Tomoyasu Tanaka

Theory of Relaxation Phenomena near the Second‐Order Phase‐Transition Point

One may expect that the observables connected with nonequilibrium processes will show abrupt changes if a substance undergoes a second‐order phase transition, since the equilibrium thermodynamic quantities appear to do so. In order to study these phenomena in a connected way the assumption is made that the long‐range order parameter and the short‐range order parameter can be treated as fluxes and forces in the sense of Onsagers theory of irreversible thermodynamics. Actual calculations are performed for two cases: an order—disorder system with short‐ and long‐range order and a system with two modes of long‐range order (antiferromagnet). The absorption of sound is calculated and its behavior near the critical temperature is analyzed. The function is continuous with a discontinuity in the slope provided the phenomenological constants are smooth functions of the temperature.

Dynamic Behavior of a Set of Weakly Coupled Ising Spins

The master equation of a set of independent equivalent spins contains only one undetermined constant, the rate constant. If one assumes the local field to be altered by the field produced by the two neighboring spins, one can formulate a set of equations for the average of one, two, three, etc., spins. On assuming an Ising interaction between the spins, weak compared to the coupling with the heat bath, we can terminate the hierarchy and solve the problem of a linear chain with periodic boundary conditions by Fourier‐transformation. The resulting secular equation determines two sets of relaxation times and two sets of eigensolutions. An explicit solution for the spin averages is given for the initial condition describing a localized excitation. Similarities with, and differences between, this and the random walk problem is pointed out.

Excitation spectra of magnetic bubble lattices

The excitation spectra of a hexagonal lattice of magnetic bubbles is calculated using the methods of lattice dynamics. The bubbles are allowed one internal zero‐mode radial degree of freedom and two translational degrees of freedom of the center of mass. This results in three branches of free oscillation, one optical and two acoustical. The equations of motion of the system are obtained from a Lagrangian with a Rayleigh dissipative function. The Fourier transform of these equations yields a secular determinant of order three corresponding to the three branches. The secular equation is a polynomial equation of sixth degree in the complex frequency. This is solved for the directions kx and ky. The interbubble potential is approximated by a dipole‐dipole potential, and restricted to nearest neighbors. The magnetostatic bubble self‐energy is replaced by the analytic approximation of Josephs and Callen. For a close‐packed lattice with no coupling between the radial and translational degrees of freedom, the opt...

Effect of n‐Spin‐Wave Interaction on the Low‐Temperature Spontaneous Magnetization

For a Heisenberg model of a ferromagnet, it is known that the commutator of the Hamiltonian H and the operator S0− = ΣjSj− is proportional to S0−. From this fact, a conjecture is made on the lowlying energy levels of two‐, three‐, …, n‐spin‐wave problems. With the aid of the conjecture and the assumption that there is no low‐lying n‐spin‐wave bound state, it is concluded that the contributions to the low‐temperature expansion of the spontaneous magnetization due to two‐, three‐, …, n‐spin‐wave problems are of O(T4), O(T13/2), …, O(T5n/2−1), respectively.

Coherent potential theory of off-diagonal randommess: Binary alloy

Abstract The single-site coherent potential approximation (SS-CPA) for a disordered binary alloy is extended in a self-consistent manner to the case of off-diagonal randomness. The density of states for a bcc lattice is calculated in the split band limit for no correlation between diagonal and off-diagonal randomness and compared with the ordered and SS-CPA density of states.

Green function theory of a spin-12 Heisenberg ferromagnet

Abstract An alternative decoupling scheme in the Green function theory of ferromagnetism is proposed. Unlike earlier first-order Green function theories, the present theory does not contain the spurious T 3 term in the expression for the spontaneous magnetization at low temperatures.

Statistical Mechanics of Uniaxial Antiferromagnet with First‐ and Second‐Neighbor Exchanges

Low‐temperature neutron‐scattering study together with the zero‐field AFMR in MnF2 furnished a reliable set of values of first‐ and second‐neighbor exchange integrals and the uniaxial anisotropy energy. It is the purpose of this paper to see to what extent one can understand the high‐temperature properties such as the Neel temperature and the anisotropic susceptibilities in terms of the known parameters obtained at low temperatures. For this purpose the cluster‐variation method is used in which up to the two‐spin correlation is taken into account. One of the main tasks of this method is to diagonalize effective one‐ and two‐spin Hamiltonians. It is noted here that the anisotropy energy could be as big, at least, as the second‐neighbor exchange energy. It is, therefore, not justifiable to treat the anisotropy energy as a small perturbation. In this paper diagonalization of the effective two‐spin Hamiltonian is achieved for the actual spin, S = 52, and the Neel temperature and the anisotropic susceptibiliti...

Complex Cluster Integrals for an Augmented Gaussian Gas

It has been shown by Montroll, Berlin, and Hart that complex cluster integrals can be evaluated very easily if the actual Ursell‐Mayer f function is replaced by a Gaussian. The possibility of using a more realistic function was suggested by the same authors. We have carried out the actual calculation of the cluster integrals for four particles and determined the dependence of the parameters on the temperature. It is shown that the integration problem is reduced to the evaluation of higher derivatives of the so called graph determinant. The method can also be used to calculate the effective interaction between two articulation points of a complicated cluster. The importance of the nonring diagrams is stressed by some numerical estimates of discrepancy between actual condensation and the divergence of the ring series.

THEORY OF SPIN-WAVES IN A DISORDERED HEISENBERG FERROMAGNET

The pair-theory of the coherent potential approximation in the presence of off-diagonal randomness (CPA-ODR) has been applied to a disordered spin-1/2 Heisenberg ferromag-netic binary alloy. The spin-wave Green’s function has been evaluated in the random phase approximation and the randomly varying exchange interaction has been treated in the CPAODR. The density of spin-wave states (DSWS) has been calculated for a bcc lattice for the entire range of impurity concentrations. It is found that the general effect of disorder is to remove the singularities, modify the band width and introduce an additional structure in the DSWS. The ferromagnetic Curie temperature Tc for all impurity concen-trations has also been investigated. For impurities which are ferromagnetic in nature, the results from the present theory are found to be close to those obtained from the mean field theory.

An alternative formulation of the theory of superconductivity

The cluster variation method of the cooperative phenomena has been applied to the study of the theory of superconductivity. The results obtained in this formulation are equivalent to the BCS results and follow in a straight forward and natural way both at zero and finite temperatures.