Rank, tensor properties

Typical electrostrictive materials include such compounds as lead manganese niobate lead titanate (PMN PT) and lead lanthanium 2irconate titanate (PLZT). Electrostriction is a fourth-rank tensor property observed in both centric and acentric insulators (14,15). 

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. 

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). 

Characterization of Molecular Hyperpolarizabilities Using Third Harmonic Generation. Third harmonic generation (THG) is the generation of light at frequency 3co by the nonlinear interaction of a material and a fundamental laser field at frequency co. The process involves the third-order susceptibility x 3K-3 , , ) where —3 represents an output photon at 3 and the three s stand for the three input photons at . Since x(3) is a fourth (even) rank tensor property it can be nonzero for all material symmetry classes including isotropic media. This is easy to see since the components of x(3) transform like products of four spatial coordinates, e.g. x4 or x2y2. There are 21 components that are even under an inversion operation and thus can be nonzero in an isotropic medium. Since some of the terms are interrelated there are only four independent terms for the isotropic case. 

For molecules containing several conjugated bonds yn becomes much larger than y°. Of course, y itself is a fourth rank tensor property (analogous to x(3)) and can be specified in the molecular or laboratory reference frames. For an isotropic medium one measures an orientational average of the hyperpolarizability... 

The components of a symmetrical second-rank tensor, referred to its principal axes, transform like the three coefficients of the general equation of a second-degree surface (a quadric) referred to its principal axes (Nye, 1957). Hence, if all three of the quadric s coefficients are positive, an ellipsoid becomes the geometrical representation of a symmetrical second-rank tensor property (e.g., electrical and thermal conductivity, permittivity, permeability, dielectric and magnetic susceptibility). The ellipsoid has inherent symmetry mmm. The relevant features are that (1) it is centrosymmetric, (2) it has three mirror planes perpendicular to the... 

Because stress and strain are vectors (first-rank tensors), the forms of Eqs. 10.5 and 10.6 state that the elastic constants that relate stress to strain must be fourth-rank tensors. In general, an wth-rank tensor property in p dimensional space requires p" coefficients. Thus, the elastic stiffness constant is comprised of 81 (3 ) elastic stiffness coefficients,... 

Spontaneous strain is a symmetrical second rank tensor property and must conform to Neumann s principle in relation to symmetry. A general spontaneous strain with all six of the independent strain components having non-zero values can be referred to an alternative set of axes by diagonalisation to give... 

As for all second-rank tensor properties, the orientational variation in chemical shift can be visualized as an ellipsoidal surface for which the symmetry and alignment of the principal axes must conform to the point symmetry at the atom position. For example. 

The signals in 2D-IR experiments are fourth-rank tensor properties with indices corresponding to the four polar vector components of the incident and detected electric fields. In an isotropic medium there are only three independent fourth... 

Equations (A.7) also show that, in general, the prediction of a property that depends on a tensor of rank I will require knowledge of orientation averages of order /. The elastic constants of a material are fourth-rank tensor properties thus the prediction of their values for a drawn polymer involves the use of both second- and fourth-order averages, in the simplest case P ico O)) and P (x>s6)), and thus provides a more severe test of the models for the development of orientation. The elastic constants are considered in section 11.4. 

In hexagonal close-packed crystals, magnetic susceptibility, as with other second-rank tensor properties, may be assigned two principal components, x, and... 

Another example of a second rank tensor property is electrical conductivity which relates the current flow j in a particular direction to the electric field ... 

Another special feature of second rank tensor properties in three dimensions is that they can be represented by a property ellipsoid, such that the value of the property in a particular direction is represented by the length of the corresponding radius vector of the property ellipsoid. A three dimensional surface representing an ellipsoid can be defined by... 

The anisotropy of liquid crystals stems from the orientational order of the constituent molecules, but the macroscopic anisotropy can only be determined through measurement of tensor properties, and macroscopic tensor order parameters can be defined in terms of various physical properties. The anisotropic part of a second rank tensor property can be obtained by subtracting the mean value of its principal components ... 

If Kajjis a. molecular second rank tensor property, the principal components of which are and defined in a molec-... 

If the liquid crystal phase is biaxial, then any second rank tensor property has three independent principal components. These are the diagonal elements of the anisotropic tensor X ap- expressed in terms... 

Using the general results for the transformation of second rank tensor properties, these... 

The principal axes of the molecular polarizability tensor are labelled /, m, n, as shown in Fig. 2. Thus the importance of order parameters in determining the anisotropy of optical properties is clearly demonstrated. Both order parameters S and D contribute to the anisotropy of second rank tensor properties even in uniaxial liquid crystals, but they cannot be separated from a single measurement of the birefringence. 

Pyroelectricity is a first-rank tensor property that relates the change in temperature to a change in electrical displacement D (or polarization P since no field is applied) ... 

Other macroscopic properties can also be related to the tensor order parameter - in fact, any second rank tensor property of the system can be expressed in terms of Q. Using (4.1), the dielectric tensor can be written as [1]... 

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