Longitudinal and vertical focusing with a field gradient spiral inflector

Traditional spiral inflectors of the Belmont-Pabot type are commonly used for axial injection of external ion beams into cyclotrons. These inflectors are designed to control the trajectory of the central path, and do not actively focus the beam in the vertical and longitudinal directions. This can introduce effects such as a large vertical divergence and a debunching longitudinal spread, making it difficult to match the injection line emittance to the cyclotron acceptance. In an attempt to overcome this, some recent inflectors have started incorporating electrodes specially shaped to produce field gradients along the central path, thereby influencing the inflector optics. This method has shown some success, and at iThemba LABS an inflector was built exhibiting good vertical focusing. However, it performed poorly longitudinally, worse than traditional spiral inflectors. In this article a generalized field gradient spiral inflector design is presented, based on a mathematical description of all possible first-order field gradients along the central path. Such a design is numerically optimized to simultaneously focus longitudinally and vertically. Experimental studies of this design show a 60% improvement in overall current extracted from the cyclotron.

Figure 1

A traditional spiral inflector (left) showing large vertical divergence, and a field gradient spiral inflector (right) with much improved vertical behavior.

Figure 2

Electrode cross sections in the transverse plane: The Dubna V shape (left) and the Sumitomo angled electrodes (right).

Figure 3

Optical coordinate system around the central path (left). Electrode cross section of a Belmont-Pabot inflector in the transverse $(u_r,h_r)$ plane (right).

Figure 4

Experimental transmission through SPC2 when using the vertically focusing C2-inflector, as a percentage of the standard C1-inflector transmission. Solid lines are with the buncher and dotted lines are without the buncher.

Figure 5

Creating normal quadrupole field gradients by using hyperbolic electrodes (left), and creating skew quadrupole field gradients by using cylindrical electrodes (right). The left-hand method is similar to the angled electrodes used by Sumitomo and by C2, while the right-hand method is similar to the V-shaped Dubna design shown in Fig. 2.

Figure 6

Creating transverse field gradients in the fringe field region by cutting both electrodes in the $(h_r,s)$ plane (left) and in the $(u_r,s)$ plane (right).

Figure 7

The designer $Q_1$ and $Q_2$ (in ${\mathrm{m}}^{-1}$) versus the path length (in mm) as used in the verification test. These curves have been chosen arbitrarily and are piecewise smooth inside the inflector.

Figure 8

Electric field gradients (in ${10}^7{\text{Vm}}^{-1}$) along the path length of the inflector (in mm), obtained from a 3D TOSCA model. The dotted lines are the numerical field gradients, and the solid lines are the gradients calculated according to Eq. (16).

Figure 9

Phase-space plots of the beam at the first acceleration gap for inflectors C1 (left column), C2 (middle column) and C3 (right column). The top row is vertical $(u,u^{\prime })$ space, the middle row is horizontal $(h,h^{\prime })$ space and the bottom row is longitudinal $(\ell ,\delta )$ space. The C2-inflector has greatly improved vertical behavior compared to the original C1 inflector, but a much larger longitudinal spread. The C3 inflector has moderately improved vertical behavior compared to C1 as well as a smaller longitudinal spread.

Figure 10

Plot of the numerically measured gradient parameters $Q_1$ (dashed line) and $Q_2$ (solid line) versus the path length (in mm) along the inflectors. Note that the C2 graph has a different vertical scale.

Figure 11

3D models of the C-inflectors, as they were manufactured: C1 (left), C2 (middle) and C3 (right). C1 has parallel electrodes in the $(u_r,h_r)$ plane, C2 has straight nonparallel lines, and C3 has quadratic lines. For C2 the entrance and exit edges are not perpendicular to the central path.

Figure 12

The transmission of C2 and C3, as a percentage of C1 transmission, without a buncher. C2 performs very well, since it was optimized for vertical focusing only, and C3 shows a small improvement.

Figure 13

The transmission of C2 and C3, as a percentage of C1 transmission, with the buncher active. The relative transmission of C3 increases when the buncher is activated, while that of C2 reduces.

Figure 14

Buncher efficiency: the extracted current when the buncher phase is varied, as a fraction of the no-buncher current. The phase has been normalized so that the buncher performance is optimal at zero degrees.