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Cooling. damping rates & FPE electron cooling stochastic cooling laser cooling thermal noise & crystalline beams beam echoes ionization cooling comparison. [MCCPB, Chapter 11]. cooling: reduction of phase-space volume & increase in beam density via dissipative forces

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Cooling

  • damping rates & FPE

  • electron cooling

  • stochastic cooling

  • laser cooling

  • thermal noise & crystalline beams

  • beam echoes

  • ionization cooling

  • comparison

[MCCPB, Chapter 11]


cooling: reduction of phase-space volume & increase

in beam density via dissipative forces

e-/e+ storage rings: cooling from synchrotron radiation

SR ‘damping rings’ for linear colliders; light sources

electron cooling & stochastic cooling of hadron beams

accumulate beams of rare particles (e.g., pbar), combat

emittance growth due to scattering on internal target,

produce high-quality beams

two types of laser cooling:extremely coldion beams &

ultra-low emittanceelectron beams

ionization cooling for muon beams & muon collider


1. damping rates & Fokker-Planck equation

dissipative force

recall

damping rate

elementary calculation shows that damping rates in the

3 planes are related:

W: rate of energy loss

p: particle momentum

in the case of synchrotron radiation this is

called the ‘Robinson theorem”


mathematically cooling can be described by

Fokker-Planck equation

simple example

f: distribution function

“noise”

cooling

stationary solution

with

I0: final equilibrium emittance due to “noise”

cooling does not produce smaller and smaller beams


2. electron cooling

proposed in 1966 by Budker

first experiments at NAP-M storage ring at INP in Novosibirsk

where an antiproton beam was cooled by many orders of

magnitude longitudinally and transversely drms~1.4x10-6, t~25 ms

idea: hadron beam & accompanying electron beam

exchange heat via Coulomb collisions

e- temperature must be lower than hadron temperature

easily fulfilled since


definition of temperature

factors 2 or 3

in the literature

equivalent definitions

for e- beam

for maximum Coulomb cross section average velocities

of ions and e-should be the same


schematic of electron cooling for an ion storage ring

velocity of cooled coasting beam equal to that of the e- beam as a result

of the cooling useful tool for tuning the ion-beam energy; for bunched

ion beam the rf frequency must be adjusted in order to match e- beam velocity



rough estimate of cooling force

collision of two particles, single ion and single electron;

reference frame where e- at rest before the collision

split collision 2 steps: (1) approach, (2) separation;

duration of either step Dt~r/u (=impact parameter /

ion velocity)

velocity after 1st step

distance moved after 1st step

collision of 1 ion and 1 electron during e- cooling


integrating over impact parameter r:

expanding in Dr

*

with

minimum impact parameter:

classical head-on collision

Debye shielding length


*

averaging over the e- velocity distribution

cooling force

cooling rate

transform to laboratory frame, obtain two factors g due to

time dilation and Lorentz contraction

cooling time if ion velocity is larger or smaller than rms

e- velocity spread:

ion

velocity

assuming Gaussian e- distribution


remarks:

  • large for large g

  • short if M small and Z high

  • ~ u3 for hot beam

  • independent of ion velocities and only dependent on

    e- temperature for cold beam

typical parameters:


cooling force Fel=Mu/telin a ‘flattened’ e- beam as a function of ion velocity in

units of rms e- velocity in beam frame ve,rms; dashed curves correspond to

asymptotic formulae from previous page; difference between transverse and

longitudinal plane is due longitudinal acceleration


two additional effects which reduce the cooling time:

  • acceleration of e- in longitudinal direction

  • e- velocity distribution is flattened in longitudinal direction

  • faster longitudinal damping

    2.longitudinal magnetic field which is employed to guide

    and confine e- beam

  • e- cyclotron motion

  • decreases effective transverse temperature of e- beam

  • reduces cooling times to values below 0.1 s

recombination

due to e- capture

also faster

cooling is faster for

highly charged ions

for higher energies cooling becomes less efficient,

rate scales as ~g2, also higher e- energy would be required


optical functions at the e- cooler & e- current

for cold electron beam

large value of b should give larger cooling rate

however, for large b also beam size is larger and ion beam

samples effect of nonlinear e- space-charge field

intermediate b is optimum!

