Post on 04-Jan-2016
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
9
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
12
“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
15
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
,
,
16
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
17
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…
20
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…
22
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
23
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
29
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%
30
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[]