The exact formula for the electrical conductivity of a metal, derived by Kubo, Greenwood, and others, is evaluated using Van Hove's methods for quantum mechanical transport problems. The evaluation is restricted to the case of elastic scattering by impurities or lattice vibrations, but completely avoids the use of the customary random phase assumption. In the limit of weak coupling our method yields formulae for the conductivity that are identical with those derived from the Boltzmann equation. The accuracy of the weak coupling approximation is investigated and it is found to be accurate as long as h/τ<<η, where τ is the collision time and η is the Fermi energy. A general treatment of the conductivity in an anisotropic metal is given and an explicit expression for the conductivity is written down in terms of the eigenvalues and eigenvectors of an integral equation. A general and rigorous derivation of the conventional Boltzmann equation is given using Van Hove's results on the quantum mechanical master equation.
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G V Chester and A Thellung 1959 Proc. Phys. Soc. 73 745
C T Chudley and R J Elliott 1961 Proc. Phys. Soc. 77 353
The incoherent inelastic scattering cross section of slow neutrons from liquids is calculated using a simple model in which the liquid is assumed to have appreciable short range order in a quasi-crystalline form. Diffusive motion takes place in large discrete jumps, between which the atoms oscillate as in a solid. The model predicts a definite, easily calculable cross section which is not dominated by diffusion effects as when continuous diffusion is assumed, but shows a characteristic variation with angle which could be looked for experimentally. The related pair correlation functions are dominated at small r and t by vibrational effects. Although simple and extreme the model explains several aspects of the observations of Brockhouse and Pope in 1959 and others. A brief discussion of the coherent scattering cross sections for the model is given although explicit formulae are not obtained.
D A Greenwood 1958 Proc. Phys. Soc. 71 585
The motion of conduction electrons in a metal in an electric field, scattered by an irregular static potential, is considered; this model is applicable to the resistance due to lattice waves at high temperatures, and to imperfections at any temperature. In §2 the Boltzmann equation is re-derived without the customary perturbation theory, avoiding the usual necessity of averaging over phases of different electron states after repeated small time intervals. The assumption that the scattering centres are distributed at random in the crystal is alone sufficient. The theory is, however, still dependent on assuming that h/τ<<kT, where τ is the collision time, as is the usual perturbation theory. §3 gives a general formula for the conductivity of the model, not subject to any assumption. This helps to justify an argument by Landau according to which the usual theory is valid provided h/τ<<η, where η is the cut-off energy of the Fermi distribution. No direct evaluation of the formula has been achieved.
S F Edwards 1965 Proc. Phys. Soc. 85 613
The probability distribution of the configurations of a polymer consisting of freely hinged links of length l and excluded volume v is studied. It is shown that the interaction of the polymer with itself can be represented by considering the polymer under the influence of a self-consistent field which reduces the problem to an equation like the Hartree equation for an atom. This can be solved asymptotically, giving the probability of the nth link of the polymer passing through the point r to be (L)exp[-27{r-(5/3)3/5(v/3πl)1/5L3/5}2(1/20Ll)] where L = nl is the length along the polymer and (L) the normalization. Thus the mean square of r, r2, is (5/3)6/5(v/3πl)2/5L6/5.
The theory is extended to polymers of finite length, to the excluded random walk problem and to n dimensions.
R S Dadson et al 1967 Proc. Phys. Soc. 92 1115
In an earlier paper some determinations of the virial coefficients of carbon dioxide and ethylene, planned as part of a continuing investigation of the (p,v,t) properties of gases and gaseous mixtures, were described. These measurements utilized the `series-expansion' technique which it was intended to use consistently throughout the series. In view, however, of the aims of the work and the consequent demands on accuracy, it was decided, as a precaution against the possible intrusion of undetected systematic error, to carry out a limited series of measurements on one gas - carbon dioxide - by an independent method. For this the `piezometer' technique developed by Michels and his collaborators was chosen.
