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The dipolar alphabet

The Hamiltonian is

{$ \hbar {\cal H} = \hbar ({\cal H}_0 + {\cal H_1}) $}

where we suppose that the Zeeman term

{$ {\cal H}_0 = - \gamma H_o \sum_j I^j_z $}

is much larger than the dipolar interaction among all nuclear spins {$\mathbf{I}^k$}

{$ {\cal H}_1 = - \hbar \gamma^2 \sum_{j<k} \frac { 3 (\mathbf{I}^j\cdot \hat{r}_{jk})(\mathbf{I}^k \cdot \hat {r}_{jk}) - \mathbf{I}^j\cdot\mathbf{I}^k } {r_{jk}^3} = - \gamma^2\hbar \sum W_{ij}$}

We distinguish six terms in this dipolar perturbation and we call them

{$ {\cal H}_1 = - \hbar \gamma^2 \sum_{j<k} A_{jk} + B_{jk} + C_{jk} + D_{jk} + E_{jk} + F_{jk} $}

Their definition is obtained writing out explicitely the dipolar interaction {$W_{ij}$} in terms of the polar angles {$\theta,\phi$} of the unit vector {$\hat r _{jk} $} (we omit for simplicity the subscripts), in a reference frame where {$\hat z$}is along the external field {$H_0$}

{$ W_{ij}= \frac 1 {r_{ij}^3} \big[ 3(I^j_z \cos\theta + \sin\theta\frac{I^j_+ e^{-i\phi}+I^j_- e^{i\phi}} 2)(I^k_z \cos\theta + \sin\theta\frac{I^k_+ e^{-i\phi}+I^k_- e^{i\phi}} 2) - \mathbf{I}^j\cdot\mathbf{I}^k \big] $}

The purpose is to isolate the terms that commute with {${\cal H}_0$} (the secular term) from those that do not and to clessify them in terms of the raising and lowering spin operators {$I_\pm$}. So, in detail, we define:

{$ \begin{eqnarray} A_{jk}&=&- I^j_zI^k_z (3\cos^2\theta-1) \cr B_{jk} & = &\frac {3\cos^2\theta -1} 2 (I^j_zI^k_z - \mathbf{I}^j\cdot\mathbf{I}^k) \cr C_{jk} &=& \frac {3\sin\theta\cos\theta} 2 e^{-\phi} (I^j_zI^k_+ + I^j_+I^k_z) \cr D_{jk} &=& C_{jk}^*\cr E_{jk} &=& \frac {3\sin^2\theta} 4 e^{-2i\phi}I^j_+I^k_+\cr F_{jk} &=& E_{jk}^*\end{eqnarray} $}

and find out by inspection that only the terms {$A_{jk}$} and {$B_{jk}$} commute with {${\cal H}_0$}.

The method of moments

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} $}

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