Showing papers in "Physical Review in 1940"
TL;DR: In this article, the intrinsic domain magnetization of a ferromagnetic with the external magnetic field was obtained, and an approximation to low temperatures and equivalent to those used by Bloch in his derivation of the ${T}^{1}$ law, were introduced.
Abstract: In this paper, the variation of the intrinsic domain magnetization of a ferromagnetic with the external magnetic field, is obtained. The basis of the treatment is the exchange interaction model amplified by explicit consideration of the dipole-dipole interaction between the atomic magnets. Approximations appropriate to low temperatures and equivalent to those used by Bloch in his derivation of the ${T}^{1}$ law, are introduced. The resultant expression for the intrinsic volume susceptibility decreases slowly with increasing field; at high fields the functional dependence is as the inverse square root of the field. The variation with temperature is linear; at room temperature and for fields of about 4000 gauss, the order of magnitude of the (volume) susceptibility is ${10}^{\ensuremath{-}4}$. The results are compared with experiment and satisfactory agreement is found.
2,884 citations
TL;DR: In this paper, it was shown that the first law of thermodynamics is a scalar equation, and not the fourth component of the energy-momentum principle, and temperature and entropy also prove to be scalars.
Abstract: The considerations of the first paper of this series are modified so as to be consistent with the special theory of relativity. It is shown that the inertia of energy does not obviate the necessity for assuming the conservation of matter. Matter is to be interpreted as number of molecules, therefore, and not as inertia. Its velocity vector serves to define local proper-time axes, and the energy momentum tensor is resolved into proper-time and -space components. It is shown that the first law of thermodynamics is a scalar equation, and not the fourth component of the energy-momentum principle. Temperature and entropy also prove to be scalars. Simple relativistic generalizations of Fourier's law of heat conduction, and of the laws of viscosity are obtained from the requirements of the second law. The same considerations lead directly to the accepted relativistic form of Ohm's law.
1,316 citations
TL;DR: In this paper, a treatment of the magnetic resonance is given for a particle with spin \textonehalf{} in a constant field and under the action of an arbitrary alternating field with circular frequency perpendicular to the magnetic field.
Abstract: A treatment of the magnetic resonance is given for a particle with spin \textonehalf{} in a constant field ${H}_{0}$ and under the action of an arbitrary alternating field with circular frequency $\ensuremath{\omega}$ perpendicular to ${H}_{0}$. A method of finding a solution, valid at any time, is given which converges the better the smaller the deviations from a rotating field or the larger ${H}_{0}$. It is shown that in the lowest order correction the shape of the resonance curve is unchanged but that it is shifted by a percentage amount $\frac{{{H}_{1}}^{2}}{16 {{H}_{0}}^{2}}$ where ${H}_{1}$ is the effective amplitude of the oscillating field. This also involves a correction in the values of the magnetic moments thus obtained towards smaller values which however in all practical cases is negligibly small.
897 citations
TL;DR: In this paper, the cross sections for different kinds of nuclear reactions are calculated as functions of the energy of the bombarding particles by means of statistical methods, and their application is restricted to heavy elements ($Ag50$) and to bombarding energies greater than 1 Mev.
Abstract: The cross sections for different kinds of nuclear reactions are calculated as functions of the energy of the bombarding particles by means of statistical methods. Their application is restricted to heavy elements ($Ag50$) and to bombarding energies greater than 1 Mev. The excitation curves of several ($p,n$)-reactions have been measured for elements with $A$ between 60 and 115; it is found that the measured cross sections and their dependence on the energy suggests a nuclear radius of $R=1.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}\ifmmode\times\else\texttimes\fi{}{A}^{\frac{1}{3}}$ cm for these elements. Section I gives a complete discussion of the calculated cross sections. Section II and III contain the derivations of these expressions. Section IV describes the new experimental material and its implications for the theory.
596 citations
TL;DR: In this paper, the relativistically invariant wave equation for free particles was studied and the necessity of Fermi-Dirac statistics for particles with arbitrary half-integral spin was shown.
