Tensor fourth rank

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] ... 

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). 

Determination of the fourth-rank tensor term F. 2 remains. Basically, F.,2 cannot be found from any uniaxial test in the principal material directions. Instead, a biaxial test must be used. This fact should not be surprising because F-,2 is the coefficient of the product of a. and 02 in the failure criterion. Equation (2.140). Thus, for example, we can impose a state of biaxial tension described by a, = C2 = c and all other stresses are zero. Accordingly, from Equation (2.140),... 

Where a, P and y are the linear polarizability, the first- and second-hyper polarizabilities, respectively, and are represented by second, third and fourth rank tensors, respectively, and is a static polarizability. 

Grad-grad of field gradient Fourth rank tensor -5 2... 

In Equation 6, n (a>.) is the intensity independent refractive index at frequency u).,.0 Tlie sum in Equation 5 is over all the sites (n) the bracket, < >, represents an orientational averaging over angles 0 and . Unlike for the second-order effect, this orientational average for the third-order coefficient is nonzero even for an isotropic medium because it is a fourth rank tensor. Therefore, the first step to enhance third order optical nonlinearities in organic bulk systems is to use molecular structures with large Y. For this reason, a sound theoretical understanding of microscopic nonlinearities is of paramount importance. 

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... 

How this additional field will alter the magnitude of the various tensor elements and the form of the rotational anisotropy has been examined by Koos et al. [122]. X0) has the properties of a fourth-rank tensor such that the third order polarization... 

The presence of the fourth-rank tensor in (7.127) and its absence in (7.11) suggests that the stress optical rule should not apply for dilute solutions of rigid rods. Unfortunately, because of the difficulty of acquiring truly rigid rods, and the problems of making measurements of stress in dilute systems, there are no data available on dilute rigid rod solutions where the stress optical rule can be investigated on this class of polymer liquids. 

On a phenomenological level the in-plane anisotropy of Hc2 cannot be explained within the (local) GL theory. In principle, non-local extension introduced by Hohenberg and Werthamer (1967) might be helpful to overcome this difficulty. In this approach, which is valid for weak anisotropies, in addition to the second rank mass tensor, a fourth rank tensor is introduced. The non-local effects were predicted to be observable in sufficiently clean materials where the transport... 

The electrostriction coefficient is a fourth-rank tensor because it relates a strain tensor (second rank) to the various cross-products of the components of E or D in the. v, y and z directions. 

The state of strain in a body is fully described by a second-rank tensor, a strain tensor , and the state of stress by a stress tensor, again of second rank. Therefore the relationships between the stress and strain tensors, i.e. the Young modulus or the compliance, are fourth-rank tensors. The relationship between the electric field and electric displacement, i.e. the permittivity, is a second-rank tensor. In general, a vector (formally regarded as a first-rank tensor) has three components, a second-rank tensor has nine components, a third-rank tensor has 27 components and a fourth-rank tensor has 81 components. 

Upon appropriate reduction in the number and nature of the independent tensorial components of i/i(s) (= j/, ), resulting from the common point-group symmetry elements of the sphere and cube (applied to fourth-rank tensors), the material tensor can be shown quite generally to be of the form (Zuzovsky et al., 1983)... 

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,... 

The in (18)-(21) are the nth-order susceptibilities. They are tensors of rank n + with 3 " components. Thus, a second-order susceptibility, is a third-rank tensor with 27 components and the third-order susceptibility, a fourth-rank tensor with 81 components. The number of independent and significant elements is (fortunately) much lower (see p. 131). 

To derive Eq. (9-44), a decoupling approximation was used so that a fourth rank tensor could be replaced by a product of second rank tensors. 

The coeflScients a, P, and y are the second, third, and fourth rank tensors and are referred to as the polarizability, first hyperpolarizability, and second hyperpolarizability, respectively. The hyperpolarizability terms are responsible for the nonlinear response of the molecule to impinging radiation. These coefiBcients are not very large, and the associated nonlinear optical effects are usually studied by taking advantage of the high optical field obtainable with laser beams. 

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