Showing papers in "Progress of Theoretical Physics in 1962"

Remarks on the unified model of elementary particles

Abstract: A particle mixture theory of neutrino is proposed assuming the existence of two kinds of neutrinos. Based on the neutrino-mixture theory, a possible unified model of elementary particles is constructed by generalizing the Sakata-Nagoya model. Our scheme gives a natural explanation of smallness of leptonic decay rate of hyperons as well as the subtle difference of G_ν's between µ-e and β-decay.

Note on Unitary Symmetry in Strong Interactions

Abstract: Assuming invariance of a theory under three-dimensional unitary group, various consequences are investigated. Both the Sakata and Gell-Mann schemes can be treated in this fashion in a simple way. A mass formula for particles belonging to the same irreducible representation is derived and compared with experiments. (auth)

Anomalous Hall effect and magnetoresistance of ferromagnetic metals.

Abstract: The anomalous Hall effect and the magnetoresistance of ferromagnetic metals are in· vestigated on the basis of the localized d·electron (or f-electron) model. First the Hamil· tonian of the interaction between conduction electrons and localized electrons is given, which is valid when the orbital angular momenta of localized electrons are not quenched. Then we find that the scattering matrix of a conduction electron depends in a complicated way on the directions of the initial and final wave vectors and of the magnetization. The transition probability calculated to the first Born approximation by using the above matrix elements leads to the electrical resistivity of the form expressed by Eq. (1. 1) in the text. To the second Born approximation we find that the transition probability from a state k to another state k 1 is not equal to that from k 1 to k and leads to the anomalous Hall effect. The magnitude and the temperature dependence of both effects are reasonable for iron and nickel when compared with experiment. In rare-earth metals, both effects can also be obtained and, besides, the electric quadrupole moments which are associated with the orbital angular momenta cause an additional scat· tering. When we calculate the normal resistivity by adding the exchange and quadrupole scatterings, we obtain a good agreement with experiment on the magnetic contribution to the resistivity of rare-earth metals with more than half-filled 4f shells

Theory of Dynamical Behaviors of Ferromagnetic Spins

Abstract: The dynamical behavior of ferromagnetic spins is studied on the basis of the statistical mechanics of irreversible processes. A macroscopic equation determining the change in time of an inhomogeneous magnetization is derived with explicit expressions for the frequency spectrum and damping constant. With the use of the general expressions thus obtained, the following problems are discussed on the basis of the Heisenberg model of ferromagnetic spins: (1) the pair correlation of spins, (2) the magnetic scattering cross section of neutrons, (3) the frequency spectrum and the clamping constant 'Of spin waves, (4) the damping of the longitudinal spin component above and below the Curie point. In the low temperature limit, a straightforward reduction of our expressions leads to the spin wave frequency and damping equivalent to Dyson's theory of spin wave interactions. A general expression is given for van Hove's parameters describing the asymptotic behavior of the spin pair correlation at large distances. It is shown that the longitudinal spin damping in the low temperature limit exhibits a variety of k dependences, depending upon the relative magnitudes of the spin wave energy Dk2, the effective magnetic field H coming from other than the exchange interaction and the thermal energy keT; in particular, for !JtJ.eH

Nuclear Magnetic Relaxation near the Curie Temperature

Abstract: A theoretical study is made on the nuclear spin relaxation in magnetic crystals mar their Curie temperatures. The exchange narrowed hyperfine broadening of the NMR line width is shown to increase due to the slowing down of a certain part of the electron spin fluctu­ ations as the transition point is approached, giving rise to the line width whose asymptotic value near the transition point is proportional to [T a! (T- T 0 ) ] 3/ 2 in cubic ferromagnets and to [TN/(T-TN)]l/2 in cubic arttiferromagnets. The spin-lattice relaxation rate is dominated by this mechanism and has the same value as the above contribution to the line width. The effect of the anisotropy and the external magnetic field on this mechanism is also dis­ cussed. The indirect nuclear spin interaction via the hyperfine interaction is treated from a general point of view by using the wavelength dependent susceptibility. The susceptibility is calculated with the use of a molecular field approximation. Since the spatial correlation between the spins becomes long ranged as the transition point is approached, the indirect nuclear spin interaction becomes long ranged at the same time. In non-cubic crystals this interaction gives rise to the line width whose asymptotic value near the transition point (both above and below T 0 ) is proportional to [T 0 /IT-T 0 1]114 both in ferro- and antiferromagnets and the coefficient of this t~mperature factor is of the same order of magnitude as those in the first mechanism. In cubic crystals this effect vanishes or is reduced by orders of mag­ nitude. The theory is compared with the F19.resonance experiments in MnF2 and the agreement is not unreasonable.

