Low-Energy Ionization “Cooling”

David Neuffer

Fermilab

2

Outline Low-energy cooling-protons ions and “beta-beams” Low-energy cooling – muons

emittance exchange

3

Combining Cooling and Heating:

Low-Z absorbers (H2, Li, Be, …) to reduce multiple scattering High Gradient RF

To cool before -decay (2.2 s) To keep beam bunched

Strong-Focusing at absorbers To keep multiple scattering less than beam divergence …

Quad focusing ? Li lens focusing ? Solenoid focusing?

ds

d

2ds

dE

E

1

ds

d2

rms

N2N

2 2 2

2 2 2

rms s

R

d z E

ds c p L

4

Low-energy “cooling” of ions

Ionization cooling of protons/ ions has been unattractive because nuclear reaction rate is competitive with energy-loss cooling rate And other cooling methods are available

But can have some value if the goal is beam storage to obtain nuclear reactions Absorber is also nuclear interaction medium Y. Mori – neutron beam source

– NIM paper

C. Rubbia, Ferrari, Kadi, Vlachoudis – source of ions for β-beams

5

Cooling/Heating equations

g

P (MeV/c)

gL

0

-1

Cooling equations are same as used for muons mass = a mp, charge = z e

Some formulae may be inaccurate for small β=v/c Add heating through nuclear interactions Ionization/recombination should be included

For small β, longitudinal dE/dx heating is large At β =0.1, gL = -1.64, ∑g = 0.36

Coupling only with x cannot obtain damping in both x and z

0.0 1.0 3.0 5.0

2

2

2 2 2e

β

γL 2 2m c β γ 2

I(Z)

2(1- )2g - +

γ ln -β

2

2

2 2 2e

β

γ2g

2m c β γ 2I(Z)

2(1- )Σ 2β +

ln - β

6

μ Cooling Regimes

Efficient cooling requires:

Frictional Cooling (<1MeV/c) Σg=~3

Ionization Cooling (~0.3GeV/c) Σg=~2

Radiative Cooling (>1TeV/c) Σg=~4

Low-εt cooling Σg=~2β2

(longitudinal heating)

~ 0

dEdx

E

2 2 222 2

2 2

214 ln 1

( ) 2e

A e e

m cdE ZzN r m c

ds A I Z

7

Miscellaneous Cooling equations

,

2 2

, 22

z a

sN eq dE

x p R ds

z E

g am c L

2

22rms 2 2 2

e e e 2

d ΔE= 4π r m c z n γ 1-

ds

2 2dE 22 2 2 2L dsE

E e e e2

gdσ β= -2 σ + 4π r m c z n γ 1-

ds β E 2

22 2 4 3

2, 2

( )( )1

2 ln[]e p

E eqL

m c am c

g

For small β:

2,

2, 2 ln[]p eq e

z a p L

m

P am g

Better for larger mass?

2 2 22e2m c γ β

ln[] ln -βI(Z)

≡

2 2 222 2 e

A e e 2 2

2m c γ βdE Zz 1 δ= 4πN ρ r m c ln -1-

ds A β I(Z) 2β

8

Example ERIT-P-storage ring to

obtain directed neutron beam (Mori-Okabe, FFAG05)

10 MeV protons 9Be target for neutrons σ ≈ 0.5 barns β = v/c =0.145 Large δE heating

Baseline Absorber 5µ Be absorber δEp=~36 keV/turn

Design Intensity 1000Hz, 6.5×1010p/cycle 100W primary beam < 1.5 kW on foil

– 0.4 kW at nturns =1000

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ERIT results

With only production reaction, lifetime is 30000 turns

With baseline parameters, cannot cool both x and E Optimal x-E exchange increases storage time from

1000 to 3000 turns (3850 turns = 1ms)

With x-y-E coupling, can cool 3-D with gi= 0.12 Cooling time would be ~5000 turns With β=0.2m, δErms = 0.4MeV, ,N = 0.0004m (xrms=2.3cm)

xrms= 7.3cm at β=2m (would need r =20cm arc apertures)

(but 1ms refill time would make this unnecessary)

10

3-D mixing lattice For 3-D cooling, need to mix with both x and y Possible is “Moebius” lattice (R. Talman)

Single turn includes x-y exchange transport– solenoid(s) or skew quads

– in zero dispersion region for simplicity

– Solenoid: BL = π Bρ

– For 10 MeV p : BL = 1.44T-m

Complete period is 2 turns

11

Emittance exchange parameters

with gL,0 = -1.63, η = 0.5m,

need G = 3.5 to get gL=0.12

– (LW =0.3m)

Beam Center

Geometry of wedge absorber

LW=1/G

L L,0 L,00

ηρg g + = g + Gη

ρ

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“Wedge” for thin foil Obtain variable thickness by bent

foil (Mori et al.)

