On the effects of microstress on macroscopic diffusion processes

Summary The effects of internal microstress fields are neglected in a usual simulation of the diffusion of a small solute trough solid heterogeneous media. However, when a heterogeneous material is used in a structure undergoing external mechanical loading, highly nonuniform stress fields can arise locally due to the variable microstructure. In this paper a simple quasi-Fickian model is studied which employs a spatially variable stress-dependent diffusivity,D . The structure ofD stems from an assumption that the stresses nonuniformly open and close the pores of the material microstructure, thus providing preferential sites for accumulation of the diffusing solute. When =0, the usual stress-free Fickian diffusivity,D ⁰, is recovered. Because of the highly oscillatory stress fields on the micro level, when employing numerical methods, such as the finite element or finite difference method, the distance between discretization nodes must be far smaller than the microstructural oscillations to obtain accurate simulations. This fact makes direct numerical simulations involvingD virtually impossible without computationally intensive, and complicated, special techniques. In this paper upper bounds are developed for the difference between solutions produced when usingD and alternativelyD ⁰ in the body under analysis. The general case, whenD ⁰, and consequentlyD , are spatially variable, is considered. The bounds are a function of onlyD ⁰ and and do not require any knowledge of the stress-dependent solution, and can thus be used as an a-priori check to determine whether potentially expensive computations are necessary.Dedicated to Prof. Dr. Peter Haupt on the occasion of his 60th birthday     

Momentum distributions and the impulse approximation in helium

The momentum distributionn(p) of atoms in condensed matter may be determined from neutron inelastic scattering measurements. However, scattering at high momentum transfers Q (high enough that the impulse approximation holds) is required. A model of solid helium is used to assess the error introduced inn(p) by assuming the impulse approximation holds at severalQ values. AtQ=20 Å^–1 the error inn(p) is 5–10%. This error can be reduced by a factor of ten using a symmetrization procedure.     

Numerical solutions of the singular integral equations in the crack analysis using the body force method

In this paper, numerical solutions of the singular integral equations of the body force method in the crack problems are discussed. The stress fields induced by two kinds of displacement discontinuity are used as fundamental solutions. Then, the problem is formulated as a hypersingular integral equation with the singularity of the form r ². In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density function and the Chebyshev polynomials. As examples, the stress intensity factors of the oblique edge crack, kinked crack, branched crack and zig-zag crack are analyzed. The calculation shows that the present method gives accurate results even for the extremely oblique edge crack and kinked crack with extremely short bend which has been difficult to analyze by the previous method using the approximation by the products of the fundamental density function and the stepped functions etc.     

Electron interference oscillations and spin-density-wave energy gaps at the Fermi surface of antiferromagnetic chromium

A chain model incorporating the periodicity of the incommensurate spin density wave is used to describe the Fermi surface of antiferromagnetic chromium, and is found to give excellent semiquantitative agreement with much of the de Haas-van Alphen data obtained by other investigators. New experimental data are presented which confirm the anomalous features of the transverse magnetoresistance oscillations observed in antiferromagnetic chromium with the spin density wave vector Q[001], the current J[010], and the magnetic field H in the (010) plane near [100]. These anomalous features are explained with the help of the chain model as arising from a quantum interference effect in the transport of charge along open orbits in the direction of Q. Estimates are obtained from our experimental data for the size of the magnetic breakdown fields that characterize the interferometer trajectories, and hence for the energy gaps on the Fermi surface associated with the spin density wave. Previous theoretical estimates of the magnetic breakdown fields obtained by a perturbation calculation for a model of tight-binding s electrons are in poor agreement with our values.Work supported by the Natural Sciences and Engineering Research Council.     

