4.1 Argonne Wakefield Accelerator facility
4.1.4 AWA homemade octupole magnet
One of the experiment targets is to demonstrate nonlinear LPS manipulation using the octupole. How- ever, AWA did not have the octupole so that we had to design and manufacture a new one. Procedure of the octupole manufacture is the following: 1. Decide location and find required maximum octupole strength. 2. Physics and engineering design and 3. Manufacture at AWA. The engineering design was done by an AWA engineer Scott Doran, so procedure of the engineering design is not included in this section.
First of all, we need to decide the octupole location in the middle section. Ideally, the octupole can be placed at any position in the middle section, but it has some practical issues. The location of octupole can provide appropriate transverse beam size. The octupole imparts third-order kick on both horizontal and vertical beam and we only want to deal with horizontal beam. Hence, the vertical beam size should be small to minimize the octupole effects. In addition, horizontal beam size should be large enough to cancel out the third-order correlation with the reasonable octupole strength. Next, we have to consider the horizontal jitters at the octupole. The first EEX beamline converted the timing and energy jitters to the horizontal phase space. When the beam is drifting, horizontal divergence jitter is appeared in the horizontal position. In addition, the middle section quadrupoles have to match horizontal phase space to the second EEX beamline for the LPS manipulation. If the octupole is placed downstream of the middle section, the quadrupoles are difficult to generate small vertical beam size and match the second EEX beamline conditions. Accordingly, the octupole is better to be put close to the first EEX beamline. Due to these reasons and physical dimension issues, best position of the octupole is in between the first and the second middle section quadrupoles.
Secondly, we calculated the maximum octupole strength at the design position. Figure 13 shows the DEEX beamline with the octupole magnet. Using Eq. (3.31) the integrated octupole strengthK to eliminate the third-order correlation in final longitudinal position is,
K=6h3 R(1)16R(2)51 +R(1)26R(2)52 R(2)52
R(1)15 +h1R(1)16
3, (4.1)
where h1 andh3 are the first- and third-order coefficients of initial LPS before the DEEX beamline,
Figure 13: Illustrate when the octupole is located in the designed position.
respectively. Transfer matrix elements R(1) and R(2) are indicated in Fig 13. Note, we assumed zero thickness TDCs of the DEEX beamline. Here,R(1)16R(2)51 +R(1)26R(2)52 is equal toR56of the DEEX beamline.
The bunch compression is related to R55 and R56 of the DEEX beamline. We want to eliminate the third-order correlation effect during the bunch compression.
Figure 14: Required integrated octupole strength maps based on Eq. (4.1). Each map has the different bunch compression conditions. Red dashed box shows reasonable beam parameters (h1andh3) region.
Figure 14 is the required integrated octupole strength with the different bunch compression condi- tions. In case of critical bunch compression RR55
56 =−h1, the final bunch length relies on theR56. Through the middle section quadrupole optimization, achievable minimumR56was 0.05. Reasonable beam pa- rameters ranges are 5<h1<15 andh3<1×107, and corresponding maximumKis∼8000 m−3. The beam parameters range was achieved from the injector optimizations.
The integrated octupole strengthKis a parameter for beam dynamic aspect. The octupole also uses a field strength parameter Baoct3 , whereBoct is magnetic field on the pole surface andais the bore radius.
TheKis defined as,
K= 1 Bρ
6Boctloct
a3 , (4.2)
where Bρ is the beam rigidity andloct is the effective length. For K =8000 m−3, the corresponding octupole field strength is∼2000 T/m3.
After the decision of maximum octupole strength, we carried out physics design of the octupole magnet. However, we had a mistake during the physics design; the maximum field strength is 8000 T/m3which is the maximum integrated strength. Accordingly, the manufactured octupole has stronger field strength than the estimation.
