accelerator physics and ion optics damping and...
TRANSCRIPT
sb/AccPhys2007_6/1
accelerator physics and ion optics
damping and cooling
Sytze Brandenburg
sb/AccPhys2007_6/2
outline
• emittance conservation: theorem of Liouville
• damping
• adiabatic
• synchrotron radiation (electrons)
• cooling
• electron cooling
• stochastic cooling
• laser cooling
sb/AccPhys2007_6/4
theorem of Liouville
• example: particle in potential
( )2p
2VH
mx= +
• canonical conjugate variables
dx H p dp H dV
dt p m dt x dx
∂ ∂= = = − = −
∂ ∂
• external forces (magnets etc.) conservative
particle motion described by hamiltonian H
phasespace density of particles constant
i i
i i
dx dpH H
dt p dt x
∂ ∂= = −
∂ ∂
• coordinates of phase space
• canonical conjugate variables of hamiltonian H
sb/AccPhys2007_6/6
• conservation of particles
r pj
t t
∂ ∂ = ρ + ∂ ∂
• particle density in six-dimensional phase space
( )r,pρ = ρ
r p
r p
r p
r
p
p
r
r pj
t t
p
t
r pr
t
H
p
t
H
r
t
r p
t t
∂ ∂ ∇ = ∇ ρ + ∇ ρ ∂ ∂
∂ ∂= ∇ ρ + ρ + ∇ ρ + ρ
∂ ∂
∂∇
∂
∂∇ ∂
∂
∂∇
∂
∂= ∇ ρ + ∇ ρ + ρ ∂ ∂
∂− ∇
∂
= 0
j 0t
∂ρ+ ∇ =
∂
sb/AccPhys2007_6/7
( ) ( )r p
r d0
t t
p
dtj
t t
∂ ∂∇ ∇ ρ + ∇ ρ
∂
∂ρ ∂ρ ρ+ = + = =
∂ ∂∂
• total derivative vanishes ρ invariant
6-dimensional phasespace area conservedr pσ σ
sb/AccPhys2007_6/9
• internal mechanism for energy loss
• e.g. friction in pendulum
damping
22
02
dx
d
d x0
t tx
d+ + ω =λ ( )
22
0 0
tx t x exp cos
2 4
λ λ = − ω −
time [a.u.]
0 500 1000 1500 2000 2500 3000
am
plit
ude [
a.u
.]
-1.0
-0.5
0.0
0.5
1.0
sb/AccPhys2007_6/10
• internal mechanism for energy loss
• e.g. friction in pendulum
damping
amplitude [a.u.]
-1.0 -0.5 0.0 0.5 1.0
mo
men
tum
[a
.u.]
-1.0
-0.5
0.0
0.5
1.0
22
02
dx
d
d x0
t tx
d+ + ω =λ ( )
22
0 0
tx t x exp cos
2 4
λ λ = − ω −
sb/AccPhys2007_6/11
adiabatic damping
• acceleration: damping term
x2 0
2 d dym
dt dt
d ym q B 0R
dtγ + − ω =
γ
s x
d dym B
dt dtvq
γ =
• equation of motion vertical betatron motion
+ + ω =2
2
02
1 dE dy
E dt
d yn y 0
dt dt
= − ω − 0
2
2
0
1 dE 1 1 dEy y exp( t)cos n t
2E dt 4 E dt
typical value 10-3 ω0
sb/AccPhys2007_6/12
adiabatic damping
−
= ω − 00
2
2 1 1 dEy cos n t
4 E d
1 dEy exp( t)
2 d tE t• envelope
( )( )
( )( )
max
maxmax
max ma
m
x
ax
1 dEd y dt
2E dt
y t E 0dy dE
y 2E y 0 E t
y = −
= − =
• emittance ε = ymax y’max ; y’max ∝ ymax/vs = ymax/βc
1 1
Eε ∝ ∝
β γβ
emittance shrinks during acceleration
sb/AccPhys2007_6/13
adiabatic damping and Liouville’s theorem
• Liouville: phasespace area conserved in ( )r,p
( )r,r '
• emittance normally defined in
y yx xp pp p
x ' sin y ' sinp p p p
= ≅ = ≅
• during acceleration px and py constant, |p| increases
x’ and y’ decrease
emittance εx = πσxσx’ decreases with 1/|p|
p2p1
sb/AccPhys2007_6/14
adiabatic damping and Liouville’s theorem
• Liouville: phasespace area conserved in ( )r,p
( )r,r '
• emittance normally defined in
y yx xp pp p
x ' sin y ' sinp p p p
= ≅ = ≅
• during acceleration px and py constant, |p| increases
x’ and y’ decrease
emittance εx = πσxσx’ decreases with 1/|p|
• |p| ∝ βγ
εβγ = εN εN independent of energy
