Kramers multiplets

Any set of energetically well-isolated levels can be described by an effective spin Hamiltonian operator by choosing S to match the corresponding number of levels. This can be just one isolated Kramers doublet of a high-spin multiplet if the... 

The application of a magnetic field to the wavefunctions obtained by the procedure described in the previous sections results in the complete removal of the degeneracy of the / multiplet, either pertaining to Kramers or non-Kramers ions, and yields a temperature-dependent population of the different 2/ + 1 components (Figure 1.2) Thus, at low temperatures, large deviations from the Curie law are observed. The effect of the magnetic field is described by the Zeeman Hamiltonian ... 

If the Ln3+ centre is a Kramers ion, the spectra can be interpreted in terms of a doublet with largely anisotropic effective -values. If one neglects the admixture of higher lying/ multiplets and considers an axial symmetry, the effective g values will be... 

In lanthanide complexes, the ligand field created by the surrounding ligands will split the atomic /-multiplets into several components. The latter are doubly degenerate (Kramers doublets (KDs)) for systems with odd number of electrons and non-degenerate (in the absence of symmetry) for systems with even number of electrons. 

Table 6.2 Energies of the lowest spin-free states originating from the SH multiplet and the energies of the low-lying Kramers doublets of the DyZn3 complex.

Atomic multiplet Spin free states J multiplet Kramers Doublets... 

The inclusion of the crystal field destroys the rotational symmetry of the ion and lifts the degeneracy of J levels (except of course Kramer s degeneracy) the only good quantum numbers will be T s, the irreducible representations of the point-group symmetry operation. If the crystal field interaction is comparable to J-J splitting (and we see from Table 2 that this is the case of actinides) it will also cause an admixture of different J multiplets. 

With a weaker axial CF the ground multiplet is quasidegenerate, 7T, r2, with the linear Zeeman coefficient Zz = gzMj = 1.71. This is followed by a true Kramers doublet r5 separated by 5 = - 3D. The nonmagnetic singlet /3 lies at 513 = -4D above the ground multiplet. 

ForNd3+ion, J = 9/2 and hence the ground state multiplet is tenfold degenerate. Under the influence of a crystal field of hexagonal symmetry, the degeneracy is partially removed. Five Kramer s doublets are formed with the eigenstates, 1/2),... 

There are six (6) Stark States evident here, each having a half-width of about 7 to 9 A. These have arisen because of the crystal field splitting, but more importantly represent transitioiis between Stark states of the Il5/2 and Ss/2 multiplets. This ion belongs to the so-called "Kramers Ions which consist of atoms having an "odd", i.e.- uneven ("u") number of electrons. 

The degeneracies are given in parentheses. Multiplets of Kramers ions are at least twofold degenerate. Each CEF state can be given as a linear combination of free-ion J,M) states. For Ce (7= ) one has for the Kramers doublet and quartet (cubic point group),... 

As discussed in [22], the spherical symmetry of is destroyed when these ions are situated in solids, so that a multiplet term level can be split up to 2/ + 1 crystal field levels for a non-Kramers ion. Due to the parity selection mle for pure electronic transitions in solids, the 41 (i) 4 (f) transition between states i and f is ED forbidden to first order. Parity describes the inversion behavior of the wavefunction of an electronic orbital, so that s,d... orbitals have even parity whereas p,f... orbitals are odd. The spectral feature representing the pure electronic transition is termed the electronic origin or the zero phonon line. An ED transition requires a change in orbital parity because the transition dipole operator (pe) is odd, and the overall parity for the nonzero integral involving the Einstein coefficient of spontaneous emission, A(ED) ... 

The ESR measurements on lanthanide ions in Lap3 indicate six magnetically inequivalent sites having Cj/, symmetry or lower [Baker and Rubins (1961)]. This result is not in agreement with the NMR measurements of Andersson and Proctor (1968) which indicate only three inequivalent sites (see section 2.3.2). In any event, each of the six ESR sites is described by identical spin Hamiltonian parameters, the only difference being in the orientation of the principal axes of the g-tensor with respect to the crystalline c axis. Because of the low symmetry of the CEF, one expects that all degeneracy in the ground state multiplet will be completely lifted in the case of non-Kramers ions so that in these cases ESR will not be detectable [Schulz and Jeffries (1966)]. The measured g-tensor components are listed in table 18.28. 

On the Structure of Multiplet 5-States in Diatomic Molecules. II H. A. Kramers... 

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