Table of contents

This paper, which is the written version of the 1962 Guthrie Lecture, describes the physics underlying the radio astronomical researches at Jodrell Bank. The following topics are covered: (i) the physical theory of the scattering of radio waves from the ionized columns of meteor trails based on the classical expression for the scattering cross section of a free electron; the effects of resonance and polarization, the manifestation of the diffraction pattern, the measurement of velocities, and the application to the measurement of certain physical parameters in the high atmosphere; (ii) the possible alternative processes occurring in the scattering of radio waves from auroral formations; (iii) the lunar scattering of radio waves, the effects of libration which give rise to the short period fading and the Faraday rotation in the ionosphere which is responsible for the long period variations in signal strength; (iv) the origin of radio waves from the galaxy and their explanation in terms of synchrotron emission from relativistic electrons in the interstellar field, and thermal emission from ionized hydrogen clouds; (v) the problem of the energy requirements in the localized galactic supernova sources and in the radio sources which are extragalactic at great distances; (vi) the line emission from neutral hydrogen on 21 cm as it affects the work at Jodrell Bank, particularly the attempt to measure the interstellar field by observation of the Zeeman effect which has recently given a positive result. The paper ends by reference to the evolution by Hanbury Brown of the intensity interferometer at Jodrell Bank and to the problem of the scintillation of the radio stars.

The form of the electron density surrounding a spherically symmetric defect in a Fermi gas at non-zero temperature is calculated. At large distances and moderate temperatures the density decay contains a factor (πk_FkT /2E_F) csch(πkT/E_F) k_Fr and reduces at zero temperature to the well-known inverse cube dependence of amplitude on distance.

The validity of the nearly-free electron (`weak-binding') approximation is considered, for the problem of a single electron in a one-dimensional chain of potential wells lacking long-range order (a `liquid' chain), over an energy range forbidden for a periodic chain of the same mean spacing.

Central limit theorems for sums of dependent variates are used to obtain the probability distribution of the Fourier components of the potential. It is shown that the findings justify a definite evaluation of the potential `power spectrum'.

A second-order expression for the energy, in weak-binding approximation, is then evaluated. The implied eigenstate density is compared with the results of Monte Carlo calculations for chains of δ-function wells. The comparison shows the fit to be good for large disorders, but very poor for small disorders.

The findings are explained in the light of knowledge regarding localized states in this energy region. The trial wave function used in the weak-binding approach can approximate to a localized state, but only for disorders large enough to destroy the gap. The relevance of this finding to work with real liquids is discussed.

The second-quantized equivalent of the general particle-conserving operator in the configuration space of Slater determinants is found by a direct calculation. The usual one- and two-particle operators are found as special cases and their equivalence with the corresponding configuration-space operators is thus shown without a lengthy comparison of either signs or matrix elements.

The mass polarization or specific isotope shift of the 1s²2s ²S and 1s²2p ²P levels of Li and B²⁺ is calculated using open-shell type wave functions and the shifts of the resonance lines of ⁷Li relative to ⁶Li and of ¹¹B²⁺ relative to ¹⁰B²⁺ are estimated to be 0.100 cm^-1 and 0.688 cm^-1 respectively compared with the observed values 0.157 cm^-1, 0.930 cm^-1 (ΔJ = 0) and 0.970 cm^-1 (ΔJ = 1). The results are also compared with the values obtained using an alternative form of the mass polarization operator.

The hyperfine splitting of the 1s² 2s ²S states of the three-electron systems is calculated using the simplest form of open-shell wave functions. The result for lithium is in good agreement with the very accurate calculation of Nesbet and with experiment.

The values for the isoelectronic sequence can be closely represented by an expansion in powers of Z^-1 of the form <f> = a₀³{0.5Z³ -1.472Z²+0.838Z+0.616} where f is the operator introduced by Nesbet.

The Schwartz method has been extended to calculations of the x-ray scattering factor, <r> and <1/r²> for the ground state of helium and to <δ(r)> for the triplet state of helium, all with good accuracy. It is shown that the Schwartz method fails for a trial wave function that does not satisfy the virial theorem.

The asymptotic forms of the positive energy solutions of the radial Schrödinger equation for a modified Coulomb field are investigated, using both the Jeffreys or WKB method, and the analytic properties of the solutions for an unmodified Coulomb field. The neutral case is also treated. Analytic expressions for the phases associated with modifying potentials of the form αr^-2 + βr^-3 + γr^-4 are obtained, so that the phases and amplitudes of solutions may easily be determined, to sufficient accuracy for most purposes, for any potential of physical interest. A short table of regular and irregular Coulomb functions is given.

Using the theory of the nature of lattice vibrations around an impurity of different mass in a cubic crystal, detailed calculations are made of the optical absorption expected from compensated group III and group V impurities in Si due to creation of single phonons. Provided good compensation can be achieved it appears that this absorption should be clearly visible in concentrations of about 10¹⁹ cm^-3. The calculated frequency spectrum of Johnson is used and it is shown that localized and band modes will give comparable effects, which change considerably with the mass of the impurity.

