damping rates

1
Cooling

• damping rates FPE
• electron cooling
• stochastic cooling
• laser cooling
• thermal noise crystalline beams
• beam echoes
• ionization cooling
• comparison

MCCPB, Chapter 11
2
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 cold ion beams
ultra-low emittance electron beams ionization
cooling for muon beams muon collider
3
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
4
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
5
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
drms1.4x10-6, t25 ms idea hadron beam
accompanying electron beam exchange heat via
Coulomb collisions e- temperature must be lower
than hadron temperature easily fulfilled since
6
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
7
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
8
former electron cooling system at LEAR
9
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 Dtr/u
(impact parameter / ion velocity)
velocity after 1st step
distance moved after 1st step
collision of 1 ion and 1 electron during e-
cooling
10
integrating over impact parameter r
expanding in Dr

with
minimum impact parameter classical head-on
collision
Debye shielding length
11

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
12
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
13
cooling force FelMu/tel in 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
14
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
15
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
16
for higher current Ie-, increase in ve,rms which
degrades the cooling force
optimum Ie-
17
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 Dx at the cooler
finite large area is due to betatron oscillations
18
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
19
3. stochastic cooling
off-axis particle gives signal of length Ts1/(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
20
test particle x, applied correction -lx
sum over other particles in the sample
or
gain
ignore other particles and set g1
more rigorously
U noise-to-signal ratio
mixing term
21
simplified
const 1/10 in practice
typical time t1 s for N107 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
22
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
23
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
24
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
25
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
26
evolution of ion momentum distribution during
laser cooling of a bunched ion beam
27
ions with transition A-gtB so that wABw 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
28
example 100 keV 7Li beam transition at
548.5 nm CW dye laser
t43 ns lifetime single absorption DE12
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, drmslt10-6
so far only longitudinal cooling, transverse
cooling via coupling, e.g., with momentum
dispersion at rf cavity, near linear resonance
29
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
30
schematic of a laser-electron storage ring
31
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
32
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

33
5. crystalline beams
cold beams have unusual noise spectra
ordinary beam
Schottky noise
interaction with beam environment
coherent frequency shift
where
!
Wnndw
spread in revolution frequency
34
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
35
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)

36
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
37
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)
38
two-particle model of signal recoherence after
applying first a dipole kick and then a
quadrupole kick, essential tune change with
amplitude
39
evolution of distribution function
solve Vlasov equation
OR exploit Liouvilles theorem
tricks used
centroid displacement
dipole kick
40
average displacement as a function of time
following a dipole kick (G. Stupakov)
41
echo signal of the beam after a second
(quadrupole) kick was applied (G. Stupakov)
signal proportional to product of kicks
42
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 h10, h9 (harmonic
number) -gt echo at h1 time of echo
time between excitations
proportional to kick strength
collision or diff. rate
diffusion destroys reversibility of the
decoherence
determine n!
TeV Acc. n3x10-4 Hz measured
techo time from1st kick to centre of echo
43
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
44
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
45
5s kick
0.25s kick
ltxgt/s
echo
simulated echo signal for m-2x10-4
46
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!
47
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 d2 lng
only longitudinal momentum is restored by rf
transverse damping
48
average muon energy loss per unit length in Be
49
transverse cooling
damping (energy loss accleration)
heating (multiple scattering)
preferred small beta function and large
radiation length LR (low Z)
50
schematic of ionization cooling in the
transverse phase space using a series of low-Z
absorbers and re-acceleration
51
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
52
8. comparison of cooling techniques
technique stoch. electron synchr. rad. laser (ions) laser (e-) ioniz.
species all ions e,e- some ions e,e- muons
beam velocity high medium 0.01ltblt0.1 very high ggt100 any glt5 very high ggt100 medium
intensity low any any any any any
cooling time Nx 10-8 s 1-0.01 s 10-3 s 10-4- 10-5 s lt10-7 s 10-2 -10-5 s
favored temp. high low any low any any