Selection rules time-reversal

According to the time-reversal selection rules, time-odd interactions with a magnetic field will be based on the symmetrized square. The spin-operator of the Zeeman Hamiltonian transforms as Tig, which is indeed included in the symmetrized square. The present case is, however, special since the Tig irrep occurs twice in the product. The multiplicity separation cannot be achieved on the basis of symmetrization since both Tig irreps appear in the symmetrized part. One way to distinguish the two products is through the subduction process from spherical symmetry. The addition rules of angular momenta give rise to... 

Bradley s (1966) work has removed the uncertainties about the subgroup method. The only comprehensive alternative seems to be a method described by Birman (1962, 1963) that uses the full group G. The effect of time-reversal symmetry on selection rules in crystals has been described by Lax (1962, 1965). 

The truly remarkable feature of these selection rules is that they only depend on the time-reversal and scalar character of the one-electron operator. This results in very strict quantum conditions which are of paramount importance for spectroscopy and magnetism of the (t2g )3 systems - as we intend to show in the next sections. 

The s.o.c. operator is a one-electron operator which is even under time reversal, and non-totally symmetric in spin and orbit space. The trace of the spin-orbit coupling matrix for the t2g-shell thus vanishes. As a result the s.o.c. operator is found to transform as the MK = 0 component of a pure quasi-spin triplet (Cf. Eq. 26). Application of the selection rule in Eq. 28 shows that allowed matrix elements must involve a change of one unit in quasi-spin character, i.e. AQ = 1. Since 4S and 2D are both quasi-spin singlets while 2P is a quasispin triplet, s.o.c. interactions will be as follows ... 

The interaction is time-even (K = 1). Hence (from the time-reversal selection rules) within a manifold with Q = g, K G [gX Q QKq an the requirement that QKQ = +1 the diagonal matrix elements will vanish if the interaction is not a totally symmetric scalar in spin and orbital angular momenta. 

In a very similar vein to the time reversal selection rule as stated by Stedman (see Ref. [12 page 97]), the Ceulemans selection rules for half-filled shell states are a set of very simple, broad statements with remarkably wide ranging application and influence. 

Let us use these selection rules for investigating the main features of the Rayleigh and pure rotational Raman scattering by spherical-top molecules in the lowest vibronic states (Ogurtsov et al., 1978). The polarizability tensors dif2i and can be expanded into components of irreducible tensor operators that in cubic groups transform as E, T2, and T, respectively. Here the behavior of the operators dir 71 with respect to time reversal 0 has to be taken into consideration. To do this, we use the explicit form of the operator djj(a>) in Cartesian coordinates ... 

It is well established that the principal results of the generalized kinetic theory, especially the functional form of the slow portion of the memory function, can be obtained also by a direct mode-coupling approach [18, 19, 20]. The basic idea behind the mode-coupling theory is that the fluctuation of a given dynamical variable decays, at intermediate and long times, predominantly into pairs of hydrodynamic modes associated with quasi-conserved dynamical variables. The possible decay channels of a fluctuation are determined by selection rules based, for example, on time-reversal symmetry or on physical considerations. 

The coupling coefficient on the right-hand side of Eq- (6.57) restricts the symmetry of the nuclear displacements to the direct square of the irrep of the electronic wave-function. This selection rule is made even more stringent by time-reversal symmetry. The Hamiltonian is based on displacement of nuclear charges, and not on momenta, so as an operator it is time-even or real. For spatially-degenerate irreps, which are... 

For complex variables, variable and derivative have complex-conjugate transformation properties. The general time-reversal selection rules are discussed in Sect. 7.6. 

The argument used in Eq. (7.59) can be generalized to describe selection rules that depend on time reversal [12]. We first introduce two parities, r and n], which describe the time-dependence of the state and of 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 ... 

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