Tensor third-rank

Because of the symmetry of the indices j and k one can replace these two by a single index (subscript) m. Consequently the notation for the SHG nonlinear coefficient in reduced form is where m takes the values 1 to 6. Only noncentrosymmetric crystals can possess a nonvanishing tensor (third rank). The unit of the SHG coefficients is m/V (in the MKSQ/SI system). 

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] ... 

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

Deformation leading to a change in the director, where the distortion is described by a tensor of third rank... 

Gradient of field gradient Third rank tensor -4 1... 

The third-rank tensors, as occur in the expression for the anharmonic temperature factor (chapter 2), the restrictions may be derived by use of the transformation law ... 

In addition to g tensor anisotropy, EPR spectra are often strongly affected by hyperfine interactions between the nuclear spin I and the electron spin S. These interactions take the form T A S, where A is the hyperfine coupling tensor. Like the g tensor, the A tensor is a second-order third-rank tensor that expresses orientation dependence, in this case, of the hyperfine coupling. The A and g tensors need not be colinear in other words, A is not necessarily diagonal in the coordinate systems which diagonalize g. 

The dual axial vector in 4-space is constructed geometrically from the integral over a hypersurface, or manifold, a rank 3-tensor in 4-space antisymmetric in all three indices [101]. In three-dimensional space, the volume of the parallelepiped spanned by three vectors is equal to the determinant of the third rank formed from the components of the vectors. In four dimensions, the projections can be defined analogously of the volume of the parallelepiped (i.e., areas of the hypersurface) spanned by three vector elements < dl, dx and dx". They are given by the determinant... 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

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... 

Piezoelectric materials are materials that exhibit a linear relationship between electric and mechanical variables. Electric polarization is proportional to mechanical stress. The direct piezoelectric effect can be described as the ability of materials to convert mechanical stress into an electric field, and the reverse, to convert an electric field into a mechanical stress. The use of the piezoelectric effect in sensors is based on the latter property. For materials to exhibit the piezoelectric effect, the materials must be anisotropic and electrically poled ie, there must be a spontaneous electric field maintained in a particular direction throughout the material. A key feature of a piezoelectric material involves this spontaneous electric field and its disappearance above the Curie point. Only solids without a center of symmetry show this piezoelectric effect, a third-rank tensor property (14,15). 

The third-rank tensors Zafir od a1 aft describe non linear response of the electron cloud to first order in E. These quantities are sometimes referred to as hypermagnetizabilities, which are related to Cotton-Motton effect (4-6), (CME- the birefringence of light in gases in a constant magnetic field) and as shielding polarizabilities (7-14). 

These equations use Cartesian tensor notation in which a repeated Greek suffix denotes summation over the three components, and where ay7 is the third-rank antisymmetric unit tensor. For a molecule composed entirely of idealized axially symmetric bonds, for which [3 (G )2 = /3(A)2 and aG1 = 0 [13, 15], a simple bond polarizability theory shows that ROA is generated exclusively by anisotropic scattering, and the CID expressions then reduce to [13]... 

The first selection rule is a consequence of the fact that the transition dipole moment has negative parity. The second reflects that the quantization axis is parallel to the polarization of the electric field. The third follows from the fact that the transition dipole moment is a tensor of rank 1 (corresponding to an angular momentum with quantum number 1). 

The piezoelectric coefficients are third rank tensors, hence the piezoelectric response is anisotropic. A two subscript matrix notation is also widely used. The number of non-zero coefficients is governed by crystal symmetry, as described by Nye [2], In most single crystals, the piezoelectric coefficients are defined in terms of the crystallographic axes in polycrystalline ceramics, by convention the poling axis is referred to as the 3 axis. 

Ordinary Raman scattering is determined by derivatives of the electric dipole-electric dipole tensor ae, and ROA by derivatives of cross-products of this tensor with the imaginary part G,e of the electric dipole-magnetic dipole tensor (the optical activity tensor) and the tensor Ae which results from the double contraction of the third rank electric dipole-electric quadrupole tensor Ae with the third rank antisymmetric unit tensor s of Levi-Civita. The electronic property tensors have the form ... 

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