The physics design was referred to Refs. [49,50] and performed with Poisson-Superfish [51] which is 2D electric and magnetic simulation code. Figure 15 shows the simulation results. Because of symmetry, the code allows 1/16 part of whole dimension. The pole surface was designed to follow the equipotential line of the magnetic field. NI in the Fig. 15 is Ampere·turn that determines magnetic field inside the octupole. Here, the NI was chosen to 662.45 A·turn to generate strong magnetic field. When the N is 160 turns with American wire gauge (AWG) 12 wire, total length of coil is 327.98 m. Because AWG 12 provides resistance per length of 5.211 mΩ/mm, the octupole resistance due to the wire is 1.71Ω. We assume that additional resistance from transmission line is 2Ωand AWA magnet power supplies can apply maximum 15 V. Then maximum current on the octupole is 4.04 A and NI is 646.9 A·turn. The current density on wire is 1.1 A/mm2which is fine to use air cooling system according to Ref. [52]. The octupole strength of the simulation was 9643.6 T/m3 that satisfied our requirement. The cubic fitting was applied to magnetic field (Fig. 15 right) to check third-order trend line. The third-order coefficient of the fitting was same with our octupole field strength value.
Figure 15: First octupole simulation using Poisson-Superfish. Left is dimension used for the simulation and right is vertical magnetic field along x-axis. Magenta lines are magnetic field lines of the octupole.
Figure 16: Round shape yoke simulation.(a) Initial yoke design. (b) Round yoke design. (c) Magnetic field comparison between (a) and (b) results on horizontal axis. (d) Difference between magnetic fields of (a) and (b) results.
Figure 17: Round shape pole surface simulation. (a) Overview of the octupole with the round pole surface. (b) Magnetic field difference on horizontal axis.
Next, we simulated the round shape yoke and pole surface. Round shape steel fabrication is much cheaper and faster than the shape in Fig. 15. However, these shapes affect the magnetic field inside the octupole, and it may not have third-order field line depending on transverse position.
Figures 16(a) and (b) are angular and round yoke shape octupole magnets, respectively. Both sim- ulations have same NI of 600 A·turn. The angular shape has flat region so that the coil can be placed as close as possible, while the round shape interrupts the same coil region and has space between them.
The simulation results are shown in Figs. 16(c) and (d). Their magnetic field difference is less than 2 G which is an ignorable level.
The ideal pole surface follows the equipotential line to generate third-order magnetic field. Although, the ideal pole shape is close to round shape, the magnetic field is definitely different from the ideal pole.
We compared the round pole and ideal pole simulations at NI = 600 A·turn. Except for pole shape, the dimension is same. Figure 17(a) is an overview of the round pole octupole. The comparison between the two poles is shown in Fig. 17(b). The field difference is a few G near the center of octupole and near pole surface it is∼20 G. The magnetic field at the ideal pole surface is∼1100 G, and 20 G is 1.8%.
Here, we cannot generate rms beam size more than 5 mm due to horizontal jitters. This means beam will experience a few G different field compared to the ideal magnetic field.
Figure 18: AWA homemade octupole. (a) Yokes and poles before assemble. (b) Assembled yokes and poles with coils. (c) Assembled AWA homemade octupole. (d) Magnetic fields along the applied current at x = 2.5 mm and y = 0 mm. (d) Magnetic fields along x-axis at current of 3 A.
Parameters Values
Bore radius 25.654 mm
Length 0.1 m
Yoke inner radius 145 mm
Yoke thickness 25.4 mm
Yoke material Zinc plating 1018 steel Pole thickness 17.99 mm
Pole material Zinc plating 1008 steel
AWG 12
Wire turn per pole 150
Table 2: Specification of the homemade octupole magnet.
Based on the physics design, Scott finished the engineering design and ordered the yoke and poles fabrication [see Fig. 18(a)]. Using a wire winding machine, we winded the eight poles with AWG 12 wire. Assembled homemade octupole magnet is shown in Fig. 18(c) and detailed specifications are presented in Table 2. Before the homemade octupole installation in the beamline, we measured saturation current [Fig. 18(d)] and the magnetic field on the horizontal axis [Fig. 18(e)]. The octupole seems to have saturation current at 2 A, but it is close to linear line until the current of 3 A. When applied current is 3 A, the field has nice third-order trend compared to the simulation and its third-order coefficient from the cubic fitting is 5566 T/m3. Note, we had a mistake on maximum octupole field strength. The required strength was only∼2000 T/m3while our homemade octupole has the 3 times stronger field strength.
The stronger field strength is no problem in our experiment.