The Exercise

Two nuclei of spin $\tfrac12$, with gyromagnetic ratios $\gamma_1$ and $\gamma_2$, sit a distance $r$ apart, and $\hat n$ is the unit vector from the first to the second. Their magnetic moments interact through the dipolar coupling, which ties each Pauli component of one spin to each component of the other. A pair of pulses given by the Pauli matrix $Z$ on a spin, one before and one after a period of free evolution, turns the coupling during that period into its conjugate by $Z$ on that spin.

The exercise averages the coupling over the four cases: no pulses, pulses on the first spin, on the second, and on both. What survives is a single term, the product $Z_1Z_2$ of the $Z$ operator on the first spin and the $Z$ operator on the second, with a strength that depends on $\hat n$ only through the angle between $\hat n$ and the $z$ axis.

Two Spins and Their Coupling

The Pauli matrices $X$, $Y$ and $Z$ are the complex $2\times2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},\qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(1) and the Pauli vector $\vec\sigma = (X, Y, Z)$ is the triple of them. For a real vector $\vec n = (n_x, n_y, n_z)$ in three dimensions, $\vec n\cdot\vec\sigma$ is the matrix $n_x X + n_y Y + n_z Z$.^✦

A subscript on $\vec\sigma$ names the spin it acts on. The isotropic coupling $\vec\sigma_1\cdot\vec\sigma_2$ of two spins pairs each Pauli matrix on spin 1 with the same matrix on spin 2, and their anisotropic coupling $(\vec\sigma_1\cdot\vec n)(\vec\sigma_2\cdot\vec n)$ along $\vec n$ is the tensor product of $\vec n\cdot\vec\sigma$ with itself: $$\begin{aligned} \vec\sigma_1\cdot\vec\sigma_2 &= X\otimes X + Y\otimes Y + Z\otimes Z,\\ (\vec\sigma_1\cdot\vec n)(\vec\sigma_2\cdot\vec n) &= (\vec n\cdot\vec\sigma)\otimes(\vec n\cdot\vec\sigma). \end{aligned}$$(2)

The dipolar coupling $H^D_{1,2}(\vec n)$ of two nuclei of spin $\tfrac12$ with gyromagnetic ratios $\gamma_1$ and $\gamma_2$, a distance $r$ apart along $\vec n$, is a multiple of the dipolar bracket, the isotropic coupling minus three times the anisotropic one: $$H^D_{1,2}(\vec n) = \frac{\gamma_1\gamma_2\hbar}{4r^3} \Bigl[\vec\sigma_1\cdot\vec\sigma_2 - 3(\vec\sigma_1\cdot\vec n)(\vec\sigma_2\cdot\vec n)\Bigr],$$(3)^✦ with $\hbar$ the reduced Planck constant. For the nuclei of the exercise, $\vec n$ is the unit vector $\hat n$.

The Refocusing Cycle

The refocusing pulse $Z_1$ on the first spin is the operator of the two spins that is $Z$ on that spin and the identity $I$ on the other, $$Z_1 = Z\otimes I,$$(4) the pulse on the second spin is the reverse, $$Z_2 = I\otimes Z,$$(5) and the joint pulse is $Z$ on both, $$Z_1Z_2 = Z\otimes Z.$$(6)

For a Hamiltonian $H$ of the two spins, the average Hamiltonian $\overline H$ of the refocusing cycle, the four cases of no pulse, a pulse on either spin and a pulse on both, is the mean of $H$ and its conjugates by the three pulses: $$\overline H = \tfrac14\bigl(H + Z_1 H Z_1 + Z_2 H Z_2 + Z_1Z_2\, H\, Z_1Z_2\bigr).$$(7)^✦

The Claim

The effective coupling $J_{\mathrm{eff}}$ of two nuclei of spin $\tfrac12$ with gyromagnetic ratios $\gamma_1$ and $\gamma_2$, a distance $r$ apart along a real vector $\vec n$, is the real number $$J_{\mathrm{eff}} = \frac{\gamma_1\gamma_2\,(1 - 3n_z^2)}{r^3}.$$(8)^✦ For the unit vector $\hat n$ at an angle $\theta$ to the $z$ axis, $n_z = \cos\theta$ and the factor is $1 - 3\cos^2\theta$.

What Is Checked

The claim is stated over the dipolar coupling (3), built from the Pauli matrices (1) through the matrix $\vec n\cdot\vec\sigma$ and the two couplings (2); over the pulses (4), (5) and (6) and the average (7) they define; and over the effective coupling (8).