Crystal field, definition

Except to note that the occurrence of the coefficients 15 and 10 in 155 and lOD obviate the need for fractions here or elsewhere in crystal-field theory thus, they are there for reasons of convenience and definition only. 

Crystal field theory has its origins in Hans Bethe s famous 1929 paper Splitting of terms in crystals. In that paper Bethe demonstrated what happens to the various states of an ion when it is placed in a crystalline environment of definite symmetry. Later, John Van Vleck showed that the results of that investigation would apply equally well to a transition-metal compound if it could be approximated as a metal ion surrounded by ligands which only interact electrostatically with the... 

Note that, of course, the cubic crystal field splitting dzi, dx2 y2 (eg) at 6Dq and dxz, dyz, dxy at -4Dq is reproduced if Ds and Dt are both zero. Note also that the centre of gravity of neither of the eg nor of the t2g sets is maintained as the symmetry departs from cubic. This means that, in low symmetry, the concept of the cubic field splitting is not clearly defined. For small departures from cubic symmetry the lack of definition is not serious in practice, but to maintain the concept in, say, a square-planar complex, as is sometimes done, requires care. 

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). 

Crystal field theory gives a survey of the effects of electric fields of definite symmetries on an atom in a crystal structure. 

Here Ai are inverse mass parameters in WZ structure, corresponding to Luttinger parameters in ZB structure, me11 and me1 are k-dependent electron effective masses. D , aic and a2c are Bir-Pikus deformation potentials. Ai and 3A2,3 correspond to the crystal-field and spin-orbit splitting energies, respectively. The definition of several operators is given as L+ = (U iLy)/V2, a+ = (ax iay)/2,... 

Today, we do not know the physical reason why there may be more than one asymmetric unit in the unit cell, nor do we have definite answer to the question When is the symmetry of the free molecule reduced by the condensation into a solid and when is this symmetry preserved Typical examples are the substituted 2,4,6-trichloro-l-X-benzenes. The symmetry of the gaseous molecule is preserved in the solid in so far as the Cl atoms in positions 2 and 6 are crystallographically equivalent for X = N02 and X = Br. Sites 2 and 6 of the molecule, however, are inequivalent in the solid for X = I, Cl, OH, and all other compounds for which the NQR spectrum is known. In the vast majority of molecules, the symmetry is reduced by condensation. We shall take the frequency splitting of NQR resonances due to this breakdown of symmetry as a qualitative measure of the crystal field effect. 

It should however be noted that for the lanthanide and actinide series numerous different definitions of the crystal field parameters have on occasion been adopted and care should therefore be taken in comparing quantities from different literature sources. In this review the crystal field parameters are throughout used in the sense defined above, but see for example Hutchings (36) for a survey of some other conventions. 

It should also be noted here that the torsional mode assignment for CH3 CC13 has recently been confirmed in the low-temperature solid phase by Durig and co-workers,19 who have demonstrated in an extensive series of infrared and Raman studies of molecular torsional vibrations that forbidden or weakly active modes can very often be definitively observed in crystal phases due to the lower symmetry of the crystal field. 

With the availability of faster computers, BPTI was simulated in aqueous solution and in a solvated crystal with a more realistic (three-center) water model.92 The simulations were limited to 8 ps of equilibration and 12 ps of analysis, somewhat short for definitive conclusions to be drawn recently, a crystal simulation of BPTI that extended over 40 ps has been reported.322 The average structures obtained from the various simulations are compared in Table VII. In the three calculations made with the same empirical potential, the van der Waals solvent and static crystal field results yielded an average structure closer to the experimental crystal structure than did the vacuum calculation. The full crystal simulations, including crystal waters, gave an average structure still closer to the X-ray result, while the deviation from the crystal structure of the average structure obtained from the aqueous solution simulation was similar to the earlier vacuum result. 

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