Progress toward high energy electron cooling Page: 2 of 5
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longitudinal magnetic field is also considered beneficial at
high energies for two reasons:
* the solenoidal magnetic field allows to combine
strong focusing with the requirement (for efficient
cooling) of low electron transverse temperature in the
cooling interaction region;
* cooling rates with a "strongly" magnetized electron
beam are ultimately determined by the electron
longitudinal energy spread only, which can be made
much smaller than the transverse one.
An electron beam is considered magnetized when its
radius of transverse Larmor oscillations is much smaller
than the beam radius. If the solenoidal field lines are
perfectly parallel, these oscillations (among other
conditions) can increase the duration of an electron-ion
interaction thus increasing the friction force (assuming
that there are at least several Larmor oscillations in the
cooling section). Such a magnetization is considered
"strong" as far as the cooling process is concerned. At
high energies (y = 10 - 100) this requires a continuos
solenoidal field of 1 - 10 kG and the cooling section of at
least 20 - 30 m. On the other hand, it is believed possible
to preserve electron transverse temperatures at a low
(thermal) level and to employ only weak magnetic field
(but strong enough to provide focusing against electron
beam space charge). Such a field can be about 100 G or
less. Both of these two field options are being currently
considered for high energy electron cooling projects.
The most serious question is how to produce a long
solenoid of required field quality and how to measure it.
Generally speaking, the field quality (defined as the ratio
of the transverse field component to the longitudinal one)
requirement at high energies is more severe (by about a
factor of y) than at low energies.
2.1 High field option (RHIC cooling system)
The researchers at Budker INP and BNL, who proposed
electron cooling of gold ions in RHIC, have encountered
an interesting problem related to the ion recombination
during the cooling process if the electron beam is cold.
To suppress this recombination it is proposed to increase
the electron temperature to 1000 eV and rely on
"strongly" magnetized cooling. The electron cooler and
beam parameters under consideration are:
Beta-function in the cooling section: 6= 60 m
Ion beam normalized rms emittance: e = 1 pm
Ion beam rms angular spread: Oi= 1.3x10-5 rad
Electron beam momentum: pc = 50 MeV
Relativistic parameter: y= 100
Electron beam transverse temperature: Te = 1000 eV
Electron beam rms angular spread: Be = 6x10-4 rad
Magnetic filed in the cooling section: Ho = 1 T
Solenoid length: L = 30 m
Larmor radius (at full energy): p = pc/eHo = 16.6 cm
Larmor period: -27cp =100 cm
To realize the conditions of the magnetized cooling in a
non-perfectly parallel magnetic field one needs to restrict
the transverse field components such as to limit the drift
of the electron Larmor "circle" away from the ion duringthe interaction. Quantitatively, the "slow" drift angle of
electrons due to the transverse field errors should not be
larger than Bi. What solenoidal field quality does this
correspond to? To answer this question, I will start with
the equation of motion for an electron in the longitudinal
magnetic field.
Suppose that there are transverse fields in a solenoid:
H(s), Hy(s) <<H=const. For small transverse
oscillations, the electron equation of motion can be
written as:,,1 , H,
SH
y"= - -
-- f, H x(1)
where ' is d/ds. After introducing new variables: z = x +
iy and B = Hy - iH, the equation (1) can be rewritten as:_ _i B
p Hop(2)
where (p - z' is a complex variable representing the
electron trajectory angle in the solenoid. The solution of
this defferential equation is as follows:(3)
B(s')exp i s
HopIt is important to note that in a solenoid a typical scale
of its transverse field variation is about equal to the
solenoid diameter, D.
The equation (3) was solved numerically (using
MathCAD) for a proposed RHIC cooler parameters: p =
16 cm, D = 30 cm. A 1-T, 10-m long, 30-cm diameter
solenoid was simulated by 1001 identical current loops,
placed 1 cm apart from s = 0 to 10 m. Starting from s =
3.5 m each current loop was randomly (with a gaussian
distribution) displaced transversely to create a transverse
field error. The rms displacement was chosen such as to
limit the field error, H_/HO, by a value of about 1x10-5.
The transverse fields were then calculated by adding
transverse fields from each current loop on the solenoid
axis. Figure 1 shows the result of such calculations. A
number of random distributions were calculated and the
results presented here are quite representative of all the
runs.I1-
Hx,
H-
Hy,
H.300 400 500 600 700 800
Distanc along a-is s (cm)
Figure 1: The horizontal and vertical field errors. The
solenoid diameter is D = 30 cm.- -I
I II.to- 5
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Nagaitsev, Sergei. Progress toward high energy electron cooling, article, July 20, 2001; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc721758/m1/2/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.