Table of contents

The problem of obtaining the mean density matrix and the Boltzmann equation for an electron in the presence of a random array of weak scatterers of arbitrary density is studied. The density matrix has previously been studied by recombining its averaged perturbation expansion, but this is criticized on the grounds that it violates sum rules which are shown to apply to the mean density matrix. A new method is given which gives an integral equation for the mean density matrix having a form very similar in structure to the Boltzmann equation and automatically satisfying the sum rules. This method can be applied at arbitrary densities of scatterers and it is shown that the mean density matrix is Lorentzian near the mean free electron curve E = h²/2m k² + Nu but Gaussian far from this curve, and the new form can be applied near band edges. The Boltzmann equation can be easily obtained from the density matrix, and its kernel can be expressed as a graphical series which is not based on the density of scatterers.

The energy of the nuclear ferromagnetic state in the presence of conduction electrons, calculated by using the hyperfine interaction between the electron and the nuclear spins as perturbation, is compared with the energy calculated earlier avoiding the use of perturbation theory. It is shown that the two values do not agree, which means that perturbation theory cannot be applied in this case.

The effect of correlation between open shell and continuum electrons on the photodetachment cross section of C^- is examined in a modified central field exchange approximation. The results show improved agreement between the dipole length and dipole velocity formulations, but continue to display an incorrect energy behaviour at low energies.

In a recent paper, Grimley has revived interest in a paper by Bopp. Our paper shows the connection between a modified Bopp's method and a thirty-year old method by Bacher and Goudsmit. We apply this method to the nuclear energy levels of ²⁰⁵Pb and ²⁰⁴Pb. Our results are fairly good in some cases, but in others it is clear that one needs to improve the basic method. We believe that the modified Bopp's method is superior to the Bacher-Goudsmit method in that one can reformulate it in a straightforward way to remove its deficiencies.

The fully coupled Hartree-Fock approximation is used to calculate the dipole and quadrupole polarizabilities and shielding factors of Be, B⁺ and C²⁺.

The second Born approximation is used to calculate the cross sections for the excitation of the 2p states of atomic hydrogen by proton, electron and positron impact, allowance being made for the effect of virtual transitions involving the 1s, 2s and 2p₀, ± 1 intermediate states.

In the case of proton impact, provided coupling to the 2s intermediate state is excluded from the second Born approximation calculations, good agreement is obtained at high energies with the cross section given by the exact numerical integration of the coupled differential equations connecting the 1s and 2p states. Satisfactory accord is also obtained with the results of close-coupling calculations in which full allowance is made for the 1s, 2s and 2p₀, ± 1 states, but exchange is neglected, for the scattering of slow electrons. On the other hand, it is found that the agreement with the cross section resulting from analogous calculations for positron impact is not good.

Methods hitherto used for allowance for exchange in the treatment of electron impact ionization of atomic hydrogen are discussed. A simple method based upon Ochkur's simplification of the Born-Oppenheimer approximation is also studied and found to give very good results.

Seaton's semi-classical impact parameter meod is extended to electric quadrupole transitions. It is justified for Δl = 2 but not Δl = 0 transitions, though it is more sensitive to choice of cut-off than for electric dipole transitions. Results are presented for several transitions in hydrogen and helium. Where they can be compared with experiment (He 1¹S → n¹D) they are in reasonable accord, except close to threshold.

The Born approximation is used to obtain differential and total cross sections for inelastic electron scattering from benzene. The three lowest excited singlet states are treated (i.e. those of symmetry ¹B_1u, ¹B_2u and ¹E_1u). Simple π-electron antisymmetric molecular orbitals are used as the electronic wave functions, and the cross sections are averaged and summed over rotational and vibrational states. The effects of vibronic interactions on the cross sections are included.

A method, based on Fourier analysis of harmonics, has been developed for measuring the rate of decay of the fundamental mode of the thermal neutron density distribution which is established in a cylinder of water after thermalization of an injected burst of fast neutrons. Measurement of this decay for a range of water depths enables determination, by extrapolation, of the decay constant vΣ_a of an infinitely large system. The 2200 m sec^-1 thermal neutron microscopic absorption cross section of hydrogen has thus been measured. The value obtained, 326 ± 3 mbn, is in good agreement with that obtained by most previous workers by a variety of other methods. The thermal neutron diffusion coefficient was also measured, although the experiment was not optimized for this. The value obtained, D = 35 200 ± 400 at 22 °C, also agrees very well with previous results.