adiabatic damping no contradiction of Liouville’s theorem
sb/AccPhys2007_6/15
electrons: synchrotron radiation
• moving charged particles electromagnetic radiation
• rest frame:2
1E ;B 0
r
∗ ∗∝ =
• lab frame: x x y y s s
x y y x s
E E ;E E ;E E
B E ;B E ;B 0
∗ ∗ ∗
∗ ∗
= γ = γ =
= γβ = −γβ =
( )2 2 2
x s x y s y s x y
S E B
S E E ; S E E ; S E E∗ ∗ ∗ ∗ ∗ ∗
= ×
= γβ = γβ = −γ β +
• energy transport
• S ∝ E2 ∝ r-4; P = 4πr2S ∝ r-2
• energy transfer in vicinity of particle, no radiation to infinity
no energy loss
sb/AccPhys2007_6/16
electrons: synchrotron radiation
• accelerated charged particles electromagnetic radiation
Larmor, Liénard
( )
γ = − πε
2 22 2
rad 2 22
0 0
q c dp 1 dEP
dt c dt6 m c
• propagation of increasing distortion of fieldlines
sb/AccPhys2007_6/17
electrons: synchrotron radiation
• accelerated charged particles electromagnetic radiation
Larmor, Liénard
( )
γ = − πε
2 22 2
rad 2 22
0 0
q c dp 1 dEP
dt c dt6 m c
• propagation of increasing distortion of fieldlines
• acceleration γ
2
2
1 dp
dt=
dE dpv
dt dt
• circular motion
2dp
dt=
dE0
dt
sb/AccPhys2007_6/18
losses due to acceleration
( )
ε=
π
2
rad 22
0 0
2q c
P6 m c
dp
dt
• = =dE dp
Fds dt ( )
=
πε
22
rad 22
0 0
q c dEP
ds6 m c
• fractional losses
( )=
πε β
2
rad 22
0 0
dEdE
d 6 m ct
qP
ds
• maximum value radiation losses negligible8dE
10 eV /mds
=
• for electrons (β = 1)
−= × 21
rad
dEdEP 3.65 10
dt ds
sb/AccPhys2007_6/19
losses due to bending
γβ= ω =
ρ
2 2
0m cdpp
dt
( )β γ
= ≅πε ρ ρπε
2 4 4 2 4
rad 42 220 0 0
q c q c EP
6 6 m c
( )
γ
=πε
2
rad 22
0 0
2
2q cP
6 m c
dp
dt
• energy loss per turn
( )πρ β γ
∆ = = ≅β ε ρ ρε
2 3 4 2 4
radrad 4
20 0 0
P 2 q q EE
c 3 3 m c
[ ][ ]
∆ =ρ
4 4
rad
E GeVE keV 88.5
m
sb/AccPhys2007_6/20
electrons vs. protons
• example:
• LEP: 100 GeV electrons
• γ = 1.95×105; ρ = 3100 m
• ∆ =radE 2.85 GeV / turn
• at same energy
4
p 13rad,e
rad,p e
mP1.13 10
P m
= = ×
• limiting factor for maximum electron energy
radE 0.011MeV / turn∆ =
• LHC : 7700 GeV protons
• γ = 8.21 ×103; ρ = 2586 m
•
sb/AccPhys2007_6/21
characteristics synchrotron radiation
• strongly forward peaked angular distribution: top angle cone 1/γ
• Lorentz transformation
sb/AccPhys2007_6/22
characteristics synchrotron radiation
• strongly forward peaked angular distribution: top angle cone 1/γ
• Lorentz transformation
• energy distribution
• scales with
• LEP: Eγc = 0.37 MeV
γ
γ=
πρ
3
c
3 h cE
4
sb/AccPhys2007_6/23
• many applications
• material science
• molecular biology (protein structure)
dedicated accelerators e.g. ESRF Grenoble
• special tools to manipulate energy distribution:
wigglers and undulators
characteristics synchrotron radiation
• strongly forward peaked angular distribution: top angle cone 1/γ
• Lorentz transformation
• energy distribution
• scales with
• LEP: Eγc = 0.37 MeV
γ
γ=
πρ
3
c
3 h cE
4
sb/AccPhys2007_6/24
damping by synchrotron radiation
• two effects
• energy dependence Prad damping in equation of motion
reduces phase space
• quantum nature statistical fluctuations
increases phase space
• balance : equilibrium phase space
sb/AccPhys2007_6/25
