Higher Rank Tensors

In this application of the BWR theory, Hudson and Lewis assume that the dominant line-broadening mechanism is provided by the modulation of a second rank tensor interaction (i.e., ZFS) higher rank tensor contributions are assumed to be negligible. R is a 7 X 7 matrix for the S = 7/2 system, with matrix elements written in terms of the spectral densities J (co, rv) (see reference [65] for details). The intensity of the i-th transition also can be calculated from the eigenvectors of R. In general, there are four transitions with non-zero intensity at any frequency, raising the prospect of a multi-exponential decay of the transverse magnetization. There is not a one-to-one correspondence between the... 

One can consider any higher rank tensor similarly to the cases of 2nd and 3rd rank tensors. For instance, the calculations have recently been performed for 4th rank tensors [64], to which elastic modulus and striction coefficients belong. In particular, new (additional to three elastic moduli Cn, C22, C44) nonzero components appear for cubic symmetry. Namely, for the surfaces of (100) type these are C33, C13, Cee and for the surfaces of (111) type these are Cm, 24= —Cm, Cs6 = Cm, which equal zero for ordinary bulk cubic symmetry. Here we use Voight notations, i.e. Cy written as C, /, = 1, 2, 3,..., 6. 

The direct effect coefficients are defined by the derivatives (5D/SX) = d (piezoelectric strain coefficient), (5D/5x) = e, -(5E/5X) = g (piezoelectric voltage constant) and -(5E/5x) = h. The converse-effect coefficients are defined by the derivatives (8x/5E) = d, (5x/5D) = g, -(5X/5E) = e, and -(5X/5D) = h. As the piezoelectric coefficients are higher-rank tensors, their mathematical treatment is rather tedious. Fortunately, in higher symmetric crystals the number of tensorial components will be drastically reduced due to symmetry constraints. An example is shown below. 

This is the standard transformation rule for components of a rank-2 covariant tensor, differing from (10.2.7) only through use of the matrix U in place of U. The process may obviously be continued, leading to the dehnition of higher-rank tensors of each type and of mixed tensors with, for example, one degree of covariance and one of contravariance. 

In general, the higher-rank tensors give more precise point group information than do the simpler properties. The number of nonzero coefficients and the number of independent coefficients for various... 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

Higher rank ZFS tensors (D4,Z)6,. .., etc.) and Zeeman splitting tensors (G3, G5,. .., etc.) for complexes with moderate and strong spin-orbit coupling. 

Tensors of higher rank are defined in the same way, for example, a mixed tensor of rank four is... 

Symmetry restrictions for a number of crystal systems are summarized in Table B.l. The local symmetry restrictions for a site on a symmetry axis are the same as those for the crystal system defined by such an axis, and may thus be higher than those of the site. This is a result of the implicit mmm symmetry of a symmetric second-rank tensor property. For instance, for a site located on a mirror plane, the symmetry restrictions are those of the monoclinic crystal system. 

The piezoelectric constant eik is a third-rank tensor and vanishes when the material has a center of symmetry. When the strain is not uniform, however, a higher order piezoelectricity appears in proportion to the... 

The problem above can also be solved analytically using tensor methods—the preferred technique when higher accuracy is required. In general, any homogeneous deformation can be represented by a second-rank tensor that operates on any vector in the initial material and transforms it into a corresponding vector in the deformed material. For example, in the lattice deformation, each vector, Ffcc, in the initial f.c.c. structure is transformed into a corresponding vector in the b.c.t. structure, Vbct, by... 

For tensors of higher rank we must ensure that the bases are properly normalized and remain so under the unitary transformations that correspond to proper or improper rotations. For a symmetric T(2) the six independent components transform like binary products. There is only one way of writing xx xx, but since xx x2 = x2 xx the factors xx and x2 may be combined in two equivalent ways. For the bases to remain normalized under unitary transformations the square of the normalization factor N for each tensor component is the number of combinations of the suffices in that particular product. F or binary products of two unlike factors this number is two (namely ij and ji) and so N2 = 2 and x, x appears as /2x,- Xj. The properly normalized orthogonal basis transforming like... 

The method described above is more powerful than earlier methods, especially for tensors of higher rank and for groups that have three-fold or six-fold principal axes. 

In order to determine the physical meaning of moments of higher rank, one may keep in mind the fact that the tensor operators Tq are proportional to the angular momenta operator /W [73]. Thus, for instance, the three components of the operator are defined as... 

Although there is a one-to-one correspondence between the first-rank cartesian and spherical tensors, the same is not true for second- and higher-rank cartesian tensors. 

Here Ctju are the stilfness constants and Sijki are the compliance constants. They form two symmetric fourth-rank tensors with 81 elements inverse one to another. For the triclinic symmetry only 21 elements are independent because the strain and stress tensors are symmetric. Consequently the indices i,j and k, I can be permuted and also can be permuted one pair with another. For a crystal symmetry higher than triclinic the number of independent elastic constants is less than 21. 

Similar expressions exist for the tensors of higher rank and can be used to rewrite Eq. (A2.87) term by term... 

Usually the dependence of the nonlinear polarizability tensors on the wavevec-tors k, k jk", etc. is neglected. In this approximation in crystals with an inversion center the third-rank tensor (u>, uiu>"), which changes sign by inversion, vanishes and nonlinear processes involving three photons are absent. Explicit expressions for the nonlinear polarizability tensors, similar to the tensor e u , k), can be obtained within a microscopic theory. Above we presented various methods for calculation of the tensor k). The important point was that to obtain this tensor it was necessary to establish the crystal polarizability in the linear approximation with respect to the electric field. The nonlinear polarizability tensors can be obtained in a similar way, but now the crystal polarizability must be determined by taking into account higher order terms with respect to the electric field (see (16)). 

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... 

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