Spin-orbit interaction in the field of an optical vortex of a few-mode fiber

A quantum-mechanical analog of the spin-orbit interaction operator is constructed for the fields of optical CV vortices and TE and TM modes in an optical fiber. It is shown that the polarization correction δβ to the propagation constant, which is the mean value of this operator, is a measure of the “level splitting” of the propagation constant in the scalar case. The difference in the operation of the individual parts of the operator on the fields of CV vortices and on the fields of TM and TE modes is indicative of the presence of two different physical processes — circular and linear birefringence in the locally isotropic optical fiber. The conversion of the “scalar” field to a vector field e ₁ as a consequence of the spin-orbit interaction operator can be regarded as resulting from a re-radiation of the additional field e ₁ by the vortex field , which rotates around the optical axis of the fiber. In this picture the additional field e ₁ can be regarded as being a “relativistic” correction to the vortex field for the distortions of the main field and arising as a result of the rotation of the field of the optical vortex in the medium of the few-mode fiber. Zh. Tekh. Fiz. 24, 87–93 (October 26, 1998)     

On the thermodynamics of the inhomogeneous mixed phase of an ideal type II superconductor

The macroscopic tensor S _ik is defined so as to characterize changes in the structure of the vortex system. This tensor contains only three independent components instead of the six of the strain tensor u _ik which is used commonly to describe deformations of the vortex system. All components of S _ik are independent of the magnetic induction B and can be considered as thermodynamic variables of the system together with B. The stress tensor for an inhomogeneous mixed phase is obtained in these variables. It is shown on the basis of the Ginzburg-Landau current equation that the mixed phase tends to be uniform; inhomogeneities (e.g., macroscopic currents) can penetrate to the uniform phase only up to some distance that diverges at H H _c2. The free energy of a weak inhomogeneous mixed state is discussed; nonlocal features of the vortex system and its special magnetic symmetry are taken into account. This symmetry allows for the existence of contributions to the free energy that are linear in the magnetic field and strains. The equilibrium conditions differ in general from the simple relation j × B = 0, so that the force-free configurations do not correspond to the thermodynamic equilibrium of an ideal material. The connection between the fields B and H is obtained in the form of a differential equation that also includes the structural parameters S _ik. These fields need not be parallel in general. The current distribution in a slab of an ideal type II material in a transverse field is calculated.     

The Runaway of Electrons and the Formation of Beams in Glow Discharges

Analytical methods are used to obtain the criterion defining the boundary between the beam and hydrodynamic modes of motion of electrons in uniform and nonuniform electric fields for Townsend and high-voltage glow discharges. The beam approximation is used to study the electron energy distribution function in view of electrons being produced in gas. The hydrodynamic approximation is used to obtain the criterion which characterizes the runaway of electrons and the criterion of transition to the beam mode of motion. It is demonstrated that the curve of discharge ignition on the U(pd) diagram differs significantly from the boundary line between the beam and hydrodynamic modes of motion of electrons.     

Spin singlet-triplet states of superfluid Fermi liquids in the presence of strong magnetic fields

The unitary singlet-triplet states of superfluid Fermi liquids are investigated in the presence of strong magnetic fields. The static spin susceptibility is calculated in the limith 0 in the framework of the weak coupling approximation. It is predicted that BS (a mixture of the Balian- Werthamer anisotropic state andD-wave singlet state) and 2DS (a mixture of the planar 2D state andD-wave singlet state) spin singlet-triplet states become stable at some temperatures and magnetic fields.     

On the Bounds of the Bilinear Complexity of Multiplication in Some Finite Fields

From the existence of a tower of algebraic function fields, we improve upper bounds on the bilinear complexity of multiplication in all the extensions of the finite fields and where p is a prime 5. In particular, we improve asymptotic upper bounds on this complexity for prime finite fields of characteristic p>5.     

Magnetic field dependence of ultrasound attenuation and sound velocity in superfluid3He-A

In the presence of a magnetic field the gap amplitude of the axial state is found to split atall temperatures if particle-hole asymmetry is taken into account. The relative gap splitting is of the order of the relative splitting of the transition temperature and leads to splittings of the frequencies of the clapping and flapping modes. The ultrasonic attenuation and sound velocity are calculated for different sound frequencies, fields, and orientations of the wave vectorq with respect to the axisl of the energy gap. The predicted splittings of the clapping and flapping mode peaks and of the cusp in the attenuation and of the step in the sound velocity at=2₀(T) should become observable at sufficiently high fields. The results are discussed in connection with ultrasonic impedance, attenuation, and velocity measurements in the A₂ and A₁ phases.     

