Table 1.

Spherical harmonic shim functions

l	m	Shim name	$f_l^m(\mathrm{x},\mathrm{y},\mathrm{z})$	$f_l^m(\mathrm{r},\theta ,\varphi )$
1	0	Z	z	rcosθ
1	1	X	x	rsinθcosϕ
1	−1	Y	y	rsinθsinϕ
2	0	Z 2	z2 – (x2 + y2)/2	$\frac{1}{2}{r}^2(3{\mathit{cos}}^2\theta -1)$
2	1	ZX	zx	r2sinθcosθcosϕ
2	−1	ZY	zy	r2sinθcosθsinϕ
2	2	X2 – Y2	x2 – y2	r2sin2θcos2ϕ
2	−2	XY	2xy	r2sin2θsin2ϕ
3	0	Z 3	z3 – 3z(x2 + y2)/2	$\frac{1}{2}{r}^3(5{\cos }^3\theta -3\cos \theta )$
3	1	Z2X	z2x – x(x2 + y2)/4	$\frac{1}{4}{r}^3\sin \theta \cos \varphi (5{\cos }^2\theta -1)$
3	−1	Z 2 Y	z2y – y(x2 + y2)/4	$\frac{1}{4}{r}^3\sin \theta \sin \varphi (5{\cos }^2\theta -1)$
3	2	Z(X2 – Y2)	z(x2 – y2)	r3sin2θcosθcos2ϕ
3	−2	ZXY	zxy	r3sin2θcosθsin2ϕ
3	3	X 3	x3 – 3xy2	r3sin3θcos3ϕ
3	−3	Y 3	3x2y – y3	r3sin3θsin3ϕ