Table 7.

Irreducible spherical tensor operators (ISTOs). This table shows the relationship between ISTOs T _lm (l = 0 to 3 by steps of 1, and m = −l to l by steps of 1), and Cartesian spin operators. The set of 16 ISTOs is used for spin-3/2 sodium signal simulations in multicompartment ²³Na quantifications. Acquired MR signal are represented by the T _1±1 operator. Abbreviations: SQC, DQC and TQC stand for single, double and triple quantum coherences, respectively. The anticommutator for the operators A and B is defined as [A,B] ₊  = A × B + B × A.

T lm	Cartesian Decomposition	Description
T 00	1	Identity
T 10	I z	Longitudinal Magnetization
T 1±1	$\mp \frac{1}{\sqrt{2}}{I}_{\pm }$	Rank 1 SQC
T 20	$\frac{1}{\sqrt{6}}\mathrm{(3}{I}_z^2-I(I+\mathrm{1))}$	Quadrupolar Order
T 2±1	$\mp \frac{1}{\sqrt{2}}{[I_z,I_{\pm }]}_{+}$	Rank 2 SQC
T 2±2	$\frac{1}{2}{I}_{\pm }^2$	Rank 2 DQC
T 30	$\frac{1}{\sqrt{10}}\mathrm{(5}{I}_z^3-\mathrm{(3}I(I+\mathrm{1)}-\mathrm{1)}{I}_z)$	Octupolar Order
T 3±1	$\mp \frac{1}{4}\sqrt{\frac{3}{10}}{\mathrm{[5}{I}_z^3-I(I+\mathrm{1)}-\frac{1}{2},I_{\pm }]}_{+}$	Rank 3 SQC
T 3±2	$\frac{1}{2}\sqrt{\frac{3}{4}}{[I_z,I_{\pm }^2]}_{+}$	Rank 3 DQC
T 3±3	$\mp \frac{1}{2\sqrt{2}}{I}_{\pm }^3$	Rank 3 TQC