Tensor, irreducible

Components of a second-rank tensor Irreducible Cartesian components... 

J. S. Griffith, The Irreducible Tensor Method for Molecular Symmetry Groups, Prentice-Hall, Englewood Cliffs, NJ, 1962, p. 20. 

Here D(Q) = D(a,f, y), Euler angles a, (5 and y being chosen so that the first two coincide with the spherical angles determining orientation e = e(j], a). Using the theorem about transformation of irreducible tensor operators during rotation [23], we find... 

It is noteworthy that dq(e,t) does not satisfy this relation, as equality [J,x, dq] = 2 C q dq+ll (the definition of an irreducible tensor operator) does not hold for it [23]. Integration in (7.18), performed over the spherical angles of vector e, may be completed up to an integral over the full rotational group due to the axial symmetry of the Hamiltonian relative to the field. This, together with (7.19), yields... 

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

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). 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

The components of the translation and rotation vectors are given as Tx> Ty, T and RX Ry, Rz, respectively. The components of the polarizability tensor appear as linear combinations such as axx + (xyy> etc, that have the symmetry of the indicated irreducible representation. 

A complete decomposition of the ab initio computed CF matrix in irreducible tensor operators (ITOs) and in extended Stevens operators. The parameters of the multiplet-specific CF acting on the ground atomic multiplet of lanthanides, and the decomposition of the CASSCF/RASSI wave functions into functions with definite projections of the total angular momentum on the quantization axis are provided. 

The spin operator S is an irreducible tensor of rank one with the following transformational properties... 

Table 1 NMR interactions in irreducible spherical tensor form ...

Addressing first the heteronuclear case (i.e., the DCP experiment introduced by Schaefer and coworkers in 1984), the dipolar coupling may be expressed as in (10) that upon heteronuclear truncation of the transverse terms may be recast in irreducible tensor operator form (7Z = T[ 0, Sz = Tf 0) as... 

T%0 are irreducible tensor operators of rank X. The Hamiltonian of a static powder sample can be easily derived from the MAS Hamiltonian by setting a>r = 0 ... 

The spin interactions, dipole-dipole (D), nuclear electric quadrupole (Q) and chemical shielding (C.S), may be formally written in terms of irreducible tensors of rank l34 in Hz ... 

The coupling tensor Rlm in the laboratory frame is time dependent due to the motions of spin-bearing molecules. It can be expressed in terms of the rotational transformation of the corresponding irreducible components pln in the principal axis system (PAS) to the laboratory frame by... 

The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A. The quantum numbers are conveniently arranged in patterns (or tableaux), called Young tableaux. Tensor representations of Lie algebras are characterized by a set of integers... 

As discussed in Ref. [1], we describe the rotation of the molecule by means of a molecule-fixed axis system xyz defined in terms of Eckart and Sayvetz conditions (see Ref. [1] and references therein). The orientation of the xyz axis system relative to the XYZ system is defined by the three standard Euler angles (6, (j), %) [1]. To simplify equation (4), we must first express the space-fixed dipole moment components (p,x> Mz) in this equation in terms of the components (p. py, p along the molecule-fixed axes. This transformation is most easily done by rewriting the dipole moment components in terms of so-called irreducible spherical tensor operators. In the notation in Ref. [3], the space-fixed irreducible tensor operators are... 

In this picture, the correspondence between irreducible representations of F (except the trivial representation) and irreducible components of the exceptional set becomes concrete. It is realized by the tautological bundles V s. In [66, 5.8], we have shown the correspondence respects the multiplicative structures, one given by the tensor product and one given by the cup product. In fact, using (4.11), we can show that two matrices... 

Ho is the unperturbed Hamiltonian, describing the Zeeman interaction and Hx(t) is the dipole-dipole Hamiltonian, time-dependent through variation in the orientation of the r/s vector. The DD Hamiltonian can conveniently be expressed using scalar contraction of irreducible tensors (16). 

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

If one tries to apply the definition (39) naively to degenerate states, one is faced with the problem that, for example, and have a different transformation behavior with respect to the spin rotation group SU2, hence would have no acceptable transformation behavior at all. The way out of this dilemma is to define the irreducible tensor components of in terms of those of yPf and y. ... 

The forced electric dipole mechanism was treated in detail for the first time by Judd (1962) through the powerful technique of irreducible tensor operators. Two years later it was proposed by Jorgensen and Judd (1964) that an additional mechanism of 4/-4/ transitions, originally referred to as the pseudo-quadrupolar mechanism due to inhomogeneities of the dielectric constant, could be as operative as, or, for some transitions, even more relevant than, the forced electric dipole one. 

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