Showing papers in "Physical Review in 1945"

Demagnetizing Factors of the General Ellipsoid

Abstract: Charts and tables of the demagnetizing factors of prolate and oblate spheroids are readily available; however, demagnetizing factors of ellipsoids of three different axes are incompletely tabulated and laborious to calculate. This article presents charts and tables which make possible easy determination of the demagnetizing factor for any principal axis of an ellipsoid of any shape. Formulas for the demagnetizing factors of the general ellipsoid are included together with supplementary formulas which cover a large number of special cases.

The Multiple Scattering of Waves. I. General Theory of Isotropic Scattering by Randomly Distributed Scatterers

Abstract: While the problem of the multiple scattering of particles by a random distribution of scatterers has been treated classically through the use of the Boltzmann integro-differential equation, the corresponding problem of the multiple scattering of waves seems to have received scant attention. All previous treatments have considered the problem in the "geometrical optics" limit, where the rays are regarded as trajectories of particles and the treatment for particles is then applied, so that the interference phenomena in wave scattering are neglected. In this paper the problem of the multiple scattering of scalar waves by a random distribution of isotropic scatterers is considered in detail on the basis of a consistent wave treatment. The introduction of the concept of "randomness" requires averages to be taken over a statistical ensemble of scatterer configurations. Equations are derived for the average value of the wave function, the average value of the square of its absolute value, and the average flux carried by the wave. The second of these quantities satisfies an integral equation which has some similarities to the corresponding equation for particle scattering. The physical interpretation of the results is discussed in some detail and possible generalizations of the theory are outlined.

A Theorem of Larmor and Its Importance for Electrons in Magnetic Fields

Abstract: The importance of a well-known theorem, originally due to Larmor, is emphasized. It enables a definition of "momentum" and "moment of momentum" for electrons in a magnetic field, hence the possibility of writing the conservation of these quantities when the geometry of the structure is convenient. As typical examples of the method, two special cases are discussed: a plane electron beam and a cylindrical electron beam with longitudinal magnetic field. In both cases it is found that the space-charge density of the beam is entirely controlled by the magnetic field and that the maximum current is obtained for a suitable optimum magnetic field.

Theoretical Continuous X-Ray Energy and Polarization

Abstract: By the theory of Sommerfeld, relativity effects and retardation of potential being neglected, matrix elements and associated components of continuous spectrum x-radiation are computed for a variety of electron-nucleus collisions and for a distributed series of positions in the spectrum. The calculations cover values of $\frac{V}{{Z}^{2}}$ from 0.06128 to 3.356, where $Z$ is the atomic number of the nucleus, and $V$ is the bombardment potential in electrostatic units. Accuracy of calculation is 1 percent. Screening is neglected except at the long wave limit of the spectra where it is taken into account by Sauter's method. Empirical algebraic formulas are found which closely represent the rigorously calculated results. Intensity and polarization predictions for any direction of emission and any excitation conditions within the range of applicability of the theory may be readily drawn from the computed results. Elwert's proposed correction factor for rectifying the approximate spectral intensities of Sommerfeld and Maue is found effective within the limits of its restricting assumptions. Theoretical efficiencies of continuous x-ray production are calculated by combining theoretical intensities with known rates of electron energy loss in traversing matter. Thick target efficiency (a ratio, not percent) is given by $1.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}Z$ kv. Thin target efficiency is found to be approximately twice the thick target efficiency for any given $Z$ and kv.

A New Approach to Kinematic Cosmology

Abstract: The kinematical aspect of relativistic cosmology is examined on the basis of three postulated requirements: The constancy of the velocity of light, spatial isotropy, and homogeneity. Three distinct types of cosmological models are obtained, characterized by different motions of nebulae. The metric of any universe is conformal to Minkowski space and Maxwell's equations are the same for all possible universes. In Part II, it is shown that the cosmological models are metrically, though not topologically, equivalent to those of H. P. Robertson. Next, special models are examined and their line elements brought into the conformal-Minkowskian form. The problem of the displacement of the lines of nebular spectra is discussed; formulas are obtained and applied to some special cosmological models. Finally, idealized experiments are described which indicate the physical content of the cosmological coordinates.

