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Status of HFOFO Snake for 6D Ionization Cooling. Y. Alexahin (FNAL APC). Nomenclature:  HFOFO = Helical FOFO channel of alternating solenoids (ASOL)  FOFO-xyz  = FOFO with xyz resonance phase advance per focusing cell consisting of one solenoid + all other stuff (RF cavities etc.)

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Status of HFOFO Snake

for 6D Ionization Cooling

Y. Alexahin (FNAL APC)

Nomenclature:

 HFOFO = Helical FOFO channel of alternating solenoids (ASOL)

 FOFO-xyz = FOFO with xyz resonance phase advance per focusing cell consisting of one solenoid + all other stuff (RF cavities etc.)

Talk outline:

 HFOFO-60 (6 cell period, Q1) - G4BL simulations

 HFOFO-120 (6 cell period, Q2) - smaller beta @ absorbers

 Really low-beta FOFO:

 FOFO-180 (2 cell period, Q1)

 HFOFO-270 (4 cell period, Q3)

 Plans

3rd Muon Collider Design Workshop, BNL, December 1-3, 2009


Basic Idea of HFOFO (6 Cell Example)

2

alternating solenoids

absorbers

RF cavities

• The idea: create rotating B field by periodically tilting solenoids, e.g. with 6-solenoid period.

• Periodic orbits for μ+ and μ- look exactly the same, just shifted by a half period (3 solenoids).

• With tune Q>1 (per period) rD>0

 muons with higher momentum make a longer path  longitudinal cooling achieved even with planar absorbers

B [T]

Bx50

Bz

By50

z [cm]

y

x, y [cm]

y

x

x

z [cm]

Periodic orbit for p=200MeV/c

In the subsequent plots all values in cm

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Beta-functions & tunes (200MHz HFOFO-60)

3

x

y

Solenoids: L=24cm, Rin=60cm, Rout=92cm, pitch 7mrad, Bzmax=2.35T (p=200MeV/c)

RF: 200 MHz pillbox 2x36cm, Emax=16MV/m

Absorbers: 15cm LH2 planar

at the absorber locations

<  > ~ 70cm

- compare with MICE’s

<  > ~ 45cm

The best results with 7mrad pitch angle, no absorber wedge angle:

mode I II III

tune 1.239+0.012i 1.279+0.007i 0.181+0.002i

_eq (mm) 3.2 4.5 6.9

ImQ=0.007  cooling rate d log  / dz = 22/L ImQ = 1/70m

Bmax=2.3T  j=58A/mm^2, Itot=4.4MA/solenoid

There was a difficulty in equalization of damping rates of the transverse modes

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Tune “repulsion” from integer resonance

4

no absorber, no RF

QI, II

Nice surprise:

Large 2nd order chromaticity due to nonlinear field components keeps both tunes from crossing the integer !

p/200

orbit length/L0

Momentum compaction factor:

- in contrast to classical HCC with homogeneous absorber where

to ensure longitudinal damping.

p/200

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Excursion - Longitudinal Hamiltonian

5

K

URF

=0.007

=0.014

p

p

 -0 (rad)

Kinetic energy in quasi-static approximation

Contour plot of the longitudinal Hamiltonian for =0.014 - kinematic nonlinearity limits momentum acceptance, not insufficient RF bucket depth.

where  = orbit length/L0

Longitudinal separatrix reaches maximum at  0.007

It seems I underestimated URF by a factor of 2 

RF phase angle can be increased from the present 27  faster cooling!

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Tracking with G4BL

6

(p-p0)/p0

px/p0

py/p0

x [cm]

t [ns]

y [cm]

Stochastic processes on, but no decays

mechanical momenta, p0=200MeV/c

blue - initial (the same ensemble), red – after 20 periods (122m)

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Analysis of G4BL Tracking Data

7

N/N0

no decays!

With decays transmission over 20 periods (122.4m) is > 85%

Initial/final N [cm]=1.7, 1.8, 2.1 / 0.37, 0.7, 1.1

6D cooling efficiency (following R.Palmer’s definition)

Q6  20

G4BL stochs. off

MICCD

G4BL stochs. on

Transmission vs period #

N [cm]

Recipe for emittance calculation:

 compute normal mode amplitudes am for each muon using lattice eigenvectors

subtract average values

find distribution in action variables Jm =| am |2

fit with exponential function

3N

2N

1N

Normal mode emittances (normalized) vs period # (_eq = 0.32, 0.45, 0.69)

multiply m by 00

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Damping Rate Equalization with Quadrupole Field

8

y (cm)

Dy (cm)

Dx (cm)

z

x (cm)

z

Most efficient “equalizer” is constant quadrupole field superimposed on the main solenidal field.

