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					Dependence of the cooling force on the
    electron transverse velocity

             MI Department Meeting
                 1 September 2010


Khilkevich Andrei, Belarusian State University, AD/MID

   Lionel Prost(Supervisor), Alexander Shemyakin
                                Electron Cooler
Electron beam generated and accelerated in a Van-de-Graaff type electrostatic accelerator
(Pelletron).

                              Cooling section consist of 10 solenoids, 2 m-long where electrons
                              mix with anti-protons; B≈100 G
                              There are 20 dipole correctors in each solenoid




                                                                                            1
Figure 1. Schematic picture of electron cooler at Fermilab
                               Cooling force
                          Non- magnetized model
 Assumptions:

-the influence of the magnetic field is neglected

- the electrons velocities are assumed to be described by a Gaussian distributions

- the electron density and velocity spreads assumed being constant
the electron density distribution is close to parabolic, the transverse velocity spread tend to
increase towards the edge of a beam


                                                                                           2
                                     Cooling force
                              Non- magnetized model
                                                                                                    Vx 2 Vy2 Vz 2
                    
                                  Vp  Ve                                   1
                                                                                                  -       +      +
F  4ne me re c   Lc f Ve 
                                                                                                    2σ t 2 2σ t 2 2σl2
                                              d 3 Ve (1) f (Ve )=
              2 4
                                                                                              e                          (2)
                                            3                              3/ 2           2
                                Vp  Ve                           (2π)          σlσ t

n e – an electron density in beam rest frame              me – electron mass

re – an electron classical radius               V p – the velocity of pbar beam

Lc – the Coulomb logarithm                      Ve – the velocity of electron beam
f Ve  – electron velocity distribution        –   a ratio of ring length, occupied by Cooling Section
                                                     to the Recycler ring circumference
Vt – transverse electron velocity
                                                 l – rms longitudinal velocity spread                                   3
Vl – longitudinal electron velocity
                                                 t – rms transverse velocity spread
                                         Purpose of studies

     How well could a non-magnetized model be used?

     F(Vt) – ?
                                         Applied oscillations

            y                               1) Kick the beam at first solenoid, futher motion is
                  2A                       determined by longitudinal magnetic field.
         und                 (2
                              )           2) Kick the beam with each dipole corrector in each
           v                                solenoid to create a helix-like trajectory with desired
                           A
                                            amplitude and period.
                                     z
                                             t   c 0 (3)      0 – rms angle spread
                 v||
                                                      2A
                                                                                                      4
 x                                           osc          (4)   Coherent angle along the beam
                                                      
Figure 2. Illustration of helix motion
                    Side benefits: possible use at BNL

    BNL is considering of using the electron cooling for RHIC’s low energy run

possible problem : beam loss caused by recombination of heavy ions with electrons


 Idea (ref. Fedotov et al, PAC 2007 ) : suppress recombination by coherent
 oscillatory motion of electron beam, and recover the cooling efficiency by
 an increase of the electron beam current
  works only if a contribution of “far collisions” is significant(see next slides)


              The idea had never been tested before experimentally                   5
                Non-magnetized model in oscillatory case
 without oscillations                     with oscillations
                                                                                          Typical values


                                                                                         r_max    1.8 mm
                r_max                                          r_osc
                                                    pbars
                                                                                         r_osc     0.1-0.5
                                                                                                    mm
               pbars
                                                   q                                             2 nm
                                                       r_max




Figures 3,4. trajectories of electron beam in both cases                                                   6
 –   the impact parameter, at which the scattering angle is equal to /2
r_max – maximum impact parameter (Debae radius)             r_osc – oscillation radius
             Non-magnetized model in oscillatory case

    1                        1) 0<r<r_osc , close collisions
                                                             3σ t     3σ t      3σ t
                                                                                             L c1f (Ve )(ΔVz + vz)
             r_osc          F(ΔVz ) = 4πn e m e re c η2 4
                                                              ∫ ∫ ∫[(vx) + (vy + U)
                                                             3σ t     3σ t      -3σ t
                                                                                         2               2
                                                                                                           + (ΔVz + vz) 2 ]
                                                                                                                           dvxdvydvz

                  pbars
                              Applied oscillations are taken into account
        q
                                      r _ osc
               r_max                      ρ                                      ρ⊥2 + r _ osc2
                               L c1 = ∫ 2       dρ =
                                      0
                                        ρ + ρ⊥2                                      ρ⊥
                             2) r_osc<r<r_max , far collisions
                                                      3σ t     3σ t      3σ t
                                                                  L c 2 f (Ve )(ΔVz + vz)
                            F = 4πn e m e re c η2 4
                                                       ∫ 3∫-3∫[(vx)2 + (vy)2 + (ΔVz + vz) 2 ]dvxdvydvz
2                                                      3σ t σt σt
                                                                                                                            7
                                      ρ + r _ max Applied oscillations are neglected
             r _ max
                     ρ                  2                    2

    Lc 2 =      ∫ρ2 + ρ⊥2 dρ = ln
             r _ osc
                                       ⊥

                                      ρ + r _ osc2 ref. Fedotov et al, PAC 2007
                                       ⊥
                                         2
Contributions of close and far collisions to the cooling force
                           Close collisions               Far collisions             Total
                      30


