Crystal tensor operators

When the matrix elements are calculated for states built from /-electron configurations it is always found that the constants A% (these quantities are related to the strength of crystal field) always occur with (the sharp brackets denote integration with respect to 4/ radial function). A parameters play an important role in crystal field calculations and can be used as parameters in describing the crystal field. For the lowest L S J state they can easily be determined by using the operator equivalent technique of Elliott and Stevens [545—547] and with the help of existing tables of matrix elements. Wybotjbne [548], however, feels that a better approach is to expand Vc in terms of the tensor operators,, as... 

In this section, we follow the symmetry-adapted approach put forward by Acevedo et al. [10], and introduce the vibronic crystal coupling constants Av y(i, t), the tensor operators 0 (Txr i, t) and the general symmetry-adapted coefficients to give a master formula to evaluate the relevant reduced matrix elements as given below ... 

Lanthanide complexes with axial symmetry (i.e., possessing at least a threefold axis, see sect. 2.4.2) are exclusively considered because the principal magnetic z axis coincides with the molecular symmetry axis (Forsberg et al., 1995) and the c 2 spherical tensor operators do not contribute to the crystal-field potentials (Gorller-Walrand and Binne-mans, 1996). The rhombic term of Bleaney s approach V6B Hi (eqs. (42), (46)) thus vanishes and the crystal-field independent methods (eqs. (51), (53)) can be used without complications. 

The C are tensor operators, whose matrix elements again can be calculated exactly, whereas the crystal-field parameters Bk are regarded as adjustable parameters. The number of parameters for this potential is greatly reduced by the parity and triangular selection rules and finally by the point symmetry for the f-element ion in the crystal. Detailed information about the crystal-field potential has been given for example by Gorller-Walrand and Binnemans (1996). 

Here Bk s stand for the crystal field parameters (CFP), and Ck(m) are one-electron spherical tensor operators acting on the angular coordinates of the mth electron. Here and in what follows the Wyboume notation (Newman and Ng, 2000) is used. Other possible definitions of CFP and operators (e.g. Stevens conventions) and relations between them are dealt with in a series of papers by Rudowicz (1985, 2000,2004 and references therein). Usually, the Bq s are treated as empirical parameters to be determined from fitting of the calculated energy levels to the experimental ones. The number of non-zero CFP depends on the symmetry of the RE3+ environment and increases with lowering the symmetry (up to 27 for the monoclinic symmetry), the determination of which is non-trivial (Cowan, 1981). As a result, in the literature there quite different sets of CFP for the same ion in the same host can be found (Rudowicz and Qin, 2004). 

The crystal field interaction can be treated approximately as a point charge perturbation on the free-ion energy states, which have eigenfunctions constructed with the spherical harmonic functions, therefore, the effective operators of crystal field interaction may be defined with the tensor operators of the spherical harmonics Ck). Following Wyboume s formalism (Wyboume, 1965), the crystal field potential may be defined by ... 

Crystal field energies by the tensor operator method We may expand the potential in terms of the tensor operators Cq to give... 

Although we have insured the correctness of our crystal field matrix elements by calculating them by two different methods (determinantal and tensor operators), there are three interesting checks that one can make to insure that there are no errors in the calculated crystal field matrix elements3. 

In calculations involving higher J multiplets, matrix elements of the crystal field Hamiltonian between states belonging to different J multiplets are needed. Although these can be calculated by the method of operator equivalents extended to elements non-diagonal in J, it is convenient to use a more general approach, utilizing Racah s tensor operator technique (26). In this method the crystal field interaction may be written as... 

Using the operator equivalent technique of Stevens (32), and following Elliott and Stevens (33), one may also express the splittings in terms of the better known A parameters (c.f. also Abragam and Bleaney (34)), which are in turn simply related to the Bk parameters which occur in the tensor operator treatment of crystal field splittings. Using the relationship given by Wyboume (35), the crystal field and Bk parameters are found to be connected by the expressions... 