The electronic energy levels of Bi₂Te₃ at symmetry points in the Brillouin zone are classified. An augmented plane waves calculation is carried out to try to determine the possible positions and symmetries of the band edges. The calculation is not self-consistent and several approximations are made in evaluating the potential. It is found that an energy gap arises essentially because of spin-orbit interaction. The bottom of the conduction band, of p-type symmetry, appears to be at or near the centre of the four-sided faces of the Brillouin zone. The valence band appears to have several extrema. Since the experimental evidence so far is weak, the main value of these theoretical conclusions is as a framework for the interpretation of future experimental results.

An isobaric spin analysis of the capture of K^- mesons in a two-nucleon system has been carried out. The production ratios of pion-hyperon resonances Y*'s produced in these reactions are evaluated, and some relationships between the cross sections of these absorption processes are found. The analysis is then extended to the capture of K^- mesons in light nuclei, helium and deuterium.

The authors have investigated the non-linear interaction of oppositely propagating electromagnetic waves at a plasma-free space interface by obtaining second-order solutions of the Boltzmann kinetic equation and the wave equation. An interesting result of the analysis is the fact that the reflected component of the wave incident from the freespace side has a portion which carries the modulations imposed on the other wave.

It is shown that the introduction of the quantum of length l₀ and of the universal constant c, the velocity of light in vacuo, into a physical theory is equivalent to the hypothesis that the space-time has a discontinuous structure.

The method used by Somerville and Williams to study the development of a spark column by passing through it a delayed current pulse has been extended so that the whole evolution and decay of the channel may be examined. The column produced by a 32 A spark in air with duration less than 100 nsec is found to expand initially in close association with a cylindrical shock wave; a period of slower expansion follows in which, presumably, conditions within the column approach pressure equilibrium. Finally the column cools and contracts by a process of radial thermal conduction. This third stage has been treated theoretically and an expression for the radius of the column as a function of time has been derived which gives satisfactory agreement with experiment.

It has been suggested that the equivalent pressure concept for the influence of a transverse magnetic field on electrical breakdown in uniform fields can be extended to explain the phenomena observed recently in electrodeless discharges. It is pointed out that these measurements are possibly unreliable because of a restricted E × H drift of electrons and the consequent influence of the tube walls. A comparison of theory and experiment on the basis of such measurements is therefore unsatisfactory.

Domains have been observed in a variety of ferrous materials for grain diameters in the range 6 × 10^-1 cm to 10^-3 cm. The results are in agreement with the theoretical values of maximum and minimum domain sizes as derived for ideal grains bounded by (100) faces in the sample surface and either (100) or (111) faces in the interior of the sample. The range of domain sizes was found to be greater than that indicated by previous measurements.

In magnetohydrodynamic systems where the electrical conductivity increases with temperature, a temperature instability may occur due to preferential ohmic heating of the hotter parts of the plasma. The rate of growth of this instability is discussed with particular reference to magnetohydrodynamic generators. It is shown that different results are obtained on the single-fluid and two-fluid (electrons treated separately from the atoms and ions) models but in both cases the rate of growth is too slow to be important in practical magnetohydrodynamic generators. A related problem of the rate of obtaining enhanced conductivity in magnetohydrodynamic generators is also discussed and it is shown that the conductivity increases in the first fraction of the duct.

Rouse and his collaborators published in this journal in 1953 their measurements of convection over parallel sources of heat. In this paper it is proved that they performed their experiments in the region of laminar flow.

The refractive indices of PbTe and PbSe have been measured by observing the minimum deviation through prisms of these materials. Within experimental error, the refractive index was found to be the same for n- and p-type PbTe containing about 10¹⁸ cm^-3 donors or acceptors. Free carrier effects were negligible over the wavelength range of the experiment. Both materials showed normal dispersion behaviour, the extrapolated values obtained for the infinite wavelength refractive indices being 5.64 ± 0.03 for PbTe and 4.70 ± 0.02 for PbSe. These refractive index data have been used to recalculate the effective electron mass in these materials from previous Faraday effective experiments.

In the absorption spectrum of a mixture of silver and tellurium vapours, heated to a temperature of about 1900 °c, a large number of bands occurring in the three regions (i) λλ 3650-3400, (ii) λλ 2640-2530 and (iii) λλ 2475-2360 of the spectrum have been observed and analysed for the first time. All these bands are degraded to the longer wavelength side and belong to three different systems designated as A <- X, B <- X and C <- X systems respectively. The equations which represent all the observed bands satisfactorily are as follows: A <- X system ν = 29 167.7 + 124.0 (v' + ½)-0.01 (v' + ½)²-195.3 (v" + ½) + 0.30 (v" + ½)² B <- X system ν = 38 764.5 + 146.2 (v' + ½)-0.20 (v' + ½)²-195.3 (v" + ½) + 0.30 (v" + ½)² C <- X system ν = 41 079.2 + 138.0 (v' + ½)-2.50 (v' + ½)²-195.3 (v" + ½) + 0.30 (v" + ½)².