Numerical Method of Determining the Characteristics of the Spectral Sensitivity of Television Image Receivers

A new method of determining the characteristics of the spectral sensitivity S() of television image receivers is proposed. The method is based on approximating the spectral sensitivity by a previously known function with unknown parameters. Using a set of wide-beam filters, the response of a television image receiver to the action of radiation from a source with known spectral distribution is measured. From the results obtained, the approximation coefficients are found and the required function S() is recovered.     

Decay of wide-angle helical texture in superfluid3He-A

In³He-A there exist two helical textures: one withl almost aligned with the flow for small magnetic fields compared to the current (roughly in dipole units) and another withl almost orthogonal to the flow for larger fields [h(40/7)^1/2 j1]. Here we calculate the ratio of the decay rates of the second current, namely the wide-angle helix, to smaller or larger values of the opening angle, via quadratic fluctuations around an extremal excursion from equilibrium.     

Natural convection in a rectangular enclosure in the presence of a magnetic field with uniform heat flux from the side walls

Summary A numerical study is presented for magnetohydrodynamic free convection of an electrically conducting fluid in a two-dimensional rectangular enclosure in which two side walls are maintained at uniform heat flux condition. The horizontal top and bottom walls are thermally insulated. A finite difference scheme comprising of modified ADI (Alternating Direction Implicit) method and SOR (Successive-Over-Relaxation) method is used to solve the governing equations. Computations are carried out over a wide range of Grashof number, Gr and Hartmann number, Ha for an enclosure of aspect ratio 1 and 2. The influences of these parameters on the flow pattern and the associated heat transfer characteristics are discussed. Numerical results show that with the application of an external magnetic field, the temperature and velocity fields are significantly modified. When the Grashof number is low and Hartmann number is high, the central streamlines are elongated and the isotherms are almost parallel representing a conduction state. For sufficiently large magnetic field strength the convection is suppressed for all values of Gr. The average Nusselt number decreases with an increase of Hartmann number and hence a magnetic field can be used as an effective mechanism to control the convection in an enclosure.List of symbols Ar aspect ratio,H/L - B ₀ induction magnetic field - H ₀ magnetic field,H ₀=B ₀/ _m - g gravitational acceleration - Gr Grashof number,gq(L/k)L ³/v ² - H height of the enclosure - Ha Hartmann number, - k thermal conductivity - Nu local Nusselt number - average Nusselt number - p pressure - Pr Prandtl number, / - q heat flux - t time - T dimensionless temperature, (–₀)/q(L/k) - u vertical velocity - U dimensionless vertical velocity,uL/ - v horizontal velocity - V dimensionless horizontal velocity,vL/ - x vertical coordinate - X dimensionless vertical coordinate,x/L - y horizontal coordinate - Y dimensionless horizontal coordinate,y/L - thermal diffusivity - thermal expansion coefficient - temperature - ₀ reference temperature - density - kinematic viscosity - viscosity - _m magnetic permeability - electrical conductivity - stream function - dimensionless stream function, / - dimensionless time,t/L ² - vorticity - dimensionless vorticity, L ²/ - X grid spacing inX-direction - Y grid spacing inY-direction - time increment - ² Laplacian operator     

Very Large Rashba Spin–Orbit Splitting in HgTe Quantum Wells

A large Rashba spin splitting has been observed in the first conduction subband of n-type modulation doped HgTe quantum wells with an inverted band structure via an investigation of Shubnikov–de Haas oscillations as a function of gate voltage. Self-consistent Hartree calculations of the band structure based on an 8 × 8 k p model quantitatively describe the experimental results. It has been shown that the heavy-hole nature of the H1 conduction subband greatly influences the spatial distribution of electrons in the quantum well and also enhances the Rashba spin splitting at large electron densities. These are unique features of type III heterostructures in the inverted band regime. The k ³ dispersion predicted by an analytical model is a good approximation of the self-consistent Hartree calculations for small values of the in-plane wave vector k . This is in contrast to the commonly used k dispersion for the conduction subband in type I heterojunctions.     

