Spin-orbit coupling time-reversal symmetry

Due to the inclusion of the spin-orbit coupling there is no need to distinguish between rotations in real and spin space. As can be seen in Table 5.2, the operations c r can be accompanied by the time reversal T. Because the operator T reverses the direction of the magnetic moments, time reversal T itself is of course no symmetry operation. 

In the non-relativistic domain one-electron operators can be classified as triplet and singlet operators, depending on whether they contain spin operators or not. In the relativistic domain the spin-orbit interaction leads to an intimate coupling of the spin and spatial degrees of freedom, and spin symmetry is therefore lost. It can to some extent be replaced by time-reversal symmetry. We may choose the orbital basis generating the matrix of Hx to be a Kramers paired basis, that is each orbital j/p comes with the Kramers partner = generated by the action of the time-reversal operator We can then replace the summation over individual orbitals in (178) by a summation over Kramers pairs which leads to the form... 

Term Hj, (spin-orbit coupling) is equal to zero for symmetry reasons (for the ground state). In the Darwin term, the nucleus-electron vs electron-electron contrihution have reversed magnitudes about 1 10, as compared to 100 1 in retardation. Again, this time it seems intuitively correct. We have the sum of the particle-particle terms in the Hamiltonian =... 

For completeness the phonon modulation of the spin-orbit coupling should also be mentioned as a possible source of spin-lattice relaxation. However, the spin-orbit coupling is weak in polymers that contain only light atoms. Furthermore, in ideal 1-D systems this relaxation route is forbidden by time reversal and inversion symmetry. It was suggested by Soda et al. [19] that in pseudo-one-dimensional systems this selection rule can be overcome by interchain hopping, in which case the relaxation rate becomes proportional to the inverse interchain transfer integral /x ... 

The principal properties of the energy as a function of k are as follows. Within the Brillouin zone E(k) is a continuous function. It is, of course, a multiple-valued function in the reduced zone scheme. At a Brillouin zone plane the gradient of (k) must be in the plane, except in certain exceptional cases. Finally, the band structure must be symmetric under inversion, k — k. This is usually referred to as time-reversal symmetry, but for simple Hamiltonians without spin-orbit coupling it follows simply from complex conjugation of Schrodinger s equation. 

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