Showing papers in "Progress of Theoretical Physics in 1960"

A New Approach to the Theory of Classical Fluids. I

Abstract: An exact integral equation is found for the pair distribution function. The integral equation is of somewhat different nature from the usual ones known in the theory of classical fluids, in the point that it involves an infinite series. The Helmholtz free energy is expressed as a series expansion which may be more rapidly convergent than the usual one. It is shown that the integral equation can be derived also by means of a variational principle from the expression for the free energy. It is pointed out that the theory of classical fluids may be constructed with the knowledge of the pair distribution function alone, even if a form of the pair interaction potential is not known.

Origin of the Magnetic Anisotropy Energy of Cobalt Ferrite

Abstract: The magnetic anisotropy of cobalt ferrite is considered to arise from the cobaltous ions in the crystalline field of a low symmetry. The crystalline field due to the averaged-out charge distribution of C02+ and Fe3+ ions in the octahedral sites gives the lowest-lying twofold degenerate orbital level of the C02+ ion and to this level are associated four spin levels, corresponding to 8=3/2, as each spin is subject to exchange field. Each of these orbitally doubly degenerate levels is further split into two by the low symmetry field arising from the difference of charges of C02+ and Fe3+ ions and spin-orbit coupling. The magnitude of this energy splitting depends on the direction of the exchange field, thus on the direction of the magnetization, and from this origin the anisotropy energy arises. The relation between the magnitude of the anisotropy energy and the strength of the low symmetry field is discussed. Two kinds of configuration, A and B, of the C02+ and Fe3+ ions neighbouring each C02+ ion are considered as having low energy values. By properly choosing the relative numbers of C02+ ions in these A and B configurations and the strengths of the respective low symmetry fields, the calculated temperature dependence of the cubic anisotropy constant Kl can be fitted with that measured by Shenker. The values of the parameters thus determined seem to be reasonable, as they can be compared with those calculated on the assumption of the point charge model. The magnetic moment consisting of the spin and orbital moments is calculated to be 3.4~3.5 J1.g per C02+ ion. Finally, the dependence of the anisotropy constant of the mixed Fe-Co ferrite on the concentration of the cobaltous ions is discussed.

Theory of Classical Fluids: Hyper-Netted Chain Approximation. III A New Integral Equation for the Pair Distribution Function

Abstract: The expansion formulae for the free energy and the radial distribution function by means of the \"hyper-netted chains\" are also presented. They contain the results in the hyper-netted chain approxi­ mation as their leading terms. Another set of the formulae in the hyper-netted chain approximation is given and compared with the theories for ionic systems in the past. Applications to practical problems will be given in forthcoming papers.

Excitations in a High Density Electron Gas. I

Abstract: A new systematical methcd for the description of an electron gas in terms of bosons is developed. This boson corresponds to an "exciton," i.e., a pair of an electron outside the Fermi sphere and a hole inside. The formalism is particularly suitable to the system at high density, as suggested by Sawada's discussion of the same system. As a straightforward application, the effect of electron exchange on the plasma frequency is calculated. The result coincides with that of a Rartree-rock treatment. (auth)

Spin-Orbit Splitting and Tensor Force. II

Abstract: The second-order effect of the tensor force on spinorbit splitting In light nuclei through the perturbation theoretic treatment was calculated, using the mesontheoretic potential of Serber, It was found that half the experimental value of the P-state doublet splittirg in He5or N15 is explained through this calculation. Previous calculations have lead to splittings of the wrong sign or of too small a magnitude, while Feingold's variatioral calculation gave toe small a magnitude with the right sign. The comparison of this calculation with Feingold's is discussed. Further results will be reported along with the effects of the tensor forces on O17 and Ca41.

A Unified Model for Elementary Particles

Abstract: By extending the Sakata model, a unified model for elementary particles is proposed, the basic particles in the Sakata model are assumed to be constructed of the lepton and B+, which is regarded as a new kind of matter. The full symmetry among three basic particles and the symmetrical property of the weak interactions which was recently pointed out by Gamba, Marshak and Okubo come automatically of this model. The nature of B+ and the new mechanics which accounts for the binding of B+ to the lepton will be the central problem to be studied in the future. very interesting results which are in good agreement with experimental data. In spite of these successes, the Sakata model has been concerned only with the strongly interacting particles, and nothing has been done with the lepton. TvIoreover, even in the case of strong interactions, the charge independence or the full symmetry (which -we shall call the 1.0.0. symmetry) among the three basic particles was introduced in a formal \vay, and it has not yet been clear why the strong interaction should have such invariance properties. In order to describe both baryon and lepton in a unified way, and to find out a profound meaning hid­ den behind the charge independence or the 1.0.0. symmetry, we must go further from the Sakata model. For this purpose let us turn our attention to the weak interaction. The theory of yveak interaction has now come to a very definite point after the great success of the current-current interaction theory proposed by· Feynman and Cell-Mann. 8 ) Furthermore, it has recently become clear that the Feynman-Gell-Mann current derived from the Sakata model is quite suf--ficient to account for the experimental facts concerning the weak processes. 9 ),10) Especially, it should be noted that the terms which obeyed the conditions J S/ LiQ = - 1 and I LlS/ LlO I > 1 automatically dropped out of the current. Thus, by adopting the Feynman and Cell-Mann theory and

