Magnetic Design of the Commutational Magnet and Quadrupoles for PERLE Accelerator

PERLE (Powerful Energy Recovery LINAC for Experiment) is a three turns high-power Energy Recovery LINAC (ERL) facility with 20 mA beam current and beam energy from 250 MeV to 500 MeV. It is a testbed to validate multi-turn high current ERL operation for the future LHeC and will be the first ERL for nuclear applications. In this work, the design and optimization of the commutational magnet (B-com) used to spread/combine the three beams and one series of quadrupole magnets is discussed. The B-com magnet is optimized for a 30° bending angle with harmonic content of 0.036%. The quadrupole magnets generate a field gradient of 34.15 T/m. Further studies to suppress saturation at a maximum gradient of 44.1 T/m are undergoing.


Introduction
Energy Recovery LINAC (ERL) [1] provides LINAC-like beam characteristics, and high average beam power in a more compact footprint than the storage ring.Up to date ERL facilities have been limited to ≤ 2 MW beam power in a single pass ERL [1].Some new ongoing ERL facilities are Cornell-BNL Test Accelerator (CBETA) (USA) [2] and bERLinPro (Germany) [3].PERLE [4] is three acceleration and three deceleration passes ERL accelerator with design parameters similar to those of the LHeC [5] (see Table 1).
PERLE is composed of three sections: the LINAC section at each straight line, the arc sections for beam recirculation, and the spreader/combiner section connecting the LINAC to the arcs.Figure 1 shows the 500 MeV first-order lattice of PERLE.
Vertical field dipoles and two families of quadrupole magnets are considered.The spreader/combiner sections accommodate commutational dipole (B-com magnet) [6] with horizontal magnetic field and a 30 o bending angle to split/combine the beam vertically at three different energies: 171 MeV, 336 MeV, and 500 MeV.A 8.842 cm vertical shift is required for the lowest energy beam.Normal conducting, Iron-dominated electromagnet technology is adopted to design PERLE magnets.Laminated steel (< 0.1% Carbon content) is used for the yoke.
The fundamental harmonic of the magnetic field produced by a pure vertical field dipole magnet is B y = C 1 , B x = 0 and that produced by a quadrupole is B y = C 2 x, B x = C 2 y.In  practice, any fabricated magnet will produce a magnetic field with higher harmonics due to the finite geometry of the material called allowed harmonics.These are the 3, 5, 7, . . .harmonics for dipole magnets and the 6, 10, 14, . . .for quadrupole magnets.Non-allowed harmonics are present due to manufacturing tolerances.The symmetry of the yoke geometry can restrict the presence of higher-order harmonics.Higher-order harmonics degrade the field distribution and must be reduced for stable beam trajectory.Dipole field errors lead to undesired beam oscillation and quadrupole field errors lead to a tune shift [7].
The solution of Laplace's equation is used to calculate the pole shape.The excitation current N I required to drive the magnetic field in an electromagnetic dipole is N I Dipole = Bh 2µ 0 with N the number of coil turns, I the current, h the gap between the poles, µ 0 the magnetic permeability of free space.The excitation current in a quadrupole magnet is N I Quadrupole = Gr 2 2µ 0 with G the field gradient, r the magnet aperture.Dipole magnet pole shape follows y = ± h 2 .Quadrupole pole shape is a hyperbolic contour xy = ± r 2 2 [7].

B-com Magnet Design
The design was carried out with Opera 3D, a finite element analysis software [8].From the beam dynamics study of PERLE lattice, the beam size does not exceed 2 mm at 3σ (see Figure 2).Taking into account the thickness of the vacuum chamber, an aperture of 4 cm was chosen.The yoke is an H-shaped magnetic steel.The coil is made of a hollow Copper conductor based on water cooling.A blend radius of 0.8 cm was introduced to the pole edges to reduce the harmonic content.Figure 3 shows the B-com magnet design and Table 2 shows the main parameters of the coil.Figure 3 shows a saturation in the pole edges.It can be suppressed by considering chamfer.Figure 4a shows the horizontal magnetic field component B x along the pole width.It gives a field of 0.72 T at the center of the magnet.The field integral along the beam trajectory is 0.3 T.m. Figure 4b shows the field homogeneity (the variation of B x along the pole width with respect to its value at the center B 0 ).B x is homogeneous along ±10 cm. Figure 5 shows that the lowest energetic beam exits the steel at 8.85 cm.The harmonic components of the magnet were calculated on a circle of 1 cm radius along the trajectory of the particles with 0.25 cm intervals.The square root of the sum of the harmonic coefficients squared is 0.036% on the lowest energy beam.The order of magnitude of the field homogeneity 10 −4 meets the requirements for a successful beam separation/combination of 8.85 cm. Figure 6 shows the value of the harmonic coefficients with respect to the fundamental harmonic b 1 .The most dominant components are the quadrupole and sextupole ones which will be dealt with in the subsequent lattice elements of quadrupoles and sextupoles.

Quadrupole Magnet Design
Figure 7a shows the multi-coil design of the 15 cm length quadrupole magnet.The yoke is a cylindrical tube of 250 mm outer diameter with 35 mm thickness considering the 450 mm vertical separation between the arcs.Aperture radius of ±20 mm is used to construct the hyperbolic pole profile of 44 mm width.Three hollow Copper conductor coils connected in series are considered to provide a less magnetic field in the pole base which is under 2 T (see Figure 7b).The current density in the three coils is 1.5 times less than that in a one-coil design of the same parameters.Table 3 shows the coil parameters.

Figure 2 :
Figure 2: Electron beam envelope and dispersion function in the first arc section.Blue bars represent dipole magnets and red bars represent quadrupole magnets.The beam size is in mm corresponding to 3σ.

Figure 3 :Figure 4 :
Figure 3: B-com magnet design with Opera 3D.The magnetic field is in Gauss.Areas with B > 1.8 T are not shown.Asymmetrical geometry was chosen for enhanced mechanical stability.

Figure 5 :
Figure 5: Trajectory of the electron beam through the B-com magnet at three energies: 171 MeV, 336 MeV, and 500 MeV.Initial position of the beam is (4.4,0,-60) cm.

Figure 6 :
Figure 6: Higher harmonics with respect to the dipole component b 1 for electron energy of 171 MeV.

Figure 8 Figure 7 :
Figure 8 shows the vertical field component B y as a function of the horizontal direction x.The gradient generated by the three-coil configuration is G = 34.15T/m.

Figure 8 :
Figure 8: Horizontal distribution of the vertical magnetic field of the quadrupole.

Table 1 :
Input Design Parameters of PERLE Accelerator

Table 2 :
Coil Design Parameters of PERLE B-com Magnet

Table 3 :
Coil Design Parameters of PERLE Quadrupole Magnet