Further Contribution to the Theory of the Line-Shape of the Exciton Absorption Band

Abstract: The line-shape of the exciton absorption band, in the case that k=O is at the bottom (or the top) of the exciton energy band and that there are no states with the same energy in other exciton bands, is investigated in the limit of weak exciton-phonon coupling, with the use of the damping theory. The equations for the energy dependent shift and broadening are solved with the aid of graphic calculation. Except for the low temperature region, one can assume the exciton-phonon scattering to be elastic, and the line-shape in the main part of the absorption band is determined essentially by the properties of long wavelength excitons and phonons. The half-value width is rather small, and is proportional to (gT)2 where g is the exciton-phonon coupling constant and T is the absolute temperature. The line-shape is strongly asymmetric, with a tail which is due to the indirect transition, on the high or low energy side according as ff:>==O is the bottom or the top of the exciton band. The line-shape of the exciton absorption band in in~mlating crystal was investigated in a previous paper/> with the method of the generating function. The absorption corresponds to the electronic transition from the ground state of the crystal to that excited state in which a bound pair of an electron and a hole propagates through the crystal with translational pseudo-momentum hk transferred to the electron system from the incident photon. Since the wave number lr, of the photon is much smaller than the reciprocal of lattice constant, we can put k~O for usual purposes. The absorption spectrum should consist of a series of sharp lines corresponding to various quantum states of the relative motion of the electron and the hole (ground and excited states of the exciton), if there were no perturbing effects such as lattice vibrations and lattice imper­ fections which scatter the exciton, and the (spontaneous) radiation field which may annihilate the exciton. 2 > The effects of lattice vibrations and lattice imperfections on the width of the absorption line were discussed in the previous paper, which will be referred to as I in the present paper. The following conclusions were obtained in I, as regards the effect of lattice vibration. vVhen the coupling constant g between the exciton and lattice vibration is sufficiently weak, the absorption line-shape

Magnetization Process of a Screw Spin System

Abstract: Effect of external magnetic field on the screw structure of spin arrangement at absolute zero is studied in detail'for the case where the field is applied perpendicularly to the screw axis. First, the anisotropy energy which confines the spin vectors in the plane perpendicular to the screw axis is assumed to be so large that for any strength of the field the spin vectors remain in that plane. For weak fields, the change of the screw structure due to the field is calculated as a power series of the field strength and the calculation of the energy of the system is carried out up to the fourth power. A singularity appears for turn angles of 90" and 180". For high fields, expansion in powers of the deviation angles of the spin vectors from the field direction is made, and it is shown that there is a critical field, H 0, above which the vectors are all parallel to the field direction and below which they oscil­ late sinusoidally about the field direction with an amplitude which is proportional to (H- H 0) 112. Phase transition between the screwing and sinusoidally oscillating structures at an intermediate field ·strength is discussed. In the neighbourhood of turn angles of 90", 180", and 60", there appear regions of stability of doubly antiferromagnetic, antiferromagnetic, and triangular arrangements. Second, the assumption for the anisotropy energy is relaxed,_ and phase transitions between. the screwing, cycloidal (more strictly, elliptically conical), and sinusoidally oscillating structures are studied for small turn angles. § l. Introduction After the discovery of the screw spin structure by Y oshimod> (first reported by Nagamiya2> at the International Conference on Magnetism at Gre­ noble, 1958) and later by Villain3> and Kaplan/> a number of substances with this structure have been found by neutron diffraction experiments. Of particular interest are modifications of screw structure found in rare-earth metals by an Oak Ridge group, including Koehler, Wilkinson, Wollan and Cable.5> Modifications of screw structure come about through two causes : external field and anisotropy energy. The former is the topic to be dealt with in the present paper. Herpin and Meriel6> observed by a neutron diffraction experiment an interesting change in the screw spin arrangement in MnAu2 wlien the magnetic field applied to this crystal was varied, and they gave a theoretical explanation for this change. Enz7> developed a similar theory. The present work, done independ­ ently of them, dea~s with this problem more generally and extensively and with more mathematical rigour.

