# Stochastic Cooling 14 Dec

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```					                                                                  Paul Derwent
14 Dec 00
1

Stochastic Cooling

Paul Derwent
14 Dec 00

http://cosmo.fnal.gov/organizationalchart/derwent/cdf_accelerator.htm
Paul Derwent
Idea Behind            14 Dec 00
Stochastic Cooling        2

   Phase Space compression
Dynamic Aperture: Area
x’
where particles can orbit

Liouville’s Theorem:
x   Local Phase Space Density
for conservative system
is conserved
x’
Continuous Media
Discrete Particles

x

Swap Particles and Empty
Area -- lessen physical
area occupied by beam
Paul Derwent
Idea Behind           14 Dec 00
Stochastic Cooling       3

   Principle of Stochastic cooling
 Applied to horizontal btron oscillation

Particle Trajectory

Kicker

   A little more difficult in practice.
   Used in Debuncher and Accumulator to cool
horizontal, vertical, and momentum distributions
   COOLING? Temperature ~ <Kinetic Energy>
minimize transverse KE
minimize DE longitudinally
Paul Derwent
Stochastic Cooling            14 Dec 00
in the Pbar Source            4

   Standard Debuncher operation:
 108 pbars, uniformly distributed
 ~600 kHz revolution frequency
   To individually sample particles
 Resolve 10-14 seconds…100 THz bandwidth
   Don’t have good pickups, kickers, amplifiers in
the 100 THz range
 Sample Ns particles -> Stochastic process
» Ns = N/2TW where T is revolution time and W
bandwidth
» Measure <x> deviations for Ns particles
 Higher   bandwidth the better the cooling
Paul Derwent
Betatron Cooling                      14 Dec 00
5

With correction ~ g<x>, where g is gain of system
 New position: x - g<x>
 Emittance Reduction: RMS of kth particle
x k  gx 2  x k  2gx k  g 2  x 2
2

 x 
1
Ns      xi 
1
Ns
xk 
1
Ns      x     i
i                          i k
Average over all particles and do lots of algebra
d x 2 2g x 2  g 2 2
               x ,where n is ' sample'
dn      Ns        Ns

 Cooling Time
1


2W
N

2g  g 2    
   Add noise (characterized by U = Noise/Signal)
 Randomization effects M = number of turns
to completely randomize sample
 Cooling Time
1


2W
N
    2
2g  g M  U  
Paul Derwent
Momentum Cooling                   14 Dec 00
6

   Momentum Cooling explained in context of
Fokker Planck Equation
                           
        C E   D E 
t        E                 E 
N
where  = density function
E
C E  is energy gain function
D E represent diffusion terms (noise, mixing, feedback)
   Case 1: Flux = 0 Restoring Force a(E-E0)
Diffusion = D0
a E  E 2 
0
   0 exp              
    2 D0     

   Cooling of momentum distribution (as in
Debuncher)
   ‘Small’ group with Ei-E0 >> D0
 Forced into main distribution
 MOMENTUM STACKING
Paul Derwent
Stochastic Stacking    14 Dec 00
7

Gaussian Distribution
 CORE



C(E)

D(E)

‘Stacked’

E0

   Injected Beam (tail)
 Stacked
Paul Derwent
Pbar Storage Rings           14 Dec 00
8

   Two Storage Rings in Same Tunnel
 Debuncher
» ~few x 107 stored for cycle length
• 2.4 sec for MR, 1.5 sec for MI
» ~few x 10-7 torr
» RF Debunch beam
» Cool in H, V, p
 Accumulator
»   ~1012 stored for hours to days
»   ~few x 10-10 torr
»   Stochastic stacking
»   Cool in H, V, p
   Both Rings are ~triangular with six fold
symmetry
Paul Derwent
Debuncher Ring          14 Dec 00
9

   ßtron cooling in both horizontal and vertical
planes
   Momentum cooling using notch filters to define
gain shape
   4-8 GHz using slot coupled wave guides in
multiple bands
   All pickups at 10 K for signal/noise purposes
Paul Derwent
14 Dec 00
Accumulator Ring               10

   Not possible to continually inject beam
 Violates Phase Space Conservation
 Need another method to accumulate beam
   Inject beam, move to different orbit (different
place in phase space), stochastically stack
   RF Stack Injected beam
 Bunch with RF (2 buckets)
 Change RF frequency (but not B field)
» ENERGY CHANGE
 Decelerates   ~ 30 MeV
   Stochastically cool beam to core
 Decelerates ~60 MeV
Power
(dB)
Core
Stacktail

Injected                      Frequency
Pulse                       (~Energy)
Paul Derwent
14 Dec 00
Stochastic Stacking               11

   Simon van Der Meer solution:

 Constant Flux:     constant
t
 
   , where E d characteristic of design
 Solution:     E Ed
E  Ei 
 =  0 exp
 E d 
         

 ExponentialDensity Distribution generated
by Exponential Gain Distribution
 Max Flux = (W2|h|Ed)/(f0p ln(2))
Gain                                Density

Stacktail
Core
Stacktail

Core

Energy                         Energy

Using log scales on vertical axis
Paul Derwent
Implementation in        14 Dec 00
Accumulator            12

   Stacktail and Core systems
   How do we build an exponential gain
distribution?
   Beam Pickups:

 Charged  Particles: E & B fields generate
image currents in beam pipe
 Pickup disrupts image currents, inducing a
voltage signal
 Octave Bandwidth (1-2, 2-4,4-8 GHz)
 Output is combined using binary combiner
boards to make a phased antenna array
Paul Derwent
14 Dec 00
Beam Pickups            13

Pickup disrupts image currents, inducing a voltage
signal

3D Loops               Planar Loops
Paul Derwent
14 Dec 00
Beam Pickups           14

   At A:

I
A

Current induced by voltage across junction splits
in two, 1/2 goes out, 1/2 travels with image
current
Paul Derwent
14 Dec 00
Beam Pickups             15

   At B:

I
B

Current splits in two paths, now with
OPPOSITE sign
 Into load resistor ~ 0 current
 Two current pulses out signal line

DT = L/bc
Paul Derwent
14 Dec 00
Beam Pickups                    16

   Current intercepted by pickup:
-w/2       +w/2
y

x                      d
Dx

Current Distribution

 Use      method of images
I beam  1       w                w 
I             
tan sinh   Dx           tan 1 
sinh   Dx          
        d     2 
          d     2 

I beam       Dx 
            exp         for large Dx
          d 

 In areas       of momentum dispersion D
D DE
Dx =
b2 E

 Placement    of pickups to give proper gain
distribution
Paul Derwent
14 Dec 00
Accumulator Pickups          17

   Placement, number of pickups, amplification are
used to build gain shape

Core = A - B
Stacktail

Energy

Gain

Core
Stacktail

Energy
Paul Derwent
AntiProton Source        14 Dec 00
18

   Shorter Cycle Time in Main Injector