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. Author manuscript; available in PMC: 2010 Feb 1.

Published in final edited form as: Prog Nucl Magn Reson Spectrosc. 2009 Feb 1;54(2):69–96. doi: 10.1016/j.pnmrs.2008.04.001

Table 2.

$W_{l m}^{j}$ coefficients used in FASTMAP optimization

	j	1	2	3	4	5	6
	(θ, ϕ)	xy $(\frac{π}{2}, \frac{π}{4})$	yx $(\frac{π}{2}, - \frac{π}{4})$	xz $(\frac{π}{4}, 0)$	zx $(\frac{3 π}{4}, 0)$	yz $(\frac{π}{4}, - \frac{π}{2})$	zy $(\frac{3 π}{4}, - \frac{π}{2})$
Shim name	(l, m)
Z	(1,0)	0	0	$\frac{1}{\sqrt{2}}$	$- \frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$- \frac{1}{\sqrt{2}}$
X	(1,1)	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	0	0
Y	(1,−1)	$\frac{1}{\sqrt{2}}$	$- \frac{1}{\sqrt{2}}$	0	0	$- \frac{1}{\sqrt{2}}$	$- \frac{1}{\sqrt{2}}$
Z ²	(2,0)	$- \frac{1}{2}$	$- \frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{4}$
ZX	(2,1)	0	0	$\frac{1}{2}$	$- \frac{1}{2}$	0	0
ZY	(2, −1)	0	0	0	0	$- \frac{1}{2}$	$\frac{1}{2}$
X² – Y²	(2,2)	0	0	$\frac{1}{2}$	$\frac{1}{2}$	$- \frac{1}{2}$	$- \frac{1}{2}$
XY	(2,−2)	1	−1	0	0	0	0
Z ³	(3,0)	0	0	$- \frac{1}{4 \sqrt{2}}$	$\frac{1}{4 \sqrt{2}}$	$- \frac{1}{4 \sqrt{2}}$	$\frac{1}{4 \sqrt{2}}$
XZ ²	(3,1)	$- \frac{1}{4 \sqrt{2}}$	$- \frac{1}{4 \sqrt{2}}$	$\frac{3}{8 \sqrt{2}}$	$\frac{3}{8 \sqrt{2}}$	0	0
YZ ²	(3,−1)	$- \frac{1}{4 \sqrt{2}}$	$\frac{1}{4 \sqrt{2}}$	0	0	$- \frac{3}{8 \sqrt{2}}$	$- \frac{3}{8 \sqrt{2}}$
Z(X² – Y²)	(3,2)	0	0	$\frac{1}{2 \sqrt{2}}$	$- \frac{1}{2 \sqrt{2}}$	$- \frac{1}{2 \sqrt{2}}$	$\frac{1}{2 \sqrt{2}}$
XYZ	(3,−2)	0	0	0	0	0	0
X ³	(3,3)	$- \frac{1}{\sqrt{2}}$	$- \frac{1}{\sqrt{2}}$	$\frac{1}{2 \sqrt{2}}$	$\frac{1}{2 \sqrt{2}}$	0	0
Y ³	(3,−3)	$\frac{1}{\sqrt{2}}$	$- \frac{1}{\sqrt{2}}$	0	0	$\frac{1}{2 \sqrt{2}}$	$\frac{1}{2 \sqrt{2}}$