Linear Tensor Research Articles

Response of a relativistic anisotropic thermal plasma. Part 2. Magnetized particles

The linear response tensor for a relativistic counterpart of a bi-Maxwellian distribution is calculated in closed form for a magnetized plasma. The method used involves the following steps: (1) a covariant form of the response tensor for an arbitrary distribution of particles is derived using both a forward-scattering method and a covariant version of Vlasov theory; (2) the response tensor is evaluated for a strictly perpendicular, thermal distribution using a method due to Trubnikov; (3) the parallel distribution is built up by applying a Lorentz transformation to sum over a weighted, continuous distribution of parallel speeds. A convenient starting point for a detailed investigation of waves in such a plasma is the small-gyroradius approximation to the general expression for the response tensor, and the relevant approximation is derived and discussed briefly. The expansion of the general expression for the response tensor in Bessel functions in given in an appendix.

Response of a relativistic anisotropic thermal plasma. Part 1. Unmagnetized particles

Expressions for the linear response tensor are derived using a covariant forward-scattering method for unmagnetized relativistic thermal distributions restricted to one-dimensional (strictly parallel) and two-dimensional (strictly perpendicular) motions. The freedom to make Lorentz transformations is used to generalize the strictly perpendicular distribution to a counterpart of the nonrelativistic Dory–Guest–Harris (DGH) distribution.

Constrained free-energy functional of deformed polymer systems

The constrained free-energy functional for deformed polymer systems is derived in a regime of small fluctuations. The free energy contains an elastic contribution due to chain deformation as well as the usual Ginzburg–Landau form. Up to terms quadratic in fluctuations, the linear elastic modulus tensor is obtained. We find that the shear modulus scales with the concentration in the same way as that suggested by scaling arguments in the mean-field regime. The elastic Poisson ratio is 1/2 for the dilute system and decreases as the volume fraction increases. In the melt limit, where the volume fraction is close to unity, the Poisson ratio is down to 0.2.

Piezo-optical properties of Ge

The piezo-optical properties of Ge, for pressure along the [001] and [111] directions, are calculated with the use of a realistic empirical tight-binding model. The real and imaginary part of components ${P}_{11}(\ensuremath{\omega})$, ${P}_{12}(\ensuremath{\omega})$, and ${P}_{44}(\ensuremath{\omega})$ of the linear piezo-optical tensor for Ge are evaluated. The main structures in ${P}_{\mathrm{ij}}$ are located around the ${E}_{1}$ and ${E}_{2}$ critical points. Deformation potential constants ${D}_{1}^{1}$, ${D}_{1}^{5}$, ${D}_{3}^{3}$, and ${D}_{3}^{5}$, for the ${E}_{1}\ensuremath{-}{(E}_{1}+{\ensuremath{\Delta}}_{1})$ transitions are also evaluated. The theoretically obtained results are in good agreement with existing experimental values.

New mode of domain imaging: Second harmonic generation microscopy (abstract)

Magnetization induced second harmonic generation (MSHG) is a novel magneto-optical technique that has an enhanced surface/interface sensitivity and demonstrates very large magneto-optical effects.1 It is described by a generalized nonlinear optical tensor χ(2) which has different transformation properties as compared to the linear tensor χ(1). As a consequence, in the same geometry MSHG may be sensitive to other magnetization components than linear magneto-optics. Therefore linear and nonlinear magneto-optics should work best in a combination. Here we report the development of a magneto-optical microscope that is sensitive simultaneously to the linear (Faraday or Kerr) and nonlinear (MSHG) magneto-optical effects. MSHG, on the one hand, is able to visualize some exotic domain structures, e.g., 180° domains in antiferromagnetics and ferroelectrics. On the other hand, because of its interface sensitivity it might help to visualize the magnetic structure at the interface between two different magnetic materials (including metals). Last but not least, the microscope evidently has an enhanced resolution due to the frequency doubling of detected light. First, we apply the microscope to the study of domains in magnetic garnet films, where a considerable difference between Faraday and MSHG images in the same configuration is demonstrated. A complicated noncollinear domain structure is straightforwardly derived from the MSHG images. Next, we consider the application of our technique to the magnetic metal surfaces and thin films (in reflection). The utilization of the microscope to correlate the surface/interface morphology with its magnetic properties is discussed.