                      25


                      20
F, MeV/c*hr




                      15


                      10


                      5


                      0

                                                                                                    8
              -0.05            0.05           0.15     0.25            0.35   0.45           0.55
                                                     a, mrad
              Figure 5. Contributions of close and far collisions to the cooling force
                                 Realization of oscillatory trajectories

                       x         y    Y BPM       X BPM         x goal         y goal

        0.060
        0.040
                                                                                                     solenoid
x,y [mm]




        0.020
        0.000
       -0.020
       -0.040                                                                                 BPM(X,Y)          BPM(X,Y)
       -0.060
                0          100       200       300        400            500            600
                                               Z [cm]

  Figure 6. Goal and calculated trajectories of electron beam
           Obtained series:
           1 oscillation per solenoid,  und _ m ax  0.5mrad                                     sine
           1 oscillation per 2 solenoids,  und _ m ax  0.4mrad
                                                                                                                    9
           1 oscillation per 3 solenoids,  und _ m ax  0.5mrad
                                                      Undulatory motion
Undulatory trajectory is created by kicking the beam with each corrector in certain direction and with
certain force. In result beam three-dimensional trajectory will be helix with specified period in Z
direction and amplitude in X and Y directions.
                                                             BPM(X,Y)    solenoid   BPM(X,Y)
                     One oscillation per two solenoids

                0.4                                             Y BPMs               Y
                0.3
    Trajectory, mm




                0.2
                0.1                                             X BPMs
                  0
               -0.1                                                                               X
               -0.2
               -0.3
               -0.4
                       0      500         1000        1500
                                    Z direction, cm                                                      10
Figure 7. Electron beam differential trajectory at X and Y
                                             Drag-rate measurements
In every experiment of studies the drag-rate was measured to determine the cooling force.
pbars~ 1*1010
                                                                                                           dP
                            0.4                                                4.3165
                                                                                                      F
Momentum deviation, MeV/c




                            0.3                                                                            dt
                                                                               4.316
                            0.2
                                                                                                      Errors:




                                                                                       Voltage, MeV
                                                                               4.3155
                            0.1
                                                                                                      – statistical ~ 10%
                              0                                                4.315
                                                                                                      – uncertainty of choosing
                            -0.1
                                                                               4.3145                   the points for fit (small)
                            -0.2
                                                                                                      – uncertainty of pbar
                                                                               4.314
                            -0.3                                                                        emittance
                            -0.4                                               4.3135
                                                                                                                                    11
                               9.20   9.21     9.22              9.23   9.24                           Measurement were made
                                                      time, hr                                         with the help of a program
                                                                                                       R111 , written by Dan
Figure 8. Pbar momentum change during voltage jump                                                     Broemmelsiek
                                                                       Results
               period is 2m, 17 July          period is 2m, 13 July           period is 4m, 17 July
               period is 6m, 17 July          period is 1m, 27 April          period is 13m, 13 July
              45
                                                                                                              – large inconsistency in
              40
                                                                                                              day-to-day measurements
              35                                                                                              at zero osc. angle
              30                                                                                              drift of the cooling section
                                                                                                              and BPM calibration
F, MeV/c*hr




              25
                                                                                                              – influence of beam offset
              20
                                                                                                                         2πA
              15                                                                                               α osc =       (4)
                                                                                                                          λ
              10
                                                                                                              contribution is rather weak < 6%
               5

               0                                                                                                                             12
  -0.05                 0.05           0.15          0.25              0.35        0.45                0.55
                                                     a, mrad
    Figure 9. The dependence of cooling force on oscillatory angle for all series of data.
                                                                      Results
                    period is 2m, 13 July   period is 13m, 13 July     fit for period = 13m   fit for period = 2m

               45

               40
                                                                                                                           The fitting parameter
               35                                                                                                          is a transverse velocity
                                                                                                                           spread in an electron
               30
                                                                                                                           beam
               25
F, MeV/c*hr




               20

               15

               10

                5

                0
                                                                                                                                             13
       -0.05                0.05            0.15              0.25             0.35           0.45                  0.55
                                                            a, mrad
    Figure 10. The dependence of cooling force on oscillatory and comparison to the theory prediction.
                                                                Results
                                     0.3 mrad                                                  0.4 mrad
                         period = 1m             period = 2m                             period = 2m                period = 4m
                         period = 4m             period = 6m                             period = 6m                theor dependance
                         theor dependance        period = 13m                            period = 13m
           16.5                                                           9

           14.5                                                           8

           12.5




                                                                       F, MeV/c*hr
                                                                          7
 F, MeV/c*hr




           10.5                                                           6

               8.5                                                        5

               6.5                                                        4

               4.5                                                        3
                     0           5                 10             15                 0        5                     10                 15
                                     Period, m                                                          Period, m

Figure 11. The dependence of electron-cooling force on the period of oscillations for                                                  14
different angles.
                                    Conclusion
The dependence of the measured drag rates on the oscillatory angles
qualitatively agrees with the non-magnetized model
    For a given angle, the drag rate increases for shorter wavelengths

There is a large experimental inconsistency in the measured day-to day
drag rate without additional oscillation
    Related to drift of the cooling section and BPM calibration
         The max/min ratio of the drag rates is 1.7
    Makes difficult to compare the results from different days
    Should weakly affect measurements at large oscillatory angles

The measured dependence of the drag rates on the oscillation period is
described by presented formulae within ~50%
    For angles of 0.3 – 0.5 mrad, where the effect of inconsistency in the initial value
    of the drag force should be low                                                        15