Quasi-spin

In 1957, Bardeen, Cooper and Schriffer proposed a microscopic theory of superconductivity [2], This theory was isotropic (s-wave pairing). In 1958, Bogoliubov [3] and Valatin [4] introduced a transformation that made the treatment of superconductivity simpler (quasi-particle transformation). Still in 1958, Anderson [5] addressed the same problem by introducing the algebra of SU(2) to describe the properties of the system (quasi-spin algebra). In the same year, Bohr, Mottelson and Pines applied BCS theory to... 

Further exploration of the internal symmetries of the Kf multiplets introduces us to the concept of quasi-spin. For a given / shell one defines an operator 1 with components M+,, lz as given in Eq. 17. 

These operators obey commutation rules which are identical to the commutation rules for angular momentum or spin operators, hence the name quasi-spin... 

The scalar product of the resultant quasi-spin vector 2 is given by ... 

Focusing now our attention to our set of (t2g )3 states we observe that 2D and S are the MQ = 0 components of a quasi-spin singlet while 2P is the Mc = 0 component of a quasi-spin triplet. The full quantum structure of the (t2g)3 multiplets thus involves seven labels QMQSLTMsMry, albeit the Me label is redundant since all states share the same MQ value. 

The quasi-spin label is perhaps the most exotic quantum characteristic of these states. It is in any case a true shell characteristic since it extends over several configurations. Most importantly it gives rise to very strong selection rules as we will demonstrate in the next section. 

In this expression the F operator is broken up into its irreducible quasi-spin parts. The behavior of F under time reversal thereby proves to be the determining factor. Two cases are possible. 

The first term in this expression is proportional to the trace of the F matrix. This is a scalar and therefore has the three quantum numbers Q, S, L equal to zero. The second term is clearly symmetric under exchange of the associated mq partners in the operator part. It thus corresponds to a quasi-spin triplet. 

Clearly this sum is made up of antisymmetric terms only. It thus corresponds to an irreducible quasi-spin singlet. 

Knowing the quasi-spin properties of F we can now turn our attention to the associated selection rules. According to the Wigner-Eckart theorem acting in quasi-spin space the selection of an interaction element of an operator K0 ... 

Finally it should also be mentioned that the quasi-spin treatment can be extended to the two-particle Coulomb interaction. This will not be considered here since in the case of the (t2g )3 multiplets the quasi-spin characteristics of the Coulomb operator do not give rise to additional selection rules. 

The first coordination sphere of acido-pentamine [Cr(NH3)5X]2+ complexes has C4v symmetry and this leads to a tetragonal resolution of the 2Eg state into 2At and 1Bl components. The classical ligand field model predicts that these components will be virtually degenerate. This is based on a combination of pseudo-spherical and quasi-spin selection rules of the shell and will be discussed later on in Sect. 5.2. At present we welcome this example of a pseudo-degeneracy as another opportunity to observe fine details of the interelectronic repulsion interaction which have the proper anisotropy to induce a splitting of the 2Eg term. 

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 ... 

Quasi-Spin and Pseudo-Cylindrical Selection Rules... 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

In the perspective of shell-theoretic selection rules the two types of emission clearly expose a difference in quasi-spin characteristics. The -emitting levels... 

One thus would predict that such complexes are characterized by a pronounced orthorhombic perturbation of the emitting state, which thereby loses its parent quasi-spin structure. Possibly this is the most efficient mechanism to make an emitting doublet state with a hole in the tlg shell. Interesting photophysical and even photochemical properties might result. In any case such complexes should constitute sensitive probes for the phase-coupling ligand properties. 

Further properties of the g shell can be explored by introducing the notion of quasi-spin <2, in analog to its use in atomic shell theory [12]. We define... 

The eigenvalue of Qz is N — 2 for a state of gN. We can now consider that the identical components of g1 and g together form a quasi-spin tensor of rank 1/ 2, whose array of ranks we can now indicate by writing G(l - - -]. The e, operators can be broken down into parts that have well-defined quasi-spin ranks however, it turns out that e2 is a quasi-spin scalar, which can be used to explain some similar matrix elements of e2 in g 2 and g 4 [10]. 

We should also mention that other symmetries can be produced by interchanging spin and quasi-spin, an operation that has been referred to as complementarity [12]. 

All the neutral single donors without d or f electrons have spin 1/2 while the double donors and acceptors have spin 0 in the ground state, but in some excited states, they have spin 1 and optically forbidden transitions between the singlet and triplet states have been observed. The spins of the neutral acceptors in the ground state depend on the electronic degeneracy of the VB at its maximum. For silicon, the threefold degeneracy of the valence band results in a quasi spin 3/2 of the acceptor ground state. 

A similar analysis can be carried out for The various contributions to T " require 7 operators z,- (i = 5, 6,. .., 11) a combined operator z that belongs to <110 0>(110)(11) but for which the quasi-spin rank K (see section 11.1) is a mixture of 0, 1, and 2 and an operator z that possesses matrix elements within a given configuration 4f proportional to those of the ordinary spin-orbit interaction Wjo given in eq. (32). The last can be dropped. The eight remaining, taken with the four from were independently parameterized and used to fit the levels of Pr 111 4P as well as those of Nd 4f and Er 4f" in LaClj (Crosswhite et al. 1968). Mean errors of the order of only lOcm were obtained. 

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