A novel construction of the response tensors for a collisionless unmagnetized plasma

The linear and nonlinear response tensors describe the response of a plasma to an electromagnetic field disturbance. A novel construction of the linear and quadratic response tensors for an unmagnetized collisionless plasma is presented here using a convenient 4-tensor notation. The construction uses the equations of charge continuity and gauge invariance, together with symmetry properties and dimensional arguments, to constrain the coefficients of an arbitrary linear combination of tensors. A new interpretation of the response tensors follows as expressions uniquely determined by fundamental physical constraints, and not simply as coefficients in a perturbation expansion of induced plasma response.

Dependence of Linear Electro-optic Coefficient on Difference in the Atomic Sizes in Zinc Blende Crystals

Covalent radii of the bonding elements have strong effects on the linear electro-optic coefficients of zinc blende crystals; these effects can be quantitatively determined by investigating the relation between the difference in the atomic sizesρand the magnitude of the linear electro-optic tensor coefficientr41. It is interesting to note that for the same cation Zn2+, Ga3+, or In3+the magnitude ofr41increases with increased covalent radius of the bonded anionrβ. Especially with the increasing tendency of the parameterρ, the magnitude ofr41of crystals that have a same cation will increase suddenly when the value ofrβbecomes larger.

Anisotropic Damage Effect Tensors for the Symmetrization of the Effective Stress Tensor

Based on the concept of the effective stress and on the description of anisotropic damage deformation within the framework of continuum damage mechanics, a fourth order damage effective tensor is properly defined. For a general state of deformation and damage, it is seen that the effective stress tensor is usually asymmetric. Its symmetrization is necessary for a continuum theory to be valid in the classical sense. In order to transform the current stress tensor to a symmetric effective stress tensor, a fourth order damage effect tensor should be defined such that it follows the rules of tensor algebra and maintains a physical description of damage. Moreover, an explicit expression of the damage effect tensor is of particular importance in order to obtain the constitutive relation in the damaged material. The damage effect tensor in this work is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. In this work, tensorial forms are used for the derivation of such a linear transformation tensor which is then converted to a matrix form.

The linear bi-spatial tensor equation ?i j AiXBj= C

A linear bi-spatial tensor equation which contains many often encountered equations as particular cases is thoroughly studied. Explicit solutions are obtained. No conditions on eigenvalues of coefficient tensors are imposed.

Non-commutative geometry of finite groups

A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first-order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of `extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a `dual connection' which exists on the dual bimodule (as defined in this work).

Spin-unrestricted time-dependent Hartree–Fock theory of frequency-dependent linear and nonlinear optical properties

A self-consistent time-dependent unrestricted Hartree–Fock (TDUHF) theory of linear and nonlinear optical properties is presented. Expressions are derived to calculate the elements of the linear polarizability tensor α, the first-hyperpolarizability tensor β, and the second-hyperpolarizability tensor γ, in terms of spin-adapted perturbed density matrices. For the hyperpolarizability tensors, β and γ, expressions are also derived from the lower-order solutions to the TDUHF equations. A novel feature of the present formulation is that it automatically allows for the separation of the contributions to (hyper)polarizability tensors from individual spin. Results obtained from the calculations of α(ωσ;αa) for H, C, N, O, F, Si, P, S, Cl, O2, NO, and OH and of β(−ωσ;ωa,ωb) corresponding to various second-order nonlinear optical processes for NO and OH radicals are presented. The present results for α show excellent agreement with the literature data. The calculated result for β∥ in the case of the NO radical is too small and has a different sign in comparison to the published experimental data. For both NO and OH, the β tensor for different second-order nonlinear optical processes show the order: β(−2ω;ω,ω)≳β(−ω;0,ω) ≡β(0;ω,−ω)≳β(0;0,0).

