Property tensor

Each second-rank tensor represented by the square matrix can be transformed by the similarity transformation to its diagonal form 

Generally, properties of liquid crystals depend on direction, they are tensorial. Some of them (like density in nematics) may be scalar. A scalar is a tensor of rank 0. It has one component in a space of any dimensionality, 1 = 2° = 3 ... = 1. Other properties, like spontaneous polarization P (e.g., in chiral smectic C ) are vectors, i.e., the tensors of rank 1. In the two-dimensional space they have 2 = 2 components, in the 3D space there are 3 = 3 components. For instance in the Cartesian system P = iP + jPy + kP. Such a vector can be written as a row (Px Py, Pz) or as a coluirm. Properties described by tensor of rank 2 have 2 = 4 components in 2D space and 3 = 9 components in the 3D-space. They relate two vector quantities, such as magnetization M and magnetic induction B, M = xB, where % is magnetic susceptibility. Each of the two vectors has three components and, generally, each component (projection) (a = x,y,z) may depend on each of Bp components (p = x,y,z)  

The matrix representation can be written in a more compact form 

Here a, p = 1, 2, 3 and, following Einstein, the repeated index p means summation over p. 

The magnetization was only taken as an example. Many other properties (dielectric susceptibility, electric and thermal conductivity, molecular diffusion, etc.) are also described by second rank tensors of the same (quadrupolar) type Microscopically, such properties can be described by single-particle distribution functions, when intermolecular interaction is neglected. There are also properties described by tensors of rank 3 with 3 = 27 components (e.g., molecular hyperpolarizability Yijk) and even of rank 4 (e.g., elasticity in nematics, ATiju) with 3 = 81 components. Microscopically, such elastic properties must be described by many-particle distribution functions. 

As physical properties of the matter are independent of the chosen frame, suffixes a and p can be interchanged. Therefore, Xap = Xpot and only 6 components of x p are different, three diagonal and the other three off-diagonal. Such a symmetric tensor (or matrix) can always be diagonalized by a proper choice of the Cartesian frame whose axes would coincide with the symmetry axes of the LC phase. In that reference system only three diagonal components Xii, X22 and X33 are finite. 

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From the point of view of tensor properties, the surface nonlmear susceptibility /, J-(.is quite analogous to the... 

The electric moments are examples of tensor properties the charge is a rank 0 tensor (which i the same as a scalar quantity) the dipole is a rank 1 tensor (which is the same as a vectoi with three components along the x, y and z axes) the quadrupole is a rank 2 tensor witl nine components, which can be represented as a 3 x 3 matrix. In general, a tensor of ran] n has 3" components. 

Intensities of Raman transitions are proportional to R and therefore, from Equation (6.13), to (da/dx)g. Since a is a tensor property we cannot illustrate easily its variation with x instead we use the mean polarizability a, where... 

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). 

The dipole polarizability, the field gradient and the quadrupole moment are all examples of tensor properties. A detailed treatment of tensors is outside the scope of the text, but you should be aware of the existence of such entities. 

In general, NMR properties of a molecule are tensor properties. That is, their value depends upon the spatial relationship of the molecule to the applied magnetic field. As such, each property can be described using three principal components, plus three angles to specify the orientation... 

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. 

In this section we provide a brief review of topics in linear algebra and tensor property relations that are used frequently throughout the book. Nye s book on tensor properties contains a complete overview and is also a valuable resource [6]. 

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

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... 

It should be noted that because [TCNQ-TTF] crystallizes in monoclinic form and because conductivity is a tensor property, four independent pieces of conductivity data are required to completely define the conductivity in the principal directions. However,... 

There is a first-order splitting pattern common to all 3n states, independent of the physical content. All the molecule-dependent physical information is contained in the parameter Aso- These facts are, of course a consequence of the tensor properties expressed in the Wigner-Eckart theorem. 

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... 

There are various publications reporting measurement from single-crystal or liquid-crystal NMR of parameter tensor properties. They all assume these matrices to be symmetric. These works include Refs. 11,31,138,139. 

J. A. Weil, T. Buch and J. E. Clapp, Crystal point group symmetry and microscopic tensor properties in magnetic resonance spectroscopy. Adv. Magn. Reson., 1973, 6,183-257. 

D. R. Lovett, Tensor Properties of Crystals. Institute of Physics Publishing, Bristol, UK, 1989,1994. 

The flows may have vectorial or scalar characters. Vectorial flows are directed in space, such as mass, heat, and electric current. Scalar flows have no direction in space, such as those of chemical reactions. The other more complex flow is the viscous flow characterized by tensor properties. At equilibrium state, the thermodynamic forces become zero and hence the flows vanish... 

From tensor algebra, the tensor property relating two associated tensor quantities, of rank / and rank g, is of rank (/-b g). Hence, the physical property connecting /, and aj is the third-rank tensor known as the piezoelectric effect, and it contains 3 = 27 piezoelectric strain coefficients, dyk. The piezoelectric coefficients are products of electrostriction constants, the electric polarization, and components of the dielectric tensor. 

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,... 

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