1. Beam dynamics and linac optics studies for medical proton accelerators
R. Apsimon, G. Burt, S. Pitman
Lancaster University / Cockcroft Institute
H. Owen
Manchester University / Cockcroft Institute

2. Overview Brief introduction to cyclinacs Beam dynamics Linac optics
Transmission through a cavity
Optimisation of beam parameters
Linac optics
Spatial considerations
Quadrupole considerations

3. Brief introduction to cyclinacs I
Cyclotron protons accelerated through linac
Allows final beam energy to be varied
Allows treatment and imaging in single machine
~250 MeV protons for treatment
~350 MeV protons for imaging
Due to Bragg peak, protons deposit most of their energy in a very small range
Minimises dose to healthy tissue
Ideal for cancer treatment

4. Energy deposition through tissue

5. Brief introduction to cyclinacs II
𝑓 𝑐𝑦𝑐𝑙𝑜𝑡𝑟𝑜𝑛 = 𝑒𝐵 2𝜋𝑚 =15.25𝐵 MHz
cyclotron
RF linac
ProBE cyclinac:
MeV cyclotron extraction
330 MeV exit of RF linac
(50 12 GHz)

6. Brief introduction to cyclinacs III
Typical frequencies
𝑓 𝑐𝑦𝑐𝑙𝑜𝑡𝑟𝑜𝑛 ~ 30 −100 𝑀𝐻𝑧
𝑓 𝑅𝐹 ~ 3 −12 𝐺𝐻𝑧
𝜎 𝑧,𝑐𝑦𝑐𝑙𝑜 ≈ 2 3 𝜆 𝑐𝑦𝑐𝑙𝑜𝑡𝑟𝑜𝑛 ≫ 𝜆 𝑅𝐹
Need to optimise linac and beam parameters for all RF phases (before RF capture)

7. Christie Hospital proton therapy centre
230 MeV cyclotron provides protons for cancer treatment in treatment rooms.
3m of space for linac in R&D room to allow protons for imaging
3 treatment rooms
1 R&D room 1 2 3 R

8. R&D 4th room

9. Christie Hospital: R&D room
3m of beam line to allow for tests of linac for future upgrade of R&D room into an integrated imaging and treatment cyclinac.
~1.8 – 2 m for RF cavities
~1 – 1.2 m for matching sections between cavities
Matching sections must be at least 20 cm to allow for quadrupoles, bellows, flanges and diagnostics; limiting the total number of cavities to 6.
Need to determine transmission through cavity to determine maximum cavity length and therefore minimum number of cavities in linac.
Need ~10% transverse transmission through a cavity for scheme to be feasible.

10. Transmission through a cavity
2 𝑟 𝑐𝑎𝑣
𝐿 𝑐𝑎𝑣
𝑆 0
𝑆 1

11. Transmission through a cavity
𝑥 0 = −𝑟 𝑐𝑎𝑣
𝑥 0 = 𝑟 𝑐𝑎𝑣
𝑥 1 = 𝑟 𝑐𝑎𝑣
Maximum area phase space ellipse inside parallelogram touches the midpoint of each line of the parallelogram. This allows us to determine the beam parameters
𝑥 1 =− 𝑟 𝑐𝑎𝑣

12. Transmission through a cavity
Taking into account the energy gain through the cavity the equations for the lines of the parallelogram can be expressed in terms of the transfer matrix through the cavity:
𝑥 0 =± 𝑟 𝑐𝑎𝑣
𝑥 1 =± 𝑟 𝑐𝑎𝑣 = 𝛽 𝑟0 𝛾 𝑟0 𝛽 𝑟1 𝛾 𝑟1 𝑅 11 𝑥 0 + 𝑅 12 𝑥′ 0
And the maximal phase space ellipse touches the parallelogram at:
𝑥 0 =± 𝑟 𝑐𝑎𝑣 𝑥′ 0 =∓ 𝑅 11 𝑟 𝑐𝑎𝑣 𝑅 12
And
𝑥 0 =0 𝑥′ 0 =± 𝛽 𝑟0 𝛾 𝑟0 𝛽 𝑟1 𝛾 𝑟1 𝑟 𝑐𝑎𝑣 𝑅 12
The normalised emittance for the maximal phase space ellipse is:
𝜀 𝑛,𝑎𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 = 𝛽 𝑟0 𝛾 𝑟0 𝛽 𝑟1 𝛾 𝑟1 𝑟 𝑐𝑎𝑣 2 𝑅 12

13. Transmission through a cavity
The normalised 1σ emittance of the beam from the cyclotron is:
𝜀 𝑛,𝑐𝑦𝑐𝑙𝑜𝑡𝑟𝑜𝑛 ~5 mm mrad
If we assume that the beam is Gaussian, then we can estimate the transverse transmission, 𝑇 𝑡𝑟𝑎𝑛𝑠 , of the cavity:
𝑇 𝑡𝑟𝑎𝑛𝑠 =erf 𝜀 𝑛,𝑥 𝜀 𝑛,𝑦 𝜀 𝑛,𝑎𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒 2 𝜋 𝜀 𝑛,𝑥

