Showing papers in "Progress of Theoretical Physics in 1959"

Effects of s-d Interaction on Transport Phenomena

Abstract: The effects of the so-called s-d interaction on transport phenomena of ferromagnetic metals and alloys are discussed. Electrical and thermal conductivities and thermoelectric effect are calculated with the simple molecular field approximation, and an anomalous thermoelectric effect is expected in these substances. In sufficiently low temperatures, the disordering of spin system is described by the collective mode of spin wave. The calculations of transport coefficient by using the spin wave approximation are also carried out.

Dynamical Properties of s-d Interaction

Abstract: The dynamical properties of two spin systems composed of conduction electron spins and' localized d-spins interacting by exchange are investigated in connection with the electron-spin resonance of d-spins with the free electron g-value. It seems to us that some basic consider­ ations are necessary for the problem along the line of the widely accepted idea that the s-d' interaction itself does not lead to any relaxation time and shift. The usual adiabatic approx­ imation is applied to this idea. A modified damping equation of the Bloch type as well as. Bloch's relaxation theory are also useful, but these are not enough for a complete solution of the problem. A certain controversy involved in recent experimental and theoretical work at Berkeley concerning the electron spin resonance in Cu-Mn dilute alloy is pointed out and examined.. We believe that the possibility could not be eliminated that the spin-lattice re­ laxation time of conduction electrons in the alloy due to mechanisms other than the sod' interaction is longer than previous authors have assumed.

Electrical Resistance of Ferromagnetic Metals

Abstract: We use a model for ferromagnets such that the electrical current is carried by 4s (or 6s) electrons which are assumed to be described in band scheme and the unpaired electrons (3d or 4J) are assumed to be localized on the lattice points. In the temperature region far below the Curie point the spin-disorder in spin orientation, which can be naturally described in terms of spin waves, gives rise to the scattering of conduction electrons through the s-d interaction and accordingly contributes to the anomalous part of the resistivity of type 1'2, which agrees fairly well with the experimental results both in order of magnitude and in temperature dependence. The effects of the lattice vibration are also discussed and turn out to give only the small additional contribution of type 1'5.

Giant Cluster Expansion Theory and Its Application to High Temperature Plasma

Abstract: The conventional virial expansion of thermodynamic functions is converted into a new expansion scheme, similar to the former but more powerful. The new method is particularly suitable to dealing with the interaction of long-range character, such as Coulomb potential, since it suffers from no divergence difficulties contrary to the conventional one. As an application of the method, the equilibrium properties of high temperature plasma is studied and the term of next higher orders than the Debye-Hiickel limiting law is obtained exactly. The order estimation indicates that the Debye-Hiickel law is accurate within the error of a few per cent in this case. A possible extension of the present method to the theory of non­ equilibrium properties of plasma or to quantum statistics is suggested.

An Alternative Approach to the Ergodic Problem

Abstract: The usual approach to the ergodic problem makes use of the idea of "coarse-graining". That is, a macroscopic observer is supposed to be limited to "coarse-grained" experiments, and the resultant lack of complete information about the system gives rise to the irreversible increase of the (coarse-grained) entropy. It is shown that this approach is untenable, since macroscopic observers are not restricted in principle to coarse-grained experiments, and in fact one "fine-grained" experiment has already been carried out in practice. An alternative approach is presented which avoids these difficulties. The irreversible increase of entropy is due to molecules outside the system proper, which collide with the outside of the box enclosing the system; this leads to a truly random, in principle unpredictable perturbation, which can be treated only stochastically. The number of particles within the system is irrelevant for this purpose, and in particular need not be large.

