in the region 4000-3000 Angstrom by using a King furnace. The (0, 1) band of BaH and the (1, 0) and (0, 1) bands of BaD corresponding to the F²Σ -> X²Σ system of BaH have been observed and analysed. In addition, a new band of BaH at 3163 Angstrom has been observed and analysed, and it has been classified as the (0, 0) band of a new transition which may be called G²Σ -> X²Σ. The rotational constants have also been calculated.

The intensity of the first negative bands in a microwave discharge in flowing nitrogen is reduced much more than those of other systems by traces of oxygen: this is ascribed to removal of N₂⁺. Experiments on the afterglows are interpreted in terms of recombination of N atoms catalysed by N₂⁺ in the pink afterglow.

Arrays of Franck-Condon factors (q_{νprimeν−or+}) and r centroids (r_{νprimeν−or+}), which are closely related to relative vibrational transition probabilities, have been computed upon a Morse model for the B - X and C - X systems of the MgF molecule. This molecule is of importance to astrophysicists as it occurs in the spectra of sunspots. The r centroids for a given (νprime, ν−or+) band have been found to decrease smoothly with increase of corresponding wavelength λ_{νprimeν−or+}. The value of Δr identical with r_νprime + 1,ν−or+ + 1 -r_{νprimeν−or+} in a sequence has also been found to remain constant. The computed values of the r centroids indicate that the potentials are neither very anharmonic nor very wide.

A detailed study of the L emission spectrum of thulium 69 has been made with the help of a 40 cm curved crystal spectrograph of the transmission type. As a result five diagram lines for this element have been measured and reported for the first time. The line λ 1441.20 X.U. corresponds to the dipole transition L_IIIO_I(β₇), while the lines λ 1518.84 X.U. and λ 1497.32 X.U. correspond to the octupole and quadrupole transitions L_IIM_V and L_IIIN_II, respectively. The remaining two new lines λ 1230.59 X.U. and λ 1341.15 X.U. are due to the non-quadrupole transitions L_IO_I and L_IIN_II, respectively. Four transitions, viz. L_IIO_IV(γ₆), L_IM_IV(β₁₀), L_IM_V(β₉) and L_IO_IV,V already reported only by Sakellaridis have been confirmed. However, the lines L_IIIO_IV,V(β₅) and L_IIIM_I(l) also reported only by Sakellaridis could not be confirmed. The energy of the O_I level of thulium has been re-evaluated.

The model of two non-degenerate equations coupled by a potential of the order of α/r² is considered. It is shown that the cross sections for excitation tend to zero following laws enunciated by Wigner. However, it is possible to give the next term in the dependence of the cross section on energy, which is determined by a long-range potential.

It is suggested that previously unidentified lines in the solar spectrum between 170 Angstrom and 70 Angstrom are due to transitions of the type 3p^n-13d-3p^n-14p and 3p^n-13d-3p^n-14f in Fe IX-XIV, and that the lines are formed by spontaneous radiative decay following collisional transitions from 3pⁿ to 3p^n-14p and 3p^n-14f. The contribution of cascades from the 4p and 4f levels to the population of the 4s and 3d levels is estimated to be approximately 12% and 4% respectively.

An error in some earlier reported work on orthopositronium collisions with helium and on the pick-off quenching of orthopositronium by helium is pointed out, and corrected results are presented. The scattering length is decreased by 13% to 1·88 a₀, and the theoretical pick-off quenching rate is doubled. The revised quenching rate is still much smaller than experimental observations.