Product operators spin 1 dynamics

Kupriyanov studied relaxation of spin 1/2 in the scalar coupled spin system AMX with quadrupolar nuclei in the presence of cross correlation effects. Khaneja et a/. presented optimal control of spin dynamics in the presence of relaxation. Eykyn et a/. studied selective cross-polarization in solution state NMR of scalar coupled spin 1/2 and quadrupolar nuclei. Tokatli applied the product operator theory to spin 5/2 nuclei. Mahesh et a/. used strongly coupled spins for quantum information processing. Luy and Glaser " inves-... 

The spin dynamics of the deuteron (spin /= 1) are more complex than those of the spin 1/2 nuclei, and the simple vector model used in other chapters, derived from the Bloch equations, provides no particular insight into deuteron spin dynamics. However, some of the geometric simplicity of the Bloch equations is present in a product-operator formalism, used to describe spin 1 NMR [117]. This formalism can provide a visual understanding of the deuteron pulse sequences in terms of simple precession and pulse rotations, albeit among a greater number of coordinate axes. The formalism can be used to understand the production of quadrupole order and the T q relaxation time (Figure 8.2(b)) and the two-dimensional deuteron exchange experiment (section 8.5). 

Because H is dynamically dependent on spin and space variables, the expression in parentheses in the r.h.s. of Eq. (3) involving integration over the latter defines a spin operator. This is just the effective Hamiltonian of interest to us. By virtue of point (iii), when the integrations are to be performed for the H" term in the Hamiltonian, only the unit operator in A need to be retained. The resulting expression will thus have the form (Ap H"l ). If one takes into account that the space state 1 ) is a product (or a combination of products, see above) of localized, one-particle states, one can immediately see that upon integrating over the spatial variables r , n= 1,2,...,AI, the spatial parts of the individual spin-dependent terms will be replaced by the corresponding quantum mechanical averages. Thus, for the entire expression in Eq. (3) is none other than one of the matrix element of the standard NMR Hamiltonian, Wnmr, between two spin-product basis states,... 

Operator equations have been employed by George and Ross (1971) to analyse symmetry in chemical reactions. In order to preserve the identity of electronic states of reactants and products, these authors worked within a quasi-adiabatic representation of electronic motions. By introducing a chain of approximations, going from separate conservation of total electronic spin to complete neglect of dynamics, they discussed the Wigner-Witmer angular momentum correlation rules, Shuler s rules for linear molecular conformations and the Woodward-Hoffmann rules. 

For mixing operations which require the continuous production of finely dispersed solids, emulsions, stable foams, etc., the in-line dynamic mixer in one of its several forms could be used. These usually consist of a rotor which spins at high speed inside a casing and the feed materials are pumped continuously to the unit. Inside the casing the fluid is subjected to extremely high shear forces which are required for the dispersing operation, see Figure 7.10. 

It is clear then that one needs to describe all relevant operators (related to the spin components) using their matrix representations in the four-fold vector space generated by the state vectors mj, ms). These matrices can be constructed by evaluating each matrix element or they can be built by the direct tensorial product of the corresponding 2x2 matrices that describe the dynamics of the separate spin 1 /2 systems [5,12], The direct tensorial product of two nxn matrices A and B lead to a x matrix C = A B whose elements are... 

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