The starting point for the optimisation study was single cell pillbox simulations with infinite periodic boundary conditions. Although you can not make a cavity like this, it is useful to inform preliminary design decisions. The model overlooks end
3.2 Single Cell Pillbox Cavities 73
cells, a power coupler and cell to cell coupling. The cavity features an asymmetric blend which is shown in Figure 3.1, because typically structures of this kind are manufactured with multiple machined cups stacked upon one another. The machined cups can have blends on one side, but the other side is just the bottom of the next cup, which is a flat surface with no blend.
Fig. 3.1 A pillbox cavity vacuum model (left) Electric field profile (centre) and magnetic field profile (right).
Figure 3.1 shows the vacuum model of one cell of the standing wave cavity on the left with some cavity parameters labelled. The centre field map is a cut plane through the centre of the cavity and shows the electric field pattern inside, the phase flips byπ from one cell (blue) to the next (red) and a shielded area where
there is no field can be seen in the beam aperture between the cells (green). The right hand model is in isometric view and shows the magnetic field pattern on the surface of the cavity, the maximum magnetic field being on the outer wall.
The aperture radius of the pillbox cavity was swept from 0.5 mm to 5 mm
to investigate how the main cavity figures of merit vary with the iris aperture. Figure 3.2 shows the peak surface electric field, and the peak surface magnetic field inside the cavity normalised to the accelerating gradient. For the same accelerating gradient, the peak fields increase with frequency and aperture radius. The cavity simulated is almost completely featureless; there are no nose cones or coupling slots to enhance the field. The septum thickness’ are scaled with frequency, 1 mm for 12 GHz and 4 mm for 3 GHz. The wavelength at 12 GHz isλ12=25 mm and the
wavelength at 3 GHz is λ3=100 mm, which is why the aperture radii have different
effects on the peak fields depending on frequency.
The left hand plot of Figure 3.3 shows the shunt impedance per unit length values for a range of different apertures at the four preliminary frequencies. The shunt impedance includes the transit time factor defined in equations 2.51 & 2.52. The shunt impedance is a measure of the cavity’s ability to concentrate the accelerating field around the beam, so it is a desirable figure of merit to maximise. At a 5 mm aperture radius the S-band structure has the highest shunt impedance per unit length, but when you consider a 2 mm aperture radius the S-band structure
74 Small Aperture High Gradient Cavity Optimisation
Fig. 3.2 The peak surface electric field (left) and peak surface magnetic field (right) both normalised to the accelerating gradient.
has the lowest shunt impedance. The higher X-band frequencies have higher shunt impedance than the S- & C-band curves at smaller apertures. This is due to the size of the aperture radius relative to the wavelength. Figure 3.4 shows the same shunt impedance plot but this time with the aperture radii normalised to the wavelength. Here one can see the range of apertures investigated are from a much larger range of apertures for X-band than the lower frequencies. A 1 mm aperture radius is small by typical RF cavity standards, a high energy physics machine like the LHC for example has a large beam current and thus needs a large aperture to contain it. However, proton imaging does not require an intense beam to deliver the required dose, a small beam current of pA is sufficient. It is possible that a small aperture X-band structure could provide the high shunt impedance and thus high gradient required for this application.
The plot on the right of Figure 3.3 shows the square root of the modified Poynting vector (Sc) normalised to the accelerating gradient. Unsurprisingly it
follows the same pattern as the peak fields, as the Poynting vector is the cross product of the electric and magnetic fields. The power flow model suggests that the modified Poynting vector represents the available power to feed the field-emission mechanism which is believed to trigger RF breakdown [82]. Thus it is a quantity we wish to minimise.
These initial simulations have shown minimising the aperture also minimises the ratio of peak fields to gradient. It has also demonstrated a crossover whereby smaller apertures yield higher shunt impedance at X-band and larger apertures yield higher shunt impedances below 6 GHz. Knowing that proton imaging requires only minuscule beam current we will attempt to take advantage of the high shunt impedance at 12 GHz, with an aperture radius of 1.75 mm.
3.2 Single Cell Pillbox Cavities 75
Fig. 3.3 The shunt impedance per unit length and √Sc normalised to Eacc.
Fig. 3.4 The shunt impedance per unit length and the aperture radius divided by the wavelength of each respective frequency.
3.2.1
Beam Aperture
In this chapter a small aperture radius of 1.75 mm is considered because initial
beam dynamics calculations prior to this thesis suggested it would provide sufficient transmission through the structure. Figure 3.3 showed that for apertures below 2 mm X-band structures have the highest shunt impedance per unit length. The aperture is very small and for some applications it would be challenging to get enough transmission through it, but for proton imaging specifically very low beam current is necessary. A beam dynamics study is presented in Chapter 5 which investigates the feasibility of using an aperture radius this small for the ProBE project specifically. The study concluded that an aperture this small was not feasible for the Christie hospital application because the required quadrupole strengths
76 Small Aperture High Gradient Cavity Optimisation
and matching cell lengths are too high to be practical. However ADAM [114] with AVO [115] are developing a linac only solution for proton therapy called LIGHT [116]. The system does not use a cyclotron as a proton source and thus has no degrader making the beam size smaller right from the source. Next the beam passes through an RFQ which focusses the beam while also accelerating it, minimising the beam size further. A small aperture solution would serve as an upgrade for such a facility. Therefore the minimum aperture scheme is still relevant to the question of whether or not a linac upgrade is feasible for existing proton therapy centres.