Abstract: In the following paper we conclude for the relativistically invariant wave equation for free particles: From postulate (I), according to which the energy must be positive, the necessity of Fermi-Dirac statistics for particles with arbitrary half-integral spin; from postulate (II), according to which observables on different space-time points with a space-like distance are commutable, the necessity of Einstein-Bose statistics for particles with arbitrary integral spin. It has been found useful to divide the quantities which are irreducible against Lorentz transformations into four symmetry classes which have a commutable multiplication like $+1,\ensuremath{-}1,+\ensuremath{\epsilon},\ensuremath{-}\ensuremath{\epsilon}$ with ${\ensuremath{\epsilon}}^{2}=1$.
596 citations
TL;DR: In this paper, the relaxation times for spin-lattice coupling in titanium and chrome alum are computed on the basis of a specific model obtained by combining the thermodynamic theory of Casimir and du Pr\'e with the writer's previous treatment of the normal modes of a cluster of the form X-6${\mathrm{H}_{2}$O, where X contains an incomplete shell.
Abstract: The relaxation times for spin-lattice coupling in titanium and chrome alum are computed on the basis of a specific model obtained by combining the thermodynamic theory of Casimir and du Pr\'e with the writer's previous treatment of the normal modes of a cluster of the form X-6${\mathrm{H}}_{2}$O, where X contains an incomplete shell. The calculation includes both the first-order or direct processes important at low temperatures, and the second-order or "Raman" type of term predominant in the liquid-air region. There is no difficulty in understanding the observed absence of dispersion in titanium alum at liquid-air temperatures, but, barring crystal imperfections, it is hard to understand this absence at helium temperatures unless the nearest excited states are unreasonably deep. The agreement between the orders of magnitude of the calculated and experimental relaxation times is adequate in chromium both at high and low temperatures. The calculations predict, in agreement with experiments, that at liquid-air temperatures the relaxation time should increase when a constant field ${H}_{0}$ is applied and should be independent of the direction of ${H}_{0}$. The computed increase, however, is apparently not great enough. At helium temperatures, $\ensuremath{\tau}$ is theoretically not quite isotropic, and $\frac{d\ensuremath{\tau}}{d{H}_{0}}$ has the wrong sign, unless one abandons the usual formula ${\ensuremath{\rho}}_{\ensuremath{\omega}}\ensuremath{\sim}{\ensuremath{\omega}}^{2}$ for the density of lattice oscillators. The calculations on chromium should also apply qualitatively to iron alum, discussed at the very end.
595 citations
527 citations
TL;DR: In this paper, it was shown that the loss of energy of a fast charged particle due to ionization of the material through which it is passing is considerably affected by the density of material.
Abstract: It is shown that the loss of energy of a fast charged particle due to the ionization of the material through which it is passing is considerably affected by the density of the material. The effect is due to the alteration of the electric field of the passing particle by the electric polarization of the medium. A theory based on classical electrodynamics shows that by equal mass of material traversed, the loss is larger in a rarefied substance than in a condensed one. The application of these results to cosmic radiation problems is discussed especially in view of the possible explanation on this basis of part of the difference in the absorption of mesotrons in air and in condensed materials that is usually interpreted as evidence for a spontaneous decay of the mesotron.
394 citations
385 citations
TL;DR: In this article, the breakdown of a uniform field is considered to occur by the transition of an electron avalanche proceeding from cathode to anode into a self-propagating streamer, which develops from anode to cathode, forming a conducting filament between the electrodes.
Abstract: The breakdown of a uniform field is considered to occur by the transition of an electron avalanche proceeding from cathode to anode into a self-propagating streamer, which develops from anode to cathode to form a conducting filament between the electrodes. A criterion is put forward for such a transition, viz., a streamer will develop when the radial field about the positive space charge in an electron avalanche attains a value of the order of the external applied field. For then photoelectrons in the immediate vicinity of the avalanche will be drawn into the stem of the avalanche and will give rise to a conducting filament of plasma, and a self-propagating streamer proceeds towards the cathode. The theory thus depends on ionization by electrons and photo-ionization in the gas and dispenses with the classical assumption of ionization by positive ions in the gas or secondary actions at the cathode. On this basis an equation for breakdown is developed, and reference to $\frac{\ensuremath{\alpha}}{p}\ensuremath{-}\frac{X}{p}$ curves enables the potential required for breakdown to be determined. Satisfactory agreement between calculation and experiment is found in air for values of pressure times gap length down to $p\ensuremath{\delta}\ensuremath{\sim}100$ mm Hg\ifmmode\times\else\texttimes\fi{}cm. The theory does not conform absolutely to Paschen's law, but the deviations are within the present day margins of experimental error. For lower values of $p\ensuremath{\delta}$ the deviation between calculation and experiment is explained by the fact that the density of photo-ionization becomes small, and the secondary mechanism, $\ensuremath{\gamma}$, is observed in this region, so that classical theory applies. The theory is indicated to be consistent with all the requirements so far established in connection with sparks for large $p\ensuremath{\delta}$ up to the lightning discharge.
