accelerator physics and ion optics damping and...

46
sb/AccPhys2007_6/1 accelerator physics and ion optics damping and cooling Sytze Brandenburg

Transcript of accelerator physics and ion optics damping and...

Page 1: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

sb/AccPhys2007_6/1

accelerator physics and ion optics

damping and cooling

Sytze Brandenburg

Page 2: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

sb/AccPhys2007_6/2

outline

• emittance conservation: theorem of Liouville

• damping

• adiabatic

• synchrotron radiation (electrons)

• cooling

• electron cooling

• stochastic cooling

• laser cooling

Page 3: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 4: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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+ ∇ =

Page 5: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 6: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 7: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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 −

Page 8: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 9: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 10: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 11: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 12: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 13: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 14: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 15: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 16: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 17: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 18: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

sb/AccPhys2007_6/21

characteristics synchrotron radiation

• strongly forward peaked angular distribution: top angle cone 1/γ

• Lorentz transformation

Page 19: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 20: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 21: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 22: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 23: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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 = ∝

Page 24: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 25: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 26: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 27: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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= −

Page 28: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 29: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 30: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 31: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 32: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 33: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 34: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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)

Page 35: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

sb/AccPhys2007_6/39

scheme electron cooler (TSL Uppsala)

Page 36: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 37: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 38: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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)

Page 39: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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)

Page 40: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 41: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 42: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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=

+

Page 43: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 44: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 45: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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

Page 46: accelerator physics and ion optics damping and …brandenburg/lecture06/dampingcooling2007.pdfaccelerator physics and ion optics damping and cooling Sytze Brandenburg. ... emittance

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