A Possible Symmetry in Sakata's Model for Bosons-Baryons System. II

Abstract: In the full symmetry theory a physical system can be characterized by six quantum numbers. Following the original intention of the composite model, an attempt is made to replace these quantum numbers by others that are easier to understand by intuition. This study leads to a new picture of the particle, which contrasts with the usual picture in the perturbation theoretical treatment. The realistic aspects developed not only promote a deeper understanding of the full symmetry theory of the composite model, but also throw a new light on the structure of elementary particles. (auth)

Equation of State of High Temperature Plasma

Abstract: It is generally accepted that the equation of state of the Debye and HUckel theory, originally developed for strong electrolytes basing on the classical statistical mechanics, is applicable to the high temperature plasmas in thermal equilibrium. However, if we were to apply the classical statistical mechanics to the fully ionized plasmas, the partition function would diverge because of the short range attraction between a pair of positive and negative charges, and there is a doubt whether the contribution of this infinite attraction may not overcome the contribution of the Debye·HUckel term. In this paper the equation of state of high temperature plasmas is investigated in. consideration of the quantum mechanics; and it is shown that the Debye-HUckel approximation. surely applies to the plasmas of low density of the order 101~_1017 or so at high temperatures where Ae::;'ao/Z, where Z is the charge of a nucleus, A. the de Broglie wave length of an electron and ao the Bohr radius. This result is obtained by reducing the problem to that of a suitable classical gas and confirming that the contribution of the watermelon terms--which is considered as the leading correction to the Debye-HUckel approximation--is negligible compared with that of the ring terms considered in the Debye-HUckel approximation.

On the Kirkwood Superposition Approximation

Abstract: The Kirkwood superpositIOn approximation is investigated by.making use of the expan­ sion theorem for the potential of average force and its correction term is found in a form of the expansion in powers of particle number density p. The lowest order term of this correction is calculated for a special configuration of three particles in a fluid consisting of hard spheres and the Kirkwood approximation is shown to overestimate the distribution function of triplets in this case. The Kirkwood integral equation with correction is solved by expanding the radial distribution function in powers of p. The solution thus obtained is shown to be exact up to the order of p2, while the Kirkwood approximation itself does not yield the correct term of the order of p2. It is shown that the distribution function of triplets can, in principle, be expanded in terms of radial distribtution functions and a few terms of this expansion are calculated explicitly, of which the first term just corresponds to the Kirkwood superposition approximation.

Equation of State of Classical Electron Gas

Abstract: An exact evaluation is presented of the free energy term which is expected to be the next higher order above the lowest one, the Debye-Huckel law. It is concluded that the Debye-Huckel is exact with respect to the deg F. Cores o order for purely Coulomb interactions as well as for the one with hard cores. (C.J.G.)

Radial Dependence of Imaginary Part of Nuclear Optical Potential

Abstract: The imaginary part of tht nuclear optical potential for neutrons is calculated by means of a method of the volume direct processes, on the basis of the statistical independent particle model with a eiffuse nuclear surface. It is shown that the localization of the absorption near the surface, which was found by Bjorklund and Fernbach, can be reproduced using reaso­ nable values for the relevant parame~ers. The effect of the Pauli exclusion principle, of the energy dependence of the two-body collision cross section, of the mass number dependence of the parameters, and of the correlation between nucleons arc also discussed.

Effect of Hard Cores on the Binding Energies of H3 and He3 Corrected Values

Abstract: There was an error in the kinetlc energy formula of a previous paper by Ohmura (formerly Kikuta), Morita, and Yamada. The binding energy of H/sup 3/ as well as the Coulomb energy of He/sup 3/ are recalculated by using the correct value of the kinetic energy. The results obtained are almost the same. (auth)

Ordinary and Anomalous Thresholds in Perturbation Theory

Abstract: Ordinary and anomalous thresholds of the matrix element corresponding to the general Feynman graph are rigorously investigated in detail. Necessary conditions for the ordinary threshold are obtained. (auth)

On the Analogy between Strong Interaction and Electromagnetic Interaction

Abstract: According to Sakata's composits hypothesis of heavy particles, the fundamental particles do not change their kind through strong interactions. A formulation of these interactions is given which requires invariance under the gauge transformation of the first kind for each baryon field. Assuming the phase function depends on the spacetime coordinate, this leads to the existence of a neutral vector meson analogous to the electromagnetic field, the mass of which need not always vanish. Its consequences are examined and found to be quite favorable for the interpretation of the strong interaction. In particular, it leads to the conservation law of parity in strong interactions and the condition for the existence of the composite state.

