Theory of magnetostriction in amorphous and polycrystalline ferromagnets: II. Numerical results

Using the basic equations derived in part I of the paper it is shown that magnetostriction of amorphous and polycrystalline ferromagnets is determined by the conventional mechanism via the strain derivative of the local magnetic anisotropy tensor as in crystalline materials, whereas the reorientation mechanism due to the stiff rotation of structural units most probably plays only a negligibly small role. The effective magnetostriction constant λ^eff_s is calculated and conditions for zero magnetostriction are discussed, as well as changes of λ^eff_s during annealing or application of external stress.     

Scaling exponents for barkhausen avalanches in polycrystalline and amorphous ferromagnets

We investigate the scaling properties of the Barkhausen effect by recording the noise in several soft ferromagnetic materials: polycrystals with different grain sizes and amorphous alloys. We measure the Barkhausen avalanche distributions and determine the scaling exponents. In the limit of vanishing external field rate, we can group the samples in two distinct classes, characterized by exponents tau = 1.50+/-0.05 or tau = 1.27+/-0.03, for the avalanche size distributions. We interpret these results in terms of the depinning transition of domain walls and obtain an expression relating the cutoff of the distributions to the demagnetizing factor which is in quantitative agreement with experiments.     

Theory of shear magnetostriction in amorphous and crystalline ferromagnetic metals

The theory of shear magnetostriction (SMS) in ferromagnetic metallic systems is formulated in terms of Green functions in real space for a tight binding model. This is general enough to include amorphous (“glassy”) alloys, as well as crystalline materials. It is shown that the SMS coefficient λ_s(E_F) must have at least four zeros as a function of band filling E_F through the d-band, which explains the change in sign between Fe alloys and Co, Ni alloys. A method is presented for computing the indefinite integral of the imaginary part of the product of two Green functions expressed as continued fractions, not necessarily over the same band width, and some preliminary calculations explore the importance of various terms.     

Effect of elastic stress on the magnetostriction of ferromagnets

An experimental study has been made of the effect of elastic tensile stresses on the magnetostriction curves of polycrystalline nickel, transformer steel, an equiatomic cobalt-platinum alloy, and an Invar iron alloy with 30–40 wt. % nickel. The sign of the magnetostriction changes during the rotation process under the effect of sufficiently high stress. The effect is explained on the basis of the concept of a magnetic rotation texture, developed earlier by one of the authors. A change in the magnetostrictive susceptibility for the paraprocess of the Invar alloy is found to be caused by the stress; the signs of the magnetostrictive constants are determined for the hardened cobalt-platinum alloy.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 12, No. 2, pp. 89–94, February, 1969.     

Theory of irreversible processes in nonequilibrium quantum systems. I. General formalism

The theory of Van Hove for nonequilibrium quantum statistical mechanics is extensively reformulated in terms of a superspace (a kind of operator space). This reformulation enables us to introduce a diagrammatic method which makes it convenient to deal with practical problems in physical systems. In our formalism, quantum statistical effects are considered on the basis of a systematic rule for the contraction technique. A complicated statistical effect in boson or fermion systems can be treated by starting with a simple unsymmetrized formalism in the Boltzmann statistics.     

Theory of two-magnon dissipation in disordered ferromagnets I. Relation to sample statistics

The theory of two-magnon dissipation in disordered ferromagnetic materials is developed. Fluctuations of one-ion anisotropy energy due to the disorder are assumed to be the source of the linewidth broadening (relaxation of magnons by two-magnon scattering). Ordinary two-magnon theory is reconsidered on the basis of the sample statistics. The main idea of it is to describe the macroscopic properties of materials by means of quantities, whose relative fluctuations over the ensemble of configurations are negligibly small for sufficiently large sample. The energies of magnons — modes with well defined wave vectork — and their lifetimes are shown to be such variables. Further, possible models for a two-component ferromagnet are considered in the approximation of high external field. The energies of magnons in mixed crystal are obtained as linear combinations of those in parent crystals. Finally, the dependence of magnon lifetimes on the ordering is separated from the dependence on the magnitude of anisotropy variations.     