General Theory of Pressure Broadening of Spectral Lines

Abstract: A new, more consistent, shape has been given to the theory of pressure broadening of spectral lines recently published by the author. In contradistinction to other theories, the present theory constitutes a very close analogy to the theory of intensity distribution in molecular spectra. There is no doubt that both phenomena are due to the same cause, i.e., to the relative movements of atomic nuclei. Thus, the theoretical treatment of both must be identical as far as possible. The method used by James and Coolidge for the calculations of intensity distribution in ${\mathrm{H}}_{2}$ and ${\mathrm{D}}_{2}$ continuous spectra can be adapted to the calculations of the profiles of broadened lines. In this case, presumably, it will not be possible to represent the intensity distribution in a closed form. In order to obtain a closed form, Condon's method (the quantum mechanical form of the Franck-Condon principle) is applied, and the Wentzel-Kramers-Brillouin approximate eigenfunctions are used for nuclear motions. The limitations of applicability of this approximation are discussed (the same limitations apply $a$ fortiori to the applicability of every theory based on the classical description of nuclear motions). Because of the above simplifications, the resulting intensity distribution formula must be considered as an asymptotic one only, valid in a restricted region of frequencies of the broadened line and only in the case of heavy atoms and high temperatures (it certainly fails in the case of broadening by light gases such as He and ${\mathrm{H}}_{2}$ or electrons), though it still constitutes a better approximation than that previously published. Apart from a correction which is in most cases insignificant, it is identical with Kuhn's intensity distribution obtained on the basis of the primitive form of the Franck-Condon principle. The present paper is drafted so as to be comprehensible to the reader without knowledge of the preceding papers of the author, the main results of which are being included here.

A Generalization of the Dielectric Ellipsoid Problem

Abstract: The classical problem of the dielectric ellipsoid involves the determination of the field within a homogeneous, isotropic, dielectric ellipsoid when it is placed in a uniform electric field. In the present generalization, both the ellipsoid and the medium in which it is placed, although still homogeneous, are anisotropic and also possess conductivities which are anisotropic. The principal axes of the ellipsoid, of the two dielectric tensors, and of the two conductivity tensors, may all be differently oriented. The external field, although uniform in space, varies sinusoidally with time. The condition specified in the last sentence is consistent with the electromagnetic field equations only in a region whose maximum dimension is small compared with $\frac{\ensuremath{\lambda}}{2\ensuremath{\pi}}$ where $\ensuremath{\lambda}$ is the wave-length which corresponds to the frequency in question. Thus the solution given here is restricted by the condition that the maximum dimension of the ellipsoid must be small compared with $\frac{\ensuremath{\lambda}}{2\ensuremath{\pi}}$.

High Frequency Vibrations of Thin Crystal Plates

Abstract: The boundary conditions of free vibration can be satisfied on the major surfaces of a plane-parallel plate if the displacement components are assumed to be products of trigonometric functions. In addition, the boundary conditions can be approximately satisfied on the minor surfaces when the plate is thin. The theory leads to a frequency equation $\ensuremath{ u}=\frac{1}{2}{(\frac{c}{\ensuremath{\rho}})}^{\frac{1}{2}}{[{(\frac{n}{2b})}^{2}+k{(\frac{m}{2a})}^{2}]}^{\frac{1}{2}}$ which has been found empirically to satisfy observations. The theoretical values of the constant $k$ are 3.7 and 1.8 for the $\mathrm{AT}$ and $\mathrm{BT}$ quartz plates, respectively, while the observed values are 3.9 and 1.7, respectively.

