Superconducting spoke cavities for high-velocity applications

To date, superconducting spoke cavities have been designed, developed, and tested for particle velocities up to ${\beta }_0\sim 0.6$, but there is a growing interest in possible applications of multispoke cavities for high-velocity applications. We have explored the design parameter space for low-frequency, high-velocity, double-spoke superconducting cavities in order to determine how each design parameter affects the electromagnetic properties, in particular the surface electromagnetic fields and the shunt impedance. We present detailed design for cavities operating at 325 and 352 MHz and optimized for ${\beta }_0=0.82$ and 1.

Figure 1

325 MHz, ${\beta }_0=1$ double-spoke cavity.

Figure 2

CST Microwave Studio© (MWS) view of a racetrack-shaped spoke. (1) base or aperture width, (2) base or aperture length, (3) aperture height.

Figure 3

Dependence of the normalized magnetic field on the spoke base length for a single- and double-spoke cavity with the same spoke and end-cap geometries.

Figure 4

Cut-away view of a two-spoke 325 MHz, ${\beta }_0=1$ cavity.

Figure 5

(a) Dependence of $B_p/E_{\mathrm{acc}}$ and (b) $R{R}_s$ on spoke base length normalized to rf wavelength for a 325 MHz, ${\beta }_0=1$ double-spoke cavity at a constant base width of 200 mm. The blue curves represents a longitudinal base geometry, while the red curves are for an identical spoke base, but oriented transversely to the beam line. In both plots, the leftmost point is common to both curves since it represents a cylindrical spoke base.

Figure 6

Dependence of energy content on spoke base length for longitudinal and transverse orientations at a constant accelerating field of $1\mathrm{MV}/\mathrm{m}$. The leftmost point is common to both curves since it represents a cylindrical spoke base.

Figure 7

Dependence of normalized electric field on spoke base length for longitudinal and transverse orientations. The leftmost point is common to both curves since it represents a cylindrical spoke base.

Figure 8

(a) Dependence of $E_p/E_{\mathrm{acc}}$ and (b) $B_p/E_{\mathrm{acc}}$ on the spoke base width at a constant base length of 480 mm.

Figure 9

(a) Dependence of $R{R}_s$ and (b) energy content (at $1\mathrm{MV}/\mathrm{m}$) on the spoke base width.

Figure 10

$B_p/E_{\mathrm{acc}}$ vs $E_p/E_{\mathrm{acc}}$ for various spoke base shapes. Those shapes within the circled region are candidates for further investigation.

Figure 11

Different transverse spoke base shapes with different ratios of width to length. (a) 0.30, (b) 0.42, and (c) 0.52.

Figure 12

Dependence of (a) $E_p/E_{\mathrm{acc}}$ and (b) $B_p/E_{\mathrm{acc}}$ on spoke base size. Three different transverse spoke base shapes are presented, identified by the ratio of base width to length. For each data point, both the base width and length are changing, but the ratio remains fixed.

Figure 13

Dependence of (a) $R{R}_s$ and (b) energy content (at $1\mathrm{MV}/\mathrm{m}$) on spoke base size. Three different transverse spoke base shapes are presented, identified by the ratio of base width to length. For each data point, both the base width and length are changing, but the ratio remains fixed.

Figure 14

Dependence of the cavity diameter on the spoke base length. A cylindrical base has the highest surface fields, but the smallest cavity diameter.

Figure 15

(a) Dependence of the normalized electric and magnetic fields and (b) $R{R}_s$ and energy content (at $1\mathrm{MV}/\mathrm{m}$) on the spoke aperture length.

Figure 16

Dependence of (a) $E_p/E_{\mathrm{acc}}$ and $B_p/E_{\mathrm{acc}}$ and (b) $R{R}_s$ and energy content (at $1\mathrm{MV}/\mathrm{m}$), on the spoke aperture width.

Figure 17

Dependence of (a) $E_p/E_{\mathrm{acc}}$ and $B_p/E_{\mathrm{acc}}$ and (b) $R{R}_s$ and energy content (at $1\mathrm{MV}/\mathrm{m}$) on spoke aperture height.

Figure 18

Peak magnetic field vs peak electric field for various spoke aperture shapes.

Figure 19

Dependence of the (a) normalized electric and (b) magnetic fields on spoke aperture shapes.

Figure 20

Dependence of (a) $R{R}_s$ and (b) energy content (at $1\mathrm{MV}/\mathrm{m}$) on spoke aperture shapes.

Figure 21

Design of a ring-shaped aperture we are using to study possible reduction in multipole effects. This design is adapted from 37.

Figure 22

End view of a single spoke showing the spoke angle $\varphi$.

Figure 23

On-axis electric field profiles (at 1 J) for different spoke angle $\varphi$.

Figure 24

Dependence of the normalized electric and magnetic fields on spoke separation, for a spoke angle of 36°.

Figure 25

Dependence of $R{R}_s$ on spoke separation.

Figure 26

Velocity acceptance for two different spoke separations.

Figure 27

Dependence of the normalized magnetic field and $R{R}_s$ on the parameter h_out.

Figure 28

Dependence of the normalized magnetic field and $R{R}_s$ on the parameter H.

Figure 29

Dependence of the normalized electric field and $R{R}_s$ on the parameter H.

Figure 30

Dependence of the normalized electric field and $R{R}_s$ on the parameter h_in.

Figure 31

Cavity with no rounding at (1) the outer conductor edges and (2) the spoke base.

Figure 32

Dependence of $B_p/E_{\mathrm{acc}}$ on the outer conductor edge rounding.

Figure 33

Dependence of $B_p/E_{\mathrm{acc}}$ on the spoke base rounding radius.

Figure 34

Resonant electrons and their energies for field levels from 0.5–10 MV.

Figure 35

Resonant electron impact energy in an end cell of the cavity in Fig. 34 for gradients up to $10\mathrm{MV}/\mathrm{m}$. The inset shows the impact energies for stable multipacting electrons for two different end-cap outer rounding radii.

Figure 36

CST MWS© cut-away view showing the surface (a) electric and (b) magnetic fields of the fundamental mode of the optimized 325 MHz, ${\beta }_0=0.82$ cavity.

Figure 37

Dependence of $R{R}_s$ on the ratio of cavity surface area to volume.

Figure 38

Coupling $k$ between cells of an optimized ${\beta }_0=0.82$, 325 MHz two-spoke cavity. The leftmost point is common to both curves since it represents a cylindrical spoke base.

Figure 39

On-axis electric field profile for equal end cell lengths and an imbalance of 5 and 10 mm, for an optimized 325 MHz, double-spoke cavity. The imbalances are created by adding length to one end cell while keeping the spokes fixed.

Figure 40

On-axis electric field profile for equal end cell lengths and an imbalance of 5 and 10 mm for a double-spoke cavity with cylindrical spokes, operating at 325 MHz. The imbalances are created by adding length to one end cell while keeping the spokes fixed.