Decomposable tensors

Elementary tensors are also known as decomposable tensors. 

In a non-Abelian theory (where the Hamiltonian contains noncommuting matrices and the solutions are vector or spinor functions, with N in Eq. (90) >1) we also start with a vector potential Af, [In the manner of Eq. (94), this can be decomposed into components A, in which the superscript labels the matrices in the theory). Next, we define the field intensity tensor through a covaiiant curl by... 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

Since the deformation tensor F is nonsingular, it may be decomposed uniquely into a proper orthogonal tensor R and a positive-definite symmetric tensor U by the polar decomposition theorem... 

If we introduce the electromagnetic field tensor operator F ix), which can be decomposed as follows ... 

The shielding tensor, and its diamagnetic and paramagnetic components, are not necessarily symmetric in the Cartesian indices [25-29], and the shielding tensor can in general be decomposed into a symmetric and an antisymmetric component, i.e. 

It was proven [96] that the molecular polarizability can be written as a sum of intrinsic atomic polarizabilities of the atoms in the molecule and a charge delocalization term. Thus, the xy element of the molecular polarizability tensor of molecule A can be decomposed as... 

Hyperfine tensors are given in parts B and C of Table II. Although only the total hyperfine interaction is determined directly from the procedure outlined above, we have found it useful to decompose the total into parts in the following approximate fashion a Fermi term is defined as the contribution from -orbitals (which is equivalent to the usual Fermi operator as c -> < ) a spin-dipolar contribution is estimated as in non-relativistic theory from the computed expectation value of 3(S r)(I r)/r and the remainder is ascribed to the "spin-orbit" contribution, i.e. to that arising from unquenched orbital angular momentum. 

We will use Cartesian sums and tensor products to build and decompose representations in Chapters 5 and 7. Tensor products are useful in combining different aspects of one particle. For instance, when we consider both the mobile and the spin properties of an electron (in Section 11.4) the state space is the tensor product of the mobile state space defined in Chapter 3)... 

This nonsymmetric second-rank tensor can be decomposed to a symmetric (i.e., ay = o-ji) and an antisymmetric tensor through a symmetrization process (28). 

In order to apply group theory to the physical properties of crystals, we need to study the transformation of tensor components under the symmetry operations of the crystal point group. These tensor components form bases for the irreducible repsensentations (IRs) of the point group, for example x, x2 x3 for 7(1) and the set of infinitesimal rotations Rx Ry Rz for 7(1 )ax. (It should be remarked that although there is no unique way of decomposing a finite rotation R( o n) into the product of three rotations about the coordinate axes, infinitesimal rotations do commute and the vector o n can be resolved uniquely... 

Any Cartesian tensor, represented by a 3 x 3 square matrix, can be decomposed into a symmetric and antisymmetric parts as follows... 

The spatial velocity gradient a = grad va can be decomposed into symmetric and skew-symmetric parts as la = sym a + skw 1 = dQ + wQ., where da and wa are the deformation rate and the spin tensors, respectively. 

Equations (5.139) to (5.142) are the basic equations for a gas-solid flow. More detailed information on both the fluid-particle interacting force Fa and the total stresses T and Tp must be specified before these equations can be solved. One approach to formulate the fluid-particle interacting force FA is to decompose the total stress into a component E representing the macroscopic variations in the fluid stress tensor on a scale that is large compared to the particle spacing, and a component e representing the effect of detailed variations of the point stress tensor as the fluid flows around the particle [Anderson and... 

A fluid in motion may simultaneously deform and rotate. Decomposing the velocity gradient tensor into two parts can separate these motions ... 

We will focus in this chapter on the basic formalism of Raman and ROA scattering, and on the understanding of ab initio computed vibrations, electronic tensors, and Raman and ROA scattering cross-sections. The usefulness of decomposing ab initio computed data will be demonstrated in the context of their comparison with the measured spectra of (+)-(P)-l,4-dimethylenespiropentane [40] which exhibits an unusual dependence on the solvent environment. 

Equivalently, we can decompose Eij into a divergence-free and trace-free tensor Eij, a divergence-free vector Ei and a scalar E ... 

