# NMR Studies of Semiconductor Nanocrystals, Solids [thesis]

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12) is the spectral density of the motion and ( ) explicitly indicates that an ensemble average over all initial and final configurations must be performed. ,,. because the other terms are oscillating too rapidly to contribute to the integral of this equation. This equation gives us the rate of change of the density matrix, however, we do not directly observe the density matrix in an experiment. We measure the rate of change of some observable, Q, which is given by 0 Ot =Tr (:ol =-__E_. , p • (t) _ p,q(t))}, (eq.

There are three distinct experiments that can be performed. First, the I spin can be inverted while the S spin is saturated. Under these conditions, the equation of motion of the I spin is ( 8 = -P < IZ " - < IZ "eq ) ' (eq. 28) " and the magnetization recovers with a rate p. 27, must be integrated. This is done by diagonalizing the rate matrix and then solving the resulting differential equation. The rate matrix can be diagonalized in the two spin case with the unitary transformation The inverse of this transformation is = 2" 1 1 " (=q.

14 and we identify the secular terms as those with re+n=0. _Oo _, '[H-'_' Itm] , m>O (eq. ,,,,o_ m(m - n) ) (eq. 16) Higher-order corrections, though more complicated, can be computed in a similar way. Expressions for the Van Vleck transformation expansion to second order have already been found, 22 but the tilting operator, S <1),was not given and the method could not be easily extended to higher orders. Transformation operator expansion homogeneously coupled spin-l/2 A method similar to the Van Vleck was previously 23 used to compute nuclei the second-order' term, S<'.