TD-13-009

Multiple stem spoke cavity design to minimize transverse field asymmetry

Paolo Berrutti, Timergali Khabiboulline, Leonardo Ristori, Vyacheslav Yakovlev

Introduction

SSR2 is a spoke resonator under development at Fermilab, it is part of Project X and it is meant to accelerate particle in the low β range. SSR2 has been initially designed as a single spoke resonator, which means that the cavity geometry has an inner electrode shaped as straight spoke. That is the simplest design for a cavity derived from a coaxial line approximately half wave length long. The electric field lines of the accelerating mode are directed from the electrode to the cavity walls, the magnetic lines are spinning around the central post. In figure 1electric field lines are represented in green (left) and magnetic ones in orange (right).

Figure 1: on the left the electric field lines are sketched in green, the magnetic field lines are shown on the right in orange color.

Let’s assume the particles are traveling along z axis, the spoke is lying on y axis and it is perpendicular to x. The symmetry on the x-y plane has been broken by the electrode; the electromagnetic field will reflect this asymmetry and the transverse kick will show a quadrupole component. The electric field asymmetry can be mitigated using a donut shape for the central part of the spoke electrode, since the electric field lines are directed from the inner conductor to the outer walls. The magnetic lines spin around the central electrode, it implies that the magnetic field will have asymmetric transverse components; consequently the magnetic kick will have different X and Y components. If the magnetic field contribution to the transverse momentum gain is not negligible, the only possible way to get symmetric transverse kick is by modification of the single stem electrode into an X-shaped or a Y-shaped spoke. These modifications will provide different cavity symmetry, reducing the quadrupole effect on the transverse plane.

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SSR2 Cavity

• SSR2 will be the third family of spoke resonators used in Project X front-end: this accelerator section will accelerate the bunches from the end of RFQ section (2.1 MeV) to the beginning of the five-cell cavity part (160 MeV)

• SSR2 will be inserted after HWR (β=0.11 design by Argonne) and SSR1 (spoke resonator β optimal=0.21 by Fermilab), the particle velocity range covered by SSR2 cavity goes from 40 to 160 MeV.

• The single spoke resonator having β optimal 0.47 is the best choice for Fermilab Project X front-end because it minimizes the number of cavities needed.

• Double and triple spoke have too narrow transit time factor vs β, they would require a higher number of cavities for Project X front end.

• All coaxial resonators (quarter wave, half wave and spoke resonators) show transverse field asymmetry in the region where the beam travels.

Field asymmetry

• The nature of the fundamental TEM mode implies that: a quadrupole component is always present in these cavities because the inner post breaks the azimuthal symmetry of the structure.

• The electric field can be symmetrized easily modifying either the shape of the central electrode or the beam tubes, for example HWR resonator by Argonne has a ring shaped electrode to compensate the electric field asymmetry.

• SSR2 single spoke cavity has a high quadrupole component due to the magnetic field which spins around the spoke electrode and interacts with particles traveling through the structure.

• SSR2 single spoke shows a considerable transverse field asymmetry, due to the magnetic field.

• This field asymmetry leads to an asymmetry in the transverse kick (quadrupole), which increases the beam envelope along the accelerator length in SSR2 section.

• The only solution to symmetrize the transverse magnetic field is to modify the spoke post, giving symmetry to the cavity: instead of a straight spoke one can use X or Y shaped inner conductors.

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• Single spoke cavity:

Figure 2: SSR2 single spoke cavity (left) and transverse electric and magnetic field on a 10 mm offset (right).

SSR2 single spoke transverse fields were evaluated with a 10 mm offset, in both X and Y directions, with respect to the center of the beam pipe (right side of the figure above). The electric field components (first column in the plot) look quite similar one to the other: patterns and amplitudes |Ex| and |Ey| are very close. The transverse magnetic fields Hx and Hy differ quite significantly by pattern and amplitude: their patterns along Z axis are substantially different, and Hy component is three times smaller than Hx.

SSR2 Y-spoke cavity design:

Figure 3: SSR2 Y-spoke cavity (left) and transverse electric and magnetic field on a 10 mm offset (right).

