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. Outline. Assumptions Space-charge effect? Two possible options Summary. Assumptions. Muon generation with 40,000 muons per bunch

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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

Initial consideration on the low- section of muon linac

4th meeting on muon g-2/EDM experiment

Jan. 30, 2009

Masanori Ikegami

KEK


Outline

Outline

  • Assumptions

  • Space-charge effect?

  • Two possible options

  • Summary


Assumptions

Assumptions

  • 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.


Kinetic parameters

Kinetic parameters

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


Transverse distribution

Transverse distribution

assuming

Analogously,


Longitudinal distribution

Longitudinal distribution

with

being the longitudinal position

of the design particle.

with

being the arrival time of the design

particle to a certain position.

assuming


Longitudinal distribution cont

Longitudinal distribution (cont.)

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.


Comparison with j parc mebt

Comparison with J-PARC MEBT

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

Envelope equation

Envelope equation for an axial symmetric bunched beam is;

; distance along the beam line

; external focusing strength

; unnormalized rms emittance


Envelope equation cont

Envelope equation (cont.)

; generalized perveance for a bunched beam

; number of particle per bunch

; classical radius of the particle

m for proton

m for muon


Envelope equation cont1

Envelope equation (cont.)

with

; peak current,

; bunch frequency,

; elementary electric charge

For J-PARC linac 30 mA operation,

For muon linac

103 difference in


Envelope equation cont2

Envelope equation (cont.)

; form factor

; aspect ratio in the beam frame

For  > 1,

 ~ 40 in muon linac@=0.08


Beam envelope for drift 0 08

Beam envelope for drift (=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


Two options

Two options

  • 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  210-5 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.


Transverse focusing

Transverse focusing

  • 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


Option 1 no longitudinal focusing with low frequency

Option 1: No longitudinal focusing with low frequency


Instability

Instability

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.


Tentative plan for option 1

Tentative plan for Option 1

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.


Tentative plan for option 1 cont

Tentative plan for Option 1 (cont.)

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.


Issues for option 1

Issues for Option 1

  • 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

  • Long total length

  • RF defocusing from RF errors


Option 2 with longitudinal focusing matched to the injected beam

Option 2: With longitudinal focusing matched to the injected beam


Matching condition

Matching condition

  • 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.


Matching condition cont

Matching condition (cont.)

Neglecting the space-charge term,

Then,

Similarly,

; Phase advance per meter for betatron oscillation

; Phase advance per meter for synchrotron oscillation


Matching condition cont1

Matching condition (cont.)

We here assume acceleration up to  = 0.7.

To keep ,

As , we should

satisfy the following condition,

At  = 0.7,


Matching condition cont2

Matching condition (cont.)

To keep ,

As ,

we should satisfy the following

condition,

or

At  = 0.7,


Longitudinal focusing force

Longitudinal focusing force

; Average accelerating field

; Synchronize phase

; Transit time factor

; RF wave length

; Rest mass


Rf defocusing

RF defocusing

; Focusing strength from quadrupole magnets

; Strength of RF defocusing

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


Case study 1

Case study 1

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.


Case study 1 cont

Case study 1 (cont.)

Ratio

Red: Requirement

Blue: Conventional DTL

Required

It may be possible to increase

assuming a short pulse.


Case study 1 cont1

Case study 1 (cont.)

Another possible way to vary is to adjust .


Tentative plan for option 2

Tentative plan for Option 2

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.


Case study 2

Case study 2

Higher frequency option

for  = 0.08

or

for  = 0.7

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


Case study 2 cont

Case study 2 (cont.)

Ratio

Red: Requirement

Blue: Conventional DTL

Required

It may be possible to increase

assuming a short pulse.


Case study 2 cont1

Case study 2 (cont.)

Another possible way to vary is to adjust .


Tentative plan for option 2 cont

Tentative plan for Option 2 (cont.)

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.


Issues for option 2

Issues for Option 2

  • 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?


Other possibilities

Other possibilities

  • 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


Summary

Summary

  • 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


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