On the Collective Motion In Even-Even Spherical Nuclei*

Abstract: A method formally analogous to that developed by Bogoliubov, Tolmachev and Shirkov to investigate the collective excitations in superconductors is applied to even-even spherical nuclei in order to investigate the mechanism of the nuclear collective motion from the standpoint of particle excitations. Our theory leads, in principle, to the same results as those obtained by Belyaev. However, the method of description of the nuclear collective motion is quite different from that of Belyaev's paper in which the .. cranking model" of Inglis is employed, and the various physical parameters used by Belyaev can be derived uniquely from the "first" principle. Thus, in so far as the vibrational motion is concerned, the physical implication underlying the nuclear collective model proposed by Bohr and Mottelson is made clear. It is outside the 'scope of this paper to relate the effecti ve intet-particle interactions used in this paper with the nuclear forces known from the two nucleon problems.

Quadrupole Interaction in Crystals

Abstract: The quadrupole array for the lowest energy of quadrupole interaction in crystals is obtained by generalizing Luttinger and Tisza's theory of dipole interaction. The theory is presented for the two kinds of problems: the simplest case of quadrupole pertaining to a .doubly degenerate orbital eg and the other general cases both for the cubic crystals. In the former case, the quadrupole interaction is written in terms of fictitious spins, whence we can get the solution of lowest energy classically, serving to obtain some informations of the orbital ordering in compounds with Mn3+, Cu2+ and Cr2+. In the latter cases, the five com­ ponents of the quadrupole are totally effective, leading to a rather complicated problem. The classical solution is, however, easily obtained, though somewhat complicated. The quadru­ pole array of lowest energy in face-centered cubic lattice proves to be realized in molecular .crystals N2, N 20, CO2 and CO, which is accompanied with large quadrupole but without or with small dipole moment.

Scattering of Electrons by the Thermal Motion of Impurity Ions

Abstract: The scattering of electrons by the thermal motion of the excess electronic potential of impurity ions is discussed. The excess potential is assumed as a screened Coulomb interaction of electrons with excess charges of impurity ions. The change in the electrical resistance of non-magnetic and substitutional dilute alloys due to this type of scattering is calculated at high temperature. It is assumed that the thermal motion of solute ions is quite similar to that of the perfect lattice. Calculated results are compared with the experimental deviations of the electrical resistance from Matthiessen's rule in cases of Cu-, Ag- and Au-based alloys. It is shown that the agreement between them is fairly good in most cases.

Lagrangian Formalism for a Classical Relativistic Particle Endowed with Internal Structure

Abstract: The problem of the classical motion of a relativistic particle is treated with classical mechanics in variatinual form; a single particle traveling in an external field along a definite curved line in space-time is assigned a Lagrang ian and internal variables to express its proper rotation. General conservation equations are derived and applied to the case of the relativistic spinning particle moving classically in an electromagnetic field. Equations are also derived for the precession velocity of the spin in an electromagnetic field. (D.L.C.)

Two Nucleon Potential with Full Recoil. I General Formalism and One-Pion-Exchange Potential

Abstract: Nonstatic nuclear potential due to a one-pion-exchange process is derived in the momentum space without using the expansion with respect to the inverse of the mass of the nucleon. Close examinations of its properties show that it can be expressed with good accuracy by a local potential in x-space at low energies (below 300 Mev). Nonstatic parts of the potential have different signs according as whether the assumed coupling between the pion and the nucleon is pseudoscalar or pseudovector. Comparison with the experimental phase shifts is briefly discussed. It is shown in the case of the pseudoscalar coupling that the nonstatic one-pion-exchange potential derived is almost exact from the theoretical point of view when the distance between two nucleons is far enough (large compared with one third of the pion Compton wavelength). It is also argued on some physical assumptions that the potential will in fact be exact in the case of the pseudovector coupling too. (auth)

Symmetrical Properties of Two-Dimensional Ising Lattices

Abstract: Some symmetrical properties on the statistics of two-dimensional Ising lattices are studied. It is shown that such considerations give us very useful suggestions even for cases in which we know no exact solution. We investigate especially the cases of the square, triangular, honeycomb and generalized square lattices, with respect to the expressions of spontaneous magnetizations and susceptibilities.

