Showing papers in "Progress of Theoretical Physics in 1968"

Classical Spins in Superconductors

Abstract: It is shown that there exists a localized excited state in the energy gap in a superconductor with a classical spin. At finite concentration localized excited states around classical spins form an "impurity band". The process of growth of the "impurity band" and its effects on observable quantities are investigated. § I. Introduction Since the monumental paper by Abrikosov and Gorkov 1 ) many works have been done experimentally and theoretically concerning the thermodynamic and transport properties of superconductors with small amount of paramagnetic impUrItIes. Recently several authors 2 ),3) have been interested in whe'ther loca­ lized excited states in the energy gap exist in a superconductor with a paramag­ netic impurity or not. As is well known, a paramagnetic impurity in a normal metal or a superconductor brings about the so-called Kondo effect 4 ) (a kind of quantum mechanical effects of spins) and this makes complete solutions of the problem quite difficult. In this paper we restrict our discussions to the classical spins in supercon­ ductors. In the s~d interaction between conduction electrons and a localized spin

Critical Fluctuation of the Order Parameter in a Superconductor. I

Abstract: The fluctuation of the order parameter in a superconductor in the temperature slightly above the transition temperature is studied in the framework of the current microscopic theory. It is convenient to divide the temperature region where the fluctuation of the order parameter becomes important into two regions; the classical and the' critical region. In the classical region the spatial correlation of the fluctuation is of the Ornstein-Zernike type, while in the critical region it depends on the relative distance like r-3/2. We show that the elec­ trical resistivity, the nuclear spin lattice relaxation time and the ultrasonic attenuation co­ efficient decrease like 1-ai1li- 1I2 and 1-a'i1li- 1I3 in the classical and in the critical region . respectively, where 1l= (TITe) -1 and a and a' are constants of the order (Telp.) 1/2 (lPo)-3/2 and (Tel p.) 1/3 (lpo) -1 respectively. Here p. is the chemical potential, l is the electronic mean free path and Po is the fermi momentum. It is also shown that the thermal conductivity has no singular term at the transition temperature.

Anomalous Spin Relaxation near the Magnetic Transition

Abstract: The time-scales characterizing spin relaxations near the magnetic transitions in isotropic Heisenberg spin systems are discussed.

On the Green's Function in an Expanding Universe and Its Role in the Problem of Mach's Principle

Abstract: In view of an equal qualification of Brans and Dicke's gravitational theory to Einstein's general theory of relativity on an experimental level, the problem of Mach's principle, fun­ damental in the former theory, is studied by the use of the retarded bi-~calar Green's func­ tion for their inertial field ifJ in a homogeneous and isotropic universe. After deriving con­ cretely the relevant Green's function, we show that Brans and Dicke's cosmology is compatible with their stipulation of Mach's principle, with some proviso about the radiation universe in which the trace of the energy-momentum tensor is vanishing. Four dimensional Green's functions play an essential role in the treatment of the propagation of action, not only in the Minkowski space-time, but also in a curved space-time. As regards a massless scalar field and. the electromagnetic field, a formal procedure to derive the Green's functions of these fields in any Riemannian space-time has been given by De Witt and Brehme/) but it is in general difficult to find their exact expressions by means of that procedure. Several years ago, however, the author and Kimura 2 ),*) derived concretely the Green's functions in a homogeneous and isotropic universe .to deal with the quantum nature of gravitons and photons in the universe. On the other hand, the recent discovery3) of the cosmic background radia­ tion of T= 3°K provides us with a renewal of inter~st in the worl9. models with the big-bang origin. Among various cosmological theories permitting the big­ bang models, the most interesting will be the a-/1-r-theory4) based on general relativity and Brans and Dicke's theory5) based on their gravitational theory with variable gravitation "constant". The reason is that, according to' Dicke,6) the two gravitational theories alone can reasonably explain the so-called three crucial tests and give results which are in accord with astrophysical observations ex­ amined in the light of modern stellar evolutionary theory. The underlying philosophy of Brans and Dicke's gravitational theory IS, however, considerably different from that of Einstein's, because they have con­ sidered that the E6tv6s experiment gives an experimental support of the" weak

Ultrasonic Attenuation near the Magnetic Critical Point

Abstract: Theoretical results on the attenuation of longitudinal sound in ferro- and antiferromagnets near the critical points are reported, and recent experiments in MnF 2 are discussed.

Dynamics of Critical Fluctuations. II

Abstract: Dynamics of critical fluctuations associated with the liquid-gas transition are studied em­ ploying the method developed in the previous works of this series. The CDV within which dynamics can be asymptotically closed have been found which correspond to the sound wave modes. However, these CDV are not sufficient to account for all the anomalies (divergences) . of the transport coefficients. Rapid random motions also contribute to the divergences through various transport modes, 'thus leading to partial violation of the dynamical scaling law. The results for the anomalies essentially confirm the work of Kadanoff and Swift. The sound propagation is also studied. In some high frequency regions the sound speed is found to exceed the usual adiabatic value by a constant factor, and the attenuation does not depend upon frequency. The case of the critical mixture is also briefly discussed.