longitudinal: damping effect
( ) ( )2
2
s2
d2 E
dt
dE E 0
dt∆ + + Ω ∆∆ =α
( )t 2 2
0 sE E e cos t−α∆ = ∆ Ω − α
• damping term in equation of synchrotron motion
• RF acceleration compensates Prad(Es)
• continuous energy loss: friction
• E > Es: E decreases |∆E| decreases
• E < Es: E increases |∆E| decreases
( ) ( )= =
ρπε πε
2 4 4 32 2
rad 4 422 2
0 0 0 0
q c E q cP E B
6 m c 6 m c
sb/AccPhys2007_6/26
• small contribution from dispersion effects
• LEP at 100 GeV
• τ = 2.3 ms
• Ωs= 6.9 × 104 rad/s Ts = 0.09 ms
• damping takes many oscillation periods
• damping time( )πε
τ = =
42
0 0s
4 3 2
rad
6 m cE 1
P q c EB
• α from energy dependence of Prad
( )rad srad
s
2P EdPdE E E 2 E
dt dE E∆ = − ∆ = − ∆ = − α∆
rads
s
PE
Eα = ∝
sb/AccPhys2007_6/27
longitudinal: quantum excitation
• rate of emission photons
• full calculation γ
= κ rad
c
PN
E15 3
38
κ = ≈
• emittance grows with A2
γ γ=2 2
c
11E E
27
γ γ= c
8E E
15 3
• LEP at 100 GeV
• Eγc = 0.37 MeV
• ∆E = 2860 MeV/turn
• n = 25100 /turn
• one dimensional random walk problem
• stepsize
• amplitude grows with : γ=2
2
c2 2
1 d A 11 NE
A dt 27 An
γcE
γ=2 2A n E
sb/AccPhys2007_6/28
longitudinal: balance
• equilibrium: quantum excitation + damping = 0
γ =2 radc2
s
P11 NE
27 A E
2γ=2 2 sc
rad
E
2
11A NE
27 P
•γ γ
= =πρ π ρ
4 3 2 22 s s s
s s
h mc E11 15 3 3 55 hA
27 8 2 2 2 mc32 3
1
2
• energy spread
22
sE
s s
55 h
E 2 mc64 3
γσ=
π ρ
• additional emittance growth factor: space charge
sb/AccPhys2007_6/29
transverse: damping
• particle coordinates
( )
( )
z
22 2
z
z A cos
Az' sin
s
A z s z '
= φ
= − φβ
= + β
• photon emitted parallel to particle momentum
• true to angle 1/γ
• longitudinal momentum: restored by acceleration
• transverse momentum: not restored by acceleration
sb/AccPhys2007_6/30
transverse: damping
• small angle approximation: sin x = x; cos x = 1p E
z' p =z' p z ' z 'p E
⊥⊥
δ δδ = − δ δ δ = −
( ) ( )
( ) ( )
2 2 2 2 2 2
z z
2 2 2
z z
A z z ' s s z '
EA A s z ' z ' s z '
E
δ = δ + δ β = β δ
δδ = β δ = β
• emission anywhere along orbit use 2z '2
2
2
z
A A Ez'
A 2E2
δ δ= = −
β
• time derivative
rad
s
P1 dA
A dt 2E= −
sb/AccPhys2007_6/31
transverse: quantum excitation
• betatron oscillation around dispersion orbit for its ∆p = ∆E/c
• photon emission
• ∆p changes by δp = Eγ/c different dispersion orbit
• y and y’ continuous (photon emitted in direction y’)
betatron amplitude increased by D(s) Eγ/E
“emittance” increase δε = D2(s) (Eγ/E)2
• complete analysis
( )
( ) ( )
2 2
2 2
2 2
2 2
y 2 yy ' y '
E ED 2 DD' D' H s
E E
γ γ
δε = δ γ + α + β
= γ + α + β =
γε=
π ∫2
2
C
N Ed 1Hds
dt E 2 R
sb/AccPhys2007_6/32
cooling
• motivation for beam cooling
• increase luminosity
• compensate interaction with “target” (rings)
• experiments with secondary particles (e.g. antiprotons)
• high-precision
• transfer of energy from beam to external medium
• electron cooling low energy, moderate emittance
• stochastic cooling high energy, large emittance
• laser cooling
sb/AccPhys2007_6/33
electron cooling
• “heat” exchange between “hot” ions and “cold” electrons
• mediated by Coulomb interaction
