Showing papers in "Philosophical Transactions of the Royal Society A in 1959"

Coulomb Gauge in Non-Relativistic Quantum Electro-Dynamics and the Shape of Spectral Lines

Abstract: An examination is made of the Green dyadic for the propagation functions of non-relativistic quantum electrodynamics in the Coulomb gauge div A = 0. The existence of false precursors is shown to be associated with singularities at the origin in certain momentum space representations of the propagators and this allows a prescription to be given to eliminate such precursors. This prescription is proved equivalent to an exact consideration of the near fields associated with quantum transitions. The analogous treatment for the state vectors in decaying systems is carried through and is applied in detail to the problems of the shape of natural spectral lines and the transition rate for stimulated emission. It is shown that the expected line shape for the Lamb 1057 Mc/s line in hydrogen is Lorentzian within the context of Lamb’s experiments—intensity against Zeeman splitting energy. This is in contradiction to the predictions of Arnous and Heitler. The distribution in k (circular frequency) for a natural line, on the other hand, has a factor k a times the resonant denominator. The definitive position between the conflicting transition rates in the literature is given. The relationship of the calculations to a gauge-independent Hamiltonian, considered in the paper, is investigated in detail. This Hamiltonian, transformed canonically from the conventional one, is shown to be valid for internal, dynamical, quantized electromagnetic fields; its analogue in the case of external applied fields being well known. It is demonstrated that the transformation eliminates many of the difficulties associated with the false near fields and that it also allows certain radiative effects, where the atom acts as a whole, to be computed in a straightforward manner.

Electrical and thermal resistivity of the transition elements at low temperatures

Abstract: The results of measurements on 20 transition elements are reported giving values for the thermal resistivity, W , from 2 to about 140 °K and for electrical resistivity, p , from 2 to about 300 °K. Values of the ‘ideal’ resistivities, W i and p i { (due to scattering of the electrons by thermal vibrations), are deduced from these and tabulated for various temperatures. Comparisons are made with values for Cu, Ag, Au and Na and with the predictions of the ‘standard’ theory, i.e. solutions of the transport equation developed by Bloch, Gruneisen, Wilson, etc. Excepting Mn, p i follows a Bloch—Gruneisen function tolerably down to op5, although slight anomalies are shown by V, Cr, Fe, Co and Ni; at low temperatures behaviour is varied but below 10 °K in Mn, Fe, Co, Ni, Pd, Pt and perhaps in W and Nb, p i appears to vary nearly as T 2 . The parameter, piM 6 & (at 273 °K) has rather similar values for different members of each group, e.g. for Ti, Zr and Hf of group IV A. The ideal thermal resistivity, Wif can generally be approximated by the relation, WiIW ao = 2(Tld)2J 5(dlT), although for many elements, W i falls more rapidly than T 2 below010. Measurements on the relatively poor conductors, e.g. Ti, Zr and Hf, suggest the presence of an appreciable lattice conductivity, which affects the confidence with which values can be deduced for W i in these elements.

