# Accelerator

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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
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
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
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
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
Figure 10. The dependence of cooling force on oscillatory and comparison to the theory prediction.
Results
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
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

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