Tensor Force of the Pion-Theoretical Potential and the Doublet Splitting in n-He4 Scattering

Abstract: To investigate whether the spin-orbit force in the theory of nuclear shell structure is due to the tensor force of the pion-theoretical potential, the doublet splitting of the pphase shifts in low energy n-He/sup 4/ scattering was analyzed. It is concluded that the major pant of the experimental doublet splitting can be reproduced by the strong tensor force of the pion-theoretical potential. It is shown qualitatively which features of the pton-theoretical potential are important to the binding energy of He/sup 4/ and the discontinuity of the binding energies between He/sup 4/ and the system of He/sup 4/ plus one nucleon. (auth)

Notes on Megneto-Hydrodynamic Equilibrium

Abstract: An effect of rational surfaces in a stellarator was studied with varying rotational transform ratio. It can be shown that lines of current describe rapidly oscillating curves in the neighborhood of the rational surface. The diffusion velocity of fluids across magnetic lines increases due to the finite conductivity of fluids and the long paths of current lines in that region. A vontex flow may occur under some conditions. The dlffusion effect will be small In practical cases, according to this semiquantitative treatment. (B.O.G.)

A Possible Symmetry in Sakata's Composite Model

Abstract: In Sakata's model of strongly interacting particles, it is assumed that the proton, neutron, and LAMBDA -particles are the basic particles which combine in various manners to make up mesons and other baryons. A certain quality which exists among these three particles is investigated. The view is taken that there exists a certain origin of the mass of the particle which can never be redueed to the self-action of the usual interactions. The symmetry of the interaction among the three basic particles is of primary importance and charge independence has a secondary meaning. (A.C)

Nuclear Photo-Reaction and Excitation Modes of Nuclei

Abstract: particle excitation and the core excitation. The careful comparison of the predictions by the single particle model with experimental data shows that these two types of excitation mode are distinguished in the photo-reaction of the light nuclei. It is indicated that the resonances observed in the (r, n) reactions of CI3, NI4 and FIg and NI4(r, p)CI3 in the energy region of 10~15 Mev are essentially due to the extra particle excitation. In the case of medium weight and heavy nuclei the resonances due to the extra particle excitation should appear, in many cases, at the energies lower than the thresholds for the (r, n) or (r, p) reactions. On the other hand, the giant resonance is characterized by the core excitation mode.· For the description of the core the possibility that the collective model may be rather reasonable than the simple independent particle model is indicated. And further, the distinction between these two models are discussed in detail.

On the Space-Time Formulation of Non-Relativistic Quantum Mechanics

Abstract: For the case of the quantization of the usual non-relativistic classical Lagrangian function quadratic in the velocity the validity is demonstrated of the non-canonical space­ time formulation of quantum mechanics proposed recently by the author, which aims to evaluate, without appealing to the Schrodinger equation, the transformation function K(x, til; y, tl) in the space representation on the basis of the composition rule K(x, til; y, tl) = ~ K(x, til; z, t)dzK(z, t; y, tl) (1) coupled with the supposition that it is approximated to zeroth order In the quantum of action h by the so-called semi-classical kernel K" (x, til; y, t l ) = [(i/ h) 82S/8x8y] 1/2 exp [(i/h) S (x, til; y, t l )] (2) written in terms of the classical action S ex, til; y, t l ) alone. In the first place the action function corresponding to the above Lagrangian is expanded in power of the interval of time T=tll-tl. Then the deviation of the semi-classical kernel (2) from the unitary transformation function is shown to be of the third order in T, and the corresponding correction term is evaluated by solving the integral equation (1). It is also shown that the semi-classical kernel is unitary for a free motion of a particle with its mass being a function in the space coordinate.