The Electronic Absorption Spectra of NH$_{2}$ and ND$_{2}$

Abstract: The absorption spectra of $^{14}$NH$\_{2}$, $^{15}$NH$\_{2}$ and $^{14}$ND$\_{2}$ have been photographed in the region 3900 to 8300 angstrom with a 21 ft. concave grating spectrograph. The radicals are produced by the flash photolysis of $^{14}$NH$\_{3}$, $^{15}$NH$\_{3}$ and $^{14}$ND$\_{3}$ respectively. A detailed study of the $^{14}$NH$\_{2}$-$^{15}$NH$\_{2}$ isotope shifts suggests that the molecule has a linear configuration in the excited state and that the spectrum consists of a long progression of the bending vibration in this state. These conclusions have been confirmed by detailed rotational and vibrational analyses of the $^{14}$NH$\_{2}$ and $^{14}$ND$\_{2}$ spectra. The spectra consist of type C bands for which the transition moment is perpendicular to the plane of the molecule. For NH$\_{2}$, sixteen bands of the progression (0, v$\_{2}^{\prime}$, 0) $\leftarrow $ (0, 0, 0) have been identified with v$\_{2}^{\prime}$ = 3, 4,..., 18. In addition four bands of a subsidiary progression (1, v$\_{2}^{\prime}$, 0) $\leftarrow $ (0, 0, 0) have been found; these bands derive most of their intensity from a Fermi-type resonance between (0, v$\_{2}^{\prime}$, 0) and (1, v$\_{2}^{\prime}$-4, 0) levels in the excited state. The interaction constant W$\_{\text{ni}}$ is 72 $\pm $ 3 cm$^{-1}$. For ND$\_{2}$, fourteen bands of the principal progression (v$\_{2}^{\prime}$ = 5 to 18) and one band of the subsidiary progression have been identified. The upper state vibration frequencies $\omega \_{1}^{0\prime}$ and $\omega \_{2}^{0\prime}$ are 3325 cm$^{-1}$ and 622 cm$^{-1}$ for NH$\_{2}$ and 2520 cm$^{-1}$ and 422 cm$^{-1}$ for ND$\_{2}$ respectively. The bending frequencies are unusually low; moreover, the anharmonicities of the bending vibration are unusually large and negative (x$\_{22}^{0\prime}$ = 11$\cdot $4 cm$^{-1}$ for NH$\_{2}$ and 8$\cdot $1 cm$^{-1}$ for ND$\_{2}$). The origin of the system lies in the region of 10 000 cm$^{-1}$. Ground-state rotational term values have been derived from observed combination differences; values for the rotational constants A$\_{000}^{\prime \prime}$, B$\_{000}^{\prime \prime}$ and C$\_{000}^{\prime \prime}$ and for the centrifugal distortion constants D$\_{A}^{\prime \prime}$, D$\_{B}^{\prime \prime}$ and D$\_{C}^{\prime \prime}$ have been determined. The bond lengths and bond angles for NH$\_{2}$ and ND$\_{2}$ agree and are 1$\cdot $024 $\pm $ 0$\cdot $005 angstrom and 103 degrees 20$^{\prime}$ $\pm $ 30$^{\prime}$ respectively. Small spin splittings have been observed. In the excited state an unusual type of vibronic structure has been found. Successive levels of the bending vibration consist alternately of $\Sigma $, $\Delta $, $\Gamma $,... and $\Pi $, $\Phi $,... vibronic sub-levels with large vibronic splittings. The origins of the vibronic sub-bands may be represented by the formula $ u \_{0}^{K}$ = $ u \_{0}$-GK$^{2}$, where G is $\sim $ 27 cm$^{-1}$ for NH$\_{2}$ and $\sim $ 19 cm$^{-1}$ for ND$\_{2}$. The rotational levels show both spin and K-type doubling. No simple formula has been found to fit the energies of the $\Pi $, $\Delta $, $\Phi $ and $\Gamma $ rotational levels; the $\Sigma $ levels fit the formula F(N) = BN(N+1)-DN$^{2}$(N+1)$^{2}$, though with a negative value for D. By extrapolating the B values for the $\Sigma $ levels to v$\_{2}^{\prime}$ = 0 we obtain B$\_{000}^{\prime}$ = 8$\cdot $7$\_{8}$ cm$^{-1}$ for NH$\_{2}$ and 4$\cdot $4$\_{1}$ cm$^{-1}$ for ND$\_{2}$. These values are consistent with a linear configuration with a bond length of 0$\cdot $97$_{5}$ angstrom. The significance of this short bond length is discussed. An explanation of the complex vibronic structure is given. The two combining states are both derived from an electronic $\Pi $ state which is split by electronic-vibrational coupling for the reasons advanced by Renner. A detailed correlation diagram is given. A quantitative treatment of this effect by Pople & Longuet-Higgins gives good agreement with the experimental data.

On the Annual Variation of Magnetic Disturbance

Abstract: Statistical features of the annual incidence of magnetic disturbance, over a very wide range of disturbance intensity and latitude, are exhaustively investigated by means of the K index and related ‘planetary’ indices. Two distinct and physically significant components are identified: ( a ) an annual component, with summer maximum and winter minimum; ( b ) a semi-annual component with equinoxial maxima. Soth components are found in all parts of the earth. The amplitude of the annual component increases markedly with latitude, while that of the semi-annual component changes little with latitude. The physical causes of the two types of variation are finally considered. The conclusions reached are ( a ) that the annual component is probably caused by an atmospheric dynamo effect; ( b ) that the semi-annual component arises because of a systematic annual variation of the angle between the earth9s magnetic axis and the sun-earth line, along which travel the solar particles which cause magnetic disturbance.

