s. kahn 5 june 2003nufact03 tetra cooling ringpage 1 tetra cooling ring steve kahn for v. balbekov,...

S. Kahn 5 June 2003 NuFact03 Tetra Cooling Ring Page 1

Tetra Cooling Ring

Steve KahnFor

V. Balbekov, R. Fernow, S. Kahn, R. Raja, Z. Usubov


Tetra Ring Parameters

Parameter Value

Circumference 36.954754 m

Kinetic Energy at Bends 0.250 GeV

Dipole Bending Field 1.453 T

Normalized Gradient Index 0.5

Maximum Long Solenoid Field 5.155 T

RF Frequency 205.69 MHz

Accelerating Gradient 15 MeV/m

LH2 Absorber Length 1.2 m

LiH Wedge Absorber 14 cm


Tetra Ring Simulations

• Original concept for this ring comes from V. Balbekov.– Originally simulated in Valeri’s program.– Documented:

• V. Balbekov et al., Muon Ring Cooler for the Mucool Experiment, Proc PAC 2001 Conf., p. 3867.

• Updated in MUCnote 249 (2002).

• GEANT simulation of Tetra Ring.– Worked on by Z. Usubov, R. Raja, and myself.

• ICOOL simulation of the Balbekov Ring.– MUCnote 258.


Hardedge Model

• Wedge Dipole:– Combined function

• Index =1/2 =52 cm defines reference radius

– Step function s dependence.• No dependence inside• Zero outside

• Solenoids– Effect of fringe field is approximated by transverse

impulse proportional to radial position.

8

45875214365

6

14336585

4

12815

2

245

8

2293767605

6

102445

4

38425

2

81

1

1

1

1

r

y

r

y

r

y

r

yBB

r

y

r

y

r

y

r

yBB

xBB

xr

xoX

yoY

Dyo


Long Solenoid Arrangement:

Boundary Condition Coils

Actual Hardedge Coils


Short Solenoid Arrangement:

Actual Hardedge Coils



+ - - ++-

Coil configuration to represent mirror plate boundary condition in ICOOL


ICOOL Hardedge Emittances

Tetra Ring Transmission

0

200

400

600

800

1000

1200

0 5 10 15 20

Turn Number

per

mil

Decay

Transverse Emittance

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

1.40E-02

0.00 5.00 10.00 15.00 20.00

Turn Number

Em

itta

nce

Decay

6D Emittance

1.0E-08

1.0E-07

1.0E-06

1.0E-05

0 5 10 15 20

Turn Number

Em

itta

nce

decay

Figure of Merit for Icool Tetra

0

20

40

60

80

100

0 5 10 15 20

Turn Number

Fig

ure

of

Me

rit


Tracking in GEANT

• This figure shows a sample of 500 events tracked in GEANT.

• The beam is smallest in the LH2 absorber where the field is largest.

• The beam is the largest in the field flip short solenoid.

– Muons are most likely to be lost in the vicinity of the bend magnets.


Emittances from GEANT

Transmission

4D Emittance

6D Emittance


Toward a Realistic Muon Cooling Ring

• The hardedge field description of this cooling ring violates Maxwell’s equations.

– It is likely that smoothing out a step function to a tanh or Enge function would solve this, but this has to be demonstrated.

• There is no free space in the lattice.

– This space would be necessary for flux returns for the solenoids and field clamps for the dipole magnet.

• Flux returns and field clamps are necessary to separate the function of the different lattice elements.

• Difficult engineering issues like how to inject (eject) beam into (out of) this ring.

– These kind of issues will be ignored at this point.


Saturation in Dipole Magnet

• Figure shows the permeability for the vertical midplane of the magnet.

<10 on inner edge of the aperture.


By Off Vertical Symmetry Plane

By on Different Planes

0

5000

10000

15000

20000

30 40 50 60 70

Radius, cmBy

, gau

ss

0 degrees

6 degrees

11 degrees

17 degrees

22 degrees

Angle Position index

0 0.473

5.625 0.469

11.5 0.516

17.125 0.584

22.5 0.746

Index Calculated on Difference Planes:


Dipole Field along Reference Path

Field Along Reference Path

0

2000

4000

6000

8000

10000

12000

14000

16000

-125 -75 -25 25 75 125

Path Position, cm

By,

gau

ss

Inside Gap

Field Clamp

Figure 4: Field components for a path displaced 10 cm vertically from the

reference path

Fields 10 cm Off Axis in Dipole Magnet

-10000

-5000

0

5000

10000

15000

-125 -75 -25 25 75 125

Path Position, cm

B,

ga

uss

By

Bz

Bx

Figure 3: By along central

reference path.