Spin-lattice relaxation times have been measured for ¹H and ¹⁹F nuclei in benzene and several benzene derivatives. In most cases the measurements have been made from the supercooled liquid temperature up to the liquid-vapour critical temperature. The results are analysed in terms of direct dipole-dipole and spin-rotation magnetic interactions. Ring and side group ¹⁹F nuclei and methyl side group protons show strong spin-rotation relaxation effects above room temperature. Ring protons in some cases show a possible weak spin-rotation relaxation near the critical temperature. Relaxation time measurements for C₆H₆ - C₆D₆ mixtures are used to calculate reliable molecular motion correlation times from the dipole-dipole relaxation for many of the benzene derivatives. A detailed account is given of the physical processes involved in the spin-rotation interaction contribution to nuclear magnetic relaxation in liquids. It is shown that the relaxation due to the spin-rotation interaction is largely independent of the details of the molecular reorientation. In several cases the molecular spin-rotation interaction tensor has been calculated from the relaxation time measurements and the values are found to be in reasonable agreement with values obtained from molecular beam measurements. Quantitative evidence is given for the validity of a simple relation (in an equation) between the spatial and angular momentum correlation times. In these liquids spatial correlation times are of order 10^-12 sec, whereas angular velocity correlation times are of order 10^-13 sec. The former decreases and the latter increases with temperature and at the critical temperature they are comparable in magnitude.

Recent experimental results for octahedral fluorine complexes of the type (MF₆)^4- indicate that the amount of unpaired spin in 2p orbits on the ligand increases quite rapidly as M is changed from Mn²⁺, 3d⁵ to Co²⁺, 3d⁷ to Ni²⁺, 3d⁸. It is suggested that this effect can be qualitatively understood by assuming that there is equal charge distribution in the different complexes.

A calculation of the relaxation of Cu²⁺ in potassium zinc Tutton salt has been made on a crystal field model. The magnitude and anisotropy predicted are in good agreement with experiment. The results imply that the tetragonal field splitting δ is about 7700 cm^-1 and the cubic field splitting Δ₀ about 8700 cm^-1.

The spin-lattice relaxation of copper in K₂SO₄(Cu_x, Zn_1-x)SO₄. 6H₂O has been measured at temperatures between 1.2 °K and 14 °K in magnetic fields parallel to the crystallographic (110) plane. The microwave frequency was 9455 Mc/s. When the copper concentration x was sufficiently small (10^-4) the relaxation appeared to be mainly by the van Vleck mechanism. It was then adequately accounted for in magnitude, and in its dependence on temperature and on the direction of the magnetic field by expressions given by Stoneham in the preceding paper.

By combining zero-field nuclear magnetic resonance techniques with specific heat measurements below 1 °K, the effective nuclear fields at the two Mn sites in ferromagnetic Mn₅Ge₃ have been uniquely determined as 195 kOe at the 4(d) site with a well-resolved quadrupole structure, and 399 kOe at the 6(g) site with a broad unresolved quadrupole structure.

e indirect evidence for the existence of the Lorentz `local time' effect, which is also a feature of the special theory of relativity, is discussed and a fairly simple direct experiment, using radio waves in the centimetre region, for testing the theorem of the relativity of simultaneity of events is proposed.

In this paper, the fluctuations of electron densities from the steady state are discussed. Expressions for the variances of the fluctuating variables (electron concentrations in the conduction band or in the traps) are calculated using a difference equation in the probabilities. For the case when there is one set of traps the difference equation is bivariate. This method is applied to a case of cyclic transitions far from equilibrium, previously considered by van Vliet. Equations for the variances are obtained, and reduce to those found earlier when certain approximations are made. In the case of very large noise these approximations may not be valid. The equations are solved for the variance of the electron density in the conduction band, but there is no need to determine the probability distribution itself.

The theory of dipole scattering in non-degenerate semiconductors has been formulated without making the Brooks-Herring approximation, namely that the potential distribution is slowly varying. The applicability of the Brooks-Herring theory is shown to depend on a parameter γ² = 5.8 × 10^-15n/T², where n is the free-carrier density and T is the absolute temperature. For moderate doping strengths, for example N_D similar N_A similar 10¹⁷ cm^-3, where N_D and N_A are the density of donor and acceptor ions respectively, the Brooks-Herring result is accurate when 0.001 ⩽ γ² ⩽ 0.07. For heavily doped samples, for example N_D similar N_A similar 10²⁰ cm^-3, the present treatment shows significant departures from the Brooks-Herring theory.

