Some Space-Charge Effects in Electron Cooling Devices
V.V. Parkhomchuk and V.B. Reva
The Budker Institute of Nuclear Physics, Novosibirsk, Russia
Abstract. The method of electron cooling uses an energy exchange between ions and electrons, both moving with the
same average velocity in a special section of the ion ring [1]. The interaction between the ions and electrons leads to the
various effects on the properties of the ion beam
INTERACTION OF THE SPACECHARGE FLUCTUATION IN THE ION
AND THE ELECTRON BEAMS
(2)
It is convenient to represent the action of the
Coulomb force on a single particle as a sum of two
parts. The first part is a short-range interaction. This
action is similar to the friction force of a fast particle
moving in a cold gas. The second, long-range part of
the interaction manifests itself as the action of a
collective electric field. This interaction is more
complicated and may lead to both decreasing ion
energy and increasing ion energy. One of the reasons
for this is related to the finite time of interaction
between the ion and electron beams in the cooling
section. In the case of a closed system the collective
oscillation is reversible in time and does not lead to
growth of the total energy of the ions. In our case,
where the system is open and the electron beam is
continuously renewed, the situation demands a special
analysis.
<%
where co0 = ^cdj + ct)f , T is the time of ion passage
through the electron cooling section, Mcool is the
transfer matrix from the state vector [sifs^] at the
point before cooling section ( s = 0 ) to the state
vector [si , S ' J at the point S = Lcool . The initial
parameters of the electron beam are zero:
se = dse /dt = 0 . The determinant value is
Det(Mcooi) < 1 for small values of the interaction
time <BbT, but the determinant Det(Mcooj) > 1 for large
G)bT>27i. The parameter 00^ (Oi M)2 defines the
difference of the Det value from 1.
This model is applicable to high-frequency
oscillations with a small wave vector.
Such
oscillations have small lifetime and fast mixing due to
Landau damping. Thus this effect can be considered as
an amplifier or a suppressor of a fluctuation. Its real
role depends on the amount of energy contained in
this short-wave fluctuation.
A simple model illustrating the features of the
finite time interaction is the the short-range
longitudinal fluctuation model. Its dynamics can be
described by the equations [2]:
(1)
The equations of transverse motion of ion and
electron beams in the magnetic field of the cooling
section cooler are [3-5]
dt2
Here Sj and se are the longitudinal shift of the ion
and the electron from an unperturbed location, co^ and
(Oi are the frequencies of longitudinal space-charge
oscillations of the given wave vectors, and (Obi and (Oie
are the oscillation frequencies of one type of particle in
the collective field of the other particle type.
2
3s
ds
+ •
1
ds
+''TT(*>''TTW-
(3)
The determinant of the ion transfer matrix
corresponding to these equations is
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329
STOCHASTIC HEATING OF THE ION
BEAM DUE TO ELECTRON BEAM
NOISE FLUCTUATIONS
Here (z,-) = (*,-)
is the centre-of-mass displacement of the ion and
electron beams,
The electron beam can affect the ion beam as a lens
with fluctuating strength. If the total space charge of
the electron beam is changing at high frequency, then
the life-time of the ion beam decreases. The noise in
the electron beam can be induced by both the noise in
the power supply and the intrinsic processes of the
electron beam. The storage of secondary electrons and
fast losses can also add space-charge fluctuations.
2=
ie
ne and nf are the densities of the ion and electron
beams in the laboratory frame, B is the magnetic
field, eZi is the ion charge, and mi and me are ion
and electron masses. We assume that the radii of the
beams are equal, the disturbance of the beams is a
long-wavelength transverse displacement, the radius of
the vacuum chamber is large so that image forces are
negligible, and the drift approximation is correct
(A He « Am, A,ei, A,ie). The matrix of one revolution
of the ion beam in the ring containing the cooling
section is
Mring=LRLMcool.
In a simple model the electron beam noise is
defined by the current fluctuation, with the fluctuation
independent for each ion turn. In this case we can
write the increment of ion betatron amplitude a as
da
dt
The cooling time of ions with betatron amplitude
equal a is [7]:
(4)
R is the Twiss matrix that is multiplied from the right
and left sides by a matrix L of adrift section with a
negative length of - Lcool / 2 .
\
(
The result of analysis in the case of small L cool
rtrt
'
a<a
a>a
(7)
where Je is the current of the electron beam, q is
the electron charge, re, ij are the classical electron and
ion radii, Lnc is the Coulomb logarithm, r| is the
fraction of the storage ring occupied by the cooling
device, p,y are beam kinetic parameters, aeff is the
effective amplitude defining the magnitude of the
cooling force, ae is the electron beam radius, p is the
betatron function at the cooling device, and f 0 is the
revolution frequency. By comparing eqs. (6) and (7),
one can see that the cooling time increases as a^ if the
ion moves inside the electron beam and as a^ if ion is
outside the electron beam. Thus it is possible that, for
some amplitudes of ion betatron oscillation, the
cooling process does not compensate the heating
process. The ion amplitudes increase and the ions are
lost from the storage ring. Fig.4 shows the heating and
cooling power of a cooling device with parameters
similar to that of ref. [6].
(5)
where A is the eigenvalue for matrix M ring . Here
(3L is the beta-function, Lcool is the length of the
cooling section.
The result is unusual. There is an eigenvalue
greater than unity, which means that there is an
unstable motion of the centre-of-mass of the ion beam.
The decrease of the beta-function J3± and the increase
of the magnetic field in the cooling section leads to a
decrease in the growth rate of this instability, but it
does not suppress it completely. The effect is
associated with the redistribution of energy between
the various ion beam oscillation modes in the cooling
section.
330
be less than 0.1%. The threshold amplitude for ion loss
is then more than 10 cm.
REFERENCES
[ 1 ] G. I. Budker, AtEnerg. 22 (1967) 246-2582.
[2] V. V. Parkhomchuk, Nucl .Instr. Meth. A441
(2000) 9-17
[3] P. R. Zenkevich, A.E.Bolshakov, Nucl. Instr.
Meth. A 441 (2000), p.36-39.
[4] A.Burov, Nucl. .Instr. .Meth , A 441 (2000), p.2327.
[5] V. V. Parkhomchuk and V. B. Reva, Journal Exp.
Theor. Phys., v.91 (2000), N 5, pp.975-982.
[ 6 ] E. I. Antokhin et al., "Conceptual project of an
electron cooling system at an energy of electrons of
350 keV", Nucl. Instr. Meth., A 441 (2000), p.87-91.
Ion betatron amplitude (cm)
FIGURE 1. Cooling and heating power as a function of ion
betatron amplitude.
One can see that a fluctuation of electron current
by about 2% leads to the situation where ions with
amplitude less than 3 mm can be cooled. The other
ions will be lost. For normal operation of a cooling
device the level of electron current fluctuation should
331