With a gradient of just 0.057 T/m:

mode I II III

tune 1.240+0.0088i 1.286+0.0087i 0.178+0.0036i

_eq (mm) 3.9 3.8 4.7

(w/o quad was 3.2 4.5 6.9)

 The sum of betas smaller - no increase in transverse emittance (despite smaller long. emittance)

 50% stronger long. damping due to larger dispersion (smaller solenoid pitch required)

 The effect does not depend on quad tilt (and polarity) as well as on the muon sign - the channel is still good for both!

 The quadrupole field does not have to be constant along the axis: it is high-beta region in solenoids that is most effective

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


G4BL Simulations with Quad “Equalizer”

9

N/N0

with quad

3N

1N

with quad

w/o quad

2N

period #

period #

w/o quad

3N

2N

1N

The ratio of 6D emittances and the cooling efficiency remain the same with quad on

period #

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Absorber Container and RF Cavity Windows

10

A MODERN AIRCRAFT ALUMINUM ALLOY (thanks to Don Summers)

Aluminum

Alloy Yield Tensile Radiation

Name Composition Density Strength Strength Length

% by weight g/cc ksi ksi cm

300K 300K 20K

6061-T6 1.0Mg .6Si .3Cu .2Cr 2.70 40 45 68 8.86

2090-T81 2.7Cu 2.2Li .12Zr 2.59 74 82 120 9.18

Supposedly 2090-T81 windows can be made 0.1mm thick (as compared to 0.18mm of 6061-T6 in MICE).

Three cases considered below:

No container nor cavity windows (original lattice)

Two 0.1mm 2090-T81 container windows per absorber, no safety nor cavity windows

Two 0.1mm 2090-T81 container windows per absorber + three 0.38mm Be windows per two RF cavities

Q1 Q2 Q31N (mm) 2N (mm) 3N (mm) ERF (MV/m)

No windows 1.239+0.012i 1.279+0.007i 0.181+0.0025i 3.2 4.5 6.9 16.0

Abs. winds 1.241+0.012i 1.281+0.007i 0.185+0.0026i 3.7 5.1 7.5 16.3

+ RF winds 1.243+0.013i 1.286+0.007i 0.191+0.0033i 4.0 5.8 7.3 17.5

Though the effect of the windows is tolerable (<30% increase in the equilibrium emittance) it would be worth while to look for a lighter material for the absorber windows (Be?, LiH??)

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


G4BL Simulations with Walls

11

3N

2N

1N

period #

no quad!

N/N0

w/o windows

with windows

period #

Final emittances for the cases 1 and 3:

1N (mm) 2N (mm) 3N (mm) 6D (mm^3)

No windows 3.67 7.01 10.96 218.2

With all windows 3.63 7.77 10.22 212.9

- Strange result, may be due to a faster cooling with 9% increase in losses and the RF gradient

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


HFOFO-60 Summary

12

 HFOFO snake with resonant dispersion generation allows to obtain large dispersion (~0.5m in the considered 200MHz case) and achieve 6D cooling even with planar absorbers.

 HFOFO snake can simultaneously cool both μ+ and μ- .

 Despite being resonant, the 200MHz HFOFO snake has sufficiently large momentum acceptance and transverse dynamic aperture to accept beam from Dave’s bunching / rotation section (G4BL simulations are in the pipeline)

 Additional quadrupole field (~constant along the axis) equalizes transverse damping rates without compromising cooling efficiency.

 HFOFO with 200MHz twin RF cavities gives T < 6mm, L ~10mm

 By reducing solenoid sizes and using single-cell RF the transverse emittance can be reduced to T ~ 3mm, while keeping L ~10mm

HFOFO characteristics can be further improved (dedicated effort is needed)

 Absorber and RF windows do not strongly affect 200MHz HFOFO performance, however, their effect on subsequent stages of 6D cooling (especially the effect of the absorber windows) will be very detrimental; studies of Be or LiH usage as the window material are necessary.

 If the answer is positive, one can project T ~ 1.5mm, L ~2.5mm for 800MHz HFOFO-60° channel

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


HFOFO-120°

13

Channel parameters (6-cell period, Q=2):

RF: 800 MHz pillbox RF 2x8cm, Emax=32MV/m

Solenoids: L=8cm, Rin=16cm, Rout=26cm,

Inclination 3mrad in rotated planes: 0, /3, 2/3, , 4/3, 5/3.