Choose x0=0.15m, LW=0.3, δo=5µ

then a=0.15, δR=2.5

for gL,0 = -1.63, η = 0.5m

Barely Compatible with β* = 0.2m

Beam energy loss not too large? <1kW Power on foil

2

2

2

2ln

2

x xy dx

a a

a x x axa x a x

a

a=0.15

( )20( ) 1 ( ) (1 ) (1 )o

W

x x xR RL ax y x

Beam

13

Other heating terms: Mixing of transverse heating with longitudinal could be

larger effect: (Wang & Kim)

dsd

21

dsd

z21

zdsdP

Lz

2rms

z

22rms

Pg

ds

d

2 2 2rms s

2 2 2a R

d θ z E=

ds β c P L

22 2 2

2 2rmse e A 22 2

a

d( ) Z z4 (r m c ) N 1

ds A P

2 22rms rms

x

dPd ddsx 1 1

x x x2 ds 2 ds

dg

ds P

At ERIT parameters: βx= 1.0m, βz= 16m, η=0.6m, Be absorber,

dδ2/ds= 0.00032, dθ2/ds= 0.0133 only 5%, 1.5% changes …At βx= 0.2m, ~25% change …

2 2T

2 1 1RF γ γ

LRF S

β PcCλ ( - )β =

2πeV cosφ

14

Beta beams for neutrinos (Zuchelli)

ν-sources

v-sources Number of possible sources

is limited (v sources easier) Want:

lifetime ~1s large ν-energy ν and ν* easily extracted atoms

Noble gases easier to extract 6He2 “easiest” Eν* =1.94MeV

18Ne10 for ν Eν =1.52MeV

Noble gas

(8B, 8Li) have E ν ,ν * =~7MeV

Want 1020 ν and ν* / year…

Shortcut to Local Disk (C).lnk

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β-beam Scenario

Neutrino Source

Decay Ring

Ion production: Target - Ion

Conversion Ring

Ion Driver

25MeV Li ?

Decay ring

B = 400 Tm B = 5 T C = 3300 m Lss = 1100 m

8B: = 80

8Li: = 50Acceleration to medium energy

RCS

8zGeV/c

Acceleration to final energy

Fermilab Main Injector

Experiment

Ion acceleration Linac

Beam preparation ECR pulsed

Ion production Acceleration Neutrino source

,

,

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β-beam Scenario (Rubbia et al.)

Produce Li and inject at 25 MeV Charge exchange injection

nuclear interaction at gas jet target produces 8Li or 8B

Multiturn with cooling maximizes ion production

8Li or 8B is caught on stopper(W) heated to reemit as gas

8Li or 8B gas is ion source for β-beam accelerate

Accelerate to Bρ = 400 T-m Fermilab main injector

Stack in storage ring for: 8B8Be + e++ν or 8Li8Be + e-+ ν*

neutrino source

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Cooling for β-beams (Rubbia et al.)

β-beam requires ions with appropriate nuclear decay 8B8Be + e+ + ν 8Li8Be + e- + ν*

Ions are produced by nuclear interactions 6Li + 3He 8B + n 7Li + 2H 8Li + 1H Secondary ions must be collected and reaccelerated

Either heavy or light ion could be beam or target Ref. 1 prefers heavy ion beam – ions are produced more forward

(“reverse kinematics”)

Parameters can be chosen such that target “cools” beam (losses and heating from nuclear interactions, however…)