Field and Geometry of the Long Solenoid


Field and Geometry of the Short Solenoid

Effect of Mirror Plate

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

-150 -100 -50 0 50 100 150

Axial Position, cm

Axi

al F

ield

, g

auss

no mirror

part mirror

full mirror

•Figure at left shows Bs for cases:

•Mirror plate boundary condition

•Partial mirror plate with 18 cm aperture

•No mirror plate. Full 29 cm aperture

cases threeallfor same thebe should dlBs


Comparison of Realistic to Hardedge Field


Inserting a Gap into the Lattice

• Part of the difficulty with the Tetra ring is that there is no extra space in the lattice for flux return, field clamps, etc.

• We have studied what is necessary to add a gap between the end of the solenoids and the dipole magnet:

Dipole MagnetLong Solenoid Field Flip SolenoidExtra Focusing Coils


Inserting a Gap

• The extra focusing coils are placed symmetrically at the ends of the solenoids into the lattice to compensate and to match into the bending dipoles.

• The requirements on the focusing coils are– They retain the focusing of the solenoid, ie is unchanged.– The value of Bs at the absorber remain unchanged.

• These requirements uniquely specifies the focusing and other solenoid currents.

• The RF frequency must be changed to account for the additional length.– The harmonic number is not changed.

• The wedge angle in the field flip solenoid should be adjusted for the focusing coil and other solenoid current changes.

dsBs 2


Field Flip Solenoid Field with Extra Focusing Coil

Original Coil Configuration Adjusted with extra focusing coil

•Difference of 5º phase between these two configurations. This is not corrected for.


Effect of Placing a Gap Between the Dipole Magnet and the Solenoids

Transmission for Fixed Momentum

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200

Focusing Coil Current

Tra

nsm

issi

on

Nominal

5 cm

10 cm

15 cm

Transverse Cooling for Fixed Momentum Muons

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200

Focussing Coil Current

Tra

nsv

erse

Em

itta

nce

Nominal

5 cm

10 cm

15 cm

Longitudinal Emittance for Fixed Momentum Muons

0

100

200

300

400

500

600

700

0 50 100 150 200

Focussing Current

Lo

ng

itu

din

al E

mit

tan

ce

Nominal

5 cm

10 cm

15 cm

•Curves show transmission, tr, L vs. extra focusing coil current.

•Cases shown are for 5cm, 10 cm, and 15 cm gaps.

•PL is held constant and no decays in this comparison.


Effect of Placing a Gap Using the Whole Momentum Range

Transmission for Whole Momentum Range

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200


Tra

ns

mis

sio

n

5 cm

Nominal

10 cm

15 cm

Transverse Cooling for Full Momentum Range

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200


Tra

ns

ve

rse

Em

itta

nc

e

5 cm

Nominal

10 cm

15 cm

Long Cooling for Full Momentum Range

0

1

2

3

4

5

6

7

8

0 50 100 150 200


Lo

ng

itu

din

al

Em

itta

nc

e 5 cm

Nominal

10 cm

15 cm

•Gaussian distribution for PL with P=18 MeV/c.

•Plots show T, tr, L vs. focusing coil current.

•Transmission drops with increasing gap


A More Realistic Description of the Solenoids in ICOOL

• As a step toward a more Maxwellian description to the solenoid fields was tried:

– Mirror plate boundary conditions are removed in solenoid regions.

– Fringe fields from solenoid sheets are superimposed on the dipole region.

• The solenoid fringe field along the reference path is the axial field. This, of course, is not correct.

– The solenoid end kicks used to describe the fringe fields are removed.

• The wedge bend magnet is still the hardedge model.

• The following transparency shows the emittance calculated in ICOOL for this scenario.


ICOOL Emittances with Real Solenoids

Transmission for Tetra with Real Solenoids

0

200

400

600

800

1000

1200

0 5 10 15

Turn Number

Ev

en

ts

Transmission

Transverse Em ittance for Realis tic Solenoids

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0 5 10 15 20

Turn Num ber

Em

itta

nce

eperp

Merit Factor for Tetra with Real Solenoids

0

20

40

60

80

100

120

0 2 4 6 8 10 12 14 16

Turn Number

Mer

it F

acto

rMerit

6D Emittance

1.E-08

1.E-07

1.E-06

1.E-05

0 5 10 15 20

Turn Number

Em

itta

nc

e

em it6D