Field stabilization of Alvarez-type cavities

Alvarez-type cavities are commonly used to reliably accelerate high quality hadron beams. Optimization of their longitudinal field homogeneity is usually accomplished by post-couplers, i.e., additional rods being integrated into the cavity. This paper instead proposes to use the stems that keep the drift tubes for that purpose. As their individual azimuthal orientations do not change the cavity’s undisturbed operational mode, they comprise a set of free parameters that can be used to modify higher mode field patterns. The latter have significant impact on the robustness of the operational mode with respect to eventual perturbations. Several optimized stem configurations are presented and benchmarked against each other. The path to obtain these configurations is paved analytically and worked out in detail through simulations. It is shown also by measurements that the method provides for flat field distributions and very low field tilt sensitivities without insertion of post-couplers.

Figure 1

Virtual division of an Alvarez-type cavity into three sections to be modeled by one transmission line each.

Figure 2

Equivalent circuit from three transmission lines for an Alvarez-type cavity.

Figure 3

Today’s first tank of the Alvarez-type DTL of the Universal Linear Accelerator UNILAC at GSI [18]. The stems are installed in the constant V-configuration.

Figure 4

Simulated current distribution on the drift tube surface corresponding to the inductance $L_t$ and electric gap field corresponding to the capacitance $C_t$ between tubes. The longitudinal electric field nearby the tubes is merged into this capacitance as well.

Figure 5

Dispersion curves of $T{M}_{01n}$-modes for an Alvarez-type cavity without stems from analytic calculation (green) and from simulations (brown).

Figure 6

Cross-configuration (a): the angle between two stems baring the same drift tube is 180 degree and at the subsequent tube the arrangement is rotated by 90 degrees. Alternating V-configuration (b): the angle between two stems baring the same tube is 90 degrees and at the subsequent tube the arrangement is rotated by 180 degrees. Both stem configurations have the same stem density in the cavity quadrant resulting in equal transverse admittances. The transverse admittance between tank and tube can be reduced to zero or even to negative values.

Figure 7

Equivalent inductances and capacitances of one cavity cell for the cross configuration and the alternating V-configuration. (a): current path inductance $L_s$; (b): capacitance $C$ from the electric field between red tank wall area and blue drift tube area.

Figure 8

Dispersion curves of stems with cross and alternating V-configuration. Solid lines are from simulations and circles are from Eq. (19).

Figure 9

Field and current pattern for two different $TE$-modes with constant V-configuration. (a): resonator between tube and upper part of the tank with the basic frequency of 149.2 MHz. (b): resonator between tube and lower part of the tank with the basic frequency of 65.0 MHz. (c): current pattern for case (b).

Figure 10

Equivalent circuit for one Alvarez-type cavity cell with two half tubes, one gap, and one pair of stems (see also Fig. 4). Capacitances between tank wall and stems are merged into $C_u$ and $C_l$.

Figure 11

Three dispersion curves from 3d simulation for the constant V-configuration, corresponding to three solutions of Eq. (13). (a) and (b) correspond to the two modes shown in Fig. 9.

Figure 12

Integration of post couplers with coaxial lines extending out of the two-stem cavity with constant V-configuration.

Figure 13

Field tilt sensitivity of the constant V-configuration without post couplers (blue) and with post couplers (red).

Figure 14

Local minimization of transverse admittance by grouping of stems. The frequency of the $T{E}_{010}$-mode depends on the stem number $m$ per group. For field stabilization $m$ must be determined such that its frequency gets close to the operating frequency ${\omega }_0$ of 108.4 MHz.

Figure 15

Frequency of the $T{E}_{010}$-mode as a function of $m$ for different cell lengths. The parameter $m$ is to be determined such that the $T{E}_{010}$-mode frequency is close to the operating frequency ${\omega }_0$ of 108.4 MHz at each section of the cavity.

Figure 16

Stem configuration $\mathbf{E}$ determined through tuning the frequency of the basic $TE$-mode. The increase of the cell length leads to a decrease of the stem density and $m$ decreases accordingly.

Figure 17

Longitudinal electrical field strength of the $T{M}_{011}$-mode as a function of the cell number for an Alvarez-type cavity for three different cases: original constant V-configuration of stems (blue), original constant V-configuration with the stem pairs of cell 14 and 34 being rotated by 180 degrees (purple), and final stem configuration after optimisation with respect to local field flatness as shown in Fig. 18.

Figure 18

Stem configuration $\mathbf{M}$ determined through tuning the local flatness of the $T{M}_{011}$-mode.

Figure 19

Simulated field tilt sensitivity of the operational mode as defined in Eq. (21) of an Alvarez-type cavity for different stem configurations. The applied perturbations are described in the text. Blue: a constant V-configuration without stabilization through stem rotation. Red: stabilization through stem configuration $\mathbf{E}$ of Fig. 16. Green: stabilization through stem configuration $\mathbf{M}$ of Fig. 18.

Figure 20

Simulated dispersion curves of an Alvarez-type cavity with different stem configurations. Blue: constant V-configuration without stabilization. Red: stem configuration $\mathbf{E}$ of Fig. 16. Green: stem configuration $\mathbf{M}$ of Fig. 18. Black: alternating V-configuration.

Figure 21

Aluminium model cavity of scale $1:3$ with 21 rf-gaps.

Figure 22

Measured (dots) and simulated (circles) field tilt sensitivities of the 21 gaps model cavity for two different stem configurations: constant V-configuration (blue) and $\mathbf{E}$ configuration (red).