but in reality saturation, again

due to e- space charge

also


for higher current Ie-, increase in ve,rms which

degrades the cooling force

optimum Ie-


already cooled

longitudinal velocity vs. horizontal position of the electron and ion beams

due to space charge the e- velocities lie on a parabola; the ion velocity

varies linearly with a slope inversely proportional to the dispersion Dxat

the cooler

finite large area is due to betatron oscillations


e- cooling of high-energy beams?

inefficient (?)

high-energy high-current e- beam needed

e- storage ring;

radiation damping preserves

e- emittance

cooling

section

proposal:

C. Rubbia, 1978

S.Y. Lee, P. Colestock, K.Y. Ng, 1997

proton or ion storage ring

bucket spacing

should be integer

ratio

other approach: RLA with energy recovery

for RHIC upgrade, BNL-BINP collaboration


3. stochastic cooling

off-axis particle gives

signal of length Ts~1/(2 W)

where W is bandwidth

of cooling system

smallest fraction of

beam that can be

observed:

“sample”

N: total no. beam particles

T: revolution time

idea: S. van der Meer, 1968

1975 first experimental demonstration at CERN

1977-83 cooling tests at CERN, FNAL. Novosibirsk, INS-Tokyo;…


test particle x, applied correction -lx

sum over other particles in the ‘sample’

or

gain

ignore other particles and set g=1:

more rigorously:

U: “noise-to-signal”

ratio

“mixing term”


simplified:

“const” ~ 1/10 in practice

typical time: t~1 s for N~107

CERN AA: factor 3x108 increase in phase-space density

e- cooling works best for cold beams

stochastic cooling works best for large (hot) beams

(and small N)

stochastic cooling for “halo cleaning”

electron “core freezing”


stochastic cooling for bunched beams has not yet been

demonstrated

need to operate at frequencies well above bunch fall-off

frequency ~bc/sz

unexpectedly strong coherent signals were observed

promising alternative: optical stochastic cooling at much

higher frequencies and bandwidth


application of stochastic cooling formalism: emittance growth

from LHC transverse damper (D. Boussard)

noise to Schottky-signal ratio

LSB ~ as quantization noise

gn: “reduced” feedback gain

“feedback via the beam”

total tune spread

~2-3.5 times rms spread

10 bit

+/- 10 s


4. laser cooling

A) ion beams

laser cooling in atomic traps well known

1981 P. Channel, application to storage rings

exploits Doppler frequency shift

in ion rest frame

ion beam

ions at different energy see

different laser frequency

selective interaction


photon absorption and

emission during laser cooling

at each absorption recoil is

added to the ion momentum

emission is isotropic

and on average does not

change the ion momentum



ions with transition A->B so that wAB=w’ will absorb photons

recoil velocity

spontaneous emission is isotropic

upper level should have short decay time to avoid

stimulated (non-isotropic) emission

=1/t

decay time

ultimate temperature


example: 100 keV 7Li+ beam

transition at 548.5 nm

CW dye laser

t~43 ns lifetime

single absorption DE=12 meV

few mW laser power, 5 mm spot

1.2x107 s-1 or 15 absorptions over 2 m IR length

or 0.2 eV/turn

ultimate temperature ~12 meV

laser cooling requires adequate energy levels & transitions

in reach of tunable lasers; so far only 4 ion species:

7Li+, 9Be+, 24Mg+, 166Er+

demonstrated experimentally in TSR & ASTRID, drms<10-6

so far only longitudinal cooling, transverse cooling via coupling,

e.g., with momentum dispersion at rf cavity, near linear resonance


B) electron beams

V. Telnov, 1996

Z. Huang, R. Ruth, 1998

e- bunch in ring interacts on each turn with intense laser beam

laser acts like wiggler magnet with peak field

Ilaser: laser intensity

Z0: vacuum impedance (377 W)

total power radiated

damping time



transverse emittance: usually from dispersion invariant H,

but here 1/g opening angle dominates

relation between photon energy and scattering angle

where

number of photons scattered into dw:

integrating the scattered angle squared over w yields

quantum excitation; balancing the result with damping

gives the transverse emittance

extremely small


momentum spread

e.g., sd ~1%, large

  • increased momentum spread

  • widens beam size in arc, reduces intrabeam scattering

    confines space-charge tune shift

  • requires good chromatic correction & high-frequency

    rf system for short bunches


5. crystalline beams

cold beams have unusual noise spectra

ordinary beam

Schottky noise

interaction with beam environment:

coherent frequency shift

where

!