333 citations
TL;DR: In this article, it was proposed that these wave functions and energies can be calculated by solving a secular equation with wave functions, which are simply plane waves made orthogonal to the core eigenfunctions.
Abstract: For many problems in the electron theory of metals none of the methods hitherto used to calculate the eigenfunctions and energy values of an electron in a crystal lattice is satisfactory. It is here proposed that these wave functions and energies be calculated by solving a secular equation with wave functions ${\ensuremath{\chi}}_{k}$ which are simply plane waves made orthogonal to the core eigenfunctions. The rapidity of convergence to be expected for such a procedure is discussed. Some methods for practical computation are suggested, and expressions are given for the matrix elements occurring in the secular equation.
TL;DR: In this paper, it was shown that the wave vector increases linearly with the time within the bounds of a single Brillouin zone, and at the boundaries of the zones transitions to other zones may take place if the accelerating field is large enough.
Abstract: The motion of an electron in a periodic potential field, and accelerated by a uniform field, can be obtained by treating the time-dependent Schroedinger equation. The result shows that the wave vector increases linearly with the time within the bounds of a single Brillouin zone. At the boundaries of the zones transitions to other zones may take place if the accelerating field is large enough.
TL;DR: In this paper, the possibility of interpreting the formalism of the general theory of relativity in terms of flat space, the fundamental tensor being regarded as describing the gravitational field but having no direct connection with geometry, was considered.
Abstract: The possibility is considered of interpreting the formalism of the general theory of relativity in terms of flat space, the fundamental tensor ${g}_{\ensuremath{\mu}\ensuremath{
u}}$ being regarded as describing the gravitational field but having no direct connection with geometry. The resulting theory in general leads to the same predictions as the Einstein theory, but there are cases where the predictions differ. The present theory may explain the principal results obtained by D. C. Miller in his "ether-drift" experiments. The implications of the theory for cosmology are briefly touched upon.
TL;DR: In this article, a theoretical treatment of multiple Rutherford scattering is given which is exact if one considers electrons with the same total path length in the scatterer, and it is shown that this distribution depends in a sensitive way upon the deviation from the Rutherford law for small angle scattering due to screening by electrons.
Abstract: A theoretical treatment of multiple Rutherford scattering is given which is exact if one considers electrons with the same total path length in the scatterer. The approximations necessary for the actual solution of the problem for a thin scatterer can be shown to have little effect on the results for small angle scattering. The distribution of scattering is expressed as a series in Legendre polynomials which has been evaluated numerically; the final result for thin scattering is approximately a Gaussian curve. It is shown that this distribution depends in a sensitive way upon the deviation from the Rutherford law for small angle scattering due to screening by electrons. This may perhaps explain the discrepancies between experiments and theory.
TL;DR: In this article, the rate of increase of the entropy of a simple viscous fluid, capable of conducting heat, is investigated in greater detail than has hitherto been customary, and it is shown that, if Kelvin's hypothesis concerning the absolute temperature be adopted, and the usual law of viscosity be assumed, the requirements of the second law are satisfied.
Abstract: The rate of increase of the entropy of a simple viscous fluid, capable of conducting heat, is investigated in greater detail than has hitherto been customary. It is shown that, if Kelvin's hypothesis concerning the absolute temperature be adopted, and the usual law of viscosity be assumed, the requirements of the second law are satisfied.