Absorbers: LiH, total width (on-axis) 1.2cm, full wedge angle 0.2rad

Total length of 6-cell period 1.68m

With p0=150MeV/c BLS=94.5T  Bzmax=11.5T

x

y

Bz/BLS

Dx

Dy

Bx/BLS

By/BLS

x

x

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


HFOFO-120°

14

Q2

Q1

p/p0

p/p0 – 1

With constant quadrupole field of just 1.45 T/m and no RF windows

mode I II III

tune 2.272+0.0052i 2.313+0.0053i 0.215+0.0029i

_eq (mm) 0.83 0.78 0.80

 HFOFO-120° channel with LiH absorbers provides ~ the same emittance as HFOFO-60° with LH2. No problem with absorber windows but more sensitive to outer RF windows due to larger beta there ( open cell on one side?)

 Momentum acceptance looks O.K., no tracking simulations performed yet but no problem expected

 800 MHz HFOFO-120° channel with LH2 absorber (if problem with windows solved) can give T ~ 0.5mm, L ~0.5mm

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


FOFO snake for final 6D cooling

15

Bz (T)

10*By

Phase advance >180/cell  small -function at absorber  smaller emittance.

With ~ half-integer tunes  PIC?

Simplest structure – just two solenoids:

Q1, Q2

=0.007

p/100

z

orbit lengthening

x,y [cm]

p/100

- no 2nd order chromaticity,

- momentum compaction too small and of the wrong sign!

z

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Helical snake for final 6D cooling

16

By increasing B-field strength it is possible to get phase advance >180/cell and small -function at the solenoid center  much smaller emittance.

Tune/period > odd_integer for resonant orbit excitation

Puzzle:

2-cell period (planar snake), Q>1

6-cell period, Q>3

4-cell period, Q>3

6-cell period, Q>5

HFOFO-270° channel parameters (4-cell period):

800 MHz pillbox RF 2x8cm, Emax=32MV/m

Solenoids: L=8cm, Rin=16cm, Rout=26cm,

Inclination: vertical +3mrad, horizontal +3mrad, vertical -3mrad, horizontal -3mrad

Absorbers: LiH, width (on-axis) 1.5cm, no wedge angle

Total length of 4-cell period 1.12m

Bzmax=18.5 T for p0=100MeV/c

p < 0

significant over-focusing required

p > 0

1

2

4

3

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


HFOFO-270°

17

y

x

x,y [cm]

Dy

Dx

- very small reduction in min at the price of significant increase in Bzmax

With constant quadrupole field of gradient 3.25T/m

mode I II III

tune 3.315+0.0083i 3.211+0.0085i 0.149+0.0025i

_eq (mm) 0.27 0.24 0.76

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


HFOFO-270° Momentum Acceptance

18

Q1

Q2

 Static momentum acceptance is fine: (-9%,+19%),

but the dynamic acceptance is just 16% (full width) – barely enough to accommodate the equilibrium momentum spread.

 To increase dynamic acceptance one can:

- reduce solenoid pitch angle (to reduce p)

- reduce p0 (to increase 1/2)

Both these methods are bad for longitudinal cooling –

 Careful optimization is necessary

 No tracking simulations yet, problems with transmission expected

p/p0

p/p0 – 1

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Return to FOFO-180°

19

Since there is no clear gain in going to 270° (just problems), more effort was applied to find solution for 180°. A superposed dipole field generated by additional coils can do the job. Such a field can not be obtained by displacing solenoids:

Bz / BLS

y

x

By / BLS

Dy

Dx

Bzmax=15.5 T for p0=100MeV/c - within the reach of Nb3Sn technology

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


FOFO-180° Momentum Acceptance

20

Q1 - 1

Q2 - 1

p/p0

p/p0 – 1

_min [cm]

This design is not finished yet

Search for a helical configuration is underway

Emittances well below 0.5mm can be expected

p/p0

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


Plans

21

 Search for the final stage HFOFO configuration (by IPAC10)

 Front end with HFOFO (by April 2010 IDS meeting)

 Determination of the number of stages and configuration for each stage (by the next MCDW)

 Optimization of each stage parameters (2010-2012)

 End-to-end simulation with matching between the stages (by 2012)

HFOFO Status - Y. Alexahin 3rd MCDW BNL December 2, 2009


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