Crystal quantum number

Due to the so-called /-mixing within the crystal field, multiplets with different / values are coupled. However, similar to the free-ion case, the levels are still designated by the principal 25+1L j component of the crystal-field wavefunction. For the further labeling of levels split by the crystal field, either the irreducible representation /j (Bethe, 1929) to which the particular wavefunction belongs or the crystal quantum number /i defined by Hellwege (1949) are most commonly used. 

Experimentalists use crystal quantum number, p, in place of the irreducible representation T, to identify the experimental energy levels obtained by electric dipole polarization selection rules [322,323]. The correspondence between crystal quantum number p and the irreducible representations T, of D3h is given in Table 8.38 and the D3h electric dipole selection rules in Table 8.39. 

TABLE 8.38 Correspondence between crystal quantum number //, and the irreducible representation T of D3[,. ... 

This situation is best illustrated by an example. Fig. 1 gives a partial energy level diagram for Nd LaCl (Dieke, 1968), which shows the splitting of the Fsp, ln/2, and I9/2 levels (all in cm" ) By the crystal field. The inp and sn levels, which are situated between the 1 3/2 and Iii/2 levels, are not shown. The crystal quantum numbers, i, are also shown these will be explained in section 2.2. The features illustrated in the figure are common to most lanthanide impurity-iop spectra, i.e. sharp levels in well-separated groups. 

Fig. 1. Energy level scheme for Nd LaCl3, showing- the Stark levels of the 19/2, In/2, mid p3/2 terms. The crystal quantum numbers are also shown.

Another alternative is to use the crystal quantum number scheme of Hellwege (1949), which has a more direct connection with the crystal-field Hamiltonian. This Hamiltonian contains parameters Bim, where the allowed values of k and m are determined by the point-group symmetry. Each point group is characterized by an integer parameter q such that allowed values of m are given by... 

The crystal quantum numbers for symmetry are shown in Table 7. 

The condition -M + q + M = 0 determines which non-diagonal matrix elements will occur in the energy matrix for a given symmetry (the -values are constrained by symmetry). The non-diagonal matrix elements are responsible for the fact that M will not remain a good quantum number for a lanthanide ion in a crystal field. The wave function of a crystal-field level will be a sum of states M + mq m is an integer). To classify the M states which can be mixed by the crystal-field Hamiltonian, the crystal quantum number jx has been introduced by Hellwege (1949) ... 

In the definition mod simply means the addition or subtraction of multiples of q to the crystal quantum number p. For q, the minimum gf-value in the crystal-field potential different from zero is taken. The crystal-field quantum numbers for a given symmetry can be found in practice by writing down different series M + mq, so that these series together contain every integer between -/ and / (/ is the highest value for the / quantum number in the system under study). The number with the lowest absolute value in each series will be considered as a crystal quantum number p. This can be illustrated for the C3 symmetry q = 3). The crystal quantum numbers for a C3 symmetry in systems with an even number of electrons are p = 0, 0" and p = l. 

For cubic groups one has to work with irreducible representations. The crystal quantum numbers are not useful to classify the states, because of the occurrence of triplet and quartet states. Quartet states are found in the double groups OJ, O and Tj. 

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