• cooling effect proportional to electron density
• Eel = Eion × mel/mion
high power in electron beam
energy recovery important
2
r
1
v∝• “interaction strength
vr relative velocity electron - ion in restframe
maximize overlap between velocity distributions
choose <βel,lab> = <βion,lab> and δβel = δβion
• restframe travels at <βion,lab> in lab frame
sb/AccPhys2007_6/34
“cooling power”
• longitudinal momentum
rest lab
rest lab labrest lab
lab lab
1p p
p p p
mc mc p
δ = δγ
δ δ δδβ = = = β
γ
• electron energyspread ∆Ee ≅ 0.5 eV
• velocity spread in restframe δβel ≅ 1.4 × 10-3
3
lab
lab lab
p 1.4 10
p
−δ ×=
β
• transverse momentum: similar analysis 3
lab
lab lab
1.4 10x '
−×=
β γ
• most effective at low ion energy
sb/AccPhys2007_6/35
beam temperature
• in restframe velocityspread δβel = δβion
• temperature related to kinetic energy
Tel ∝ mel(δβel)2
Tion ∝ mion(δβion)2
ionion el el
el
mT T T 1836A
m= = ×
• after cooling Tion = Tel
elion el el
ion
m 1
m 43 Aδβ = δβ ≅ δβ
• heated electrons dumped and replaced by new, cold ones
sb/AccPhys2007_6/37
3
22 ion el elion 2 4
m m TC 1 3
L Z e nL m2 2
τ = γ
π
• cooling time3
22 ion el elion 2 4
m m TC 1 3
L Z e nL m2 2
τ = γ
π
cooling time
• cooling force F2
r
1
v∝
• ion
ion ion
dv1 1 F
v dt mv= ∝
τ
• for vion >> vel
does not work well for tails of distribution
3
ion
1 1
v∝
τ
• typical value τ ≈ 1 s
• cooling time
sb/AccPhys2007_6/38
τ = γ
π
3
22 ion el elion 2 4
m m TC 1 3
L Z e nL m2 2
cooling time
γ
3
2
2
ion
ion el
2
3
el
Lorentz boost
Cfraction of time in cooler section
L
m m energy transfer in collision
Z strength Coulomb interaction
nL number of electrons in cooler section
T (electron velocity)
sb/AccPhys2007_6/39
scheme electron cooler (TSL Uppsala)
sb/AccPhys2007_6/40
stochastic cooling
• initial problem: how to accelerate anti-protons
• produced in in p-induced reaction at ~ 26 GeV
• large energy spread; large angular spread
• cooling needed to fit in acceptance accelerator
• electron cooling not possible (E too high, emittance too large)
need another trick
• produced at low rates
correct orbit on “individual” basis
• simple concept, but ..... 15 years hard work to get there
sb/AccPhys2007_6/41
principle of operation: single particle
• measure transverse position particle
• correction at location with betatron phase advance (n+0.5)π
• Q non-integer
orbit corrected after a few (1/|n-Q|) turns independent of initial betatron phase
sb/AccPhys2007_6/42
principle of operation: many particles
• pick-up and kicker have bandwidth W
signal originates from sample 1/2W Torbit of beam
correction applied to sample 1/2W Torbit of beam
maximize bandwidth to minimize sample size (W ~ GHz)
• same sample at pick-up and kicker next turn
• sample ≡ “one particle”
• after few turns cooling stops (no error on mean position)
sb/AccPhys2007_6/43
principle of operation: many particles
sample mixing needed
• momentum spread ∆p/p and momentum compaction η ≠ 0
• ideal situation:
• no mixing between pick-up and kicker
• correction to right particles
• complete mixing between kicker and pickup