Theory of Photoelectric Emission from Metals

Abstract: The theory of volume effect in photoelectric emission is developed. Formulae are derived for the rate of electron excitation and for the photoelectric yield. The calculated threshold frequencies for volume effect are 5.91\ifmmode\times\else\texttimes\fi{}${10}^{14}$ and 5.69\ifmmode\times\else\texttimes\fi{}${10}^{14}$ for sodium and potassium respectively. The estimated photoelectric yields for these metals are of the same order of magnitude as those calculated for surface effect and are comparable with those observed experimentally. The volume effect should not, therefore, be neglected except in the immediate neighborhood of the threshold frequency. Approxmate estimation indicates that light absorption of sodium and potassium in the visible and ultraviolet regions should be largely due to quantum excitation of electrons. Accurate calculation of electron excitation and absorption requires detailed knowledge of electron wave functions.

The Vibration-Rotation Energies of Polyatomic Molecules Part II. Accidental Degeneracies

Abstract: The first- and second-order corrections to the vibration-rotation energies of polyatomic molecules are dealt with in instances where two or more vibration frequencies are accidentally degenerate. The method of the contact transformation employed in Part I is extended and made applicable to the two types of first-order resonance interactions, i.e., the Fermi-Dennison type and the Coriolis type. The components of the energy matrix are evaluated in general, and examples are considered to demonstrate how the actual energies may be evaluated.

Detection of Metastable Ions With the Mass Spectrometer

Abstract: It has usually been customary to operate mass spectrometers with the filament and electron gun at a positive potential with respect to the analyzer section which is grounded. Under these conditions the ions pass through the exit slit and strike the ion-collector plate with the full energy acquired in the ion gun since this plate is at ground potential. Recently with an instrument of the sector type employing a magnetic deflection of 90°, the first slit in the ion gun was connected to ground while the analyzer section was made negative with respect to ground. Since the ions are formed in a region one or two volts above ground, they reach the collector plate in this arrangement with an energy of only a few electron volts.

On the Decay Process of Positive and Negative Mesons

Abstract: S OME years ago Tomonaga and Araki' pointed out that on account of the electrostatic interaction with nuclei, the capture probability should be for negative mesons in dense material much greater than the decay probability; while for positive mesons the decay probability should prevail on the capture probability. In 1941 in a direct measurement of the mean-life of mesons stopped in Al-absorber, Rasettin obtained for the ratio g between decay-processes and stopped mesons the value g =0.42 +0.15. Successively Auger, Maze, and Chaminade' found no essential lack from unity for g. Further experiments by Rossi and Nereson (Al absorber), and by ourselves' (Fe absorber) gave, respectively, q =0.4 and g =0.49&0.07. From the above mentioned works it seems reasonable to conclude that q is about 0.5. However, some doubt can be raised against the precision of such a value of ri, considering that it is deduced from the comparison between its evaluated value and the rate of the actually registered decayelectrons, which represent only a small fraction of all the decay-processes occurring in the absorber. Among other sources of error affecting the evaluation of g, one ought to consider the determination of the minimum delay of the registered electrons which depends on the size and shape of the counter-pulses. In order to avoid such a difhculty, in our previous work5 we counted the delayed coincidences with little and slightly different delays and obtained from the plotted results the point \"zero\" of the time-scale. In order to obtain an independent determination of g and to check at the same time whether, according to Tomanaga and Araki prediction, negative mesons do not undergo the decay-process in dense materials, we performed an experiment based on the possibility of concentrating positive and negative mesons by means of magnetized iron cores. The registering set—whose circuit has been already