Similar investigations on the absorption spectrum of Ag-Se vapours heated to 1800 °c have revealed the existence of about nine red-degraded bands in the region λλ 3520-3400 and three bands in the region λλ 2450-2350. As the data are insufficient, no formula has been proposed for them. However, it is apparent that the vibrational frequency in the lower and upper states for the bands in the region λλ 3520-3400 is of the order of 233 cm^-1 and 127 cm^-1 respectively.

By introducing small proportions of appropriate gases into a high-temperature hydrogen plasma produced by the toroidal discharge apparatus SCEPTRE IV, the spectra of highly ionized B, C, N, O, F, and Ne have been obtained and investigated in the wavelength range 200-7000 Å. New as well as previously observed resonance lines of Li I-like ions have been measured. Hydrogen-like transitions of O VI have been observed, the wavelengths of which agree within the experimental errors with the predictions of a polarization formula. According to this formula, the ionization limit of O VI is 1114010 cm^-1. New lines of Ne VII, Ne VI, and Ne IV have been measured and new energy levels have been obtained.

The scattering of a plane electromagnetic wave by a bounded column of plasma has been calculated treating the plasma as a dielectric medium small in diameter compared with the incident wavelength. Results are given for the dependence of position and width of the resonance found, taking into account glass walls and collisions of the electrons. The theory is developed to apply when the column is placed across a rectangular waveguide propagating the fundamental transverse electric mode.

Comparison is made with experimental measurements and the relevance to the subsidiary resonances observed experimentally is examined.

The investigation is a sequel to the spectral analysis of Kaufman, Hughes and Williams (1960). From spectrograms taken with the high-current discharge assemblies Sceptre III and IV, new lines of Ne IV, Ne V and Ne VI in the range between about 2200 Å and 3000 Å have been observed and classified. The lists of classified lines and term values, both new and revised, are given in tabular form. No intersystem combinations have been observed in any of the three spectra. A method of distinguishing the higher stages of ionization (Ne IV, Ne V, Ne VI) has been developed by varying the concentration of the neon added to the deuterium. The relative intensities of the spectra vary with the concentration in a characteristic manner and so the lines of a given spectrum can be picked out with comparative ease. A theoretical explanation of the effect is advanced on the assumption that the plasma is cooled by spectral-line radiation. Methods of identifying the stage of ionization when high-current discharges are used as spectral sources are appraised.

In the first part of the paper values of transmission T and resolution W are computed for prolate spheroidal field magnetic spectrometers with various positions of the source. A value of 15% of 4π has been found for T with W = 1%. The aberration due to size of source has been estimated by computing trajectories from non-axial points. Envelopes and caustics formed by monoenergetic families of rays are mapped for several magnetic rigidities.

The second part considers the focusing of particles in superpositions in various ratios of a prolate spheroidal and a uniform field with a common axis. It appears that the position of a point image I of an axial point object O is unrestricted except to the extent that O and I must lie between the foci of the spheroids on the axis and that a narrow zone of angles of emission is used. A β spectrometer using one such ratio is found to give a poorer aperture aberration but an improved source size aberration when compared with the pure spheroidal field instrument. In each case the luminosity is similar to that of the uniform field lens spectrometer.

The Hall coefficient and resistivity of copper-doped germanium have been measured as a function of temperature over the range 300 °K to 1.8 °K. Typical impurity conduction has been observed. The dependence of the resistivity in the impurity conduction region on the mean separation between the copper impurity centres has been used to estimate the radial extent r₀ of the wave function of the copper ground state. The 0.04 eV copper state is approximately half the size of the 0.01 eV shallow acceptor states. The activation energy in the impurity conduction region agrees well with the theory of Miller and Abrahams. For those crystals containing more than 2 × 10¹⁶ copper atoms/cm³, the resistance in the impurity conduction region decreases in the presence of a magnetic field.

The current noise spectrum of copper-doped germanium has been measured from 30 c/s to 16 kc/s and from 5 Mc/s to 50 Mc/s. From the generation-recombination noise level, the capture cross section for holes by the ionized copper atoms is 8.1 × 10^-14 cm²; the high frequency noise spectrum indicates that the carrier lifetime is single valued. In a suitably cleaned crystal no 1/f noise was observed above 30 c/s; excess 1/f noise occurred only when the surface of the crystal was contaminated with a metallic deposit, or a damaged surface layer was present.

A preliminary calculation of the specific heats of sodium and potassium has been made by Toya's method. Blackman's sampling technique with a coarse mesh of 125 points per Brillouin zone has been adopted. There is no agreement between the theoretical and the experimental values of C_v. The use of a finer mesh is not likely to improve the agreement substantially.