The mobility of photoexcited carriers in high resistivity n-type silicon has been measured from 4.2 °K to 300 °K. Carriers excited by infra-red radiation of about 10 μm wavelength showed hot carrier effects and had a temperature-independent mobility below 10 °K. The use of visible radiation caused the neutralization of ionized impurities and under these conditions photoexcited carriers had a mobility limited by lattice scattering only. This resulted in a mobility of 1.3 × 10⁶ cm² sec^-1 v^-1 at 4.2 °K.

e infra-red lattice bands of germanium have been remeasured and interpreted as multiple phonon bands involving four characteristic phonon energies. These energies are found to agree closely with the energies of the four maxima in the density of states curve calculated from the Cochran `shell' model. This observation supports the view that the form of the lattice bands is determined primarily by the summation over states rather than by specific selection rules.

e monotonic relationship between r centroid and band frequency has been investigated in detail, and the circumstances which cause r centroids to increase or decrease with increasing frequency according as r_e' < or > r_e" are discussed. Except in cases where change in molecular properties with internuclear separation is of specific interest, band frequency (or wavelength) may thus often be more conveniently used as an alternative to the r centroid as an independent variable in the discussion of the change of such molecular properties as transition probabilities from band to band in a system.

Nitrogen ions of velocity 1.6 × 10⁸ cm sec^-1 to 6.0 × 10⁸ cm sec^-1 passed through a carbon foil of thickness about 500 Å which served both to increase the degree of ionization and to leave the ions in excited states. It is shown that by measuring the relative intensities of the de-excitation emission lines as a function of ion velocity the degree of ionization of the emitting ions can be determined unambiguously, in confirmation of the suggestion made by Kay in 1963. Measurements of line intensities as a function of distance from the exciting foil yield directly the mean lives of excited states in singly and doubly ionized nitrogen. The techniques should be capable of extension to very high degrees of ionization.

The optical constants of bulk specimens of aluminium were measured over the range 0 6 to 4.6 eV. An absorption band at 1.55 eV reported for evaporated films of aluminium by Ehrenreich et al. in 1963 was confirmed for the bulk specimens.

Single crystals of indium, with both hand-polished and electro-polished surfaces, were studied over the range 0.15 to 4.0 eV, small absorption peaks being found at 0 40, 0.75 and 1.50 eV. These peaks were most sharply defined in the basal plane. The identification of these transitions is briefly discussed.

Little justification is found for subjecting the infra-red measurements of indium to a free-electron analysis although a relaxation wavelength of just over 6 micrometres, for both principal directions (compared with 11.5 micrometres for aluminium), is suggested.

First and second approximations to the change in variance of the line profile arising from dispersion and other physical aberrations are obtained by means of series expansions. To a first approximation the increase in variance (in 2θ) due to dispersion is 4μ₃ tan⁴θ/λ³, and the change in variance resulting from other physical aberrations is 4R₁μ₃ tan²θ/λ², where μ₃ is the third moment of the line profile about its centroid and R₁ is the derivative with respect to λ of the logarithm of the physical factor in question. In the first approximation the effects of the angular-velocity factor, the polarization factor, the trigonometric factor specific to the instrument, and such response variations as absorption in the filter and the quantum-counting efficiency are additive. In the second approximation cross terms appear.

e radial distribution of a stream of electrons after drifting through a gas in a uniform electric field has been theoretically analysed. The solution takes into account electron ionization and the boundary conditions imposed by the electrodes. The distribution has been experimentally verified for hydrogen, and values of the ratio of the diffusion coefficient to the mobility coefficient have been obtained for 15 < E/p₀ < 100 V cm^-1 torr^-1.

An expression is derived for the frequency spectrum of the electron current fluctuations in a gas discharge, with a uniform electric field and steady illumination of the cathode. The expression is modified for those cases in which the positive ion current must be taken into account, and it is shown how the distribution of frequencies depends on the type of secondary process present. The application of the results to experimental measurements is discussed.

The photoionization cross section of cadmium has been measured in the range 1150 Å to 1400 Å. The cross section at the series limit was found to be 0 32 ± 0 03 Mbn and the f-number contribution for the range investigated is 2 4 × 10^-3.

The first- and second-order Raman spectra of calcium fluoride have been measured. With the crystal at room temperature the structure observed is in good agreement with earlier measurements of infra-red absorption and vibronic satellites. Some additional structure observed in the second-order Raman spectrum is ascribed to overtones about k = 0 in the phonon spectrum.

A simple expression is obtained for the phase shift produced by an atom having a quadrupole moment, assuming {l(l + 1)}^1/2/k to be large compared with the mean atomic radius. This gives results which are in good agreement with Born phases for scattering by H 2p, but which are not in agreement with phases calculated by Myerscough and McDowell for scattering by carbon.