Cubic Forms in Thirty-Two Variables

Abstract: It is proved that if C(xu...,*„) is any cubic form in n variables, with integral coefficients, then the equation C{xu ...,*„) = 0 has a solution in integers xXi...,xn, not all 0, provided n is at least 32. The proof is based on the Hardy-Littlewood method, involving the dissection into parts of a definite integral, but new principles are needed for estimating an exponential sum containing a general cubic form. The estimates obtained here are conditional on the form not splitting in a particular manner; when it does so split, the same treatment is applied to the new form, and ultimately the proof is made to depend on known results.

Propagation of Elastic Wave Motion from an Impulsive Source along a Fluid/Solid Interface. I. Experimental Pressure Response. II. Theoretical Pressure Response. III. The Pseudo-Rayleigh Wave

Abstract: Parts I and II of this report compare the experimentally observed pressure response for the impulse excited fluid/solid interface problem with that derived from a corresponding theoretical investigation. In the experiment a pressure wave is generated in the system by a spark and detected with a small barium titanate probe. The output of the probe is displayed on an oscilloscope and photographed. Two cases are investigated: one where the transverse wave velocity is lower than the longitudinal wave velocity of the fluid and the other where the transverse wave velocity is higher. Both of these observed responses are shown to agree even as to details of wave-form, with exact computations made for a delta-excited line source. This comparison is justified by making an approximate calculation for the decaying point source and showing that at these distances it does not differ appreciably from the delta-excited line source. In the case of low transverse wave velocity one finds, besides critically refracted P , direct, and reflected waves, a Stoneley type of interface wave. Although the emphasis in recent years has been towards minimizing the importance of Stoneley waves, the evidence here is that a Stoneley wave can be the largest contributor to a response curve. In the case of high transverse wave velocity the critically refracted P wave is smaller, and the Stoneley wave, though it tends to maintain a rather constant amplitude, becomes compressed in time and arrives very soon after the reflexion. Between the critically refracted P wave and the direct arrivals one finds both experimentally and theoretically a pressure build-up preceding the arrival time that might be expected for a critically refracted transverse wave. In part III this pressure build-up is investigated and found to consist of the superposition of three arrivals. The most prominent of these is a pseudo-Rayleigh wave. The others are the critically refracted transverse wave and the build-up to the later arriving Stoneley wave. Detailed investigation of the pseudo-Rayleigh wave shows it to have the velocity of a true Rayleigh wave which is independent of the existence of the fluid. Furthermore, it has the same retrograde particle motion as the true Rayleigh wave. However, it is radiating into the fluid as it progresses and therefore has many of the properties of a critically refracted arrival when measurements are made in the fluid. Mathematically it differs from the true Rayleigh wave in that its origin is not from a pole on the real axis of the plane of the variable of integration, but rather from a pole which lies on a lower Riemann sheet in the complex plane. In the high transverse wave velocity case this pole is not too far removed from the real axis and the imaginary part of the pole location might be interpreted as a decay factor. The real part, however, yields only approximately the velocity of the pseudo-Rayleigh wave, for the actual velocity as pointed out above is precisely that of the true Rayleigh wave velocity. The migration of this complex pole explains why such a pseudo-Rayleigh wave was not observed in parts I and II in the low transverse velocity case. The problem under discussion is intimately related to the classic work of Horace Lamb On the propagation of tremors over the surface of an elastic solid. One need make only a minor re-interpretation of the source function in order to compare directly the wave-forms (excluding of course the Stoneley wave contribution). Finally, a method is suggested for obtaining the solid rigidity of bottom sediments in watercovered areas from in situ measurements of the pseudo-Rayleigh wave and/or Stoneley wave velocities and arrival times