The spatial fluctuations of the magnetization in amorphous ferromagnets

The spatial variation of the magnetization of amorphous ferromagnets containing transition metal and metalloid atoms is shown to depend explicitly on that of the alloy concentration and of the nearest neighbour distance. This variation will thus be in general on an atomic scale for these materials.     

Stress effect on the magnetoresistance of amorphous ferromagnets

To investigate the magnetization in amorphous ferromagnetic alloys we measured the change in the magnetoresistance (H⊥j and H 6j) simultaneously applying a tensile stress along the direction of the ribbons. With increasing stress we observed a complete alignment of the magnetic domain vectors M_s parallel andantiparallel (for λ_s > 0) or perpendicular (for λ_s < 0) to the stress axis. It is found that on average a value of σ_s = 10 kg/mm² is sufficient in all measured samples to produce this effect. The change in the magnetoresistance by stress annealing indicates that a preferred domain orientation in stress direction is induced.     

On the width of the critical regime in amorphous ferromagnets

Amorphous ferromagnets exhibit an unusually large temperature range of deviations from the Curie-Weiss behavior in the paramagnetic regime. The question is discussed whether this may be interpreted as an extended critical behavior or as a non-critical phenomenon for which scaling ideas are not valid. Comparison is made with other disordered ferromagnets.     

Magnetic small-angle scattering of neutrons in amorphous ferromagnets

In amorphous Fe₄₀Ni₄₀C₁₄P₆ alloys the field dependence of the magnetic scattering cross section and the analysis of the scattering profile indicate the presence of stress centres corresponding to quasidislocation dipoles.     

Spin wave excitations in amorphous itinerant ferromagnets II. Dynamical susceptibility

The Izuyama, Kim, and Kubo theory of dynamical susceptibility in crystalline itinerant ferromagnets is extended to the amorphous itinerant ferromagnets. Effects of amorphousness in spin wave spectrum, given by the poles of the susceptibility, is discussed.     

Pair model of anisotropy and magnetostriction in soft amorphous magnets

A pair model for amorphous alloys has been developed, considering both the spherical symmetry (isotropic alloys) and the cylindrical one (alloys exhibiting pair ordering or microstructural anisotropy). The elastic energy has been first derived. Then starting from the Heisenberg exchange, the pseudodipolar (PSD), the cluster electric field (CEF) and the simple dipolar couplings, the magnetoelastic coupling and magnetic anisotropy energies have been written in the limiting case of a very small random anisotropy.     

High-magnetic-field magnetization and magnetoresistance of some amorphous ferromagnets

Measurements of isothermal magnetization and magnetoresistance at T = 4.0-4.2 K and 15 ? H ? 132 kG on the amorphous ferromagnets Fe₈₀B₂₀, Fe₇₈Mo₂B₂₀, Fe₄₀Ni₄₀P₁₄B₆, and Fe₃₂Ni₃₆Cr₁₄P₁₂B₆ indicate a relatively large H-induced increase (3.6%) in the high-magnetic-field magnetization of the latter alloy, and marked differences in the magnitude and sign of the high-field magnetoresistance of the four alloys. The results are qualitatively interpreted in terms of internal effective field distributions which include a small fraction of atomic spins in negative field sites.     

Investigation of the impurity hyperfine field polarization in polycrystalline Rare Earth ferromagnets

Time differential perturbed angular correlation spectra of¹¹¹Cd in ferromagnetic polycrystalline Dy have been measured at 4.2 K in external magnetic fields up to 60 kG. The experimental data were well reproduced by a calculation which assumed that the angular distribution of the magnetic hyperfine fields is identical to that of the magnetic moments of the 4f-shells. The distribution of the 4f-moments was derived from magnetic anisotropy data. The results of this work seem to justify the application of the integral perturbed angular correlation technique for the determination of magnetic hyperfine fields in incompletely polarized ferromagnetic samples. The magnetic hyperfine fields of¹⁷⁷Hf:Gd and¹⁷⁷Hf:Dy have been measured by this method as:H _hf(Hf:Gd)=–375(60)kG andH _hf(Hf:Dy)=–225(45)kG.     

