Effect of solenoid field errors on electron beam temperatures in the RHIC electron cooler

Abstract

As part of a future upgrade to the Relativistic Heavy Ion Collider (RHIC), electron cooling is foreseen to decrease ion beam emittances. Within the electron cooling section, the "hot" ion beam is immersed in a "cold" electron beam. The cooling effect is further enhanced by a solenoid field in the cooling section, which forces the electrons to spiral around the field lines with a (Larmor) radius of 10 micrometers, reducing the effective transverse temperature by orders of magnitude. Studies of the effect of solenoid field errors on electron beam temperatures are reported.

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EFFECT OF SOLENOID FIELD ERRORS ON ELECTRON BEAM
TEMPERATURES IN THE RHIC ELECTRON COOLER∗
C. Montag†, J. Kewisch, BNL, Upton, NY 11973, USA
Abstract
As part of a future upgrade to the Relativistic Heavy Ion
Collider (RHIC), electron cooling is foreseen to decrease
ion beam emittances. Within the electron cooling section,
the “hot” ion beam is immersed in a “cold” electron beam.
The cooling effect is further enhanced by a solenoid field
in the cooling section, which forces the electrons to spiral
around the field lines with a (Larmor) radius of 10 microm-
eters, reducing the effective transverse temperature by or-
ders of magnitude. Studies of the effect of solenoid field
errors on electron beam temperatures are reported.
INTRODUCTION
To improve the luminosity of the Relativistic Heavy Ion
Collider (RHIC) as well as as essential part of the electron-
ion collider eRHIC currently under study, electron cooling
at high beam energies of γ ≈ 100 is being foreseen [1].
The RHIC electron cooler consists of a superconducting
energy-recovery linac to accelerate electrons to energies up
to some 55MeV, thus matching the Lorentz factor of the
stored ion beam. To enhance the cooling effect, a magne-
tized beam and a solenoidal field in the cooling section is
foreseen [2]. Figure 1 shows a schematic drawing of the
RHIC electron cooler.
Table 1 shows some key parameters of the RHIC electron
cooler.
However, topreservelowelectronemittancesfromthegun
through thelinacand transferlineandthrough thesolenoid,
exact optics matching as well as high magnetic field quality
is required. Simulation studies were performed to identify
the effect of transverse magnetic field components in the
∗Work performed under contract number DE-AC02-98CH10886 with
the auspices of the US Department of Energy
†montag@bnl.gov
γmax
B?
110
1T
30m
20000K
1 · 10−4
10nC
2mm
5cm
30min
solenoid length
cathode temperature
∆p/p
bunch charge
transv.rms beam size in cooler
electron bunch length
cooling time
Table 1: Parameters of the RHIC electron cooler
cooler solenoid on electron beam temperatures. The goal
was to determine the required field quality as a function of
the wavelength of the transverse field distortion, which in
turn defines the optimum spacing of correction coils.
SIMULATION RESULTS
To study the effect of transverse field errors on electron
beam temperatures, an ensemble of particles was tracked
through a 30m long 1T solenoid with an additional trans-
verse field component of
Bx(z)
ˆB⊥
=
=
ˆB⊥· sin(k · z),
a · B?,
(1)
(2)
where B?= 1T is the main solenoid field, z is the lon-
gitudinal coordinate along the solenoid, and k = 2π/λ is
the wave number for an error field wavelength λ. The coef-
ficient a has been scanned to determine the magnetic field
error tolerances for a given temperature increase.
Theelectronsareassumedtohavebeencreatedatacathode
with temperature T = 20000K. This beam is accelerated
to γ = 107 to match the storage energy of gold ions in
RHIC. The longitudinal momentum spread is assumed as
∆p/p = 1 · 10−4, the design value of the RHIC electron
cooler.
Tracking within the solenoid is performed by application
of the Lorentz force equation,
?F =d? p
dt= md? v
dt= e · (? v ×?B),
(3)
which is integrated using a fourth-order Runge-Kutta
method.
To calculate temperatures, all velocities are transformed to
the moving beam frame according to
v?=
v∗
?− v0
1 − v∗
?·v0
c2
(4)
and
v⊥=
v∗
⊥
γ0·
?
1 − v∗
?·v0
c2
?.
(5)
Here, v?and v⊥are the longitudinal and transverse veloc-
ity of one particular electron, respectively, while v0and γ0
are the velocity and Lorentz factor of the bunch center-of-
mass. An asterisk denotes the laboratory frame. Using the
0-7803-7739-9 ©2003 IEEE 2011
Proceedings of the 2003 Particle Accelerator Conference

