Kramers degeneracy theorem

Kramers degeneracy theorem states that the energy levels of systems with an odd number of electrons remain at least doubly degenerate in the presence of purely electric fields (i.e. no magnetic fields). This is a consequence of the time-reversal invariance of electric fields, and follows from an application of the antiunitary T-operator to the wavefunction of an odd number of electrons [51]. 

The JT effect is based on the following theorem If a non-linear molecule (or polyatomic ion) has a degenerate electronic level (apart from Kramers degeneracy) it is unstable with respect to displacements of the atoms. 

A free atom or an ion possesses spherical symmetry and each energy level is (2J+1)-fold degenerate. If this ion is placed in a crystal, each J level splits because of the electric field produced by this new environment. In crystals or compounds, the environment about the f ion possesses a well-defined symmetry (lower than spherical) and the splittings of the various J levels depends on the point symmetry of the site of the f ion. Table 3 shows the maximum number of states for various f ions and their LS terms [17]. For an ion with n (of f ) odd, Kramers theorem states that there remains a two fold degeneracy of the energy states which cannot be removed by the crystal field. This Kramers degeneracy can be split by the application of a magnetic field. 

It is seen from the values of the degeneracy number e that Kramers theorem that an even number of independent wave functions always participates in the energy level of a system with an odd number of electrons is valid. Kramers degeneracy can only... 

Exercise 13.4-6 It was shown in Exercise 13.4-5 that the dimension / of representations of type (c) is an even integer. Therefore, even though time reversal introduces no new degeneracies, l is always at least 2 and Kramers theorem is satisfied. 

For this reason, ESR is much more easily done in Kramers ions which have an odd number of electrons and at least a two fold degeneracy in the absence of an applied magnetic field (Kramers theorem). This restricts one to those paramagnetic iron materials which have half integer spin (1/2, 3/2, 5/2) and eliminates those iron materials with spins of 0, 1, 2 unless special efforts are made which we won t discuss here. 

The (3d) system (or the equivalent d hole) is subject to Kramers theorem which states that when a system is composed of an odd number of electrons (or holes) it is not possible for electric fields to remove degeneracies completely — at least two-fold degeneracies must remain. 

It is seen that Ai is no longer six-fold degenerate but is split into three doublets which have different energies. The states with -1- Ms and — Ms are still degenerate and, according to Kramers theorem, there can be no further removal of degeneracies without the use of magnetic fields. It is convenient to define... 

The Jahn-Teller theorem states that degenerate ground states are not possible There will always be a normal coordinate in the point group of the complex or molecule which provides a mechanism for lifting this degeneracy. While the theorem is valid in all symmetries with the exception of linear molecules for orbitaUy degenerate states there is a restriction in the case of in degeneracy in so far as Kramers doublets... 

The primary effect of the crystal field on any manifold of levels of a 4f ion is to raise the 2J + I degeneracy, giving a set of levels with an overall splitting of a few hundred wave-numbers. If the ion has an odd number of electrons, then by Kramers theorem a two-fold degeneracy must remain in each level, but for a non-Kramers ion there is no such restriction. All the degeneracy may be lifted by a field of low... 

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