Table of contents

Close upper and lowei bounds are found on the binding energy of a system of N identical bosons interacting in one dimension by an attractive square-well pair potential. In the limit in which the potential has the shape of a δ function the bounds are found to be consistent with McGuire's exact solution. In three dimensions the limit of vanishingly small range, in which the two-particle binding energy is kept constant, gives infinite binding to the N-particle system, in agreement with Thomas's theorem Consideration of other limiting Proc.esses eliminates an apparent contradiction between Thomas and Svartholm.

The possible occurrence of a singularity g₁₁ = 0 in the collapse of a non-homogeneous spherically symmetric dust distribution has been studied A general formula for the frequency shift of radiation emitted from the surface of such a collapsing system is also given

The Lagrangian density of a classical field is considered, which depends on the partial derivatives of the field components up to the Nth order. Explicit expressions for the density of energy, momentum and angular momentum are obtained from Noether's theorem A Hamilton formalism for this kind of field is presented.

A dynamical problem in quantum optics is considered to be that of solving Schrodinger's equation with the initial state of the radiation field as a coherent state By cairying out a particular unitary transformation it is shown how one can arrive at a representation of the radiation field operators which is particulaily convenient for Proc.esses in which the interaction with the material system only involves one photon or a few photons. It is shown in which sense this representation is a counterpart of the Fock representation. In the derivation of the representation, which is exact throughout, some features are brought out, which may be interpreted as being semi-classical.

It is shown that Rubinow's method is a reformulation of the second Hulthén method The two methods give the same results.

The O(4) symmentry of hydrogenic atoms is used to interpret angular momentum nlm and Stark [vertical line]n₁n₂m eigenstates as different realizations of irreducible representations of the O(4) group Differential expressions are obtained for [vertical line]n₁n₂m shift operators, and for the [vertical line]nlm shift operator O_-¹ which changes l and m by -1

These are used to obtain the formula <n₁n₂m|nlm'> = (-1)^2n₂+|m|-m2C[1/2(n-1), 1/2(n-1), l, 1/2(m+n₂-n₁), 1/2(m+n₁-n₂), m'] for the transformation matrix elements connecting the two sets of states where, in the notation of Rose, C(J₁J₂J, m₁m₂m) is the angular momentum Clebsch-Gordan coefficient

It is shown that the unrestricted Hartree-Fock equations may be solved by expanding the orbital wave functions in perturbation series with parameter Z^-1/2 The work of a previous paper is extended to the calculation of the orbital functions to second order, and the energy to fifth order A similar calculation is made for the restricted Hartree-Fock approximation The higher-order energy coefficients are estimated Using these estimates, the restricted Hartree-Fock energy sums to -2 861 68 au, which is the value obtained by Roothaan et al, while the unrestricted Hartree-Fock energy sums to -2 877 99 au, which is the value obtained by Froese.

A method, based upon time-dependent Hartree-Fock theory, is developed for calculating the efficiencies of two-photon processes The lifetime for two-photon decay of He(2 ¹S) is calculated with the result 2 0 × 10^-2 sec. Estimates for the cross section for the two-photon ionization of the ground state of helium are given for photon energies near the two-photon threshold. For incident photons at a wavelength of 1000 Å the cross section is 5 × 10^-31I cm¹ w^-1, where I (w cm^-2) is the intensity of the incident photons.

Positions and widths of six series of autoionizing states 2pnl of the neutral beryllium atom are calculated (i) by direct solution of 2s-2p close-coupling equations for the system Be⁺ + e^- in the elastic scattering region, and (ii) by solving the same equations above the 2p excitation threshold and using the many-channel quantum defect theory to extrapolate below threshold Results of the two methods are compared. By extrapolation below the 2s threshold, the positions of the bound states of the neutral atom are calculated allowing for configuration mixing.

