Crystal field theory matrix elements

All the off-diagonal matrix elements of the spin-orbit coupling in the >, Tl> [ basis are thus reduced by the factor y, and we use the experimentally observed quenching to calculate Ej j and the corresponding geometrical distortion (14). In the Cs2NaYClg host lattice the total spread of the four spin-orbit components of T2 is 32 cm whereas crystal field theory without considering a Jahn-Teller effect predicts a total spread of approximately 107 cm-. ... 

Several methods exist for calculating g values. The use of crystal field wave functions and the standard second order perturbation expressions (22) gives g = 3.665, g = 2.220 and g = 2.116 in contrast to the experimentaf values (at C-band resolution) of g = 2.226 and g 2.053. One possible reason for the d screpancy if the use of jperfXirbation theory where the lowest excited state is only 5000 cm aboye the ground state and the spin-orbit coupling constant is -828 cm. A complete calculation which simultaneously diagonalizes spin orbit and crystal field matrix elements corrects for this source of error, but still gives g 3.473, g = 2.195 and g = 2.125. Clearly, covalent delocalization must also be taken into account. 

The interpretation of Eq. (4-3) is direct and the formula could almost be written down immediately in terms of Fermi s familiar Golden Rule of quantum theory (see, for example, Schiff, 1968, p. 314). The absorption of light arises from the coupling, caused by the electric field, of occupied states with unoccupied states, However the absorption can only occur if the two states differ in energy by hco, so that energy can be conserved if the transition is made and one photon is absorbed. It should also be noted that in a perfect crystal the matrix elements will vanish unless the wave numbers of the coupled states in the Brillouin Zone are the same. That is, the transitions must be between states that are on the same vertical line in a diagram such as Fig. 4-7. We shall return to a more complete description of absorption later. 

Note that in the above treatment it was necessary to evaluate only two integrals of the crystal field potential, one over the d orbital [Eq. (7a)] and the second over the orbital [Eq. (7b)]. This derives from the group theory of high symmetry crystal fields, where for an octahedral or tetrahedral complex each of the d orbital expressions is a good wave function for the molecular Hamiltonian and gives an energy appropriate for all members of the 2s and g sets, respectively. For a lower symmetry complex Eq. (4) must be expanded into a perturbation secular determination [Eq. (9)1 with 25 matrix elements, H , involving all pairwise combina-... 

The Akm can be calculated from a model such as the modified point-charge model presented in section 3.2.4, the Rt l) can be calculated as well, and matrix elements of ju, can be computed between crystal-field split sublevels for a particular lanthanide ion in a particular host crystal a priori, without fitting experimental intensity measurements (Esterowitz et al., 1979a Leavitt and Morrison, 1980). However, this method is not prevalent in the literature rather, usually the theory is expressed in terms of a few adjustable parameters and a fit is made to intensity data. To this end, we consider the line strength, defined by (Condon and Shortley, 1959)... 

Substitution (101) appears to be the simplest way in which spin-dependent phenomena can be incorporated into the theory. The operator (100) commutes with S and is thus consistent with effects produced by such spin-independent interactions as the Coulomb and crystal-field terms in the Hamiltonian. Moreover, the effect of (101) turns out to be equivalent to adding to each reduced matrix element of F " a part (proportional to c,) that involves the reduced matrix element of the double tensor where = s (Judd 1977b). Such double tensors are straightforward to evaluate furthermore the proportionality = /4 holds for all terms of... 

Since the position of the levels is a function of two parameters, no simple relationship can be deduced for the relative position of the crystal-field levels. If Bl is accidentally equal to zero, an equal spacing will be found between the three levels. The degenerate I 2) level will be at the intermediate position in all cases. Because one can determine the Bq parameter from the splitting of a 7 = I level, the experimental position of two of the three crystal-field levels in the 7=2 level is in theory sufficient to determine Bq. The position of the crystal-field levels in the octagonal D4d symmetry ( =8), is the same as in the hexagonal crystal field, due to the absence of non-zero off-diagonal matrix elements. 

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