18

β-beams example: 6Li + 3He 8B + n Beam: 25MeV 6Li+++

PLi =529.9 MeV/c Bρ = 0.59 T-m; v/c=0.09415

Absorber:3He Z=2, A=3, I=31eV, z=3, a=6 dE/ds = 110.6 MeV/cm, LR=756cm

– at (ρHe-3= 0.09375 gm/cm3)Liquid, (ρHe-3= 0.134∙10-3P gm/cm3/atm in gas)

dE/ds= 1180 MeV/gm/cm2, LR= 70.9 gm/cm2

If gx = 0.123 (Σg = 0.37), β┴ =0.3m at absorber

εN,eq= ~ 0.000046 m-rad σx,rms= 1.2 cm at β┴ =0.3m, σx,rms= 3.14 cm at β┴ =2.0m

σE,eq is ~ 0.4 MeV ln[ ]=5.68, energy straggling

underestimated?

,

2 2

, 22

z a

sN eq dE

x p R ds

z E

g am c L

22 2 4 3

e p β2E,eq 2

L

(m c )(am c )β γσ = 1-

2g ln[]

19

Cooling time/power: 6Li + 3He 8B + n

Nuclear cross section for beam loss is 1 barn (10-24 cm2) or more

σ = 10-24cm2,corresponds to ~5gm/cm2 of 3He

~10 3-D cooling e-foldings …

Cross-section for 8B production is ~10 mbarn At best, 10-2 of 6Li is converted

Goal is 1013/s of 8B production then at least 1015 Li6/s needed

Space charge limit is ~1012 6Li/ring Cycle time is <10-3 s If C=10m, τ =355 ns, 2820 turns/ms 5/2820=1.773*10-3 gm/cm2 (0.019 cm @ liquid density …) 2.1 MeV/turn energy loss and regain required …(0.7MV rf) 0.944 MW cooling rf power…

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β-beams example: 7Li + 2H 8Li + 1H Beam: 25MeV 7Li+++

PLi =572 MeV/c Bρ = 0.66 T-m; v/c=0.0872

Absorber:2H Z=1, A=2, I=19eV, z=3, a=7 dE/ds = 184.7 MeV/cm, LR=724cm

– at (ρH-2= 0.169 gm/cm3)Liquid, (ρH-2= 0.179∙10-3P gm/cm3/atm in gas)

dE/ds= 1090 MeV/gm/cm2, LR= 70.9 gm/cm2

If gx = 0.123 (Σg = 0.37), β┴ =0.3m at absorber

εN,eq= ~ 0.000027 m-rad

σx,rms= 1.0 cm at β┴ =0.3m,

σx,rms= 2.5 cm at β┴ =2.0m

σE,eq is ~ 0.36 MeV ln[ ]=6.0,

Easier than 8B example ?

,

2 2

, 22

z a

sN eq dE

x p R ds

z E

g am c L

22 2 4 3

e p β2E,eq 2

L

(m c )(am c )β γσ = 1-

2g ln[]

21

Cooling time/power- 7Li + 2H 8Li +1H Nuclear cross section for beam loss is 1 barn (10-24 cm2) or more

σ = 10-24cm2,corresponds to ~3.3gm/cm2 of 2H

~9 3-D cooling e-foldings …

Cross-section for 8Li production is ~100 mbarn 10-1 of 7Li is converted ?? 10 × better than 8B neutrinos

Goal is 1013/s of 8Li production then at least 1014 Li7/s needed

Space charge limit is ~1012 7Li/ring Cycle time can be up to 10-2 s, but use 10-3s If C=10m, τ =355 ns, 2820 turns 3.3/2820=1.2*10-3 gm/cm2 (0.007 cm @ liquid density …) 1.3 MeV/turn energy loss and regain required …(0.43MV rf) 0.06 MW cooling rf power…

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Space charge limitation At N = 1012, BF =0.2, β=0.094,

z=3, a=6, εN,rms= 0.000046:

δν ≈ 0.2 tolerable ??