The electron affinities of CH, NH, SiH and PH are estimated to be 1 61, 0 22, 1 46 and 0 93 ev, respectively. These results are based on alleged Hartree-Fock results for CH^-(X ³Σ^-), NH^-(X ²II₁), SiH^-(X ³Σ^-) and PH^-(X ²II₁) molecules at the R_e (calc) values of the parent neutral systems previously reported by Cade and Huo. However, a correction due to the differential importance of the `correlation' energy for AH as against AH^- is necessary in order to obtain good estimates of the electron affinity. Two methods previously used, one by Clementi et al and the other by Cade, are employed to give values which are believed reliable to ±0 1 to ±0 2 ev There are no firm experimental electron affinities for these four systems These results, consistent with the specific electron configurations, imply that consideration of the electron affinities of AH molecules should be linked with the parent separated atom, not the related isoelectronic united atom, in those cases where the Proc.ess ωπⁿ + e^- -> ωπⁿ⁺¹ is involved.

The Vainshtein approximation for electron-atom collisions may be based on either the post or the prior wave function Both versions involve a supplementary peaking approximation In using the prior function Vainshtein and his colleagues introduced an arbitrary modification of the peaking approximation The removal of this modification alters the cross section considerably. In the case of atomic hydrogen it leads to cross sections for 1s-2s and 1s-2p excitation and for ionization which are markedly different from those based on the post function For a real effective charge the prior function gives a low-energy maximum which exceeds that of the first Born approximation, which in turn exceeds that of the post function No shaip peak appears near to threshold The accuracy of the Vainshtein approximation remains an open question It is tentatively suggested that the post function might with advantage be used for inelastic collisions and the prior for superelastic collisions

Cross sections σ₁₀ and σ₀₁ for one-electron capture and loss have been determined in the energy range 60-400 kev by studying, respectively, the charge neutralization and stripping of helium ion and atom beams during passage through `thin' targets of hydrogen and helium In the studies of electron loss, measurements made with helium atom beams prepared by charge-transfer neutralization of He<sup>+</sup> ions during passage through various `thick' gaseous targets were found to yield apparent cross sections σ₀₁ which were dependent on the particular neutralizer gas. For example, in the energy range investigated, values of σ₀₁ for helium atoms in hydrogen were found to increase by between 18% and 40% when hydrogen was substituted for helium as the neutralizer gas, an effect believed to be due to the formation of an increased fraction of metastable helium atoms in the fast beam. The apparently strong influence of a helium metastable atom component in this case suggests that a satisfactory description of the charge equilibriation of fast helium beams during passage through thick gaseous targets cannot always be given in terms of a simple three-component system Previous data on electron capture and loss obtained by the use of such a treatment must therefore be interpreted with extreme care

The asymptotic variation of the amplitudes at high impact energies for the non-resonant H⁺ + D(1s) -> H(1s) + D⁺ and symmetrically resonant H⁺ + H(1s) -> H(1s) + H⁺ Proc.esses are calculated approximately in the second Born approximation using the final-state Green function G_f Although agreement with the results of Drisko is obtained for scattering at small angles by approximating the Coulomb functions of G_f by plane waves, this agreement may be fortuitous for two reasons (I) an exact calculation might yield ambiguous answers, (II) sums of contributions from an infinite number of discrete intermediate states, each varying with the same power of impact energy, may not be finite. In the case of symmetrical resonance not only do contributions also originate from the angular region near θ = π, but all but one of these calculated partial amplitudes is dominated by the corresponding first Born amplitude, however, the same calculational difficulties, (I) and (II), also occur for this case Contributions from the 1s intermediate state diverge for each part of the post interaction V_f, but these divergences cancel when the complete interaction V_f is used

A simple technique previously used by the authors in electron-atom scattering is modified to describe a charge-transfer Proc.ess and calculations performed for the case of protons colliding with atomic hydrogen. A comparison with experimental and other theoretical results is made It is found that the method is less successful in describing charge-transfer Proc.esses than in describing exchange Proc.esses in electron-atom collisions

