Tensor triple

The triple product of three noncolinear line elements in the reference configuration provides a material element of volume dV. Another well-known theorem in tensor analysis provides a relation with the corresponding element of volume dv in the current spatial configuration... 

Anticommutation relations (14.19) can be rewritten for the components of triple tensors ... 

For triple tensors a we shall express the irreducible components of quasispin operator (15.39)—(15.41) as follows ... 

In general, the double tensor W[see (14.30)] is related to the triple tensor... 

This commutator is a generalization of (14.48)-(14.50). The commutation relations between the components of two triple tensors W KkK have the form... 

Bilinear combinations composed of creation and annihilation operators can be expressed in terms of triple tensors W KkK (see (15.59) and (15.61)). In this case, these tensors are generators of the Rsi+4 group, and relationship (15.64) determines the completeness condition for the generator set of this group in relation to the commutation operation. Out of this set we shall single out two subsets of operators - and Each of... 

The ranks of triple tensors in (15.66) can be selected so that the left side contains operators with known eigenvalues. Then in the right side, we have the desired linear combinations of irreducible products of triple tensors. 

Let us now return to the Casimir operators for groups Spy+2, SU21+1, R21+1, which can also be expressed in terms of linear combinations of irreducible tensorial products of triple tensors WiKkK To this end, we insert into the scalar products of operators Uk (or Vkl), their expressions in terms of triple tensors (15.60) and then expand the direct product in terms of irreducible components in quasispin space. As a result, we arrive at... 

We have established above that one-electron operators are expressible in terms of tensors W(kK) related to triple tensors W KkK by (15.59). Therefore, we shall find here the expansion in terms of irreducible tensors in quasispin space only for the two-particle operator that is a scalar in the total momentum. 

In Chapter 14 we derived, in the second-quantization representation, two different forms of the expressions for that operator - (14.61) and (14.63). To begin with, we consider expression (14.63) in which we, by (15.49), go over to triple tensors. Then, after some transformations and coupling the momenta in quasispin space, we arrive at... 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

Specifically, this relationship holds for the electron creation operator a(jfj which, by (15.49), is a component v = 1/2 of the triple tensor 0. If we take into consideration that the submatrix element of the creation operator is expressed in terms of the CFP from (15.21), we have... 

The operators of orbital, spin and quasispin angular momenta of the two-shell configuration are expressed in terms of sums of one-shell triple tensors (15.52) ... 

Out of the operators (18.22) we can compose the following triple irreducible tensors ... 

We write creation and annihilation operators for a state 1/1) as a and aA, so that ) = a lO). We use the spin-orbital 2jm symbols of the relevant spin-orbital group G as the metric components to raise and lower indices gAA = (AA) and gAA = (/Li)3. If the group G is the symmetry group of an ion whose levels are split by ligand fields, the relevant irrep A of G (the main label within A) will contain precisely the states in the subshell, the degenerate set of partners. For example, in Ref. [10] G = O and A = f2. In the triple tensor notation X of Judd our notation corresponds to X = x( )k if G is a product spin-space group if spin-orbit interaction is included to couple these spaces, A will be an irrep appearing in the appropriate Kronecker decomposition of x( )k. 

The dielectric tensor describes the linear response of a material to an electric field. In many experiments, and particularly in optical rheometry, anisotropy in is the object of measurement. This anisotropy is manifested as birefringence and dichroism, two quantities that will be discussed in detail in Chapter 2. The nonlinear terms are responsible for such effects as second harmonic generation, electro-optic activity, and frequency tripling. These phenomena occur when certain criteria are met in the material properties, and at high values of field strength. 

The SHF frequencies provide information on the one-body spin-orbit and spin-spin terms. If a doublet structure with a small splitting is indeed observed in the two-laser microwave triple resonance experiment, it confirms the cancellation of the scalar and tensor spin-spin terms as predicted by theory. The observed difference of gHF — shf s then rather insensitive to the magnetic moment of the antiproton. An observation of the suppressed transition z/gHF (see fig. 2) or a direct measurement of gHF or gHF, however, could reveal information on which is so far only known to 3 x 10-3 from X-ray measurements of antiprotonic Pb [14]. 

As is to be expected from non-dipolar 2D NLO-phores, the ir system of simple benzene or naphthalene derivatives is too small to show practically useful NLO responses (Wortmann et al., 1993 Wolff et al., 1997). Especially the perpendicular transition is of quite low intensity as indicated by the smaller double-pointed arrow in Scheme 25. Therefore, the tri- and tetrasubstituted blueprint structures [98]-[100] (Scheme 20) were systematically elongated in one, two or all three directions by the interpolation of phenyl-ethynyl bridges. The use of triple bonds eliminates possible conformational isomerism. The trisubstituted type [124]-[127] is shown. By a combination of EOAM and polarization-dependent EFISHG and HRS, all four independent tensor components were evaluated (Wolff et al., 1997). Results are given in Table 5. The dipole moment ju, lies parallel to the z axis, the y axis within the molecular plane and the x axis perpendicular to the molecular plane. The /3 values (at 1064 nm) are in units of 10 ° C m° V . ... 

If the coupling constants are known in advance, the total mixing time can be reduced in multiple-step selective coherence-transfer experiments by using the selective homonuclear analog of the optimized heteronuclear two-step Hartmann-Hahn transfer technique proposed by Majumdar and Zuiderweg (1995). In this technique [concatenated cross-polarization (CCP)] a doubly selective transfer step (DCP) is concatenated with a triple selective mking step (TCP). For the case of a linear three-spin system with effective planar coupling tensors, a CCP experiment yields complete polarization transfer between the first and the third spin and the total transfer... 

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