Application of the renormalization group technique to the problem of phase transition in one-dimensional metallic systems. I. Invariant couplings,vertex, and one-particle Green's function

A one-dimensional system of electrons interacting via a BCS-type interaction is investigated by renormalization group techniques in two successive approximations atT=0, keeping only a single energy variable . The first approximation is equivalent to the summation of leading logarithmic terms carried out by Bychkovet al., and correspondingly the vertex function displays a singularity at a finite value of . The second approximation accounts for the next leading logarithmic terms as well, and by this means the singularity is shown to be pushed down to =0. Due to important self-energy contributions, however, the invariant couplings behave differently and tend to a saturation value at =0.     

Correction to the formula for the London moment of a rotating superconductor

This paper gives a full quantum mechanical analysis of the magnetic field (first discussed by London) that appears spontaneously when a sample of superconductor is set into rotation. It is shown that, for slow rotation velocities and using certain approximations, the fieldB threading a cavity within a superconductor that rotates at angular velocity is given bye B=2(m _o–W/c ²), where — e is the charge on the electron,m _o is the free electron mass,W is the work function of the superconductor, andc is the velocity of light. In this calculation effects that are second order in the rotation velocity have been ignored, and the result is only strictly valid at the zero of temperature. The application of this result to experiments using practical, nonideal apparatus is then illustrated for a simple geometry.This work was supported in turn by the Science Research Council; Trinity College, Cambridge; and the U. S. Department of Commerce under contract number NB81RAC10026.     

Step Model of Erosion of Electrodes. III. Adaptation to Arbitrary Regimes of Electroerosion Treatment

The step model of erosion of the cold electrodes of electricarc heaters proposed in Inzh.Fiz. Zh., 76, No. 2, 36–42 (2003) and Ann. N.Y. Acad. Sci., 891, 36–42 (1999) has been modified for calculating the rate of electroerosion treatment of materials. The point heat source [J. Appl. Phys., 66, No. 9, 4095–4103 (1989)] for determining the effective enthalpy of erosion has been reduced to a plane surface heat source equivalent in erosion and corresponding to the step model. The approximation analytical relations in dimensionless numbers have been obtained. The calculation results are in satisfactory agreement with the experimental data of different authors.     

Green's functions for superfluid3He-A in the case of spin hydrodynamics: Influence of the spin-orbit coupling

The spin hydrodynamic equations for superfluid³He-A for the case of external, time-dependent fields are considered. The terms in equations containing these fields are found on the basis of a microscopic approach. Considering the linear response of the system to the switching on of external fields, formulas are found for suitable Green's functions constructed from operators entering the spin hydrodynamics. The Green's function connected with the order parameter operator has a 1/q ² singularity. Suitable connections between Green's functions lead to relations among kinetic coefficients (Onsager relations). A more detailed consideration of the spin-orbit coupling when the spin waves are not true Goldstone modes removes the mentioned singularity and the Onsager relations remain valid.     

D''yakonov–Perel'' Spin Relaxation Controlled by Electron–Electron Scattering

The D'yakonov–Perel' mechanism of spin relaxation is connected with the spin splitting of the electron dispersion curve in crystals lacking a center of symmetry. In a two-dimensional noncentrosymmetric system, e.g. quantum well or heterojunction, the spin splitting is a linear function of k, at least for small values of k. We demonstrate that the spin relaxation time _s due to the spin splitting is controlled not only by momentum relaxation processes as widely accepted but also by electron–electron collisions which have no effect on the electron mobility. In order to calculate the time _s taking into account the electron–electron scattering, we have solved the two-dimensional kinetic equation for the electron spin density matrix. The result has been compared with that obtained assuming the momentum scattering to occur due to elastic scattering of electrons by ionized impurities. We have also extended the quasi-elastic approximation to describe the electron–electron collision integral for a spin-polarized three-dimensional electron gas.