A new apparatus, based on the crossed-beam principle, has been used to measure the cross section of Li⁺ in the reaction Li⁺ + e -> Li²⁺ + 2e for electron energies up to 1000 eV. The cross section reaches a maximum value 4 0 × 10^-18 cm² for electrons of 350 eV energy. The maximum total errors of the measurements are assessed to be ± 25% near threshold and ± 10% at higher electron energies. Within these limits the results agree with the measurements on Li⁺ by Lineberger, Hooper and McDaniel in 1966. The present measurements also agree very closely with cross sections for the ionization of neutral helium measured by Smith in 1930 when the latter are multiplied by the classical scaling factor (χ₂/χ₁)². Here χ₁ and χ₂ are the ionization energies of Li⁺ and He respectively

Azimuth angles of shower particles produced in 24 and 27 GeV/c proton interactions have been analvsed by two methods (i) the method of consecutive angles, (ii) the `minimum fall-out' method It is concluded that azimuth angular distributions for stars with both N_H 4 and N_H 4 are consistent with isotropy and that spurious asymmetries can result from biased sampling

The angular distribution of neutrons with an incident energy of 15 7 MeV elastically scattered from protons has been measured in the angular range between 180° and 60° in the centre-of-mass system using the nuclear emulsion technique. The results are consistent with the previous measurements at 14 MeV and yield a value of σ(180°)/σ(90°) = 1 07 ± 0 02. By combining the present results with others at 14 MeV and fitting to an equation of the form dσ/dΩ = A + B cos θ + C cos² θ, values were obtained of B/A = -0 033 ± 0 022 and C/A = 0 042 ± 0 026.

The intensity of neutron inelastic scattering by phonons is considered, using the symmetry properties, by group theoretical methods (the extension to scattering by magnons is briefly discussed). Under some circumstances there are selection rules forbidding scattering at particular K values and, by examining the phonons at a series of points K = k+G for different reciProc.al lattice points G, it is often possible to make a definite symmetry identification of the various branches. It is hoped that the results will be of use in the interpretation of experimental data in more complicated crystals than have so far been studied. The fluorite and cuprite structures are considered as examples which might soon be profitably investigated. Standard projection operator techniques are employed.

The interaction of X rays and neutrons with phonons in perfect crystals is treated by the methods of the dynamical theory of X-ray diffraction. It is found that phonon scattering Proc.esses make a contribution to both the real and imaginary parts of the refractive index of the crystal The intensity of the phonon-scattered peaks is calculated, and it is found that in the absence of elastic scattering the expressions for the intensity reduce essentially to those given by the kinematic theory. When the incident wave can be elastically scattered (at a Bragg angle), then the spatial variation of the amplitudes of the incident and elastically scattered waves is the dominant factor in deciding the amplitudes of the phonon-scattered waves Thus in the Bragg reflection condition the phonon peaks are reduced in intensity owing to primary extinction, and the intensity of the inelastic scattering is asymmetrically distributed about the Bragg peak. In the Laue reflection condition the Pendellosung fringes in the elastically scattered waves are accompanied by a similar but less marked variation in the inelastically scattered waves, and the Borrmann effect of anomalous transmission also has its counterpart in the phonon scattering The anomalously low absorption coefficient in the Borrmann effect will have a temperature-dependent contribution due to inelastic scattering Some of these effects should be observable experimentally if methods are used to distinguish elastically and inelastically scattered radiation

The neutron diffraction from single crvstals of face-centred cubic K₂IrCl₆ has been investigated below the transition temperature of 3 05 K The spins are found to order in an antiferromagnetic type IIIA face-centred cubic pattern, and the measurements are consistent with the direction of the spins lying along the unique axis of the magnetic unit cell This is in agreement with theoretical predictions made from the known interaction parameters by several authors

In measuring g' by a macroscopic experiment the ratio of total magnetization to observable angular momentum is found To isolate spin and orbital effects for outer valence electrons, corrections for nuclear magnetization and for core diamagnetism are required. Formulae are given, based on an assumed knowledge of the electron density distribution for core and valence electrons, and on the nuclear magnetic moment The correction for the surface dipole charge distribution is not negligible for metals.