Space charge sets limit on number of particles in beam

2p tot

2F N,rms

z r N

4 aB

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Inverse vs. Direct Kinematics

6Li3 beam + 3He2 target 25 MeV Li

gas-jet target tapered nozzle for wedge?

product is ~ 8B 8 to 23 MeV

Stop in Tantalum absorber penetrates more (5 to 12μ) <14.5° Beam divergence ~6° (2.5σ)

Boil out to vapor neutralize to vapor, extract,

ionize, accelerate 100% efficiency …

3He2 beam + 6Li3 target 12.5 MeV He beam

Li foil target (or liquid waterfall?) fold to get wedge …

product is ~ 8B ~1 to 8 MeV

Stop in Tantalum absorber penetrates less (2 to 5μ) <30° Beam divergence is ~10° (2.5σ)

Boil out to vapor

24

X-sections, kinematics

25

Rubbia et al. contains errors

Longitudinal emittance growth 2× larger than his synchrotron oscillations reduce energy spread growth

rate but not emittance growth rate

Emittance exchange with only x does not give 3-D cooling – need x-y coupling and balancing of cooling rates

Increases equilibrium emittance, beam size increase needed for space charge, however

Includes z2 factor in dE/dx but not in multiple scattering, …

26

Low-Energy “cooling”-emittance exchange

dPμ/ds varies as ~1/β3

“Cooling” distance becomes very short:

for liquid H at Pμ=10MeV/c

Focusing can get quite strong: Solenoid:

β=0.002m at 30T, 10MeV/c

εN,eq= 1.5×10-4 cm at 10MeV/c

Small enough for “low-emittance” collider

116dP

cmP ds

22

0.3

PB

B B

Pµ

Lcool

200MeV/c

10

0.1 cm

2

, 22 s

N eq dEt p R ds

E

g m c L

100 cm

11

cool

dPL

P ds

2, !!N eq

27

Emittance exchange: solenoid focusing

Solenoid focusing(30T) 0.002m

Momentum (30→10 MeV/c)

L ≈ 5cm R < 1cm Liquid Hydrogen (or gas)

2

, 22 s

N eq dEt R ds

E

g m c L

Pµ

Lcool

200MeV/c

10

0.1cm

Use gas H2 if cooling length too short

11

cool

dPL

P dsεN,eq= 1.0×10-4 cm at 10MeV/c, 50T

-Will need rf to change p to z

28

Gas absorbers With gas absorbers can adjust energy loss rate

avoid beam loss

2 2 2 22 e e

A e 2 2 2 2

dP m c 2m c γ β1 Z 1 δ= 4πN ρ r ln -1-

P ds m c A β I(Z) 2β

2 2rms s

4 2 2 2μ R

d θ E=

ds β γ (m c ) L

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Collider ParametersParameter 4TeV(2000) 4TeV low emittance NFMCC baseline Muons, Inc. inspired

Collision Energy(2E) 4000 4000 GeV Energy per beam(E) 2000 2000 GeV Luminosity(L=f0nsnbN2/42) 1035 1035 cm-2s-1 Source Parameters 4 MW 0.9 MW p-beam Proton energy(Ep) 16 8 GeV Protons/pulse(Np) 431013 211013 Pulse rate(f0) 15 20 Hz acceptance(/p) 0.2 0.05 -survival (N/Nsource) 0.4 0.2 Collider Parameters Mean radius(R) 1200 1000 m /bunch(N±) 2.51012 21011 Number of bunches(nB) 2 2 Storage turns(2ns) 1500 2000 Norm. emittance(N) 0.610-2 2.510-4 cm-rad Geom.. emittance(t =N/) 310-6 1.310-7 cm-rad Interaction focus o 0.3 0.1 cm IR Beam size =(o)½ 3.1 0.36 m E/E at collisions 0.12 0.2%

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Summary

31

β-beams alternate: 6Li + 3He 8B + n Beam: 12.5MeV 3He++

PLi =264 MeV/c Bρ = 0.44 T-m; v/c=0.094

Absorber: 6Li Z=3, A=6, I=43eV, z=2, a=3 dE/ds = 170 MeV/cm, LR=155cm

– at (ρLi-6= 0.46 gm/cm3)

Space charge 2 smaller If gx = 0.123 (Σg

= 0.37), β┴ =0.3m at absorber εN,eq= ~ 0.000133m-rad

σx,rms= 2.0 cm at β┴ =0.3m,

σx,rms= 5.3 cm at β┴ =2.0m

σE,eq is ~ 0.3 MeV ln[ ]=5.34

,

2 2

, 22

z a

sN eq dE

x p R ds

z E

g am c L

22 2 4 3

e p β2E,eq 2

L

(m c )(am c )β γσ = 1-

2g ln[]