Fictitious spin

The analytic evaluation of the density matrix requires the diagonalization of Hi, which can be easily performed for the two extreme cases, a>r( Qg. Indeed, in the case of a low rf-field, only off-diagonal terms related to the CT are retained in the H Hamiltonian (39), which thus behaves like a fictitious spin-1/2 operator, affecting only the CT coherences /1 io- These coherences are thus selectively excited with the nutation frequency ... 

The spin-locking and CP behavior of the most commonly used SQ coherence (CT) in quadrupolar nuclei under static and MAS conditions has been described in detail by Vega using the fictitious spin-1/2 approximation [223]. In a static sample, the Hartmann-Hahn matching condition requires that co = nut where co ut is one of the nutation frequencies associated with the SQ coherence of the quadrupolar S spin (see Sect. 2.3.4). In the simple case of on-resonance SQ-CP this translates to [224]... 

See Abragam and Bleaney s book [see Ref. 151, pp. 651ff] for some discussion of the situation for matrix g, wherein it is stated that g is not a true tensor. The essence here is the difference between the concepts of actual spins and fictitious spins. 

Here the first scalar term is the isotropic splitting, and the second traceless tensor term is the anisotropic dipolar splitting. Therefore the complete spin Hamiltonian for a doublet state radical expressed in terms of spin variables only (thus it is for the "fictitious spin" of a system in which spin-orbit coupling is nonexistent) is... 

In this equation we have followed the NMR convention and set the constant fi = 1. This is equivalent to measuring energies in angular frequency units. Employing a suitable set of base functions of the Hilbert space, this equation can be converted into a set of linear differential equations for the matrix elements of p. In the case of a single parr of tunnel levels the Hamiltonian of the two levels with their tunnel splitting can be treated as a two-level system, employing fictitious spin 1/2 operators, describable by the Hamiltonian H... 

A complete operator set that permits to adequately describe the behaviour of the spin system has to contain 47(7 +1) independent operators, which constitutes eight operators for spin 7 = 1. A set of operators of the fictitious spin-1/2 7p, (p = X, y, 2 f = 1,2,3) contains 12 operators, eight of which are independent and consequently form a complete basis set of the spin-1 system. 

The fictitious spin-1/2 have to be determined in the basis set of eigenstates of operator Hq. This choice permits the simplest way of linking the components of the magnetization of the nuclear-spin systems with the mean values of the operators of the fictitious spin-1/2. ... 

Here Hamiltonian formally describes an Ising-like system under the external field At. Unlike true magnetic systems, the degeneracies of the two levels are different thus the suffix Ising-like. Within the mean field approach the single molecule can be characterised by the Hamiltonian... 

However, this appearance can be misleading because the apparent hyperfine coupling of a given nucleus, given in equation (1), would not correspond to the actual hyperfine coupling of that nucleus. This is because of the ,g-over-.g effect , which results from use of the fictitious spin S = 1/2, rather than the real spin S =2/2, 5/2. The relation between the actual and observed hyperfine coupling matrices is given as follows ... 

In this expression the fundamental la), /3) spinor resembles a quark state three quarks are coupled together to form the quartet result. The use of the quartet spin bases does not mean that our /g really corresponds to a quartet spin. It only means that we can introduce a fictitious spin operator, S, which acts on the /g components in the same way as the real spin momentum would act on the components of a spin quartet. The transformations of the /g spinor under the elements of the group O may be obtained by combining the transformation matrices for the fundamental (la) 1/3)) spins with the quartet coupling scheme in Eq. (7.69). In the group O, this irrep is denoted as ie- In Table 7.7 the results are shown for two generators of the octahedral group. These matrices can be taken as the canonical basis relationships... 

Introduce a fictitious spin operator S that recognizes these states as the components of a triplet spin, S = 1, and consider the spin Hamiltonian... 

In octahedral symmetry the fictitious S operator follows the T irrep. Its third symmetrized power transforms as the components of the /-harmonics and also subduces a T irrep, as indicated in Table 7.1. Rewrite the p- and /-parts of the IT I operators for the /g quartet state as a spin Hamiltonian of the fictitious spin. 

The action of the spin Hamiltonian in the fictitious spin basis gives then rise to the following Hamiltonian matrix (in units of... 

The action of the components of the fictitious spin operator on the /g basis is dictated by the general expressions for the action of the spin operators on the 5=1 basis functions. It is verified that the spin-Hamiltonian that generates the Jp part of the matrix precisely corresponds to... 

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