Crystal-field term symbol

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

Inspection of Equation 1.23 and consideration of the properties of 3-y and 6-j symbols confirm that only even A--values contribute to crystal field splitting. Further, it indicates that mixing between levels belonging to different / multiplets can only occur if terms with k site symmetry of the lanthanide, in much the same way as discussed above for the Stevens formalism. 

Figure 3.10 Partial energy level diagram for the Fe3+ or Mn2+ ions with 3tfi configurations in high-spin states in an octahedral crystal field. Only sextet and quartet spectroscopic terms and crystal field states are shown. Note that the same energy level diagram applies to the cations in tetrahedral crystal fields (with g subscripts omitted from the state symbols for the acentric coordination site).

In a weak field, atomic considerations dictate which orbitals are filled, as was the case in the rare earths. Atomic states described by term symbols of total L and S describe the states for d orbital occupation these are spht by the d—d electrostatic repulsions. Spin orbital interaction is small in the transition metals and usually neglected in the first-order description of the levels. A weak crystal field shifts the levels and effects a spfitting which occurs because the crystal field removes the degeneracy of an L level the L level splits into its components. Atomic free ion wave functions having the S5unmetry of the crystal field are used to calculate the splittings. 

In the development of the field of defect chemistry of inorganic compounds various systems of notation have been proposed and used to describe point defects. However, the most widely adopted system is that due to Kroger and Vink (1956) (see also Kroger (1964)), and this will be used in this book. This system describes crystals in terms of structural elements, and an imperfection is indicated by a major symbol describing its chemical content and a subscript that indicates the site that it occupies. 

The symmetry properties of the 6j- symbol limit the even values of rank A to 2,4,6. In addition, due to the triangular conditions for the non-vanishing 3j- symbols that determine the reduced matrix elements of spherical tensors in equation (10.18), the rank t of the crystal field potential operator has to be odd. Indeed, since is even, and/ = 3 for 4/ electrons, the odd parts of the crystal field potential contribute to the transition amplitude, while the terms with even values of t contribute to the energy. 

It is defined as the sum of the spherically averaged crystal potential and an additional confining potential term which is only applied to valence states (denoted by the symbolic writing S v). The atomic XC-field Bg is the spherical average of the crystal XC-field B ",... 

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