TL;DR: In this paper, the authors explore the possibilities of the "itinerant" or "collective" electron picture in the theory of metals to explain the quenching of orbital angular momentum in solids and, through the introduction of $l$,$s$ coupling, the phenomenon of ferromagnetic anisotropy in cubic crystals.
Abstract: The purpose of the present investigation is to explore the possibilities of the "itinerant" or "collective" electron picture in the theory of metals to explain the quenching of orbital angular momentum in solids and, through the introduction of $l$,$s$ coupling, the phenomenon of ferromagnetic anisotropy in cubic crystals. The approach used is that of the Bloch "approximation of tight binding" in the theory of metals, the exchange energy being treated as a Weiss internal field as in the work of Stoner and Slater, and the spin-orbit coupling being introduced as a perturbation. The anisotropy is shown to appear in the fourth approximation, and to have the correct order of magnitude for iron and nickel. The model also predicts the correct sign of ${K}_{1}$ in nickel and iron, but this prediction is not entirely satisfactory because computational difficulties prevent the inclusion of all the $d$-wave functions in the calculation. A qualitative discussion of the behavior of iron-nickel alloys is given. The chief weakness of the model is its failure to take account adequately of Russell-Saunders coupling within the atom, and the dependence of many of its predictions on details of the model which are not very well established.
TL;DR: In this paper, the penetration of uranium fission fragments through matter as revealed by cloud-chamber photographs were discussed, and a novel character which depends on the interplay between electron capture by the fragments over their entire range, and the ultimate stopping by nuclear collisions was presented.
Abstract: In two recent notes(1,2) a number of features of the penetration of uranium fission fragments through matter as revealed by cloud-chamber photographs were discussed. In particular, it was pointed out that we have here to do with a
velocity-range relation of a novel character which depends on the interplay between electron capture by the fragments over their entire range, and the ultimate stopping by nuclear collisions.
TL;DR: In this paper, the internal friction of crystalline copper, tin, lead, and zinc has been measured by the composite piezoelectric oscillator method, and it was found that the decrement of an unannealed crystal may be as large as that of the polycrystalline material, that annealing reduces the decrements to a value of the order of{10}^{\ensuremath{-}4}$ to ${10} √ 6}$, and that both Young's modulus and decrement vary with the vibrational strain amplitude at
Abstract: The internal friction of crystalline copper, tin, lead, and zinc has been measured by the composite piezoelectric oscillator method. It is found that the decrement of an unannealed crystal may be as large as that of the polycrystalline material, that annealing reduces the decrement to a value of the order ${10}^{\ensuremath{-}4}$ to ${10}^{\ensuremath{-}5}$, and that both Young's modulus and the decrement vary with the vibrational strain amplitude at strain amplitudes as low as ${10}^{\ensuremath{-}6}$. In the case of zinc crystals, a detailed study has been made of the way in which the elastic modulus and internal friction depend on the previous history of the specimen, on the vibration frequency and amplitude, and on the orientation of the vibration axis with respect to the crystal slip planes. The results suggest that the mechanism involved is a propagated "dislocation" of the sort proposed by Taylor, Polanyi and Orowan to account for macroscopic plastic flow, and that the application of a stress is accompanied by a plastic strain, together with an associated strain hardening in consequence of which the stress-strain relation on removal of the applied stress is nearly elastic.
TL;DR: The possibility of constructing a systematic theory of irreversible processes is surveyed in general terms, by utilizing some of the results established in later parts of the paper as discussed by the authors, and three assumptions underlying Gibbs' application of the second law to equilibrium problems are formulated in explicit but general mathematical form.
Abstract: The possibility of constructing a systematic theory of irreversible processes is surveyed in general terms, by utilizing some of the results established in later parts of the paper. Three assumptions underlying Gibbs' application of the second law to equilibrium problems are formulated in explicit but general mathematical form. It is shown that they restrict the equations governing irreversible changes. The theory of a general fluid mixture is developed in some detail, and is then applied to mixtures of ideal gases. It is shown that the usual equations for the velocity of chemical reactions are consistent with the second law provided that the departure from equilibrium is not too great. Mathematical complexities make it difficult to decide whether this is the case for larger deviations also. A somewhat general theory of diffusion and heat flow is considered and the requirements of the second law are formulated as the positive definiteness of a certain matrix whose elements depend on the diffusion coefficients, thermal conductivity, etc.