Wn=ndw

spread in revolution

frequency


beam noise power becomes direct measure

of beam temperature

remarkable suppression of noise spectrum for cold beam first

observed with an electron-cooled proton beam at NAP-M

in Novosibirsk

“crystalline beams”

predicted/proposed/observed new state of matter

to be reached by strong cooling

particles ’lock’ into fixed positions


Hamiltonian in convenient coordinates

inter-particle potential

relativistic shear term - can render

the Hamiltonian unbounded

in the beam frame

(accelerated frame of

reference)

this and the time-dependent focusing can melt

the crystal

  • 2 conditions for crystal beams:

  • AG lattice, below transition (kinematically stable)

  • ring lattice periodicity larger than 2x maximum

  • betatron tune (no linear resonance between

  • crystal phonons and machine lattice periodicity)


low density: 1-D crystal

when

2-D crystal in plane with weaker focusing

still larger density 3-D crystal

distance between

nearest neighbors

crystalline beams have been observed in the ESR and

SIS rings in Darmstadt


6. beam echoes

two independent pulse excitations

later coherent signal grows out of a quiet beam

slope + magnitude diffusion process

beam temperature (e.g., Dp/p)

analytical example in textbook

dipole kick followed by quadrupole kick

start with initial Gaussian distribution

use action-angle coordinates

compute time evolution with some approximations

include tune shift with action (important)


two-particle

model of

signal

recoherence

after applying

first a dipole

kick and

then a

quadrupole

kick, essential:

tune change

with amplitude


evolution of distribution function

solve Vlasov equation

OR exploit Liouville’s theorem

tricks used:

centroid displacement

dipole kick



echo signal of the beam after a second (quadrupole) kick was

applied (G. Stupakov); signal proportional to product of kicks


measurements of longitudinal echoes

2 rf kicks applied at frequencies f1 and f2; response

is observed at difference frequency f1-f2

e.g. excitation at h=10, h=9 (harmonic number)

-> echo at h=1

time of echo:

time between

excitations

proportional to

kick strength

collision

or diff. rate

diffusion destroys reversibility

of the decoherence

determine n!

TeV Acc. n~3x10-4 Hz measured

techo: time from1st kick to centre of echo


at center of echo zero response

~ slope of distribution function

shape of echo, e.g. spacing between two peaks,

contains information on shape of distribution and

on rms momentum spread

allows detection of non-Gaussian distribution

effect of longitudinal wake fields

nonlinear momentum compaction factor

L. Spentzouris, P. Colestock, ~1995


transverse echo after applying

two dipole kicks

(F. Ruggiero et al., 2000)

time 0: large kick

time t: small kick ~1/2 Dr’

(1/2 distance between

filaments)

time 2t: signal reappears


5 s kick

0.25s kick

<x>/s

echo

simulated echo signal for m=-2x10-4


echo?

measurement at SPS:

0.9s kick followed by

0.2s kick

simulation for the same

parameters of experiment,

detuning with amplitude

estimated from decay time

good agreement!


7. ionization cooling

muon collider requires reduction in m phase-space

volume by factor 106

proposed scheme: ionization cooling, similar to e- cooling

where e- beam is replaced by solid material

energy loss described by Bethe-Bloch formula

I: average

ionization energy

density

effect

d~2 lng

only longitudinal momentum is restored by rf

transverse damping



transverse cooling:

damping

(energy loss

& accleration)

heating

(multiple

scattering)

preferred: small beta function and

large radiation length LR (low Z)


schematic of ionization cooling in the transverse

phase space using a series of low-Z absorbers

and re-acceleration


energy spread can be reduced by a transverse variation

in absorber thickness at a location with dispersion

this reduces the longitudinal emittances

but increases transverse emittance

more complicated schemes,… experiments are planned



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