A nonlinear tensor field and the second metric tensor

In our preceding paper {see [L. Sh. Grigorian and S. Gottlober, Astrofizika (in press)]} we investigated a self-gravitating system consisting of a scalar field and a linear tensor field ωik= ωki with minimal coupling and with allowance for the action of vacuum polarization effects. In the present paper we investigate the case of a nonlinear tensor field ωik. The action S ψ (ɛ) of the field ωik is determined by the difference Rik — Řik, where Rik is the space-time Ricci tensor and Rik is the analogous quantity constructed using the metric γik=gik+eωik induced by ωik (e is a free parameter). Here\(\mathop {lim}\limits_{\varepsilon \to 0} \) S ψ (ɛ) coincides with the previously known expression for the action of a linear field ωik. Equations of motion are derived for ωik in curved space-time. The energy-momentum metric tensor, determining the contribution of ωik to the gravitational field equations, is calculated.

Vibrational quantum beats observed by Raman-induced optical Kerr effect

Ultra fast molecular motion in liquids may be studied in the time domain with the pump/probe optical heterodyne-detected (OHD) Raman-induced optical Kerr effect (RIKE) technique, which makes use of ultra short pulses from a mode-locked laser. The oscillatory signal observed is attributed to vibrational quantum interference, and a detailed theoretical account is provided. In the framework of linear response theory, the theoretical development of this work follows two different approaches: the density matrix formalism in the Schrödinger representation and the Hellwarth formalism involving the evaluation of the time correlation of linear susceptibility tensors. An unambiguous explanation of the decay, in addition to the oscillation, observed in OHD-RIKE is also provided.

Reductions in Intuitionistic Linear Logic

In this work we show how some useful reductions known from ordinary intuitionistic propositional calculus can be modified for Intuitionistic Linear Logic (without modalities). The main reductions we consider are: (1) the reduction of the depth of formulas in the sequents by addition of new variables, and (2) the elimination of linear disjunction, tensor and constant F. Both transformations preserve deducibility, that is, a transformed sequent is deducible if and only if the initial one was deducible. The size of the sequent grows linearly in case (1) and ≤ On8 in case (2).

On the limits of linear moment tensor inversion of surface wave spectra

Some theoretical and practical limits to linear moment tensor inversion of surface waves are analyzed in detail, in particular when one or few stations are used for rapid determination of source parameters. The theory is briefly outlined and steps of preprocessing, especially corrections for source history and propagation, are discussed in order to guarantee the validity of the moment tensor inversion. The inverse problem is first studied from a theoretical point of view. Then the feasibility of the moment tensor inversion is tested from synthetic computations and the formalism is improved in order to obtain a better system conditioning. Finally, some cases of practical nonuniqueness of the solution are shown and possible restrictions to the applicability of the method are discussed. All steps are illustrated with the example of theMs=6.9 Erzincan (Turkey) earthquake of March, 13, 1992.

Enhancement of the magneto-opticl Kerr rotation in nonlinear optical response

Using an electronic theory we calculate the frequency-dependent nonlinear magneto-optical Kerr rotation and show that the second harmonic polarization ellipsis is more strongly influenced by magnetism than in the linear case. We find for the magnetization dependence of the nonlinear and linear magneto-optical susceptibility tensor elements a linear dependence on magnetization. The influence of the symmetry of matrix elements for s or d transitions on the second harmonic intensity is discussed. Furthermore, the ab initio nonlinear Kerr spectra of Fe layers of different thicknesses are presented.

The symmetric tensor field in the relativistic theory of gravitation

The system of a self-gravitating scalar field is frequently used in inflationary cosmological models. In the present paper we study a more complicated system containing an extra linear tensor field ψik=ψki with minimal coupling. We determine five of the six free parameters that occur in the most general expression for the actionSψ of this field. In doing so we assume that in flat space-time the field ψik must be invariant under gauge transformations. In a special case theSψ found becomes a known expression for the action of a massless tensor field ψik. We compute the metric energy-momentum tensor that determines the contribution of ψik to the Einstein equations. We also exhibit the equations of motion of ψik in curved space-time.

