Crystal field symmetry

The excited state Po is singly degenerate, while the ground state has a degeneracy of 2/ + 1 = 13. Therefore, the maximum number of emission peaks that we could observe around 285 nm would be 13. Of course, this would occur only in crystals leading to a very low symmetry crystal field around the Tm + ions. The actual number of peaks depends on the crystal and, specifically, on the crystal field symmetry around the Tm + ions in this crystal. 

Comparing Eq. (50) to Eq. (47), we find that the contribution of the eg orbitals has been transferred from g to g . This difference demonstrates the sensitivity of the spin Hamiltonian to crystal-field symmetries. 

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. 

In (a) the ion is so situated as to be in a noncentrosymmetric field, even when it is not vibrating. In this case electric-dipole emission is allowed. In (b) there is inversion symmetry when the ion is not vibrating, but vibration carried it to some other point Py at which the center of symmetry is lost. It should be self-evident that, even when the ion is in a noncentrosymmetric environment, vibrations may be important. That is, changes in the crystal-field symmetry induced by the vibronic motion will lead to violations of the crystal-field-selection rules. 

A new treatment for S = 7/2 systems has been undertaken by Rast and coworkers [78, 79]. They assume that in complexes with ligands like DTPA, the crystal field symmetry for Gd3+ produces a static ZFS, and construct a spin Hamiltonian that explicitly considers the random rotational motion of the molecular complex. They identify a magnitude for this static ZFS, called a2, and a correlation time for the rotational motion, called rr. They also construct a dynamic or transient ZFS with a simple correlation function of the form (BT)2 e t/TV. Analyzing the two Hamiltonians (Rast s and HL), it can be shown that at the level of second order, Rast s parameter a2 is exactly equivalent to the parameter A. The method has been applied to the analysis of the frequency dependence of the line width (ABpp) of GdDTPA. These results are compared to a HL treatment by Clarkson et al. in Table 2. 

Symmetry of crystal field Symmetry notation Crystal field states Mineral examples... 

Copper surface, 137 Co-precipitation, 440 method, 312, 388 Core shell structure, 411, 415 Crookes, Sir William, 39 Cronstedt, Axel Frederik, 5 Crystal field symmetry, 371 Crystallographic relationships, 265 ErCuPbSj, 265 ErCuPbSej, 265 Er5CuPb3Sen, 266 Er2EuS4, 267 Er2PbS4, 267 LaCuPbSs, 265 La2S3, 265... 

The zero field splitting parameter D was found to be in the range 0.2 < D < 1 cm by a perturbation theory calculation which assumed trigonal (or tetragonal) distortion of the predominately octahedral crystal field symmetry. It was necessary to include in the calculation a lower symmetry zero field splitting E(S/ — 8 ) in addition to the axially symmetric term in order to explain the spectrum,... 

NMR spectra of Tb in Rb2NaTbF6 also suggest that the crystal field symmetry in the terbium ion site differs from cubic at liquid heliiun temperatures (Tagirov et al. 1979). 

In the closely parallel orientation Hq c small transverse components of the hyperfine field, which induce relaxation transitions, can also occur as a consequence of the local distortion of a crystal field symmetry (e.g., near impurity ions see sect. 3.5). In insufficiently perfect crystals this effect can almost entirely conceal the sharp dip in Tp in the vicinity of 0=0, which apparently is what occurs in LiTmF4 (Aminov et al. 1980). 

Crystal-field symmetry changes with site occupation for Er in LaH ... 

The many-electron states of an atom in a crystal field or a molecule can obviously not be labelled by the IRs of SO(3), since the Hamilton operator, the angular moment operator and therefore also the many-electron wave functions transform according to the IRs of a less symmetric point group. The lower symmetry may also remove the degeneracies of the LS terms. For example, the ground term of a boron atom becomes T u in an octahedral crystal field so that the three fold degeneracy is retained, but splits into two LS terms of and symmetry when the crystal field symmetry is lowered to C v ... 

The matrix element in eq. (143) is valid for transitions between two crystal-field levels. Because of the radial integrals, the calculation of the matrix element is very tedious and can in fact only be done if some approximations are made. Axe (1963) treated the quantities Aiaj3(k,X) in eq. (143) as adjustable parameters. In this expression, X is equal to 2, 4 or 6, and k is restricted to values of Ail. The values of q are determined by crystal-field symmetry constraints and lie between 0 and k. This parametrization scheme was used by Axe for the intensity analysis of the fluorescence spectrum of Eu(C2H5S04)3 9H20. Porcher and Caro (1978) introduced the notation Bxki for the intensity parameters ... 

The interaction of Fe + in Fe,H-ZSM-5 with NH3 and pyridine led to a complete disappearance of the low-field Unes at g2 = 5.65 and g3 = 6.25, and interaction with H2O to their considerable decrease. In any event, the intensity at gi = 4.27 was markedly enhanced. This was especially pronoimced with NHj and pyridine indicating an increase of the crystal field symmetry upon adsorption of these powerfiil Ugands. Interaction with O2 resulted in a considerable but reversible broadening of the Fe + ESR lines caused by dipole-dipole interaction of Fe + with O2. With NHj, the samples of Fe,H-ZSM-5 were reduced at higher temperatures (823 K) as indicated by the disappearance of the signals of Fe + and formation of Fe clusters. Reoxidation did not fiilly restore the original spec-triun. Interaction with p-xylene yielded an ESR spectrum characteristic of p-xylene cation radicals. 

In the case of the Eu(lll) ion, where ground and excited state manifolds are well-separated, this direct dependence of the number of My levels on the crystal field symmetry is often utilised to determine the point group symmetry of the metal ion in a complex or solid state material from the emission spectra. This method of descending symmetry is performed with the help of a diagram such as the one shown in Fig. 1.5 [41]. A similar analysis can also be performed on the basis of absorption spectra. 

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