Matrix crystal field

The intensity of the EPR resonance absorption is a measure of the number of paramagnetic centres present [346], while the type of line observed and the measured g factor are indications of the interactions of the paramagnetic particles and of their distribution within the matrix. Such spectra are much more sensitive to changes in crystal field and atomic orientations than X-ray diffraction and are not dependent upon crystallinity [347]. The nature of the paramagnetic particles may be discerned from the superfine structure of the spectrum. 

Mixing of LS-states by spin orbit coupling will be stronger with an increasing number of f-electrons. As a consequence, intermediate values of Lande g factor and reduced crystal field matrix elements must be used. 

Due to the intermediate coupling the sign of the crystal field matrix element 6 is reversed compared to the pure Russell-Saunders state. Thus for 8-fold cubic coordination a F7 ground state was found. From EPR measurements on Pu3"1" diluted in fluorite host lattices, a magnetic moment at T=0 K can be calculated, ranging from li ff = 1.333 (in Ce02) to y ff = 0.942 (in SrCl2) (24,... 

Accommodation of metal atoms of widely differing ionic radii into the same overall structure creates interesting possibilities for the doping of metal ions into a common matrix for spectroscopic examination under nearly constant crystal field effects. 

Theoretical analyses (75-77) of the matrix-induced changes in the optical spectra of isolated, noble-metal atoms have also been made. The spectra were studied in Ar, Kr, and Xe, and showed a pronounced, reversible-energy shift of the peaks with temperature. The authors discussed the matrix influence in terms of level shift-differences, as well as spin-orbit coupling and crystal-field effects. They concluded that an increase in the matrix temperature enhances the electronic perturbation of the entrapped atom, in contrast to earlier prejudices that the temperature dilation of the surrounding cage moves the properties of the atomic guest towards those of the free atom. 

Crystal field energy levels can be found by diagonalizing the corresponding matrix, which is made up by elements of the type ... 

Alexandrite, the common name for Cr-doped chrysoberyl, is a laser material capable of continuously tunable laser output in the 700-800 nm region. It was established that alexandrite is an intermediate crystal field matrix, thus the non-phonon emitting state is coupled to the 72 relaxed state and behaves as a storage level for the latter. The laser-emitted light is strongly polarized due to its biaxial structure and is characterized by a decay time of 260 ps (Fabeni et al. 1991 Schepler 1984 Suchoki et al. 2002). Two pairs of sharp i -lines are detected connected with Cr " in two different structural positions the first near 680 nm with a decay time of approximately 330 ps is connected with mirror site fluorescence and the second at 690 nm with a much longer decay of approximately 44 ms is connected with inversion symmetry sites (Powell et al. 1985). The group of narrow lines between 640 and 660 nm was connected with an anti-Stokes vibronic sideband of the mirror site fluorescence. 

When the ion is embedded in a matrix, the crystal field splits the ground multiplet and the crystal field levels give rise to the same problem as the free ion multiplets, to be solved in the same way. If several crystal field levels are involved, a complete treatment using Eq. (13) is needed. 

The above equations have been obtained on the assumption that no orbital states have energies close to that of the ground state. This means that they should be applicable to d3, d5, and d8 for crystal fields which are close to octahedral in symmetry. They should be applicable to d4 and d9 also, when the distortion from octahedral symmetry is tetragonal, since in this case matrix elements of are zero between the ground state and the nearby excited state, d2, d6, and d1 in octahedral symmetry must be treated in a manner similar to that used for dl in Sec. III.D. For other crystal-field symmetries, the treatment used depends on whether the crystal field gives low-lying excited states that have nonzero matrix elements of with the ground state. 

He points out that the variation of lifetime with glass matrix is due to at least two causes, the first being the changes in refractive index. If the wave functions of the ion remain essentially the same from host to host, the spontaneous-transition probability will increase with increasing refractive index because of the increase in density of final states. The second cause is configuration mixing of 4/ and 5d states, which must reflect the size and symmetry of the crystal field produced at the ion by the surroundings. 

Tn spite of the conceptual inadequacies of CFT outlined in Section 6.2.1.3, parameterization of spectral and other data within the crystal field framework has been widespread and continues to be so. The calculation of the CFT matrix of the type of equation (4) by the direct methods indicated in their preamble is tedious. Elegant and efficient procedures for performing the evalu-... 

Entirely general analytical expressions for the matrix elements of equation (4) have been listed for the d-orbital case for an almost arbitrary assembly of charges surrounding a metal atom.5,38 They are reproduced in Appendix 1. By implementing these expressions as a computer program the problem of calculating the d-orbital energies in the crystal field model for any ordinary stereochemistry is made trivial. 

There is still another type of internal solid state reaction which we will discuss and it is electrochemical in nature. It occurs when an electrical current flows through a mixed conductor in which the point defect disorder changes in such a way that the transference of electronic charge carriers predominates in one part of the crystal, while the transference of ionic charge carriers predominates in another part of it. Obviously, in the transition zone (junction) a (electrochemical) solid state reaction must occur. It leads to an internal decomposition of the matrix crystal if the driving force (electric field) is sufficiently high. The immobile ionic component is internally precipitated, whereas the mobile ionic component is carried away in the form of electrically charged point defects from the internal reaction zone to one of the electrodes. 

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... 

First, let the unitary transformation diagonalizes the interaction matrix that involves the operators of the electron repulsion, the crystal field, and the... 

Third, the transformation into the crystal-field multiplets is provided by a matrix product... 

In the AOM formalism, a is 4e A, b is 2ezA + 2e B, c is 3effA, and d is eoA + 2e B. Again, since only energy differences are determined spectroscopically, there are only three recoverable degrees of freedom. We denote this by saying that matrix M3 has three spectroscopically independent parameters. In crystal field theory these are defined as... 

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 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). 

---