This paper compares the values of the second virial coefficients obtained over the temperature range -10 to 200°C in the two series. These proved to be in very precise agreement, the discrepancies at corresponding temperatures being nowhere greater than 0.03 × 10-3 amagat (0 7 cm3 mole-1) and on the average of the order of only 0 01 × 10-3 amagat (0.2 cm3 mole-1). The significance of these results is discussed.
J P Russell 1965 Proc. Phys. Soc. 85 194
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.
E H Putley and W H Mitchell 1958 Proc. Phys. Soc. 72 193
The electrical conductivity and Hall effect have been measured over the temperature range 20°K to 500°K on single crystals of silicon with extrinsic carrier concentrations between 2 and 5 × 1012 cm-3. The Hall mobility for electrons and holes can be represented between 100° and 300°K by the expression 1.2 × 108T-2 and 2.9 × 109T-2.7 respectively. Both these results indicate a higher Hall mobility than has been previously reported, and the result for holes is greater than values reported for the drift mobility. From the results between 350° and 500°K the expression ni = 3.10 × 1016T3/2 exp -0.603/kT was obtained for the intrinsic concentration. Attempts were made to estimate the total impurity concentration in these specimens. The variation of extrinsic carrier concentration with temperature and the effect of impurity scattering at 20°K both indicate that the total concentration of impurities is less than 1014 cm-3.
B J Hiley and G S Joyce 1965 Proc. Phys. Soc. 85 493
The theory of closed-form approximations in crystal statistics developed by Yvon, Rushbrooke and Scoins, Fournet, and Domb and Hiley is extended to include long-range forces. This extension is used to obtain successive approximations for the entropy, specific heat and the susceptibility in zero field for the disordered state, for arbitrary long-range forces. The general forms of the approximations are studied to determine the shape dependence of the exact properties when the specimen is ellipsoidal and the interactions are dipolar. It is found that, in the disordered state, the susceptibility is shape-dependent while the specific heat and entropy are not. A numerical investigation of the susceptibility using the first two closed forms (a Bethe-type approximation) with an interaction potential of type r-p shows that, as the range of the force increases, the critical point depends predominantly on the dimensionality and becomes insensitive to the detailed structure of the lattice. As p → d (the dimensionality of the lattice), it is found that this approximation gives the exact behaviour in the limit. This conclusion, however, is not valid for dipolar forces.
T H K Barron and M L Klein 1965 Proc. Phys. Soc. 85 523
For an unstressed solid a set of second-order elastic constants cαβστ can be chosen such that the strain-energy density, the first-order stress-strain relations and the equations of motion all have simple forms; the cαβστ are symmetric with respect to the interchanges (α, β), (σ, τ), (αβ, στ). In the present paper different ways of generalizing the cαβστ when the solid is under stress are discussed and compared, both for a general stress and for the special case of isotropic pressure. It is suggested that the symbols cαβστ (Voigt: cij) should be used for the elastic constants defined from the stress-strain relations and Cαβστ (Voigt: Cij) for the elastic constants defined by expressing the strain-energy density as a function of the Lagrangian strain coordinates. The relation of these elastic constants to the dependence of strain energy on homogeneous strain is discussed in detail, and explicit expressions given for cubic crystals under isotropic pressure.
H Ryzko 1965 Proc. Phys. Soc. 85 1283
The avalanche current resulting from the release of a short (30 nsec) pulse of photoelectrons (5 × 104) at the cathode of a uniform field gap (5 cm) is observed experimentally. By this technique the drift velocities of electrons (v-) and of positive ions (v+) are measured. In dry air in the range E/p = 50-100 v cm-1 torr-1 at temperature 20 °c, v- = 256 × 103 E/p + 5 × 106 and in water vapour in the range E/p = 55-85 v cm-1 torr-1 at 20 °c, v- = 255 × 103 E/p + 8.9 × 106, with units cm sec-1 and v cm-1 torr-1. The values of v- obtained for dry air fit the values calculated by Heylen and are higher than those measured with the magnetic-deflection method by Townsend and Tizard.
