Irreducible tensors second order

Although the rate-of-strain tensor is irreducible, the antisymmetric vorticity tensor is not. A well-known result from tensor calculus3 is that an arbitrary second-order tensor M can... 

Here, y is the unique symmetric and traceless (irreducible) part of Q, whereas D(i) are symmetric and traceless second-order (irreducible) tensors, and v(i) and S are vectors and a scalar. It may be noted that, if we write a general third-order tensor in the form (8 33), there is no loss of generality in assuming that the second-order tensor components are symmetric. The antisymmetric part of any second-order tensor can always be represented by a vector. For example, if D = D,s + D" then the antisymmetric part can always be written as D"= e d where d = — e D" and included in the vector terms of (8-33). 

It is convenient to represent the SSC Hamiltonian in the form of a product of two second-rank irreducible tensor operators in order to take an advantage of using symmetry of a spin [106]. 

The evaluation of the SSC matrix elements is analogous to the treatment of the SOC matrix elements with exception that the SSC Hamiltonian of (49) is a product of two irreducible second-rank tensors operators. Because SSC has nonvanishing contribution in the first order, it is usually sufficient to neglect contributions from states of different multiplicities. Application of the WET theorem to SSC Hamiltonian of (49) reads (S = S ) ... 

Tensor representations, synonymous for product representations and their decomposition into irreducible constituents, are useful concepts for the treatment of several problems in spectroscopy. Important examples are the classification of the electronic states in atoms and the derivation of selection rules for infrared absorption or the vibrational Raman or hyper-Raman effect in crystals. In the first case the goal is to reduce tensors which are defined as products of one-particle wave functions, while in the second case tensors for the dipole moment, the electric susceptibility or the susceptibilities of higher orders have to be reduced according to the irreducible representations of the relevant point groups. 

Although the irreducible spherical tensor approach is valuable fix the definition of the (xdo ing tensors and for their manipulation under rotation it does not always provide a ready undostanding of the physical significance of the various components. This is sometimes available from a Cartesian representation of the ordering tensor. The most familiar example is the Saupe ordering matrix which represents the orientational ordering at the second rank level [8]. It is defined by... 

Here the bar indicates an ensemble average and /(cu) is the normalized singlet orientational distribution function for finding the director at an orientation co in a molecular frame. Of this infinite set only the second rank quantities 2, can readily be measured and these five components, corresponding to m =0, 1, 2, form the ordering tensor which is the irreducible analogue of the more familiar Saupe ordering matrix. This matrix is defined by... 

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