High gradient quadrupoles for low emittance storage rings

High gradient quadrupoles are key components for the coming generation of storage ring based light sources. The typical specifications of these magnets are: almost $100\mathrm{T}/\mathrm{m}$ gradient, half a meter long, and a vertical aperture for the extraction of the x-ray beam. This paper presents the preparation work done at the European Synchrotron Radiation Facility, from the design to the manufacture and measurements of a prototype. It demonstrates the feasibility of such magnets. Different aspects of magnet engineering are discussed, including the study of the main scale factors and the preliminary design, the pole shaping, the impact of mechanical errors, and the magnetic measurements of a prototype with a stretched-wire system.

Figure 1

Schematic view of one cell of the ESRF-EBS lattice. The storage ring will be made of 32 cells. QF, QD: quadrupoles; SF, SD: sextupole; OF: octupole magnets; DL: dipole with longitudinal gradient; DQ: dipole-quadrupoles. The fast correctors are not represented.

Figure 2

Conservation of the magnetic flux across the pole.

Figure 3

Definition of the main geometrical magnet parameters.

Figure 4

Magnet power consumption (a) and mass (b) vs total length (including coil heads), at 44 T integrated field gradient and 12.5 mm bore radius. The crosses are 3D simulation results and the black disk indicates the parameters of the prototype magnet. The lines are spline interpolations between the simulated points.

Figure 5

Magnetic efficiency $\eta$ as defined in Eq. (10), vs taper angle ${\theta }_2$, at $r_0=12.5\mathrm{mm}$, $r_2=80\mathrm{mm}$ and for a 545 mm total length (including coil heads). The black dot indicates the design value and the lines are spline interpolations between the simulated points (markers).

Figure 6

Magnetic efficiency $\eta$ vs coil taper angle ${\theta }_3$, for different values of the pole taper length $r_2$, at ${\theta }_2=20°$. The black square indicates the design value and the lines are spline interpolations between the simulated points (markers).

Figure 7

Magnetic efficiency $\eta$ vs $r_1$ pole radius. The dot indicates the design value and the line is a spline interpolation between the simulated points (markers).

Figure 8

Gradient vs gap between poles obtained from Eq. (3) and with a 3D model with Table 3 parameters ($r_1=1\mathrm{mm}$ for the simulations shown in this plot). ${(NI)}_{\text{Sat}}=7668\mathrm{A}$ turns. The current density is $3.7\mathrm{A}/{\mathrm{mm}}^2$ at ${(NI)}_{\text{Sat}}$ and $10\mathrm{A}/{\mathrm{mm}}^2$ at 2.7 ${(NI)}_{\text{Sat}}$. The bore radius is 12.5 mm.

Figure 9

Pole shapes obtained with different settings of the optimization algorithm. The dashed arc indicates the 7 mm radius GFR.

Figure 10

12-pole term vs current, for Profile 1 and Profile 2. Reference radius: 7 mm.

Figure 11

Pole field $B$ (top) and magnetization ${\mu }_{0}M$ (bottom) at 90 A, with Profile 1. All values are in [T].

Figure 12

Excitation curve of the high gradient quadrupole. The results obtained from the analytic model [Eqs. (1) and (4)] are given assuming a 493 mm magnetic length.

Figure 13

Standard deviation of the multipoles ${\sigma }_n=[\sigma (a_n)+\sigma (b_n)]/2$, expressed at 7 mm. The tolerances are $\pm 0.020\mathrm{mm}$ for the shape tolerances, $\pm 0.020\mathrm{mm}$ for the pole displacements and $\pm 0.040\mathrm{mm}$ for the complete magnet.

Figure 14

Simulated integrated field ${\int }_{-\infty }^{\mathit{s}}{\mathit{B}}_{\mathit{n}0}\mathit{d\nu }$ normalized by the integral of the quadrupole term.

Figure 15

Simulated field multipoles with quadrupole symmetry: ${\mathit{B}}_{\mathit{\theta }20}\propto \mathit{\rho cos}(2\mathit{\theta })$, ${\mathit{B}}_{\mathit{\theta }21}\propto {\mathit{\rho }}^3\mathit{cos}(2\mathit{\theta })$ and ${\mathit{B}}_{\mathit{\theta }22}\propto {\mathit{\rho }}^5\mathit{cos}(2\mathit{\theta })$.

Figure 16

High gradient quadrupole prototype installed on a stretched wire measurement bench [4].

Figure 17

Quadrupole gradient normalized by a 493 mm magnetic length. The gradient has been measured with circular measurements at 12.4 mm radius.

Figure 18

Higher order multipoles of the high gradient quadrupole prototype at 90 A and expressed at 7 mm radius. Circular measurements at different radii, ranging from 5 mm up to 12 mm, have been combined for measuring the multipoles. The standard deviations have been computed at only one radius, which is a pessimistic case. The dipole and quadrupole terms are not shown. The simulations are for interpole distances with errors equal to $\pm 0.020\mathrm{mm}$ (depending on the pole), with a symmetry that does not allow sextupole terms.