ANTECEDENTES DEL ESTUDIO
TIPOS DE INVESTIGACIÓN
The small aperture beam dynamics scheme aims to benefit from the fact that pCT requires a smaller beam current than proton therapy linacs, or high energy physics experiments. It is thought that by minimising the beam size throughout the cavities, the aperture radius can also be minimised and higher gradients can be achieved as a result of higher shunt impedances. In the small aperture scheme, higher gradients are also necessary because more space between cavities must be occupied by magnets to keep the beam size small. Figure 5.2 shows a diagram of a beam envelope through an RF cavity, with the ‘waist’ or smallest point set in the centre of the structure. If the length of the cavity is reduced, so is the maximum beam size that exists within that cavity. This alongside the need for more magnets
146 Beam Dynamics
between cavities contributes to lower active acceleration length of the linac and increases the gradient requirement.
Fig. 5.2 Typical beam envelope through an RF cavity. The beam envelope is shown in red, and the smallest part of the beam or the ‘waist’ is shown in the centre of the cavity [129].
As the bunch length is long compared to the RF wavelength the beam parameters were optimised to maximise transmission for the entire RF cycle rather than a small range of phases. The cavity was modelled as a drift length as it was assumed the transverse and longitudinal fields over an entire RF cycle would cancel. Thus the transfer matrix for cavity was given as
x1 x′1 = 1 Lcav 0 1 x0 x′0 = x0+Lcavx′0 x′0 (5.1)
As is shown in Figure 5.2, the beam size is minimised in the central cell to minimise the overall beam size in the cavity. The aperture radius rap at the beginning and
end of the cavity can be projected onto phase space as shown in Figure 5.3. The vertical dashed lines represent the cavity aperture radius (rap) at the entrance to
the cavity and the diagonal dashed lines represent the aperture at the cavity exit. Inside of the parallelogram is the largest ellipse possible at the acceptance of the cavity. From this the acceptance Twiss parameters and the emittance at the start of the cavity were determined.
5.3 Cavity Aperture Study 147
Fig. 5.3 The allowed phase space region through an RF cavity with aperture radius
rap and the largest phase space ellipse possible inside of it [129].
Using Equation 5.1 the aperture limit at the end of the cavity was written as
x1 =x0+Lcavx′0 =±rap and the limits on phase space were then written as
x0 ≤rap (5.2) x0 ≥rap (5.3) x′0 ≤ rap−x0 Lcav (5.4) x′0 ≥ rap+x0 Lcav (5.5)
This describes a parallelogram in phase space but the acceptance will be the largest ellipse that fits inside of this parallelogram. The corners of the parallelogram were next expressed as P1 : (x, x′) = (rap,0) (5.6) P2 : (x, x′) = (rap,− 2rap Lcav) (5.7) P3 : (x, x′) = (−rap,0) (5.8) P4 : (x, x′) = (−rap,2 rap Lcav ) (5.9)
The midpoints between the corners of the parallelogram are where the ellipse touches it and were then expressed as
148 Beam Dynamics M1 : (x, x′) = (rap,− rap Lcav) (5.10) M2 : (x, x′) = (0,− rap Lcav) (5.11) M3 : (x, x′) = (−rap, rap Lcav ) (5.12) M4 : (x, x′) = (0, rap Lcav ) (5.13)
The points M1 − M4 were then equated to the corresponding points on the
parametrised phase space ellipse shown in Figure 2.9. From M4
s ε βT = rap Lcav (5.14) From M1 q εβT =rap (5.15) −αT s ε βT =− rap Lcav (5.16)
and from these the acceptance Twiss parameters and the geometric emittance for the minimum aperture scheme were determined in terms of the length of the cavity
Lcav and the radius of the aperture rap
βTacc =Lcav (5.17) αTacc = 1 (5.18) εgacc = r2ap Lcav (5.19)
and as we require the beam to be be focussing in both the horizontal and vertical planes at the start of the cavity, and defocussing in both planes at the output of the cavity, βTacc = βTout and αTacc = αTout, where βTout and αTout are the output
Twiss parameters.
The magnets between RF cavities that keep the beam on track are called matching sections. Having determined the optimum beam parameters at the acceptance and output of each cavity, the matching sections were designed. In order to match the four Twiss parameters βTacc, βTout, αTaccandαTout 4 degrees of
freedom were needed, and thus 4 quadrupoles in each matching section. Permanent Magnet Quadrupoles (PMQs) were chosen for this application due to their compact size. The aperture radius of the PMQs was chosen as 6 mm to allow for a large
5.3 Cavity Aperture Study 149
beam ‘blow-up’ between cavities, helping to minimise the beam size inside the cavities. The PMQs were assumed to be 3 cm each in length and 2 cm each side of the magnet was left free for practical purposes. This made the minimum length of each matching sectionLmatch =22 cm. The total linac length should not exceed
3 m to ensure it can fit in the allocated space in the Christie beam line. If the number of RF cavities in the linac is n and length of the cavity is Lcav then the
active accelerating length is nLcav assuming no matching section is required after
the final cavity there will ben−1 matching sections and
nLcav+ (n−1)Lmatch ≤3 (5.20)
A maximum gradient of 65 MV/m was assumed for the small aperture scheme
based on the optimisation studies presented in Chapter 3. This is comparable to what has been demonstrated experimentally for medium-β 3 GHz cavities (see
Chapter 7. Using a typical synchronous phase of 20◦ the accelerating gradient seen
by the beam will be lowered to 61 MV/m thus the active accelerating length of the
linac must be at least 1.64 m to gain 100 MeV. 6×22 cm matching sections will fit
in the remaining 1.36 m non active length thus the minimum aperture scheme can
consist of a maximum of 7 ×24 cm cavities.
The acceptance Twiss parameters and emittance derived in Equations 5.17 were used with the values Lcav=24 cm and rap=1.75 mm to design the matching
sections for the small aperture scheme. MAD-X [130] is a code developed at CERN to simulate beam dynamics and optimise beam optics, and it was employed to study the matching the sections outside of this work [129]. The study indicated a quadrupole strength ofK=367 m−2 or a field gradient of 1044 Tm−1 was necessary
to make the minimum aperture scheme feasible. Neodymium PMQs magnets can produce a pole tip field of up to 1.4 T. In this study the quadrupoles were assumed
to be 3 cm in length with a bore radius of 6 mm to allow for a large beam size between cavities.TheK-strength of a magnet is calculated as
K = g pc =
cg
βE (5.21)
whereg =B/rb is the quadrupole field gradient which is the pole tip fieldB divided
by the bore radius of the PMQ (rb). This makes the maximum K-strength at
330 MeV (β= 0.6729) 82 m−2. As the required K-strength of the magnets is over 4
times larger than the feasible K-strength of normal conducting magnets, the small
aperture scheme is deemed infeasible for the ProBE project. It could be feasible for a linac with more flexibility over the total length.
150 Beam Dynamics