• no sample correlation
• no noise
• reality:
• some mixing between pick-up and kicker
• incomplete mixing between kicker and pick-up
• noise (large bandwidth comparable to signal)
sb/AccPhys2007_6/44
formalism
• correction in one turn
( ) ( )i i
22 2 2
i i i
2 2
ii
2
i
gx
2
x x
x x g gx x gx x xx x
→ −
→ − − − ∆= =−
• separate contribution of xi to x
sN
sij
j 1;j is s
N 1xx x x x
N N
∗ ∗
= ≠
−= + = ∑
( ) ( )2
22 ii i s i s2
s s
2gx gx x N 1 x x N 1 x
N N
∗ ∗ ∆ = − + − + + −
• averaging over many turns = many samples
• mean
• variance of mean
x 0∗ =
2 22
x
s
x xN
∗ = =σ
beam width
sb/AccPhys2007_6/45
formalism
• correction in one turn
( )2 2 2 22 2
s x
3
s
i i
s s
2
i 2
2gx g x
N N
g N
Nx
1− σ∆ = − + +
single particle cooling Schottky noiseheating by other particles
• average over all particles; Ns >> 1
2 2 22 x xx
s s
2g g
N N
σ σ∆σ = − +
• cooling time
( ) ( )2
2 2x2
orbit x s orbit
1 1 1 2W2g g 2g g
T N T N
∆σ= − = − = −
τ σ
sb/AccPhys2007_6/46
sample mixing and noise
• sample mixing
• momentum spread δ = ∆p/p and momentum compaction η
• sample period ∆Ts = 1/2W; orbital period Torbit
• variation in orbital period ∆Torbit = Torbit δ η
s
orbit orbit
T 1M
T 2W T
∆= =
∆ δη• number of turns for complete mixing
• imperfect mixing (M > 1) g2 term in 1/τ multiplied by M
• noise including by adding term
• U = electronic noise / Schottky noise ≥ 1 !
2 UWg
N−
• cooling time including imperfect sample mixing and noise
( )21 2W2g g M U
N = − + τ
1optimum g
M U=
+
sb/AccPhys2007_6/47
damping and cooling; what about Liouville
• synchrotron radiation damping: Prad energydependent
non-conservative “friction” force
conditions of Liouville’s theorem not fulfilled
• electron cooling: velociticy dependent interaction
non-conservative “friction” force
conditions of Liouville’s theorem not fulfilled
• stochastic cooling: conservative forces
Liouville’s theorem should hold
• beam not continuous but granular
emittance contains empty regions
• stochastic cooling “squeezes” empty space out of emittance
no contradiction with Liouville’s theorem
sb/AccPhys2007_6/48
laser cooling
• atomic excitation only applicable to ions with Q < Z
• available lasers limit applicability (alkalis, earth-alkalis)
• laser collinear with ion beam
• incident photons one direction, emitted photons isotropic
acceleration
sb/AccPhys2007_6/49
• Doppler shifted frequency in resonance at specific energy
resonant process
• frequency sweep “collects” beam at one energy
• two lasers in opposite direction
beam “prisoner” in narrow energy interval
• achievable energy spread ∆E/E = 10-6
• cooling time < 100 µs
sb/AccPhys2007_6/50
next lecture
• cyclotrons
• reading material
• CERN Accelerator School 1992, CERN report 94-01
Chapter 33 Cyclotrons Chapter 34 Injection and extraction from cyclotrons
• CERN Accelerator School 1995, CERN report 96-02 Chapter 7 Introduction to cyclotrons Chapter8 Cyclotron magnet calculations
Chapter 9 Injection into cyclotrons Chapter 10 Extraction from cyclotrons
• there is significant overlap between the different documents, so be selective