TL;DR: In this article, the authors described an approach with which electrons have been accelerated to an energy of 2.3 Mev by means of the electric field accompanying a changing magnetic field.
Abstract: Apparatus with which electrons have been accelerated to an energy of 2.3 Mev by means of the electric field accompanying a changing magnetic field is described. Stable circular orbits are formed in a magnetic field, and the changing flux within the orbits accelerates the electrons. As the magnetic field reaches its peak value, saturation of the iron supplying flux through the orbit causes the electrons to spiral inward toward a tungsten target. The x-rays produced have an intensity approximately equal to that of the gamma-rays from one gram of radium; and, because of the tendency of the x-rays to proceed in the direction of the electrons, a pronounced beam is formed
TL;DR: In this article, the Hartree ion core field has been used to calculate the binding energy of beryllium, which is shown to be consistent with the results of the present paper.
Abstract: Calculations of total energy as a function of lattice constant, and of some other properties, have been made for metallic beryllium, by a self-consistent field method. Since such calculations have not been made before for any but monovalent metals, the principal object was to find out how far the various assumptions which are usually made in them remain valid for higher valencies, and to test the practicability of certain methods of making the calculations. The theoretical values of binding energy, lattice constant, and compressibility agree reasonably well with experiment. The calculated work function, however, could be made to coincide with the experimental value only by assuming a surface double layer of over 5 volts, which seems impossibly large. This suggests a very large deviation of the exchange energy from the value for completely free electrons. An investigation of the behavior of the exchange energy yields an expression for the deviation from the free electron value which is valid for low electron densities, but not for those which occur in beryllium. The distribution of the electronic states in energy is found to be of the sort needed to account for the diamagnetism of beryllium. Concerning methods of calculation, it is shown that a rather complicated procedure is necessary to obtain quantitative results when a Hartree ion core field is used (as was done in the present case), and that construction of an empirical field is preferable. The assumption ${E}_{k}={E}_{0}+\frac{\ensuremath{\alpha}{\ensuremath{\hbar}}^{2}{k}^{2}}{2m}$, for the energy of an electron with wave vector $k$, cannot be used for calculations of lattice constant or compressibility for a divalent metal; it is therefore necessary to calculate directly the energies of states near the Fermi surface. This was done by the "orthogonalized plane wave" method, which is shown by tests to be capable of fairly high accuracy, though laborious. This method suggests a simple qualitative way of understanding a number of features of the electronic energy spectrum of a metal and its manner of variation with lattice constant. Incidental results include a proof that the interaction of the $1s$ shells is entirely negligible, and a calculation of the electrostatic interaction energy of the ions as a function of the $\frac{c}{a}$ ratio.
TL;DR: In this paper, it was shown that the Pauli-Weisskopf wave equation has complex frequencies, such that the quantized field Hamiltonian can no longer be diagonalized, and points of similarity and difference between this theory and the Dirac positron theory are discussed.
Abstract: For some electrostatic potentials, such as that represented by a sufficiently deep well, the Pauli-Weisskopf wave equation has complex frequencies. It is shown that when this happens the quantized field Hamiltonian can no longer be diagonalized. Some points of similarity and difference between this theory and the Dirac positron theory are discussed.
TL;DR: The magnetic resonance method of determinng nuclear magnetic moments in molecular beams, recently described by Rabi and his collaborators, has been extended to allow the determination of the neutron moment as discussed by the authors.