On the Collective Excitation of Spherical Nuclei

Abstract: The eigenvalue for collective excitations (vibrations) in spherical nuclei caused by the pairing interaction of the particles in the outermost partly filled shell is deduced with the j-j coupling shell modeland without the adiabatic approximation such as thnt used by Belyaev. The result is identical with that of Belyaev, and the conditions for the stability of spherical nucleiare discussed. (D.L.C.)

Collective Excitations of Electrons in Degenerate Bands. I ---Spin Waves in Stoner's Model of Ferromagnetism---

Abstract: Spin waves in the collective electron model of ferromagnetism are derived in a completely similar manner to that adopted in deriving exciton waves in insulators. The internal motion of an electron-hole pair forming the spin wave with a long wavelength is shown to be localized in the ordinary space. The frequency of the spin wave with long wavelength coincides with the result obtained by K. Y osida and T. Kasuya in the case where all electron spins are pointed toward the same direction in the ground state. It is concluded generally that spin waves break down unless there is a sufficiently large difference between the number of the electrons with up spin and the number of those with down spin.

Measurement in Quantum Mechanics

Abstract: In quantum mechanics it is well known that, if any two states are superposed, they interfere with each other. It is true, we should not deny such interference in principle, but we may assert what follows. When two states different from each other in a great many degrees of freedom are superposed, the interference effect becomes obscure. If they are dif· ferent in an infinitely many degrees of freedom, they do not interfere at all, and their superposition is nothing but a mere probability function. This assertion enables us to understand how the probability amplitude for a micro·system is converted into a probability function for a measuring apparatus in the course of measurement.

One-Particle Motions in Many-Particle Systems and the Optical Model in Nuclear Reactions

Abstract: A possible scheme of the systematic one-particle motion III a many-particle system is presented from the first principle as a time-dependent formalism. The theory is formulated and interpreted exclusively for the optical model in nuclear reactions, although the present formalism can be utilized to study various problems in solid state physics. First the one­ particle amplitude is so defined as to describe the processes of elastic scattering. Then it is shown that the systematic part of the amplitude, corresponding to the coarse-grained motion of the system, obeys the one-particle Schrodinger equation with the optical potential, and that the fluctuating part of the amplitude is governed by the Langevin-like equation with the same optical potential and by the fluctuation-dissipation theorem. This is just the scheme assumed in the previous paper from the semi-phenomenological point of view. The optical potential can be calculated from its definition given as the Fourier transform of the so-called "self­ energy" part appearing in the equation of the one-particle Green function in the medium of the target nucleus. From the definition it is easily seen that the optical potential is, in general, non-local and slightly energy-dependent. The optical,potential is decomposed into two parts, one being the static (or energy-independent) part to be observed in the target nucleus in the fixed ground state and the other representing reactions of nuclear excitations. It is inferred that the former would not be so different from the corresponding term of the one­ particle potential as expected in the ordinary shell model, and that the latter is small due to the average effect originating in the energy spread of the incident beam. The former is purely real, while the latter has an imaginary part which is responsible for the probability" dissipation of elastic scattering. The face of the optical potential may be of the same type irrespective of the question whether the incident beam is a simple short wave-packet or a !nixed beam, so far as the coarse-grained motions are pursued. Finally it is proved that the fluctuation-dissipation theorem holds for the, correlation function of the fluctuating source or amplitude if the system is excited in quasi-equilibrium.

A Solution of the Combined Gravitational and Mesic Field Equations in General Relativity

Abstract: Solutions of the combined gravitational and mesic fields are attempted. Since the mesic field cannot be solved in strictly empty space, it is solved for nonempty space. (D.L.C.)

A Semiphenomenological Proton-Proton Potential

Abstract: An energy independent potential is constructed which reproduces all available p-p data up to 310 Mev. At 310 Mev the potential predicts the MacGregor phase-shift solution. The potential includes the central, tensor, linear, and quadratic LS potentials. The quadratic LS potential is manifestly required in the singlet, even-parity state where the linear LS potential vanishes. The linear LS potential turns out to be more singular but of shorter range than previously thought. It appears now that the p-p data below 310 Mev can be understood in terms of a potential consistent in all respects with the pion theory of nuclear forces. (auth)