g-Shift and Anomalous Hall Effect in Gadolinium Metals

Abstract: It is shown that the coupling between spins of conduction and localized electrons due to a covalent mixing between them can be expressed by a usual exchange type interaction with negative (antiferromagnetic) sign. It is proposed that the mixing effect is large in gadolinium metal under the assumption that the energy level corresponding to a trivalent gadolinium ion is very near and below the Fermi surface. Then it is shown that the mixing effect gives rise to a large negative g·shift for the localized spin, which substantially cancels the positive g-shift which comes from the exchange interaction, thus leading to a good .agreement with the observation. Furthermore, by introducing a spin-orbit coupling of localized electrons, we can obtain an anisotropic interaction between conduction and localized electrons, which is found to account for a large anomalous Hall effect observed in gadolinium metal.

On the Energy Distribution of Terms and Line Arrays in Atomic Spectra

Abstract: The Coulomb interaction leads to a splitting of the different terms belonging to the same many-electron configuration. We have studied the resulting energy distributions of terms and also of line arrays for transitions between different configurations. Expressions are derived for the first two moments of the distributions. namely, the average energy shift and the mean square deviation, as a function of the number of particles. The detailed shapes of the distributions are investigated both for (d)n configurations and for a simplified two-dimensional model.

The Ground State of Liquid He4

Abstract: In a recent study of liquid He/sup 4/ (Wu and Feenberg, Phys. Rev. 122,739) the virial theorem was used to determine the range and strength parameters in several 6-n type interatomic potentials. Comments are presented on the criticism of the study by Hiroike (Prog. Theor. Phys. 27,342.). (W.D.M.)

Collective Motion of Particles at Finite Temperatures

Abstract: A quantum-statistical theory of frequency spectra and damping constants of the col­ lective motion of interacting particles at finite temperatures is presented, and used to clarify several problems. The formulation is based on the explicit recognition of the fact that a set of collective variables , properly chosen describe the collective motion in such a way that the average values of the collective variables at a time determine their average values thereafter. The frequency distribution of the density fluctuations in fluids is thus analyzed, and molecular expressions for the intensities and widths of the spectral lines are obtained. The widths are written in terms of generalized transport coefficients which depend upon the wavelength of the collective oscillation or the frequency spectrum. The expressions are valid even in the case in which the hydrodynamical description is inapplicable, and turn out to be useful for describing the sound attenuation in liquid helium II at low temperatures and the inelastic scattering of neutrons by liquids. The transport coefficients of fluids are formulated in terms of equilibrium fluctuations from a new point of view without the use of the local equilibrium ensemble. The results for the shear viscosity and thermal conductivity agree with those obtained by the author previously. A new term, however, is found to be added to the dynamical flux determining the bulk viscosity. This term arises as a result of subtracting a pressure fluctuation as­ sociated with the fluctuation of the mass and energy densities to define a random force.

Theory of Vector Interactions in Nuclear Beta Decay

Abstract: In a previous work the Siegert theorem was extended a vector interactions in nuclear beta decays on the basis of the conserved vector current hypothesis. It was shown that the experimental data of RaE are consistent with the conserved vector current hypothesis. The purpose of this note is to give the general formulae for arbitrary order of forbidden transiti ons, since it is highly desirable to investigate as many other non-unique forbidden beta decays as possible. The validity of non-relativistic approximation is assumed for the motion of nucleons in nuclei, so that the formalism is completely analogous to the conventional theory of nuclear radiative transitions. (auth) ferro- and antiferromagnets and the coefficient of this temperature factor is of the same order of magnitude as those in the first mechanism. In cubic crystals this effect vanishes or is reduced by orders of magnitude. The theory is compared with the F/sup 19/-resonance experiments in MnF/sub 2/ and the agreement is not unreasonable. (auth)

Possible Unified Models of Elementary Particles with Two Neutrinos

Abstract: Possible unified models of elementary particles are discussed assuming the existence of two kinds of neutrino accompanying with electron and muon respectively. The discus­ sions are focused on the Nagoya model which is based on the Sakata model of baryons and mesons and the Gamba-Marshak-Okubo symmetry. In its connection the following assump­ tions are taken: i) Fundamental particles among baryons and mesons have one-to-one correspondence with leptons or their linear combinations. The correspondence is realized through a kind of "matter". ii) Basic leptons do not transmute each other by the strong interaction between the fundamental baryons. There are two essentially different types of model. One depends on the existence of two neutrinos which are Dirac particles, and the other is related with two Majorana neutrinos. With regard to the models, the difference between electron and muon and the asymmetry of A-pionic decay are also discussed.

