Tensor component matrix

The tensor quantities given in this chapter are all second rank, and are sometimes referred to as matrices, according to common usage, so that the two terms, tensor and matrix, are used interchangeably. In many cases, the components (or coefficients) of second-rank tensors are represented by 3 x 3 matrices. Symbols for tensors (matriees) are printed in bold italic type, while symbols for the components are printed in italic type. In general, the base tensors are those for a rectangular Cartesian coordinate system. 

Examples for non-totally-symmetric components in the decomposition of density matrix into irreducible tensor components are the one-particle spin density matrices ... 

Using (4) to write stress tensor Pxx components as Pyy and Pz2, matrix components for the case under consideration are expressed specifically and are substituted into expressions for stress tensor components. Thus, we obrais the following relationships (details of transformation in 29)) ... 

Table 15.4. Matrix elements of S (see eq. (15.3.7)) for the group C4V and a basis comprising the symmetrical T(2) tensor components ctk defined in eq. (15.1.19).

The matrix elements of the angular momentum operators in Cartesian directions a = x,y, z form (complex) matrices La. Now we can proceed with an evaluation of the /1-tensor components in three steps. 

In order to evaluate the matrix elements of the dipole moment operator in Eq. (24), it is convenient to separate out the geometrical aspects of the problem from the dynamical parameters. To that end, it is convenient to decompose the LF scalar product of the transition dipole moment d with the polarization vector of the probe laser field e in terms of the spherical tensor components as [40]... 

M contains all the information about the anisotropic optical properties of the medium that supports the electromagnetic fields some of its 36 tensor components are the target of the ellipsometric measurements of anisotropic crystals. This optical matrix M is thus defined by... 

For more general deformations, it is necessary to represent the strain as a tensor (a matrix), with components Sy, where i and j both go from 1 to 3 (in the three-dimensional case). The diagonal elements (the ones for which i = j) still represent normal strain (elongation or compression) as in the uniaxial case [compare with equation (3)]. The off-diagonal elements (obtained when i + j) represent shear strains, which may be interpreted in terms of the angle change of two directions that are perpendicular in the reference configuration. 

In the cube shown in Fig. 3.5. the tensor components for the strain-stress relationship of a 3D-body can be seen. Neglecting the z-coordinate, the tensor reduces from a 3x4 to a 2x2 matrix. The use of the 3D-rheology for related surface problems is only valid if a 3D-analogue for the relaxation is introduced. This is the only way to learn about the surface state in the absence of ideally elastic behaviour of the adsorption layer. 

Therefore, instead of the irreducible tensor components, a proper combination of the angular momentum operators (Jz, J+, J ) is applicable. However, the expansion coefficients (the potential constants) need to be redefined to account for the proportionality factor a (k,j). A set of equivalent operators is compiled in Table 8.17. It is quite practical to handle the equivalent operators since they can be easily constructed by matrix multiplications with the help of computers. 

The last definition of function / of 9 variables (allowed by symmetry of D) permits to employ the customary tensor (or matrix) descriptions, e.g. the summation convention in component form. This is the reason for using this definition of / in (3.146), (3.147) and other formulae in this book (similar definitions may be used for skew-symmetric tensor and vector and tensor functions [7, 14, 79]). As may be seen from the definition above, the main property of / is (when D is symmetrical and this is just such a case) that is indeed symmetrical, e.g. 

A metric tensor with matrix 9pq is obviously symmetrical and regular (this last assertion is necessary and sufficient for the linear independence of gp in the basis of k orthonormal vectors in this space, we obtain det g , as a product of two determinants first of them having the rows and second one having the columns formed from Cartesian components of gp and gq. Because of the linear independence of these k vectors, every determinant and therefore also det g , is nonzero and conversely). Contravariant components gP of the metric tensor are defined by inversion... 

The use of a spherical basis for the representation of tensor components facilitates the application of (tensor) operator techniques to the calculation of matrix elements in atomic spectroscopy. 

We note for later applications of this formula that, k in (54) will be restricted to the values 1 and 0. Also, the operators u operate in different spaces, e.g., the orbital or the spin part of a wavefunction. Note, furthermore, that the reduced matrix element on the right-hand side of (54) contains only a product of tensor components and not a tensor product. 

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. 

For the tensor components pi of the density matrix this yields... 

Table 3. Symmetry Properties of Excitation Amplitudes, Density Matrix Elements, and Spherical Tensor Components of the Density Matrix for Difierently Chosen z Axis...

In this way the angular distribution is determined by three types of quantities pfc, R, and F. The tensor components characterize the excited atom, each yielding a contribution to the electron intensity that is proportional to [Pg.382]

If the autoionization process only depends on L rather than on / the excited atom has to be characterized by the L-dependent tensor components PkqiL) in place of the /-dependent ones. This makes sense anyway, since the excitation processes in heavy particle collisions are mostly spin independent. Therefore the spin can be regarded as spectator during the excitation process, and at time t = 0, when the excitation takes place, the density matrix factorizes into an L-dependent part and a spin-dependent one which is diagonal. The pkq L,T = 0) therefore characterize the excitation process and are the quantities that one would like to know in order to obtain information about excitation mechanisms. 

The formulas (40) or (48) for the description of correlated angular distributions can also be used for the uncorrelated ones by only taking the appropriate values for the tensor components p, (L) with 9 = 0. As was discussed in Section 6.1 the density matrix is diagonal for the case of noncoincident measurements and the pkq vanish for 9 0, if the projectile beam direction is taken as quantization axis. Then equations (40) and (48) can simply be written as... 

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