Abstract: The magnetic resonance method of determinng nuclear magnetic moments in molecular beams, recently described by Rabi and his collaborators, has been extended to allow the determination of the neutron moment. In place of deflection by inhomogeneous magnetic fields, magnetic scattering is used to produce and analyze the polarized beam of neutrons. Partial depolarization of the neutron beam is observed when the Larmor precessional frequency of the neutrons in a strong field is in resonance with a weak oscillating magnetic field normal to the strong field. A knowledge of the frequency and field when the resonance is observed, plus the assumption that the neutron spin is \textonehalf{}, yields the moment directly. The theory of the experiment is developed in some detail, and a description of the apparatus is given. A new method of evaluating magnetic moments in all experiments using the resonance method is described. It is shown that the magnetic moment of any nucleus may be determined directly in absolute nuclear magnetons merely by a measurement of the ratio of two magnetic fields. These two fields are ($a$), that at which resonance occurs in a Rabi type experiment for a certain frequency, and ($b$) that at which protons are accelerated in a cyclotron operated on the $n$th harmonic of that frequency. The magnetic moment is then (for $J=\frac{1}{2}$, $\ensuremath{\mu}=\frac{{H}_{b}}{n{H}_{a}}$. $n$ is an integer and $\frac{{H}_{b}}{{H}_{a}}$ may be determined by null methods with arbitrary precision. The final result of a long series of experiments during which 200 million neutrons were counted is that the magnetic moment of the neutron, ${\ensuremath{\mu}}_{n}={1.93}_{5}+0.02$ absolute nuclear magnetons. A brief discussion of the significance of this result is presented.
TL;DR: In this paper, the authors consider the multiple scattering of fast electrons by thin foils and give a new formula which represents the mean projected deflection within 1 percent over the whole range of experimental conditions.
Abstract: According to a number of observers the multiple scattering of fast electrons by thin foils is appreciably less than the scattering predicted by the theory of this effect given some time ago by the present author. The statistical part of the theory, which in the earlier work was developed with special regard to the scattering of cosmic-ray particles, has been reconsidered more closely for the conditions of the experiments on fast electrons. The results confirm within a few percent the scattering given by the earlier general formula. The discrepancies must accordingly be attributed to experimental error or to a failure of the basic collision theory. A new formula is given which represents the mean projected deflection within 1 percent over the whole range of experimental conditions. The distribution of the projected deflections is considered more closely than in the earlier paper, and a general expression for the most probable deflection in space is also given. The theory is extended to the multiple scattering of $\ensuremath{\alpha}$-particles, the quantum-mechanical collision theory in this case taking the form of classical mechanics. The results are in satisfactory agreement with the early experiments of Geiger and of Mayer. A discussion is given of recent papers on the subject. Reasons are given for the nonoperation of the interference effect considered by Wheeler, and which according to him appreciably reduces the scattering. The numerical results of a new treatment of multiple scattering by Goudsmit and Saunderson are shown to be the same as those required by the general formula given by the writer's theory. Other points raised by these authors are also discussed. A summary of the theoretical results is given in Section (7).
TL;DR: In this paper, a mass-spectrometric study of the products of ionization and dissociation of water vapor and ammonia by electron impact is given, and the possible processes responsible for the formation of the various ions observed, both positive and negative, are discussed.
Abstract: The results of a mass-spectrometric study of the products of ionization and dissociation of water vapor and ammonia by electron impact are given. The ionizing potential of the ${\mathrm{H}}_{2}$O molecule is found to be 13.0\ifmmode\pm\else\textpm\fi{}0.2 volts, and of the N${\mathrm{H}}_{3}$ molecule 10.5\ifmmode\pm\else\textpm\fi{}0.1 volts. The possible processes responsible for the formation of the various ions observed, both positive and negative, are discussed. Ions observed in ${\mathrm{H}}_{2}$O vapor are ${\mathrm{H}}_{2}$${\mathrm{O}}^{+}$, O${\mathrm{H}}^{+}$, ${\mathrm{O}}^{+}$, ${\mathrm{H}}^{+}$, $\mathrm{H}_{2}^{+}$, ${\mathrm{H}}_{3}$${\mathrm{O}}^{+}$, ${\mathrm{O}}^{\ensuremath{-}}$, and ${\mathrm{H}}^{\ensuremath{-}}$. Ions observed in N${\mathrm{H}}_{3}$ are N$\mathrm{H}_{3}^{+}$, N$\mathrm{H}_{2}^{+}$, N${\mathrm{H}}^{+}$, ${\mathrm{N}}^{+}$, ${\mathrm{H}}^{+}$, N$\mathrm{H}_{3}^{++}$, N$\mathrm{H}_{2}^{\ensuremath{-}}$, and ${\mathrm{H}}^{\ensuremath{-}}$.