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UnlikeSpins< Like Dense Spins | Index | The quadrupolar interaction > The method of moments for unlike spins The main difference in estimating the linewidths if the perturbing nuclear moments are different from the perturbed one is that the term {$B$} in the dipolar alphabet is no more an energy conserving term (it does not commute with{${\cal H}_0$}, hence it is non-secular). If one omits this term in the calculations and calls {$ \mathbf{S}^k $} the perturbing unlike spins One can see that the second moment of the resonance line {$\Delta\omega^2$} is given by:
{$ \overline{\Delta\omega^2}= - \frac {Tr[{\cal H}_1,\sum_k I^k_x]^2}{Tr \sum_k (I^k_x)^2} $} We can approximate {${\cal H}_a1$} by {$\hbar \gamma^2 \sum_{j<k} A_{jk} + B_{jk} $}. Actually, one may show that the secular Hamiltonian provides a better measure of the experimental second moment, by neglecting low outlying tails of the distribution that escape experimental detection, but would give large contributions to the second moment. The evaluation of the two traces is a good exercise and it yields {$ \begin{eqnarray} Tr (\sum_k I^k_x)^2 &=& N \frac{I(I+1)} 3 (2N+1)^N \cr Tr[\sum_{j<k} A_{jk}+B_{jk},\sum_l I^k_x]^2 &=& \frac 2 9 \gamma^2\hbar I^2(I+1)^2(2I+1)^n \sum_{j<k} \big(\frac {3(3\cos^2\theta_{jk}-1)}{2r_{jk}^3}\big)^2 \end{eqnarray}$} Since {$ \sum_{j<k} a^2_{jk} = \frac N 2 \sum_k a^2_{jk} $} and {$a_{jk}=a_{kj}$} so that the last sum is independent of {$j$}, we obtain that {$ \overline{\Delta \omega^2}= \frac 3 4 \gamma^4\hbar^2 I(I+1) \sum_k\frac{(3\cos^2\theta_{jk}-1)^2}{r_{jk}^6} $} |