On dirty superconductors.

Abstract: The purpose of this work is to study the mean free path effect of non-magnetic impurities on superconductors. It is shown by including the impurities in the electron-phonon system that, contrary to the results of the earlier theories, there is no change in the transition temperature which is of the order of 1/w0r, where w0 is the cutoff frequency of the BCS theory and 'C' the relaxation time in the normal state due to the scattering. The same effect on the superconductor with the anisotropic energy gap is studied explicitly. The scattering in this case leads to the reduction of the transition temperature which is initially linear in 1/'t' but becomes logarithmic when 't'Tc

One Boson Exchange Model in Proton-Proton Scattering*): An Approach to the Strong Interaction

Abstract: A phenomenological model for strong reactions is proposed. In the model it is assumed that the resonance states can be treated as elementary particles as the meson( e (as Fe/sup 59 ) and nucleon and the physical processes can be described by the sum of the lowest order matrix elements. As an application of the model the proton-proton scattering below st run. Concent 300 Mev is studied. It is shown that the present model can successfully explain the experimental phase shifts if the meson( e (as Fe/sup 59 ), scalar boson, and vector boson dominantly contribute to the nuclear force at this energy region. (auth)

Fermi surfaces and spin structures in heavy rare-earth metals.

Abstract: The screw-like spin structures in heavy rare-earth metals found ·by neutron diffraction experiments are investigated on the basis of the s-f exchange model. The calculation shows that a screw structure with the screw axis along the c axis is stable for a hcp lattice with three conduction electrons per atom and that the period of the screw becomes longer with decrease of the atomic number. From this and other considerations it is concluded that the spin structure has a 'close correlation with the Fermi surface for heavy rare-earth metals.

Electrical Resistivity Associated with Screw-Type Spin Arrangements

Abstract: The effect of a screw-type spin ordering on electrical resistivity, as a consequence of interaction between localized spins and conduction electrons, is calculated. The electrical resistivity of Er is calculated by this method between 0 and 130 deg K as an example, and the results are compared with experimental data.

Note on Unitary Symmetry in Strong Interaction. II Excited States of Baryons

Abstract: A classification of the baryon isobars has been investigated on basis of the unitary symmetry model which has been developed in a previous paper under the same title.

Conducitivities for Inhomogeneous Electric Fields in Space and Time

Abstract: The electron transport equation for conductors under electric fields which vary both in space and time is developed quantum-mechanically. The case where electrons are scattered elastically by static impurity centers and the case where scattered inelastically by phonons are treated separately. The equation derived has a new extra term, which represents the coupled effects of electron collisions with impurities or phonons and electron motions by the external oscillating fields. This term plays an important role in high frequency regions, though vanishes in the static limit. The conductivities in the static limit and in the high frequency limit are given from special solutions of this equation. (auth)

On the Static and Spherically Symmetric Solutions of the Yang-Mills Field

Abstract: The b field equations proposed by Yang and Mills are investigated from a classical point of view, and the general solution is obtained under the condition of the static and spherically symmetric field. It is found that the solution is reducible to the "canonical form" by means of the isotopic gauge transformation. From this canonical form it is very likely 'that the mass of the b quantum is zero in spite of the non-linearity of the field equations. § l. Introduction It was pointed out by Yang and Mills 1 l some years ago that the conven­ tional formalism of charge independence hypothesis is not consistent with the concept of the localized field that underlies usual physical theories. They explored the possibility of admitting the arbitrary orientations of the isospin axes at all space-time points. Such an arbitrary way of choosing the orientation of the isospin axes is called isotopic gauge. The isotopic gauge bears a ·close analogy with the electromagnetic gauge which represents an arbitrary way of choosing the complex phase factor of a charged field at all space-time points. In electrodynamics the introduction of the electromagnetic field enables us to counteract the gauge transformation of charged fields. In quite a similar manner Yang and Mills introduced a physical field of a new kind to ensure the invariance of the theory under the isotopic gauge transformation. This new field is called a b (or B) field, which is represented by twelve components bw Here the index p. (p.=l, 2, 3, 4) refers to the space-time coordinates and the bold face letter b indicates that it is a vector-like quantity in an isospace. Since the proposal of Yang and Mills, various investigations have been made 2 J-IO) concerning the isotopic gauge invariance and the properties of the b field from the standpoint of the classical as well as that of the quantized theory of fields. In spite of these studies, however, it is still difficult to draw any definite conclusion about their quantum properties on account of the non-linear