The mobility of positive ions in dry air, which is constant in the range E/p = 45-70 v cm-1 torr-1 and is equal to 2.2 × 103 cm2 v-1 sec-1 (at 1 torr and at 20 °c), gradually decreases at higher E/p values. In water vapour the mobility of positive ions was constant over the measured range E/p = 50-90 v cm-1 torr-1 and is equal to 0.61 × 103 cm2 v-1 sec-1 (at 20 °c and at 1 torr). In humid air the value of mobility is lower than the value calculated from Blanc's law. This suggests a clustering of ions.
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J W Tucker 1967 Proc. Phys. Soc. 92 1143
An examination is made of the decoupling procedures which have previously been used in a Green function treatment of the spin-phonon interaction in paramagnetic crystals An exact equation connecting the Green function of the phonons with those of the spin system is derived.
T A Green 1967 Proc. Phys. Soc. 92 1144
The wave function used by Rudge as a basis for modifying the Ochkur approximation is generalized The results obtained with the new function cast doubt on the validity of the arguments used by Rudge and Crothers in support of Rudge's modification.
S J Czyzak et al 1967 Proc. Phys. Soc. 92 1146
Using methods described by Saraph, Seaton and Shemming in 1966, collision strengths have been calculated for the isoelectronic sequences 2p3, 2p4, 3p2 and 3p4.
A Joseph 1967 Proc. Phys. Soc. 92 847
A general study of a new system, known as a spin dipole, is made. Its properties are contrasted with those of a point dipole, a system which has recently received some attention. The Schrödinger equation for the eigenvalue problem associated with zero energy is solved. This makes it possible to give conditions for the existence of bound states for the finite (spin) dipole.
P L Altick and E N Moore 1967 Proc. Phys. Soc. 92 853
The structure of some low-lying autoionizing D states in helium has been investigated. Eigenvectors are presented which show that a plus-minus classification is appropriate. The relationship between structure and width is discussed.
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M Ryle 1965 Proc. Phys. Soc. 85 201
The exciting contributions made by radio astronomical observations over the last decade would not have been possible without the development of radio telescopes of increasing sensitivity and resolving power. In particular the study of the `radio galaxies' has depended very much on the availability of instruments capable of determining their positions with accuracy, investigating their detailed angular structure and their radio spectrum. These sources have a radio emission very much greater than that of our own Galaxy, and the mechanism by which this intense radiation is produced is of great astrophysical interest; the socalled `quasi-stellar' sources in which the whole radio emission originates in a region only about one hundredth of the dimensions of the galaxy, represent an extreme case involving unknown sources of energy.
By using radio telescopes of great sensitivity and angular resolving power, radio galaxies can be detected at very large distances; owing to the time taken for the radio waves to reach the earth, they are consequently observed as they were some thousands of millions of years ago, and such observations offer the possibility of exploring evolutionary changes in the Universe. In order to extend these types of observation, more powerful instruments became necessary, but the construction of larger steerable paraboloids of sufficient accuracy presents very great engineering difficulties. More economical instruments have been built by using fixed parabolic cr spherical reflectors, which can be supported from the ground at many points, but even with these instruments the cost becomes very great if an angular resolution better than five minutes of arc is sought.
Alternative methods of building large telescopes have been developed, in one of which advantage is taken of the fact that for most observations the sky can be regarded as unchanging over the period of observation; it is then not necessary to have the whole of the telescope present at the same time, and a large effective instrument may be `synthesized' by using two small elementary aerials arranged successively in all the positions present in the large area. The observations may be combined in a computer to produce a map of the sky having a resolution and signal-to-noise very much greater than that associated with the small elementary aerials.
Four large instruments have been built using this principle, two of which have been in use for six years. They have been particularly valuable both in enabling large resolving powers to be attained at long wavelengths in order to establish the radio spectra of the sources, and in extending the observations to the weakest sources yet observed.