Electric Circuit Models of the Schrödinger Equation

Abstract: Equivalent circuits are developed to represent the Schr\"odinger amplitude equation for one, two, and three independent variables in orthogonal curvilinear coordinate systems. The networks allow the assumption of any arbitrary potential energy and may be solved, within any desired degree of accuracy, either by an a.c. network analyzer, or by numerical and analytical circuit methods. It is shown that by varying the impressed frequency on a network of inductors and capacitors (or by keeping the frequency constant and varying the capacitors), it is possible to find by measurements the eigenvalues, eigenfunctions, and the statistical mean of various operators belonging to the system represented. The electrical model may, of course, be replaced by an analogous mechanical model containing moving masses and springs. At first the network for the one-dimensional wave equation for a single particle in Cartesian coordinates is developed in detail, then the general case. A companion paper contains results of a study made on an a.c. network analyzer of one-dimensional problems: a potential well, a double barrier, the harmonic oscillator, and the rigid rotator. The curves show good agreement, within the accuracy of the instruments, with the known eigenvalues, eigenfunctions, and "tunnel" effects.

Shock Wave Pressures in Water Produced by Impact of Small Spheres

Abstract: Microsecond spark shadowgrams were made showing shock waves in water. The waves were produced by spheres ${\frac{1}{8}}^{\ensuremath{'}\ensuremath{'}}$ to ${\frac{1}{4}}^{\ensuremath{'}\ensuremath{'}}$ in diameter when they struck a water surface with velocities between 2000 and 4800 ft./sec. Pressures in the wave were calculated from measurements of the absence-of-light band. The pressure was found to be greatest at a point directly ahead of the sphere and to fall off to normal pressures near the surface. The waves of large pressures were observed to travel faster than a sound wave and to have velocities in accordance with the measured pressure. Because of the varying strength over the wave front, the waves have the shape of a semi-ellipse. When the entrance velocity $V$ and projection area of the sphere were varied it was found that the pressure varied as ${V}^{2.17}$. It also increased linearly with the projection area of the sphere. The pressures were shown to be in fair agreement with those calculated from the compression of the water at the sphere's entrance; in the calculation an arbitrary loss of water in the splash was assumed.

On Permanent Charges in Solid Dielectrics I. Dielectric Absorption and Temperature Effects in Carnauba Wax

Abstract: The influence of the temperature upon dielectric absorption is studied for carnauba wax. Isothermic and non-isothermic current-time curves are measured. It is shown that a considerable part of the absorbed charge can be "frozen in," if the temperature is reduced to a value sufficiently inferior to that prevailing during the charging period before the system is short-circuited. The "frozen" charge dissipates extremely slowly, if the temperature is kept low, but it is liberated rapidly if the temperature is raised again. The effect is explained by the increase of the charging and discharging rates with increasing temperature. It is closely related to the permanent moment of the electret.

Disorder Scattering of X-Rays by Local Distortions

Abstract: The disorder scattering (background) owing to local distortions is caused mainly by the elastic strain field surrounding the distorted zone rather than by the misfit atoms in the zone of distortion. As an example, the scattering caused by a distortion of spherical symmetry is calculated. The result is extended for more general types of distortions. The background intensity increases strongly in the neighborhood of a line, and it tends toward a constant value for small scattering angles. This explains previous observations on rolled copper.