Theory of Resonance Probe

Abstract: A theoretical analysis is made of the resonance phenomena in the radio-frequency probe experiments of Takayama et al. The Boltzmann-Vlasov equation is solved under the action of an external rf electric field. The solution gives the resonance peak of the de component of the electron current to the probe at the plasma frequency. For a partially ionized plasma, the peak-height IJj and the half-width ilah/2 are given by the following formulae.

Quantization of Flux in Superconductors

Abstract: It is shown that flux quantization in superconducting cylindrical tubes follows directly from a microscopic theory

The Production of Radio Isotopes in Collisions of Cosmic Ray Nuclei with Hydrogen in Space and the Effect of Their Decay on the Composition of the Radiation Observed near the Earth

Abstract: The effects arising from the decay of radioisotopes produced in collisions of cosmic-ray nuclei with hydrogen in space on the composition of the radiation observed in the neighborhood of the earth are investigated. The production cross sections for all the individual isotopes resulting from such collisions are deduced, using experimental data and an empirical relation. The calculations show that: the cosmic radiation should traverse an amount of 2.5 plus or minus 0.5 g/cm/sup 2/ of hydrogen between the source region and the earth; the calculated relative intensitles of Li, Be, and B nuclei, produced in collisions of heavy nuclei with hydrogen, after taking decay effects into account, agree with the observed relative intensities of these nuclei in the primary cosmic radiation in the vicinity of the earth; nuclei with Z = 16 to 19 are very rare near the source region compared to nuclei with Z = 10 to 15 and Z = 20 to 28; the decay effects of radioisotopes give rise to pronounced even-odd effects for nuclei with Z = 10 to 19; and with experimental data on the composition of the primary cosmic radiation at the top of the earth's atmosphere and the partial cross sections deduced,more » it is impossible to start with a pure Fe- source for the cosmic radiation and obtain the observed chemical composition near the earth by spallation processes. (auth)« less

Variational Study of Ground State of a Bose-Einstein Fluid

Abstract: Under the assumption that the ground-state wave function is expressed as a product of pair functions, the expectation value of the ground-state energy is rigorousJy expressed in terms of the pair distribution function. An integra-differential equation is derived for the pair distribution function from the variational principle that the expectation value of the energy is minimum. It is pointed out that the expression for the energy and the integra­ differential equation become identical with those c1eri vecl by Abe if the hyper-netted chain (HNC) approximation is introduced. It is proved that the HNC approximation is compatible with the virial theorem. The nature of the formulas is investigated by applying them to some cases that have been investigatecl by other theories. It is suggested that the present theory may in ge_neral predict the phonon-like excitations.

Alpha-Cluster and Nuclear Surface

Abstract: It is shown that the internal probability density of alpha particles emitted from heavy nuclei has a maximum at the nuclear surface. The radial dependence of the overlap integral of the wave functions for four nucleons at the top levels of heavy nuclei and alpha particles was calculated using the harmonic oscillator shell model.

Statistical Dynamical Behaviors of a One-Dimensional Lattice with an Isotopic Impurity

Abstract: Equations of motion of a harmonically coupled one-dimensional lattice are exactly solved in the case that a lattice contains an isotopic impurity particle of an arbitrary mass. Statistical dynamical properties of the system are also investigated by introducing an initially canonical ensemble. The time series of displacement and velocity of a particle in the lattice constitute the stationary Gaussian processes, whose statistical properties, i. e. the ergodicity, non-Markovian properties and the approach to the equilibrium distribution are discussed. It is shown that the approach to the equilibrium (Maxwellian distribution) fails when the system has a localized vibrational mode.