Transformation second-rank tensor

Combining Eqs. (9.14) and (9.15) and recalling the general rules for transforming second-rank tensors between two Cartesian coordinate frames, the following relation is obtained [155]... 

In contrast, the second term in (4.6) comprises the full orientation dependence of the nuclear charge distribution in 2nd power. Interestingly, the expression has the appearance of an irreducible (3 x 3) second-rank tensor. Such tensors are particularly convenient for rotational transformations (as will be used later when nuclear spin operators are considered). The term here is called the nuclear quadrupole moment Q. Because of its inherent symmetry and the specific cylindrical charge distribution of nuclei, the quadrupole moment can be represented by a single scalar, Q (vide infra). 

This matrix describes the transformation from x y z to xyz as a rotation about the z axis over angle a, followed by a rotation about the new y" axis over angle /), followed by a final rotation over the new z " axis over angle y (Watanabe 1966 148). Formally, the low-symmetry situation is even a bit more complicated because the nondiagonal g-matrix in Equation 8.11 is not necessarily skew symmetric (gt] -g. Only the square g x g is symmetric and can be transformed into diagonal form by rotation. In mathematical terms, g x g is a second-rank tensor, and g is not. 

These ideas may be extended to define tensors of any rank. There are three varieties of second rank tensors, defined by the transformations... 

Second-rank tensors transform according to the expression... 

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... 

Warnings (i) The Tpq do not form a second-rank tensor and so unitary transformations must be carried out using the four-index notation Tijki. (ii) The contraction of TiJki may be accompanied by the introduction of numerical factors, for example when 7(4) is the elastic stiffness (Nye (1957)). 

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... 

Equation 10.1 is a second-rank tensor with transpose symmetry. The normal components of stress are the diagonal elements and the shear components of stress are the nondiagonal elements. Although Eq. 10.1 has the appearance of a [3 x 3] matrix, it is a physical quantity that, for one set of axes, is specified by nine components, whereas a transformation matrix is an array of coefficients relating two sets of axes. The tensor coefficients determine how the three components of the force vector, /, transmitted across a small surface element, vary as different values are given to the components of a unit vector / perpendicular to the face (representing the face orientation) ... 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a( w w) is a symmetric second-rank tensor hke Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components,... 

Its S3unmetry properties are not the same as those of the electric dipole, which transforms under pure rotations as the multipole components (1,0) and (1, 1) and is antisymmetric under inversion in a centre of sjmmetry. The magnetic dipole transforms hi the same way under rotations but is symmetric to inversion. The properties are those of an antisymmetric second-rank tensor, or axial vector, and can be rationalized using the model that a magnetic field has as source a plane current loop. The current direction in the loop is invariant to inversion of the spatial coordinates but reversed by twofold rotation about an axis in the plane. 

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

These components transform under rotation like the spherical harmonic functions T]m. In general, an irreducible spherical tensor Tim transforms like the function Ylm. The spherical components of a second-rank tensor are again collected in Table 1.13. 

As is well known, explicit expressions for the polarization fields can be given such that equation (3) has the requisite properties these expressions, involving multipole series or line integrals, are by no means tmique. That this must be so can be seen from at least two levels of theory. We noted earlier that the notion of an electric field (or a magnetic field) is not invariant with respect to Lorentz transformations under such transformations however V should be an invariant scalar and this implies a definite transformation law for (P(x), M(x) that mixes them, and mirrors that for E(x), B(x) classically both pairs can be shown to be components of skew-symmetric second-rank tensors [7]. Although... 

Let us now derive phenomenological equations of the kind (5.193) corresponding to the expression (5.205). As has been mentioned before, each flux is a linear function of all thermodynamic forces. However the fluxes and thermodynamic forces that are included in the expression (5.205) for the dissipative function, have different tensor properties. Some fluxes are scalars, others are vectors, and the third one represents a second rank tensor. This means that their components transform in different ways under the coordinate transformations. As a result, it can be proven that if a given material possesses some symmetry, the flux components cannot depend on all components of thermodynamic forces. This fact is known as Curie s symmetry principle. The most widespread and simple medium is isotropic medium, that is, a medium, whose properties in the equilibrium conditions are identical for all directions. For such a medium the fluxes and thermodynamic forces represented by tensors of different ranks, cannot be linearly related to each other. Rather, a vector flux should be linearly expressed only through vectors of thermodynamic forces, a tensor flux can be a liner function only of tensor forces, and a scalar flux - only a scalar function of thermodynamic forces. The said allows us to write phenomenological equations in general form... 

Each of the three components / Fi dV of the resultant force (concerning all the internal stresses) can be transformed to a surface integral (Landau and Lifshitz, 1987). It follows from the general field theory that, in this case, the components of Fj must be the divergences of a certain second-rank tensor, i.e. 

A quantity having nine components, which transform as (2.25) is csAXoA a second-rank tensor by definition. 

We can show that a second-rank tensor transforms like the product x, ag if we use the transformation law (2.16)... 

The transformation definition of the second-rank tensor can be generalized and used for definition of tensors of higher order. 

According to this definition, a tensor of the first rank is simply a vector. As examples of second rank tensors within classical mechanics one might think of the inertia tensor 6 = (0y) describing the rotational motion of a rigid body, or the unity tensor 5 defined by Eq. (2.11). A tensor of the second rank can always be expressed as a matrix. Note, however, that not each matrix is a tensor. Any tensor is uniquely defined within one given inertial system IS, and its components may be transformed to another coordinate system IS. This transition to another coordinate system is described by orthogonal transformation matrices R, which are therefore not tensors at all but mediate the change of coordinates. The matrices R are not defined with respect to one specific IS, but relate two inertial systems IS and IS. ... 

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

We note that, sometimes the symmetric stress and strain tensors with six components are represented as six element vectors and S, where A = l/2(k + j)5p, + [9- a + j)](l - dp,), i.e., tire 11- 1 22- 2 33->3 23 = 32- 4 13 = 31 5 12 = 21 6 transformations are used. ° In this case, the piezoelectric constants are formally expressed as 3 x 6 element second-rank tensors. These notations are simpler, but much less transparent than the third-rank tensor notation, so in the following, we will keep the mathematically more transparent notation. 

What remains to be shown is that the matrix Tap is actually a representation of a second-rank tensor r to which we shall henceforth refer as the stress tensor. We need to demonstrate that the matrix representing t satisfies transformation properties under rotation of the coordinate system that constitute a second-rank tensor. To this end consider an infinitesimally... 

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