Applicability Conditions of the Hydrodynamical Model of Multiple Production of Particles from the Point of View of Quantum Field Theory

Abstract: In the framework of quantam field theory, it is attempted to investigate whether the hydrodynamical description is applicable to the meson cloud produced in extremely high energy collisions of nucleons as considered in Landau's theory of the multiple production of particles. The applicability conditions of the hydrodynamical model consist of local equilibrium and conditions for the possibilities of defining the local system in the meson cloud, which are prepared by the methods based on quantum statistical mechanics of irreversible processes. These conditions are examined by cornparison of the correlation lengths and the relaxation times of the meson fluid with a characteristic length and time, in which the thermodynamical parameters, the temperature for example, of the fluid decrease or increase by an appreciable amount on a macroscopic scale. From such examinations, it may be concluded that the hydrodynamical model holds almost everywhere except ia the front part of the cloud after the whole cloud spreads over a region whose size is the order of the correlation length. It is ernphasized that the interactions in the initisl cloud directly ater collision and in the front part of the expanding cloud can never be described by any statistical law or hydrodynamics. The factmore » that the front particles are never in any thermal equilibrium suggests that they remember some features of initial high energy interactions in the very small cloud. It is inferred that the distributions (for example, K/ pi ratio and the momentum or angular distribution) of the front particles may be informative about the interactions at very small distances. The influenees of initial interactions on the remaining cloud are only taken into account through the initial boundary conditions for the hydrodymical equation. In addition, it is pointed cut that the assumption of the perfect fluid used by Landau is not so good; it turns out that one can expect an increment of the number of particles through the final interactions. Finally it is discussed whether these characteristics may be consistent with the recent experiments. (auth)« less

The Evolution of Massive Stars. I

Abstract: The evolution of stars with mass 15.6 times the solar mass is considered in their early stage. Our model star consists of three regions; a convective core consuming hydrogen, a radiative envelope, and an intermediate zone of continuously varying composition. Radiation pressure is taken into account throughout the three regions. The hydrogen content in the core ranges from 90 per cent to 6.3 per cent by weight. It is found that the convective core retreats as its hydrogen content decreases, setting up the intermediate zone of continuously varying composition and a thin convective unstable region between the radiative envelope and the intermediate zone. The evolutionary track in the FI R diagram is compared with the stars of the h + X Persei clusters.

Electron Interaction in Very Long Linear Conjugated Molecules. I: One-Dimensional Collective Oscillation of π Electron

Abstract: Low-lying electronic states of very long linear conjugated molecules with equal CC bond ]engths is studied within the framework of Pariser and Parr's TC electron approximation, by 'means of Tomonaga's one-dimensional sound wave method. It is found that as the number N of carbon atoms increases, the frequency of the sound wave with the lowest wave number . tends to zero in proportion to vTog~NIN. It is shown that Araki and Murai's treatment which gives a result contradicting with ours contains a difficulty.

Electron Interaction in Very Long Linear Conjugated Molecules. II ---Elementary Excitations in a System with Small Energy Gap---

Abstract: are our main interests. These states are investigated in the framework of the tight-binding approximation. In usual one-dimensional systems, such as very long linear conjugated molecules, the plasmon levels sink into the level continuum given by excitation energies of one-pair states. However, it is shown that the plasma oscillations are stable when their energies are sufficiently larger than the energy gap. On the other hand, the plasma oscillations with sufficiently small momentum whose energies in case of Eg=O were smaller than the present energy gap seem to be dissolved away into the level continuum. A formulation to derive exciton solutions is given. The screening effect for the attractive force between the electron and the hole is investigated by means of the Gell-Mann and Brueckner technique and is found to grow larger as the energy gap becomes smaller. By this screening effect the possibility of getting the exciton-like bound state diminishes as Eg->O. As a collorary we have found that the potential energy of two electric charges zle and Z2e separated by a distance r in a three dimensional system with small energy gap becomes ZlZ2e2 exp( -ar) jr if r /BjABI, where E=ra-1 (IBI/(AB)2) X (e2/an).