The aim of this decomposition is that, as we shall see, the sets of scalar quantities A, B, C, E, vector quantities >,. Ei and tensor quantities Eij evolve independently from each others. Note that we are left with four scalars A, B, C, E), two vectors (Bi, Ei) which both have three components but which obey obey divergenceless constraint so that we have four independent components, and one tensor (Eij) which is a 3 x 3 symmetric matrix with one traceless constraint and three divergenceless constraints, which therefore has only two independent components. As expected, we are still left with ten independent components for the metric perturbations. As we said, four of these perturbations are in fact unphysical. Let us decompose the infinitesimal coordinate change 4 / into... 

Chemical shift spectra of PTFE obtained at 259° are shown in Figure 1. These lineshapes, for three different samples of varying crystallinity, may be seen to be a linear combination of two lineshapes one is characteristic of an axially symmetric powder pattern and the other of an isotropic chemical shift tensor. At this temperature these two lineshapes differ greatly and may be numerically decomposed. 

In order to evaluate the matrix elements of the dipole moment operator in Eq. (24), it is convenient to separate out the geometrical aspects of the problem from the dynamical parameters. To that end, it is convenient to decompose the LF scalar product of the transition dipole moment d with the polarization vector of the probe laser field e in terms of the spherical tensor components as [40]... 

The reduced matrix element in (9.124) is evaluated by decomposing the second-rank tensor into its constituent first-rank tensors one finds that... 

The problem therefore reduces to that of evaluating the left-hand side of equation (9.178). We first decompose the third-rank tensor by making use of the result... 

Let us consider the spatially periodic porous medium consisting of an infinite number of identical unit cells. The spatially periodic medium is subjected to a macroscopic deformation described by the tensor of deformation A, and the local displacement d — A x + d can be decomposed into a macroscopic deformation A x and a microscopic spatially periodic displacement d cf. Poulet et al. (1996). This decomposition introduced into elastostatic Equations (29) and (30) yields... 

The following common conventions are used complex entities have been noted by a tilde and can be decomposed into a real and an imaginary part, e.g. in the case of the polarizability tensor ... 

For a pure dipole interaction, the anisotropic term in the above equation should have the form -a, -a, la. This implies that the unpaired electron is not purely p based, but also there must be some occupancy of the orthogonal p orbitals. In other words, the anisotropic tensor is actually the result of two dipolar interactions with two radius vectors, each of which is along a coordinate axis of the molecule. Therefore the anisotropic tensor is the sum of two dipolar coupling tensors and can be decomposed into two traceless components (-a, -a, la) and (-h, Ih, -h) as follows ... 

The large isotropic component is due to the unpaired electron spin density in the carbon 2s orbital, and this value (544MHz) can be used to derive an estimate of the carbon 2 s orbital contribution to the molecular orbital. Since the theoretical isotropic coupling constant for is 3777MHz, then C2s = 544/3777 = 0.144. The anisotropic dipolar part of the hyperfine arises from unpaired spin density in the 2p orbital. However because the dipolar contribution in Equation 1.52 cannot be reduced to zero, this implies that a fraction of the spin density is allocated to the 2p orbital perpendicular to the molecular plane. Therefore, the dipolar component of Equation 1.52 must be further decomposed into two symmetrical tensors oriented along the z and x axes ... 

Prior to discussing the properties of octopolar molecules, it is instructive to consider first some of the basic properties of tensors. In general, any tensor of rank n can be decomposed in a sum of so-called irreducible tensors that are invariant under three-dimensional rotation [99] ... 

This contains an TCP of the TpaL tensor, which is derived from the electron spin and dipole-dipole interaction tensor(See equation (11)). Hence, the first question we confront is whether those tensors are correlated or not. In case they are not the total TCP can be decomposed into a product of auto correlations for the the electron spin and dipole-dipole interaction tensor, respectively. In case they are, however, it is necessary to consider the whole TCP and the electron spin has to be correlated with the dipole-dipole interaction tensor. The time dependence in the electron spin tensor can be obtained by integrating the time dependent Schrbdinger equation for the electron spin under the electron spin Hamiltonian. The electron spin is just like the nuclear spin precessing around the external magnetic field and influenced by molecular dynamics. 

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