Sodium MRI: Methods and applications

. Author manuscript; available in PMC: 2015 May 1.

Published in final edited form as: Prog Nucl Magn Reson Spectrosc. 2014 Mar 7;79:14–47. doi: 10.1016/j.pnmrs.2014.02.001

Coherence pathway

D_{\pm 1, \pm 2}^{3} (ϕ_{3}, θ_{3}, - ϕ_{3}) D_{\pm 2, \pm 1}^{3} (ϕ_{2}, θ_{2}, - ϕ_{2})

T_3,1 → T_3,2 → T_3,1

\begin{array}{l} - \frac{5}{2} e^{i (ϕ_{3} - ϕ_{2})} {cos}^{3} \frac{θ_{3}}{2} (- 1 + 3 cos θ_{3}) \\ sin \frac{θ_{3}}{2} {cos}^{3} \frac{θ_{2}}{2} (- 1 + 3 cos θ_{2}) sin \frac{θ_{2}}{2} \end{array}

T_3,1 → T_3,2 → T_3,−1

\begin{array}{l} - \frac{5}{2} e^{i (3 ϕ_{3} - ϕ_{2})} cos \frac{θ_{3}}{2} (1 + 3 cos θ_{3}) \\ {sin}^{3} \frac{θ_{3}}{2} {cos}^{3} \frac{θ_{2}}{2} (- 1 + 3 cos θ_{2}) sin \frac{θ_{2}}{2} \end{array}

T_3,1 → T_3,−2 → T_3,1

\begin{array}{l} - \frac{5}{2} e^{i (- 3 ϕ_{3} + 3 ϕ_{2})} cos \frac{θ_{3}}{2} (1 + 3 cos θ_{3}) \\ {sin}^{3} \frac{θ_{3}}{2} cos \frac{θ_{2}}{2} (1 + 3 cos θ_{2}) {sin}^{3} \frac{θ_{2}}{2} \end{array}

T_3,1 → T_3,−2 → T_3,−1

\begin{array}{l} - \frac{5}{2} e^{i (- ϕ_{3} + 3 ϕ_{2})} {cos}^{3} \frac{θ_{3}}{2} (- 1 + 3 cos θ_{3}) \\ sin \frac{θ_{3}}{2} cos \frac{θ_{2}}{2} (1 + 3 cos θ_{2}) {sin}^{3} \frac{θ_{2}}{2} \end{array}

T_3,−1 → T_3,2 → T_3,1

\begin{array}{l} - \frac{5}{2} e^{i (ϕ_{3} - 3 ϕ_{2})} {cos}^{3} \frac{θ_{3}}{2} (- 1 + 3 cos θ_{3}) \\ sin \frac{θ_{3}}{2} cos \frac{θ_{2}}{2} (1 + 3 cos θ_{2}) {sin}^{3} \frac{θ_{2}}{2} \end{array}

T_3,−1 → T_3,2 → T_3,−1

\begin{array}{l} - \frac{5}{2} e^{i (3 ϕ_{3} - 3 ϕ_{2})} cos \frac{θ_{3}}{2} (1 + 3 cos θ_{3}) \\ {sin}^{3} \frac{θ_{3}}{2} cos \frac{θ_{2}}{2} (1 + 3 cos θ_{2}) {sin}^{3} \frac{θ_{2}}{2} \end{array}

T_3,−1 → T_3,−2 → T_3,1

\begin{array}{l} - \frac{5}{2} e^{i (- 3 ϕ_{3} + ϕ_{2})} cos \frac{θ_{3}}{2} (1 + 3 cos θ_{3}) \\ {sin}^{3} \frac{θ_{3}}{2} {cos}^{3} \frac{θ_{2}}{2} (- 1 + 3 cos θ_{2}) sin \frac{θ_{2}}{2} \end{array}

T_3,−1 → T_3,−2 → T_3,−1

\begin{array}{l} - \frac{5}{2} e^{i (- ϕ_{3} + ϕ_{2})} {cos}^{3} \frac{θ_{3}}{2} (- 1 + 3 cos θ_{3}) \\ sin \frac{θ_{3}}{2} {cos}^{3} \frac{θ_{2}}{2} (- 1 + 3 cos θ_{2}) sin \frac{θ_{2}}{2} \end{array}