Contribution to the Theory of Impurity Band Conduction. II. : Hall Effect

Abstract: In this paper we discuss the Hall coefficient of heavily doped semiconductors at very low temperatures by making use of the general method developed in previous two papers (Matsubara-Toyozawa and the first part of this series). The effect of weak magnetic field on the impurity band conduction is taken into account on the basis of the Kubo formalism and the diagram method. An expression for the Hall coefficient is obtained in terms of the Green's function and it is proved that this expression is reduced in a special case to the usual one derived from the Boltzmann equation method. Some discussion of the general feature of Hall effect in a random system is given.

Free-Energy Shift of Conduction Electrons Due to the s-d Exchange Interaction

Abstract: The free-energy shift of the conduction electrons due to the exchange interaction with a localized impurity spin is calculated by perturbation expansion up to the fourth order of the exchange interaction. It is shown that the usual procedure to obtain the free-energy shift at finite temperature by first calculating the energy shift at absolute zero and then replacing the step function by the fermi distribution function involves a serious error. By a careful analysis of this point, we find a logarithmic term in the fourth order. The contribution to the entropy of this term is which is negative for both signs of J. This term represents a decrease of the entropy of the conduction electrons and may be interpreted in terms of the correlated motion of the spin polarization coupled to the localized spin. This result seems to indicate that the binding energy of the ground state in the case of negative exchange interaction is obtained by cal­ culating the free energy at finite temperature (to infinite order) and then extrapolating the result to zero temperature.

Muon Capture in C12 and Nuclear Structure

Abstract: general p-shell wave functions which were obtained by considering the dynamical properties in the A = 12 system. Our theoretical formula for the capture rate is of such a form that the effect of the finite size of the nucleus on the muon wave functions is easily taken into account. The experimentalft value of B12, which in all previous theories is used to normalize the muon capture rate, has not been adopted in our analysis. An excellent fit to the ex­ perimental data for the capture rate as well as for the ft value has been obtained with the B12 wave function proposed by Kurath to explain the magnetic moment of B12 and a C12 wave function introduced by us. These wave functions have slightly different values of configuration mixing parameters from those originally given by Cohen and Kurath. The value for the oscillator strength parameter which we use is b= 1.64 f; this is in conformity with the elastic and most of the inelastic electron scattering data. On the other hand, should b be taken as 1.90 f, the value derived from the analysis of the inelastic electron scattering data associated with the 15.1 MeV level in 0 2, it is relatively difficult to have a simultaneous fit to the data for the capture rate and the ft value. Our results are based on the assumptions of 1) uni­ versality of the V-A interaction in muon capture and beta decay, 2) the validity of the con­ served vector current theory, and 3) the induced pseudoscalar coupling constant estimated by Goldberger and Treiman.

Magnetic-Field Dependence of the Bound State Due to the s-d Exchange Interaction

Abstract: On the basis of the theory developed by Y osida, Okiji and Y oshimori on the singlet ground state of a system of conduction eleCtrons and a localized spin coupled with an antiferromagnetic exchange interaction, the magnetic-field dependence of the ground state is investigated· in logarithmic accuracy by collecting the most divergent terms in the integration kernel. The correctness of a previous result for the susceptibility obtained by an iteration method is confirmed in the weak coupling limit, and it is further concluded that in this limit the bound state does not disappear at a finite value of magnetic field, but approaches the normal state asymptotically at high field. recently succeeded in deriving a closed form ot solution for the singlet bound state by collecting the most divergent terms in the kernel of the secular integral equation. Further, by the use of this method, calculations have been made of such quantities as the normalization integral of the wave function, the kinetic energy and the charge density and more conclusive results of a series of work are derived. 6 ) In a previous paper/) as a part of a series of the work, we have calculated by the iteration method the magnetic susceptibility of the singlet bound state at OaK and obtained ,uB 2 /IEI for the susceptibility, where E denotes its binding energy. It was shown that this result does hold at any stage of approximation. However, as the calculation was restricted in that paper to a weak field, it was impossible to elucidate the behavior of the localized spin in high field. In high field the situation is considered qualitatively as follows. The singlet bound state, which is stable in the absence of the field and at OaK, will decrease its binding energy as field is increased, because field makes a spin-flip process due to the exchange interaction difficult. In this paper, using the method of collecting the most divergent terms in the integration kernel,5),6) we derive an expression for the binding energy as a function of the applied field, which is valid up to a