On the theory of localized spin and phonon excitations in amorphous ferromagnets

Within the framework of a perturbation theory and a quasicrystalline approximation we have solved the linearized equation of motion for the circular spin component S⁺_j = S^x_j + iS^y_j in a one-dimensional amorphous ferromagnet with periodic external excitation of the spin S⁺₀ at site j = 0. It is shown that localized spin modes of the simple form $\text{«S}{}^{\mathrm{+}}{}_{\mathrm{j}}\text{a}\text{?}\text{= S}{}^{\mathrm{+}}\text{(q0) exp[iq}{}_0\text{· R}{}_{\mathrm{j}}\text{- iω(q0) t] exp (-gk?R}{}_{\mathrm{j}}\text{?)}$ with fall-off-length κ^-1 are solutions of the ensemble-averaged equation of motion. On the other hand, we have a damping of extended spin waves according to exp(-Γt). A simple relation is derived between the fall-off-length κ^-1 of localized spin modes and the damping factor Γ of extended spin waves. Analogous results hold for phonons in amorphous materials.     

Repolarization studies in amorphous and polycrystalline silicon

A procedure has been developed to extract qualitative and quantitative information on the muonium fractions, in particular the Mu^* fraction, in polycrystalline and amorphous materials from their longitudinal field repolarization curves. Preliminary results for amorphous silicon suggests that both the Mu^* and Mu^* fractions here are generally lower than in crystalline silicon at temperatures below 200K, but the Mu^* fraction may survive to room temperature in this disordered host.     

Efficient electronic structure theory via hierarchical scale-adaptive coupled-cluster formalism: I. Theory and computational complexity analysis

ABSTRACT

A novel reduced-scaling, general-order coupled-cluster approach is formulated by exploiting hierarchical representations of many-body tensors, combined with the recently suggested formalism of scale-adaptive tensor algebra. Inspired by the hierarchical techniques from the renormalisation group approach, H/H²-matrix algebra and fast multipole method, the computational scaling reduction in our formalism is achieved via coarsening of quantum many-body interactions at larger interaction scales, thus imposing a hierarchical structure on many-body tensors of coupled-cluster theory. In our approach, the interaction scale can be defined on any appropriate Euclidean domain (spatial domain, momentum-space domain, energy domain, etc.). We show that the hierarchically resolved many-body tensors can reduce the storage requirements to O(N), where N is the number of simulated quantum particles. Subsequently, we prove that any connected many-body diagram consisting of a finite number of arbitrary-order tensors, e.g. an arbitrary coupled-cluster diagram, can be evaluated in O(NlogN) floating-point operations. On top of that, we suggest an additional approximation to further reduce the computational complexity of higher order coupled-cluster equations, i.e. equations involving higher than double excitations, which otherwise would introduce a large prefactor into formal O(NlogN) scaling.     

Anisotropic solutions of the Einstein-Boltzmann equations: I. General formalism

Given a choice of a timelike vector field, a particle distribution function in a general curved space-time can be analysed into spherical harmonics; the Liouville and Boltzmann equations can then be written as a set of equations relating these spherical harmonic components. We obtain these equations and the resulting equations for the spherical harmonic moments of the distribution function. An orthonormal tetrad formalism is used as an aid in our calculations; the set of moment equations used can be completed by giving Einstein's field equations as equations for the rotation coefficients of this tetrad. We discuss time and space reversal symmetry properties of the Boltzmann equation, but leave applications of the set of equations obtained to further papers.