# A comparative study of RF and induction linac approaches to phase rotation of a muon bunch in the production region of a {mu}{sup +}- {mu}{sup {minus}} collider Page: 5 of 7

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completed the energy spread is partially regenerating as the

beam moves along. Due to the head-to-tail energy variation,

the length of the beam pulse changes as it propagates through

the phase rotation linac. The transverse energy is a significant

fraction of the total energy and needs to be taken into account

when analyzing the longitudinal propagation of the pulse.3

0.8

0.6

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0

Fig. 3: Nor

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been previot

be repeated*

of the RF an

out and on w

Pions are

pulses impin

peaking in th

study we the

50 to 250 M

energy inter

spread in the

= 0.5*L. Fo

half a wave

within a wav

of the target

f - 100 MII

- 5 nsec. For

swing AB a

hysteresis l

Metglas 260

and hysteres

cAt - 40 m:

80 m from t

length is - 1

The acc

maximum edetermined by the Kilpatrick criterion relating Eoax in the

accelerating gap to the RF frequency f[4],

8.5

f(MHz) = I.64EK(MV / m)e EK(AIV/m) (1)and the enhancement factor by which the Kilpatrick field can

be safely exceeded. The enhancement factor is a function of

pulse length and is - 2 for pulse lengths 50 to 200 sec [5]

which spans the cavity fill times of interest for this study- For

the three frequencies f = 30, 50 and 90 MHz chosen for the RF

linac solution, the corresponding two times Kilpatrick fields

are 15, 19.8 and 21.8 MV/m. The acceleration gradients

averaged over the full cavity length were then calculated by

SFISH giving 2.1, 3.3 and 4.2 MV/m for the particular cavity

geometries that were specified. The practical gradient for the

lowest frequency case doesn't simply scale from the other two

and the Kilpatrick fields because in this case a folded cavity

geometry was chosen to reduce the cavity diameter, and this

0 2 4 6 8 10 causes some enhancement of surface field relative to average

field strength in the gap.

oT The accelerating gradient in the induction linac is limited

malized Fourier transform of the cavity voltage. by consideration of the properties of the induction core,

.s of the two approaches to phase rotation have dielectric breakdown, vacuum insulator surface flashover and

sly described [1,2] and for the most part will not vacuum breakdown. Details of the analysis are in ref. [2].

here. Instead we will concentrate on comparison Since voltage rise time requirements are relatively relaxed,

d induction linac solutions that have been worked t - 50 nsec, the induction cores and the vacuum insulators

hy they turned out the way they did. were stacked vertically to maximize accelerating gradient. The

gradient limiting factors were then insulator flashover and

II. DESIGN CONSTRAINTS vacuum breakdown. For pulse duration - 100 nsec, the

produced by at = 1 nsec 10 to 30 GeV proton insulator surface flashover field strength was taken to be

ging on a target and have kinetic energy spectra 50 kV/cm and the vacuum breakdown field 100 kV/cm. For

he range 100 to 200 MeV. For purposes of design induction cells occupying 40% of the axial length of the

refore chose a pion/muon kinetic energy interval accelerator the practical gradient was found to be - 1 MV/m.

eV for phase rotation. As the pions/muons in this If the breakdown limiting field strengths are increased by 50%

val propagate a distance L from the target the then the flux core would limit the practical gradient to

ir arrival times increases roughly according to cAt _ 1.5 MV/m. In summary the realizable accelerating gradients

r the RF approach cAt should not exceed about are two to three times higher in the RF linac than in the

length so the highest frequency cavities should be induction linac approach to phase rotation.

length L - X of the target, or within 2 - 3 meters For 2x1013 muons per pulse, the peak beam current in the

if the size of cavities is to be reasonable (say induction linac is less than 100 A and far less than the

z). The corresponding beam pulse lengths are induction core leakage current - 3.5 kA so waveform

the induction linac, cAt is determined by the flux distortion due to beam loading is negligible and does not

nd the maximum B that can be tolerated for introduce any design constraints. Furthermore because of the

)ss in the induction cores; cAt = c . . For relatively low beam current and a large beam tube radius

Bmax of 15 cm, beam breakup instability is not a concern in the

5SC, which has high saturation flux (AB = 2.5 T) induction linac,

is losses measured up to 5max - 20 T/ sec [3],

so the induction cavities should be placed about

he production target and the corresponding pulse

25 nsec.

elevating gradient in the RF linac is limited by the

electric field strength before breakdown. This is2

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Turner, W. C. & Kirk, H. G. A comparative study of RF and induction linac approaches to phase rotation of a muon bunch in the production region of a {mu}{sup +}- {mu}{sup {minus}} collider, article, October 1996; California. (https://digital.library.unt.edu/ark:/67531/metadc688227/m1/5/: accessed July 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.