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Figure 1: Schematic overview of the RHIC electron cooler. The magnetized electron beam is accelerated in an energy-
recovery superconducting linac and stretched to the required bunch length in in an extra loop in the arc that also serves as
a transfer line into the solenoid. At the exit of the solenoid a second arc reduces the bunch length to its original valueand
transports the beam back to the linac for energy recovery. For easier manufacturing, the 30m long superconducting
solenoid is split into two parts with an appropriate matching section in between that preserves beam magnetization and
thus low beam temperatures.
velocities in the moving frame, temperatures are calculated
from the respective rms values as
T?=
m · ?v2
2k
??
(6)
and
T⊥=m · ?v2
⊥?
2k
,
(7)
where k = 1.3806 · 10−23J/K is the Boltzmann constant.
Figure 2 shows the ratio of longitudinal electron beam tem-
peratures at the entrance, T?(z = 0m), and at the exit of
the solenoid, T?(z = 30m), as a function of the transverse
field wavelength for different values of the coefficient a,
corresponding to relative transverse field errors in the range
ofˆB⊥/B?= 1 · 10−6...5 · 10−5. The longitudinal elec-
tron beam temperature increase shows a significant reso-
nant behavior when the wavelength of the transverse field
component is in the vicinity of the Larmor wavelength of
the electrons,
λLarmor=2πγmc
eB?
≈ 1.14m,
(8)
where m and e are the electron rest mass and charge, while
c and γ denote the speed of light and the Lorentz factor,
respectively.
In contrast to this, the transverse temperature T⊥stays
practically constant, independent of the wavelength of the
0.1
1
10
100
1000
00.20.40.60.81 1.21.41.61.82
T_end/T_initial
lambda [m]
Figure 2: Longitudinal electron beam temperature increase
T?(z = 30m)/T?(z = 0m) at the end of the 30m long
cooler solenoid, as function of the wavelength λ of the
transverse field component. The lowest, red line corre-
spondstoatransversefieldamplitudeofˆB⊥= 1·10−6·B?,
while the top line (black) shows the relative temperature in-
crease for the caseˆB⊥= 5 · 10−5· B?. Lines in-between
correspondtoˆB⊥= 2·10−6·B?(green),ˆB⊥= 5·10−6·B?
(blue)ˆB⊥= 1·10−5·B?(purple), andˆB⊥= 2·10−5·B?
(turquoise), respectively.
2012
Proceedings of the 2003 Particle Accelerator Conference

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0
50
100
150
200
250
300
350
400
450
500
051015
z [m]
20 2530
T(z)/T(0)
Figure 3:
T?(z)/T?(z = 0m) in the resonant case with λ =
λLarmor≈ 1.14m as function of the longitudinal position
along the solenoid. The lowest, red line shows a transverse
field amplitude ofˆB⊥= 1 · 10−6· B?, while the top line
(black) corresponds to the relative temperature increase for
thecaseˆB⊥= 5·10−5·B?.Linesin-betweencorrespondto
ˆB⊥= 2·10−6·B?,ˆB⊥= 5·10−6·B?,ˆB⊥= 1·10−5·B?,
andˆB⊥= 2 · 10−5· B?, respectively.
Relative longitudinal temperature increase
transverse magnetic field.
electron beam temperature T?can therefore be explained
as coupling of transverse electron motion into the longitu-
dinal direction, thus effectively increasing the longitudinal
energy spread ∆p/p and hence the longitudinal tempera-
ture T?.
The relative longitudinal temperature increase as function
of the longitudinal position within the cooler solenoid in
the resonant case with λ = λLarmor ≈ 1.14m is de-
picted in Figure 3. As this picture reveals, the longitu-
dinal electron beam temperature increases approximately
linearly along the cooler solenoid.
The increase in longitudinal
CONCLUSION
The effect of a transverse sinusoidal field component
on electron beam temperatures inside the solenoid of the
RHIC electron cooler has been studied. The longitudinal
temperature shows a significant resonant effect when the
wavelength λ of the transverse magnetic field equals the
Larmor wavelength λLarmor = 1.14m, while the trans-
verse temperature remains unchanged over a large range of
transverse field amplitudesˆB⊥. To preserve the small lon-
gitudinal electron beam temperature, transverse magnetic
field components must not exceedˆB⊥= 1 · 10−6· B?for
wavelengths close to the Larmor wavelength λLarmor of
the electrons inside the solenoidal magnetic field. There-
fore, dipole correctors should be placed at distances of
λLarmor/4 to enable effective suppression of transverse
magnetic fields with these wavelengths.
REFERENCES
[1] I. Ben-Zvi, J. M. Brennan, X. Chang, J. Kewisch, W. Mackay,
C. Montag, S. Peggs, T. Roser, T. Srinivasan-Rao, D. Trboje-
vic, D. Wang, R&D towards Cooling of the RHIC Collider,
this conference.
[2] J. Kewisch, I. Ben-Zvi, X. Chang, C. Montag, D. Wang, Lay-
out and Optics for the RHIC Electron Cooler, this conference
2013
Proceedings of the 2003 Particle Accelerator Conference