Threshold Measurements on the Nuclear Photo-Effect

Abstract: The high voltage x-radiation from the betatron has been employed to produce ($\ensuremath{\gamma},n$) reactions in several elements of atomic number up to 47, the reaction observed by detection of induced radioactivity. The peak x-ray energy was controlled by integrating the voltage on the main coils with an RC circuit, which actuated the orbit expander at a predetermined electron energy. Upon irradiating samples at sequences of energies, measuring their activity with beta-counters, and plotting activities against peak energy, smooth curves were obtained from which thresholds were estimated. The accuracy of the method is limited by low activity in almost all cases and by uncertainties in energy output. The following thresholds in Mev have been measured: ${\mathrm{C}}^{11}$, 18.7 to 19.4; ${\mathrm{N}}^{13}$, 11.1\ifmmode\pm\else\textpm\fi{}0.5; ${\mathrm{O}}^{15}$, 16.3\ifmmode\pm\else\textpm\fi{}0.4; ${\mathrm{Fe}}^{53}$, 14.2\ifmmode\pm\else\textpm\fi{}0.4; ${\mathrm{Cu}}^{62}$, 10.9\ifmmode\pm\else\textpm\fi{}0.3; ${\mathrm{Zn}}^{63}$, 11.6\ifmmode\pm\else\textpm\fi{}0.4; ${\mathrm{Se}}^{79\phantom{\rule{0ex}{0ex}}\mathrm{o}\mathrm{r}\phantom{\rule{0ex}{0ex}}81}$, 9.8\ifmmode\pm\else\textpm\fi{}0.5 for the lower, short period isomer; ${\mathrm{Mo}}^{91\phantom{\rule{0ex}{0ex}}\mathrm{o}\mathrm{r}\phantom{\rule{0ex}{0ex}}93}$, 13.5\ifmmode\pm\else\textpm\fi{}0.4; ${\mathrm{Ag}}^{108}$, 9.3\ifmmode\pm\else\textpm\fi{}0.5; ${\mathrm{Ag}}^{106}$, 9.5. The first three are in fair agreement with other data, but because of exceptionally weak activities are not sufficiently reliable to afford a test of the method. Rough measurements on the uranium photo-fission threshold are also reported. Since the radiation from the betatron is continuous and its spectral distribution is unknown, the excitation curves are not susceptible to interpretation, and cross sections for the above reactions cannot be estimated at present.

Collision Cross Sections for 25-Mev Neutrons

Abstract: Collision cross sections of a number of elements for high energy neutrons have been measured. Neutrons with a maximum energy of 25.4 Mev were obtained by bombarding lithium with 10.2 Mev deuterons. The reaction ${\mathrm{C}}^{12}(n,2n){\mathrm{C}}^{11}$, which has a measured threshold energy of approximately 21 Mev, was used as an energy sensitive detector for the transmission measurements. The cross section obtained for the neutron-proton collision process was 0.39 \ifmmode\pm\else\textpm\fi{}0.03\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}24}$ ${\mathrm{cm}}^{2}$. This is higher than the cross section calculated for $s$-scattering (0.35\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}24}$ ${\mathrm{cm}}^{2}$), but agrees well with the value of 0.40\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}24}$ ${\mathrm{cm}}^{2}$ predicted by the symmetrical meson theory of Rarita and Schwinger. Measurements on other nucleii ranging from carbon to mercury show that the collision radius is given by ${R}^{\ensuremath{'}}=b+{r}_{0}{A}^{\frac{1}{3}}$, with $b=1.7\ifmmode\pm\else\textpm\fi{}0.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm and ${r}_{0}=1.22\ifmmode\pm\else\textpm\fi{}0.15\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm. These measurements are in good agreement with the inelastic cross-section measurements of Grahame and Seaborg. The value of ${r}_{0}$ is somewhat lower than the values deduced from $p\ensuremath{-}n$ reactions, Coulomb energies, and $\ensuremath{\alpha}$-particle decay.

Classical Theory of the Point Electron

Abstract: The difficulties of the classical theory of the electron are examined and methods to eliminate them are given. It is shown that the whole theory can be derived from a division of the total field created by a point charge in two parts, one which reacts on the generating particle and accounts for the emission of radiation, another which does not react on the praticle but acts on other particles. There are several types of motions of the particles depending on the kind of field they generate, fields which are always solutions of Maxwell's equations. Only three types of motions are, apparently, physically interesting: (a) motions with positive or negative kinetic energy in which the particles radiate, and (b) radiationless motions analogous to the stationary motions of quantum theory. It is shown that the field picture of Faraday and Maxwell must be revised because not all the electric actions between particles can be considered as arising from their interaction with a field. The whole theory of the particles and the field can be derived from an action principle and boundary conditions for the equations of motion of the particles and the field.