The propagation of plane irrotational waves through an elastoplastic medium

Abstract: This paper is an attempt at a systematic investigation of wave propagation in a metal, treating interactions between elastic and plastic waves, and the formation and propagation of shock waves, in the general case of motions with unidirectional strain arising from an initial smooth loading­ unloading pulse. A stress-strain relation with linear elastic paths and concave-upward plastic paths (where compression is measured as positive) is derived and used so that the elastic wave velocity is uniform, and the plastic wave velocity an increasing function of stress. The analysis is in terms of engineering stress and strain with a Lagrangian co-ordinate system. Analytic solutions to the interactions between different types of continuous waves are developed incorporating an expression for the motion of the elastic-plastic boundary. An analysis of the breakdown of a smooth plastic compression wave into a shock wave is presented, and the propaga­ tion conditions derived. It is shown that the heat dissipated is proportional to the cube of the strain jump, its low value for moderate shock strength suggests that the shock does not appreciably affect the stress-strain relation, an assumption from which a solution for the unloading of a plastic compression front by an overtaking elastic wave, while shock formation is taking place, is derived. A numerical illustration of this solution for a particular pulse in aluminum is given

Reflexion levels and coupling regions in a horizontally stratified ionosphere

Abstract: The propagation of radio waves through a horizontally stratified and slowly varying ionosphere is governed, in the case of oblique incidence, by a quartic equation (Booker 1938). Ray theory breaks down when two roots of this quartic are equal, for then coupling occurs between the characteristic waves, and full wave theory must be used. This paper is concerned with determining the conditions under which the two roots are equal; it is not concerned with the full wave theory. Values of the plasma frequency, and electron collision frequency, which lead to equal roots, are determined, and are exhibited in a set of curves. A full solution of the ‘Booker’ quartic is also given for a case of special interest. It is pointed out that the electric wave-field is unlikely to become very large in a slowly varying ionosphere, so that, if the ionosphere were irregular, scattering cannot be unduly enhanced by a plasma resonance.

The rational characterization of certain sets of relatively Abelian extensions

Abstract: Let H be a class group— in the sense of class-field theory— in the rational field P, whose order is some power of a prime l . With H there is associated an Abelian extension K of P. The purpose of this paper is to determine in rational terms and for all fields K given in the described manner, the set T(K/P) of cyclic extensions A of K of relative degree l , which are absolutely normal. In particular we shall find the ramification laws for these fields A, and the possible extension types of a group of order l by the Galois group of K, which are realized in Galois groups of fields in T(K/P). It is fundamental to the programme outlined, that we aim at obtaining purely rational criteria of determination.

A method of analysis for long-wave phenomena in the ocean, using electronic network models I. The Earth's rotation ignored

Abstract: A method of analysis for long-wave phenomena in the ocean and examples of its applications are described. A phenomenon to be analyzed is assumed to be ‘ response9 resulting from an * excitation9 on the ‘system’, and the relations between these are analyzed. The original hydraulic system is divided into a number of cells of finite size. Each cell is assumed to have a constant depth within itself, and is represented by a two-dimensional electric network consisting of many capacitors, which are proportional to the area of the mesh, variable inductors, inversely proportional to the depth of each mesh and variable resistors, proportional to the total energy loss in each mesh. The excitation and its response are represented either by a voltage which represents the water elevation of each mesh or electric current which corresponds to the total flow across each mesh. Complicated cases, like that where the excitation is changing irregularly with both time and two-dimensional space, can be treated by simple and small-scale equipment by applying the reciprocity theorem and the pulse-superposition method.

Solution of a Diffraction Problem II. The Narrow Double Wedge

Abstract: The problem of diffraction by a ‘narrow double wedge’ (width much smaller than wavelength) is investigated. Strong reflexion and quasi-static effects are the main features of this problem. The asymptotic behaviour of the solution is determined by the edge singularities. This leads to an approximate solution, which seems to be very accurate. This solution is found to be in good agreement with approximate solutions derived by different methods. The reflexion coefficient and the ‘end correction’ are evaluated. The results are compared with those obtained by other authors. It is shown that they contain a new effect, the 9evanescent mode correction’, which is very small in this region. Resonance effects in channels of finite length are analyzed.