Y-spoke cavity design has been made replicating the spoke inner electrode every 120 degrees to get a Y shape (left in figure 3). This geometry shows negligible transverse field asymmetry, compared to the single spoke geometry.

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SSR2 X-spoke cavity design:

Figure 4: SSR2 Y-spoke cavity (left) and transverse electric and magnetic field on a 10 mm offset (right).

The left picture in figure 4 shows the X-spoke cavity design, which has been made replicating the inner electrode every 90 degrees to get an X shape. The transverse field asymmetry looks negligible.

Transverse Kick Calculation

Field quadrupole asymmetry can be evaluated by calculating the kick in transverse

directions, it has been introduced a parameter Q=

Δ px (r ,0 ) c−Δ py (r , π2 )cΔ px (r ,0 ) c+Δ py (r , π2 )c

2

, the smaller

the Q parameter is the better beam quality can be achieved with the resonator. One can look at the plot of Q vs particle β to understand the difference among the various

geometries and their capability of providing low quadrupole effect in the full energy range of usage.

Single spoke Q vs β (35-160MeV):Figure 5 shows the amplitude of the Q parameter as a function of particle β, particles in SSR2 section of Project X will be accelerated from 35 MeV to 160 MeV, so the full β domain has been covered by the calculation. Q parameter is normalized so it does not depend on the peak field, gradient or stored energy in the cavity. The maximum value of Q for the single spoke design occurs at the very beginning of the β range, and it is pretty high 0.4.

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Figure 5: SSR2 single spoke cavity Q vs β plot for the full particle energy range.

Y-spoke Q vs β (35-160MeV):The Y-spoke design Q vs β plot shows a much lower quadrupole asymmetry. In the whole β domain the Y-spoke design has lower Q parameter; the maximum of the Q parameter is now 3% of the single spoke highest point.

Figure 6: SSR2 Y-spoke cavity Q vs β plot for the full SSR2 β domain.

X-spoke Q vs β (35-160MeV):Q asymmetry parameter as a function of β drops even more for the X-spoke design. The maximum of Q for the X-spoke design 0.006% of the peak for the single spoke cavity.

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Figure 7: SSR2 X-spoke cavity Q vs β plot for the full particle energy range.

Field expansion into multipoles confirms the results obtained using the Q parameter simplified approach, the table below report the RF field expansion coefficients for all of the cavities calculate at r=10 mm and at particle β equal to 0.47.

An/rn−1 Single

spokeY spoke X-spoke

1st [keV] 5.812e-14 0.2501582 2.63E-142nd [keV/mm] 3.391 0.0170884 0.0062083rd [keV/mm2] 4.331e-16 0.2362148 3.43E-144th [keV/mm3] 4.747e-4 0.0009250 0.0837175th [keV/mm4] 4.957e-18 0.0034837 4.91E-146th [keV/mm5] 2.338e-08 0.0149121 0.002517th [keV/mm6] 1.367e-19 0.0041916 2.63E-148th [keV/mm7] 2.719e-10 0.0033935 0.005494

X and Y spoke geometries have been optimized and their performances look comparable to the ones of the single spoke cavity

Single spoke Cross spoke Y spokeG [Ohm] 112.98 122.67 119.94

R/Q [Ohm] 289.94 272.26 269B/Eacc [mT/(MV/m)] 6.107 6.13 6.07

E/Eacc 3.45 3.34 3.48βopt 0.4714 0.479 0.48

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ConclusionsDespite the more complicated design multi-stem spoke resonators allow to overcome the transverse field asymmetry. Both magnetic and electric fields are symmetrized using X or Y shaped inner electrode, consequently quadrupole and octupole amplitudes drop to negligible levels. This happens because the cavity geometry recovers azimuthal symmetry, or transverse plane symmetry, which would be broken by the single spoke inner electrode. Instead of using a quadrupole corrector inside the cryomodule it is possible to modify the cavity geometry to obtain zero transverse asymmetry. A preliminary RF optimization showed that the performances achievable with all the cavities are very similar, the choice of a multi-stem spoke does not influence the electromagnetic parameters.