Table 1.

Properties of the field coils when the coil system is shielded.

N	Field coil	Field profile	$N_{\mathrm{contours}}$	l (m)	B/I ($\mathrm{\mu }$T/(Am${}^{(N-1)}$))	$\mathrm{\Delta }{B}^{\mathrm{RMS}}$ (%)	$\max (|\mathrm{\Delta }B|)$ (%)
1	$B_x$	$\mathbf{B}=\widehat{\mathbf{x}}$	30	44.8	160	0.08	0.18
1	$B_z$	$\mathbf{B}=\widehat{\mathbf{z}}$	400	270.0	718	0.33	0.84
2	$\mathrm{d}{B}_x/\mathrm{d}z$	$\mathbf{B}=(z\widehat{\mathbf{x}}+x\widehat{\mathbf{z}})$	12	16.3	136	2	–
2	$\mathrm{d}{B}_z/\mathrm{d}z$	$\mathbf{B}=(-x\widehat{\mathbf{x}}-y\widehat{\mathbf{y}}+2z\widehat{\mathbf{z}})$	10	13.5	108	6	–

Six independent coils generate three order $N=1$ uniform fields ($B_x$, $B_y$, $B_z$) and three order $N=2$ constant linear field gradients ($\mathrm{d}{B}_x/\mathrm{d}z$, $\mathrm{d}{B}_y/\mathrm{d}z$, $\mathrm{d}{B}_z/\mathrm{d}z$). The $B_x$ and $\mathrm{d}{B}_x/\mathrm{d}z$ coils have identical properties to the $B_y$ and $\mathrm{d}{B}_y/\mathrm{d}z$ coils, respectively. For each wire configuration of length l, the streamfunction is contoured with $N_{\mathrm{contours}}$ to generate a magnetic field profile B with a field strength per unit current B/I averaged along the z-axis of the optimisation region. For each coil, the difference between the normalised field in the standalone shield and the target field, $\mathrm{\Delta }B=B-B^{\mathrm{des}.}$, as a percentage of the desired field, is evaluated along the z-axis of the optimisation region and its root-mean-square (RMS) value is calculated. For the $B_x$ and $B_z$ coils, the maximum deviation from perfect field uniformity, $\max (|\mathrm{\Delta }B|)$, is also evaluated in the same region. All magnetic field values are calculated from experimentally measured data.