Tensor current density

In a crystal, the electronic and ionic conductivities are generally tensor quantities relating the current density Iq to the applied electric field E in accordance with Ohm s law. The scalar expression for the mobile-ion current density in the different principal crystallographic directions has the form... 

Notation-. T is the temperature, Vi the fluid velocity, II,j the viscous pressure tensor, Jg the heat current density, p its chemical potential, the current density of molecular species a, v J the stoichiometric coefficient (13), and Wp the speed of reaction p. 

Therefore, the vacuum charge and current densities of Panofsky and Phillips [86], or of Lehnert and Roy [10], are given a topological meaning in 0(3) electrodynamics. In this condensed notation, the vacuum 0(3) field tensor is given by... 

There has been much recent progress in the application of density functional theory (DFT) to the calculation of shift tensors, and several methods are presently available. The sum-over-state (SOS) DFT method developed by Malkin et al. (70) does not explicitly include the current density, but it has been parametrized to improve numerical accuracy. Ziegler and coworkers have described a GIAO-DFT method (71) that is available as part of the Amsterdam density functional package (72). An alternate method developed by Cheeseman and co-workers (73) is implemented in Gaussian 94 (74). 

The scheme named continuos transformation of origin of the current density , see (27t21t22t28) proves that the transformed diamagnetic current density tensor,... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

Because W/tv is an antisymmetric Lorentz tensor, the total iso vector current density must satisfy dvJvw = 0. 

It is difficult to make an exhaustive list of the applications of quantitative imaging, because a large number of parameters are quantifiable proton density, relaxation time T, T2, T2 or T 2, T p), data qualifying interaction of pools by magnetization transfer, apparent diffusion coefficients, indices characterizing diffusion phenomena from tensor estimation or a (/-space approach, temperature difference, static magnetic field, B1 field amplitude, current density or values related to dynamic MRI contrast agent uptake. 

The derivatives with respect to x sample the off-diagonal behaviour of F > and generate terms related to the current density j and the quantum stress tensor er. The first-order term is proportional to the current density, and this vector field is the x complement of the gradient vector field Vp. The second-order term is proportional to the stress tensor. Considered as a real symmetric matrix, its eigenvalues and eigenvectors will characterize the critical points in the vector field J and its trace determines the kinetic energy densities jK(r) and G(r). The cross-term in the expansion is a dyadic whose trace is the divergence of the current density. 

The second-order expansion given in eqn (6.96) recovers all of the physical quantities needed to describe a quantum system and determine its properties the charge density p and its gradient vector field Vp define atoms and determine many of their properties in a stationary state the current density determines the system s magnetic properties and the change in p in a time-dependent system and, finally, the stress tensor determines the local and average mechanical properties of the system. Thus, one does not need all the... 

Equation (8.175) is a generalization of Ehrenfest s theorem (Ehrenfest 1927). This theorem relates the forces acting on a subsystem or atom in a molecule to the forces exerted on its surface and to the time derivative of the momentum density mJ(r). It constitutes the quantum analogue of Newton s equation of motion in classical mechanics expressed in terms of a vector current density and a stress tensor, both defined in real space. 

In Bi2Tes single crystals, the general relationship between the electric field E, the current density J, and the magnetic field H is expressed, in terms of galvanomagnetic tensor components, by... 

In Section V it will be shown that the quaternion structure of the fields that correspond to the electromagnetic field tensor and its current density source, implies a very important consequence for electromagnetism. It is that the local limit of the time component of the four-current density yields a derived normalization. The latter is the condition that was imposed (originally by Max Bom) to interpret quantum mechanics as a probability calculus. Here, it is a derived result that is an asymptotic feature (in the flat spacetime limit) of a field theory that may not generally be interpreted in terms of probabilities. Thus, the derivation of the electromagnetic field equations in general relativity reveals, as a bonus, a natural normalization condition that is conventionally imposed in quantum mechanics. 

In this expression for the electromagnetic field intensity tensor, Q is a constant of proportionality with the dimension of charge, inserted on both sides of Eq. (48) (negative). The four-current density is... 

Since Fpy is an antisymmetric tensor in spacetime and since the components of the ordinary affine connection are symmetric in the indices (ap), it follows that the 4-divergence of the current density /, automatically vanishes. In other words, as in the standard formulation, the equation of continuity follows from taking the covariant divergence of Maxwell s equation (31a) ... 

We have seen in this section that the factorization of Einstein s symmetric, second-rank tensor field equations (10 relations) to a quaternion form (16 relations) not only yields the gravitational and electromagnetic manifestations of matter in a unified field theory but also reveals a feature of quantum mechanics. In particular, it was found that in the flat-space approximation to the curved-space representation in general relativity, the time component of the electromagnetic four-current density corresponds in a one-to-one way with the probability density of quantum mechanics. Its integration over all of space in this limit is found to be unity. 

The permittivity of a vacuum Eq has SI units of (C /J m). The specific conductivity (Tc (l/( 2-m)) couples the electric field to the electric current density by J= OcE. From the relations described in (6b), it becomes evident that optically generated gratings correspond to spatial modulations of n, , or Xg. The parameters AA, , and Xg are tensorial. This means that the value of Xg depends on the material orientation to the electric field (anisotropic interactions). In general, P and E can be related by higher-rank susceptibility tensors, which describe anisotropic mediums. The refractive index n, and absorption coefficient K, can be joined to specify the complex susceptibility when K (Xp) 471/Xp such that... 

Let us introduce a second-rank current density tensor, jB, through the relationship... 

Response tensors by continuous transformation of the origin of the current density... 

The definition of corresponding molecular tensors arrived at by continuous transformation of the origin of the current density, which formally sets its diamagnetic term to zero (abbreviated as CTOCD-DZ), is discussed in Section 3. Section 4 deals with the problem of origin dependence of magnetizability polarizabilities of a molecule in the presence of uniform EFG. 

RESPONSE TENSORS BY CONTINUOUS TRANSFORMATION OE THE ORIGIN OE THE CURRENT DENSITY... 

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