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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)
   Add MIXING
      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
        » Larger Radius
        » ~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
   Target Station Upgrades
   Debuncher Cooling Upgrades
   Accumulator Cooling Upgrades
      GOAL: >20 mA/hour

				
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