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Initial consideration on the low- section of muon linac

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Initial consideration on the low- section of muon linac

4th meeting on muon g-2/EDM experiment

Jan. 30, 2009

Masanori Ikegami

KEK

- Assumptions
- Space-charge effect?
- Two possible options
- Summary

- Muon generation with
- 40,000 muons per bunch
- Isotropic momentum spread of 3 keV/c
- RMS size of 2.5 mm
- RMS pulse length of 3 psec

- Ideal initial acceleration to =0.08
- Simply adding the longitudinal momentum keeping the RMS beam size, RMS pulse length, and the momentum spread.

=0.08 corresponds to 3 MeV for proton.

MUONPROTON

Mass [MeV/c2]105.7938.3

0.080.08

1.00321.0032

0.08030.0803

p0 [MeV/c]8.4875.3

Ek [MeV]0.3403.02

assuming

Analogously,

with

being the longitudinal position

of the design particle.

with

being the arrival time of the design

particle to a certain position.

assuming

We here introduce

for the longitudinal

coordinate with

the fundamental frequency.

We assume

We also introduce

with

being the kinetic energy of the design particle.

x’ [mrad]

E [keV]

MUON

MUON

0.36

0.24

-2.5

2.5

-0.35

0.35

x [mm]

-0.36

[deg]

-0.24

0.084 keV•deg

0.025 mm•mrad

0.89 mm•mrad

J-PARC

J-PARC

x’ [mrad]

E [keV]

2.6

16

-1.1

1.1

-5.8

5.8

x [mm]

[deg]

93 keV•deg

3.15 mm•mrad

-16

-2.6

2.9 mm•mrad

Envelope equation for an axial symmetric bunched beam is;

; distance along the beam line

; external focusing strength

; unnormalized rms emittance

; generalized perveance for a bunched beam

; number of particle per bunch

; classical radius of the particle

m for proton

m for muon

with

; peak current,

; bunch frequency,

; elementary electric charge

For J-PARC linac 30 mA operation,

For muon linac

103 difference in

; form factor

; aspect ratio in the beam frame

For > 1,

~ 40 in muon linac@=0.08

Red: w/ space charge

Blue: w/o space charge

Red: w/ space charge

Blue: w/o space charge

The beam envelope for drift (without external focusing)

Obtained by integrating the envelope equation

No acceleration

Drift for 12 m or 0.5 sec

- Option 1: No longitudinal focusing with low frequency
- s is set to 0 deg (on crest).
- The frequency is chosen to be 324 MHz or lower.
3 ps 0.36 deg @324 MHz 210-5 deviation in energy gain

- Optional transverse focusing

- Option 2: With longitudinal focusing matched to the injected beam
- The focusing strength is chosen to be matched with the pulse width and momentum spread to avoid emittance growth from the nonlinear nature of the RF force.
- The transverse focusing is introduced, at least, to cancel the RF defocusing force.

- In both options, transverse beam manipulation before and after the accelerating section may be effective to ease the tolerance for the good field region.
- The increased transverse momentum spread in the accelerating section should be addressed.

Quad’s

Quad’s

beam

Accelerating section

x’

x

10-4

10-4

Num. of cells

Num. of cells

Non-relativistic approximation.

Start with 0.36 deg deviation from the crest.

Might be feasible with small number of cells.

Plan 1-A: DTL without DTQ

- = 0.08 to 0.7
Energy gain: 42 MeV

Total length: 21 m

Number of cells: 58

Time of travel: 0.18 s

Comparable to J-PARC DTL

It may be possible to increase assuming a short pulse.

Plan 1-B: Higher gradient DTL without DTQ

- = 0.08 to 0.7
Energy gain: 42 MeV

Total length: 10.5 m

Number of cells: 29

Time of travel: 0.09 s

is increased assuming a short pulse.

- Instability
- Small number of cells is required.
- Effect from abrupt cell variation?

- Small margin for the momentum spread specification
- Tolerance for RF phase and amplitude
- Transverse field uniformity

- Low shunt impedance at high- side
- Unable to introduce frequency jump
ZT2 ~ 8M/m @=0.7 extrapolated from J-PARC SDTL

- Unable to introduce frequency jump
- Long total length
- RF defocusing from RF errors

- We adopt the smooth approximation (or assume continuous focusing in both transverse and longitudinal).
- We find adequate external focusing strength to make the initial beam widths the equilibrium (matched beam widths).
- The external focusing strength can be found by solving the following equations.

Neglecting the space-charge term,

Then,

Similarly,

; Phase advance per meter for betatron oscillation

; Phase advance per meter for synchrotron oscillation

We here assume acceleration up to = 0.7.

To keep ,

As , we should

satisfy the following condition,

At = 0.7,

To keep ,

As ,

we should satisfy the following

condition,

or

At = 0.7,

; Average accelerating field

; Synchronize phase

; Transit time factor

; RF wave length

; Rest mass

; Focusing strength from quadrupole magnets

; Strength of RF defocusing

We need to introduce transverse focusing force to cancel the RF defocusing.

Comparable to J-PARC DTL

for = 0.08

or

for = 0.7

Conventional DTL can provide sufficient longitudinal focusing at low- side, but not at high- side.

Ratio

Red: Requirement

Blue: Conventional DTL

Required

It may be possible to increase

assuming a short pulse.

Another possible way to vary is to adjust .

Plan 2-A: 324 MHz E0-ramped DTL

- = 0.08 to 0.7
- Energy gain: 42 MeV
- Total length: ~14 m
- Number of cells:~40
- Time of travel:0.12 s

The longitudinal focusing is adjusted by ramping .

Transverse focusing can be provided with DTQ’s.

may be too high for high- side.

Higher frequency option

for = 0.08

or

for = 0.7

648 MHz DTL can provide sufficient longitudinal focusing at high- side with reasonable .

Ratio

Red: Requirement

Blue: Conventional DTL

Required

It may be possible to increase

assuming a short pulse.

Another possible way to vary is to adjust .

Plan 2-B: 648 MHz s-ramped DTL

- = 0.08 to 0.7
- Energy gain: 42 MeV
- Total length: ~17 m
- Number of cells:~94
- Time of travel:0.14 s

It is disadvantageous to ramp E0 to obtain required ks0 with higher frequency.

It may be possible to adopt higher gradient to shorten total length.

- Feasibility of precise s- or E0- ramping
- Susceptible to the longitudinal parameter of the injected beam
- Nonlinearity of the RF force in s-ramping scheme
- Low shunt impedance at high- side
- Frequency jump?

- Long total length
- Transverse focusing to cancel RF defocusing
- DTQ or external quad?

- To match the longitudinal focusing at the injection, but gradually introduce mismatch in the downstream portion without s- or E0- ramping
- Does the beam distribution adiabatically change?

- To manipulate the longitudinal beam parameter with buncher and debuncher before and after the accelerating section.

buncher

debuncher

beam

Accelerating section

z’

z

- Low- section for muon linac are considered assuming ideal acceleration up to =0.08.
- Space-charge effects seem to be negligible.
- Two options are proposed:Option 1
No longitudinal acceleration

Option 2

With longitudinal acceleration matched to the injected beam