Similar to the ODF for texture, SODF can be subjected to a Fourier analysis by using generalized spherical harmonics. However, there are three important differences. The first is that in place of one distribution (ODF), six SODFs are analyzed simultaneously. The components of the strain, or the stress tensor can be used for analysis in the sample or in the crystal reference system. The second difference concerns the invariance to the crystal and the sample symmetry operations. The ODF is invariant to both crystal and sample symmetry operations. By contrast, the six SODFs in the sample reference system are invariant to the crystal symmetry operations but they transform similarly to Equation (65) if the sample reference system is replaced by an equivalent one. Inversely, the SODFs in the crystal reference system transform like Equation (65) if an equivalent one replaces this system and remain invariant to any rotation of the sample reference system. Consequently, for the spherical harmonics coefficients of the SODF one expects selection rules different from those of the ODF. As the third difference, the average over the crystallites in reflection (83) is structurally different from Equations (5)+ (11). In Equation (83) the products of the SODFs with the ODF are integrated, which, in comparison with Equation (5), entails a supplementary difficulty. 

A Spectroscopic Constants, Coefficients, and Matrix Elements B Irreducible Tensors and Tensor Operators C Classification of Crystal-Eield Terms and Multiplets D Calculated Energy Levels and Magnetic Parameters References... 

In tensor operator notation the crystal-field interaction is written... 

We have already seen that the crystal field potential can be rather simply expressed through a set of radial parameters and the irreducible tensor operators Tk m which refer to a certain, symmetry-predetermined combination of spherical harmonic functions (see Table 8.12). Such an expression can be written in several equivalent forms... 

In these formulae the symmetry adaptation coefficients, 3/- and 6y-symbols occur along with the set of crystal field parameters Dq, Ds, Dt, Da and Dt. In addition, there are the reduced matrix elements of the n-electron unit tensor operators, evaluable with the help of the coefficients of fractional parentage (dn lvSL dnvSL) as follows... 

The various mechanisms of mixing are thoroughly discussed by Wybourne (2). One of the most important mechanisms responsible for the mixing is the coupling of states of opposite parity by way of the odd terms in the crystal field expansion of the perturbation potential V, provided by the crystal environment about the ion of interest. The expansion is done in terms of spherical harmonics or tensor operators that transform like spherical harmonics. This can be formulated in a general Eq. (1)... 

The intensity parameters are appropriate for examining the coalesced transitions from level to level of the type J - J. The appearance of any fine structure (or, to use a more appropriate term, crystal fine structure) corresponding to transitions from sublevel to sublevel of the type n n makes it possible to extract more information about the nature of the transitions. All work in that area stems from the initial study of Axe (1963), which has been briefly mentioned in section 5.4.2. In order to appreciate the later developments, we need to recast the theory of section 5.4.2 in terms of tensor operators. We regard the even-rank tensors F as arising from the inner part of expression (71) when closure is used to remove the intermediate states Xr and when (for odd k) and D are coupled to rank t. In the electrostatic model the part is implicitly multiplied by Ff,(6>,0) of section 5.3... 

These states do not have integer values of quantum numbers L, S and/, which necessitates the use of operator techniques for a complete description of the free-ion and crystal-field interactions. An elegant description using tensor operators (see e.g. Judd 1963) is expressed as... 

The averaging of nuclear spin Hamiltonians under rotations may be easily studied when it is expressed in terms of irreducible spherical tensor operators, TL,m and In liquid crystals, the main interest is in time averag-... 

A Fermi hyperfine constant for a nucleus i bJ crystal-field parameters of rank k spherical tensor operators of rank k Cj Bleaney s factor of lanthanide j (scaled to -100... 

The orientational dependence of the crystal-field Hamiltonian V is obtained by considering the properties under rotation of the tensor cf and therefore also the tensor bf Since the terms are the components of an irreducible tensor operator of rank k, they satisfy the same commutation rule with respect to the angular momentum J as the... 

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