The Fibroblast-Populated Collagen Microsphere Assay of Cell Traction Force—Part 2: Measurement of the Cell Traction Parameter

In Part 1 of this work, we formulated and analyzed a mathematical model for our fibroblast-populated collagen microsphere (FPCM) assay of cell traction forces (Moon and Tranquillo, 1993). In this assay, the FPCM diameter decreases with time as the cells compact the gel by exerting traction on collagen fibrils. In Part 1 we demonstrated that the diameter reduction profiles for varied initial cell concentration and varied initial FPCM diameter are qualitatively consistent with the model predictions. We show here in Part 2 how predictions of a model similar to that of Part 1, along with the determination of the growth parameters of the cells and the viscoelastic parameters of the gel, allow us to estimate the magnitude of a cell traction parameter, the desired objective index of cell traction forces. The model is based on a monophasic continuum-mechanical theory of cell-extracellular matrix (ECM) mechanical interactions, with a species conservation equation for cells (1), a mass conservation equation for ECM (2), and a mechanical force balance for the cell/ECM composite (3). Using a constant-stress rheometer and a fluids spectrometer in creep and oscillatory shear modes, respectively, we establish and characterize the linear viscoelastic regime for the reconstituted type 1 collagen gel used in our FPCM traction assay and in other assays of cell-collagen mechanical interactions. Creep tests are performed on collagen gel specimens in a state resembling that in our FPCM traction assay (initially uncompacted, and therefore nearly isotropic and at a relatively low collagen concentration of 2.1 mg/ml), yielding measurements of the zero shear viscosity, mu 0 7.4 x 10(6) Poise), and the steady-state creep compliance, J0e. The shear modulus, G (155 dynes/cm2), is then determined from the inverse of J0e in the linear viscoelastic regime. Oscillatory shear tests are performed in strain sweep mode, indicating linear viscoelastic behavior up to shear strains of approximately 10 percent. We discuss the estimation of Poisson's ratio, v, which along with G and mu 0 specifies the assumed isotropic, linear viscoelastic stress tensor for the cell/collagen gel composite which appears in (3). The proliferation rate of fibroblasts in free floating collagen gel (appearing in (1)) is characterized by direct cell counting, yielding an estimate of the first-order growth rate constant, k (5.3 x 10(-6) s-1). These independently measured and estimated parameter values allow us to estimate that the cell traction parameter, tau 0, defined in the active stress tensor which also appears in (3), is in the range of 0.00007-0.0002 dyne.cm4/mg collagen.cell.(ABSTRACT TRUNCATED AT 400 WORDS)

Non-resonant optical rectification in optically active liquids

It is shown that optical rectification can be produced by linearly polarized light via the fourth-order susceptibility in optically active liquids and solutions. In order to induce a static polarization, two beams of different frequency are necessary, however. The mechanisms responsible for the effect are described in terms of both linear and nonlinear electric dipole polarizability tensors, and the magnitude of the effect is estimated.

Dependence of Electro-Optic Effects on the Orientations of Epitaxial LiNbO3 Films

AbstractEpitaxial LiNbO3 films were grown on a-cut sapphire, i.e., α-A12O3(0112), substrates by pulsed laser deposition. It is found the oxygen pressure is an important deposition parameter to influence the orientations of the LiNbO3 films. At an oxygen pressure of 10−1 torr, LiNbO3(0001) film was grown epitaxially. And, at a low oxygen pressure (i.e., 5x10−4 torr), LiNbO3(1120) film was grown epitaxially. Due to the anisotropic electro-optic (EO) constants, these films with the different orientations display large differences in their EO properties: the epitaxial LiNb3(0001) film shows little EO effects, but the LiNbO3(1120) film shows a large quadratic EO behavior with an effective coefficient of 2.38×10−5 m2/V2. The dependence of EO effects on the orientations of the epitaxial films was analyzed in terms of the linear and quadratic EO tensors of LiNbO3.