14. Transmission through a cavity

15. Transmission through a cavity
Recall that the lines defining the acceptance region of phase are:
𝑥 0 =± 𝑟 𝑐𝑎𝑣
𝑥 1 =± 𝑟 𝑐𝑎𝑣 = 𝛽 𝑟0 𝛾 𝑟0 𝛽 𝑟1 𝛾 𝑟1 𝑅 11 𝑥 0 + 𝑅 12 𝑥′ 0
And the maximal phase space ellipse touches the parallelogram at:
𝑥 0 =± 𝑟 𝑐𝑎𝑣 𝑥′ 0 =∓ 𝑅 11 𝑟 𝑐𝑎𝑣 𝑅 12
And
𝑥 0 =0 𝑥′ 0 =± 𝛽 𝑟0 𝛾 𝑟0 𝛽 𝑟1 𝛾 𝑟1 𝑟 𝑐𝑎𝑣 𝑅 12

16. Optimisation of beam parameters
Since we know the locations where the phase space ellipse touches the parallelogram, we can relate those positions to Twiss parameters:
𝜀 𝛽 = 𝛽 𝑟0 𝛾 𝑟0 𝛽 𝑟1 𝛾 𝑟1 𝑟 𝑐𝑎𝑣 𝑅 𝛽𝜀 = 𝑟 𝑐𝑎𝑣 𝛼 𝜀 𝛽 = 𝑅 11 𝑟 𝑐𝑎𝑣 𝑅 12
And we obtain the optimal beam parameters at the entrance of the cavity as:
𝛽 0 = 𝛽 𝑟0 𝛾 𝑟0 𝛽 𝑟1 𝛾 𝑟1 𝑅 𝛼 0 = 𝛽 𝑟0 𝛾 𝑟0 𝛽 𝑟1 𝛾 𝑟1 𝑅 11
𝜀 𝑛 = 𝛽 𝑟0 𝛾 𝑟0 𝛽 𝑟1 𝛾 𝑟1 𝑟 𝑐𝑎𝑣 2 𝑅 12
And at the end of the cavity as:
𝛽 1 = 𝛽 𝑟1 𝛾 𝑟1 𝛽 𝑟0 𝛾 𝑟0 𝑅 𝛼 0 =− 𝛽 𝑟1 𝛾 𝑟1 𝛽 𝑟0 𝛾 𝑟0 𝑅 11

17. Optimisation of beam parameters
As the bunch length is much greater than the RF wavelength, we need to optimise the cavity for all phases. If we average the transfer matrix of an RF cavity for all phases, we get a drift length and the optimal Twiss parameters become:
𝛽 0 = 𝛽 1 = 𝐿 𝑐𝑎𝑣
𝛼 0 =− 𝛼 1 =1
𝜀 𝑛 = 𝛽 𝑟 𝛾 𝑟 𝑟 𝑐𝑎𝑣 2 𝐿 𝑐𝑎𝑣

18. Linac optics Maximise transmission:
Need to match beam parameters through each cavity
Need quadrupole matching section
Medical proton accelerators must be as compact as possible
Need to minimise length of matching sections
Maximise accelerating gradient of cavities

19. Quadrupole considerations
Each matching section ~24 cm
Need strong quadrupole fields (250 – 400 m-2) to match optics in such small space:
Permanent magnet quadrupole
Small aperture electromagnetic quadrupoles
Quadrupole length: 3 – 3.5 cm
Bore radius: 1.5 – 2.5 mm
Drifts between quads: ~2 – 6 cm
Need minimum of ~ 2 cm for bellows, flanges etc.

20. How many quadrupoles per matching section?
Need to match 4 constraints ( 𝛽 𝑥 , 𝛽 𝑦 , 𝛼 𝑥 and 𝛼 𝑦 ) so need 4 degrees of freedom. So the options are:
2 quads + 2 drift
Relatively long drifts
Large beam size through quadrupole aperture
Large losses
3 quads + 1 drift
Drift length > 2 cm (due to bellows, flanges, diagnostics…)
Drift length < 4 cm (to prevent large beam losses through quads)
Range of drift length too small to find perfect solution
4 quads (all drift lengths fixed)
Perfect solution exists
Can tweak quad lengths and bore radii to give more favourable parameters if needed

21. 3 m
X-band cavity
Permanent magnet quadrupole (PMQ)
30 cm
24 cm
3 cm
2.4 cm

22. Conclusions
Studies of transverse beam dynamics through cavities have been undertaken:
Maximum transverse transmission through a cavity determined
Verified against tracking simulations
Determined optimal beam parameters through cavity to give maximum transverse transmission
Matching sections designed to match beam parameters between cavities within the spatial limits.
Still to do:
Study beam dynamics through full linac structure
Only single cavities investigated so far
Verify transmission studies for short bunches
Design matching sections for entire linac structure