Formulae are derived for the boundary tensions σ of the triangular and honeycomb lattices in zero magnetic field Transformations are used to obtain σ for two lattices in arbitrary field for both these lattices σ becomes zero along the transition curve `Onsager boundaries' are shown to yield the correct value of σ for the honeycomb lattice in zero field

An analysis has been made of published data on the frequencies of modes of vibration in quartz which have been measured by infra-red and Raman spectroscopy It is shown that the measurements are consistent with one another when account is taken of the fact that certain frequencies are split by the macroscopic electric fields associated with some of these modes All of the measured frequencies of the symmetric, antisymmetric and doubly degenerate modes have been used as data to determine the parameters of two new models for the lattice dynamics of quartz, one involving only short-range forces and the other including Coulomb interactions New measurements have been made by the method of inelastic neutron scattering of the frequencies of certain normal modes with wave vectors along the z axis of the crystal While both models give good qualitative agreement with these measurements, the model which includes Coulomb interaction is to be preferred since it agrees better with the spectroscopic data

The relation between the lattice dynamics of quartz and the α-β transition at 573 °c is discussed It is shown that the same force constant models give results in good agreement with observed Raman frequencies in β-quartz

The six third-order elastic constants of a pure specimen of indium antimonide have been measured at room temperature These have been obtained from measurements of the change in acoustic velocity with applied stress using an improved version of the ultrasonic `sing-around' system. Possible contributions to the constants other than those arising directly from interatomic forces are considered and are shown to be negligible in our measurements The results are interpreted and discussed in terms of a recent theory in which the third-order constants of covalent materials are dominated by three anharmonic force constants between first- and second-nearest neighbours.

In gyroptic gases and liquids the cubic susceptibility tensor contains, in addition to the expected part involving the cubic molecular polarizability tensor, an additional contribution involving the quadratic molecular polarizability tensor The additional contribution arises from the quadratic polarization through the local field corrections and will be involved in general mixing phenomena, the Raman effect and two-photon absorption, but has no influence on the Kerr effect or third harmonic generation

Plasmas generated by focusing Q-spoiled ruby lasers onto solid targets are used as spectroscopic sources for the study of far vacuum ultra-violet emission spectra of ions of astrophysical interest. New, inner-shell transitions of fluorine-, oxygen- and nitrogen-like ions are classified in S X, K XI, XII, XIII, Ca XII, XIII, XIV, Sc XIII and Ti XIV Other new classifications in Cl XII to XIV are also listed

Spectral lines of T1 VII to XII and isoelectronic transitions of wavelengths between 250 and 600 Å are classified Identifications of Fe XIV solar lines are confirmed, including the visible coronal line at 5302 86 Å The ions were excited in a plasma produced by a 500 MW laser beam focused onto metal targets. The target of the pure metal was situated at the Sirks focus of a three-metre, normal-incidence spectrograph

Absorption spectra in the wavelength region 1600-3500 Å have been obtained for thin films of frozen Ar, Kr and Xe containing low concentrations (<or= 1^o_o) of atomically dispersed Cd and Zn The spectra of lightly doped samples show only a strong band with triplet structure in the vicinity of the impurity atom ns²¹S_o-nsnp ¹P₁^o resonance transition. As the concentration of Cd or Zn is increased, a variety of additional spectral features appear It is shown that many of these bands may be attributed to the dimers Cd₂ or Zn₂ An intense band with well-defined doublet structure was present in most samples on the short-wavelength side of the ¹P₁^o band It is likely that this band is due to a transition starting from a metastable triplet state of the Cd or Zn impurity.