Instabilities of cooled antiproton beam in recycler Page: 4 of 5
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beam modes. Impedance and Landau damping just add
their terms to the matrix diagonal elements:dA, _
dt
4=-0(AL)PAP - i (Oz)p Ap
(16)
Note that the matrix T (14) is strongly degenerated: for
any finite dimension it is reduced, all its eigenvalues but
one are exact zeroes. With impedance, half of these zeroes
are getting unstable; they could be stabilized by the Landau
damping. For Gaussian distribution, the Landau damping
rate (AL)n is calculated as
AL j atscn eXp(-4 /2) , xa =Osc/Awb(n),
(17)
where the chromatic frequency spread Awb(n)/wo
r/n -$8p/p. If the distribution is not Gaussian, the correc-
tion is obvious. Note that the dimensionless energy spread
x, does not change, if the beam is adiabatically bunched:
it depends on the longitudinal phase space density. In other
words, growth of the space charge tune shift with the beam
bunching is compensated by an equal growth of the mo-
mentum spread, so that the dimensionless spread xn does
not change. As a consequence, the Landau damping grows
linearly with the bunching factor.
The described analysis predicts several times increase of
the phase space density due to the digital damper. Cur-
rently, longitudinal phase space density is typically about
twice higher than its stability threshold value without the
damper.
This section is essentially based on Ref. [6]. A different
way to present this problem was suggested later in Ref. [7].
Results of the two papers are close.
TWO-BEAM INSTABILITY AT
ELECTRON COOLING
Electron cooling is a powerful tool to increase phase
space density of hadron beams. It is successfully used at the
Recycler [10], as well as at many other storage rings; the
Recycler's beam kinetic energy is at least an order of mag-
nitude higher than anywhere else. Circulating antiprotons
are cooled because of their thermal collisions with elec-
trons of a co-moving single-pass electron beam. The same-
velocity beams share a small portion of the ring circum-
ference (20 m from 3.3 km). While individual antiproton-
electron scattering leads to cooling, a coherent interaction
of the two beams may lead to a two-beam instability. Al-
though this instability was never directly seen in the Re-
cycler, it still can be suspected to have place at high fre-
quencies or for quadrupole modes. A reason for this suspi-
cion is that there is a lifetime degradation, and sometimes
emittance growth, with increase of the antiproton density
happened either with cooling or with longitudinal squeeze.
Another possible explanation to these phenomena is an ex-
citation of single-particle resonances by a space charge ofthe cooled or squeezed antiproton bunch. A remedy de-
pends on the reason, so it was important to understand if the
antiproton-electron instability is responsible for the men-
tioned phenomena.
Two features of the beam-beam interactions are of prin-
cipal importance. First, since the beams are moving with
the same velocities, their interaction is local. Second, since
there is a solenoidal magnetic field in the cooler, electron
transverse motion is essentially a drift. Namely, a trans-
verse offset of the ion beam causes a dipole electric field,
forcing electrons to drift in the orthogonal transverse direc-
tion. This drift gives its own electric field, acting back on
the ions. Being linear and local, this electron response can
be described as a perturbation of the ion's revolution ma-
trix. At first order, this non-symplectic perturbation matrix
is proportional to a product of the electron and ion currents.
In the leading order, equations of motion for antiproton
("ion") and electron complex offsets 2,e =2,e + iy2,e are
reduced to the following set:C2 -2kede
(18)
0;
0.with k12e and k e6 as wave numbers, describing the beams in-
teraction. Here, modification of the beam-beam interaction
by the antiproton Larmor rotation is neglected. Also, the
beam-beam phase advances pe k1e6, Ved k dl over
the cooler length l are assumed to be small: pe, ed < 1.
Solution of Eqs. (18) leads to the cooler's matrix for an-
tiproton beam, perturbed by its interaction with the elec-
trons; the perturbation is scaled by the interaction parame-
ter
z
e~ed
proportional to both antiproton and electron currents. The
beam-beam interaction can be described by means of the
perturbed revolution matrix R, its bare value RC0> and the
perturbation P:R _ RC0) + P - Ca(I + P) -R",
Complex shifts of the phase advances 8pn _ pi
then follow by means of the perturbation theory:
=P
2"(19)
(o)
(fi
(20)where Vn, n= 1, 2, are the optical eigenvectors (1). This
yields growth rates An Im(fp~)/To, with To as the rev-
olution time. In the leading order, the perturbation 4 x 4
matrix P, calculated by means of Eqs. 18, has skew struc-
ture; in terms of 2 x 2 blocks it has only anti-diagonal el-
ements of equal values and opposite signs. The skew way
of the two-beam interaction leads to a conclusion that this
two-beam instability, if reveals itself at all, has to be highly
sensitive to x - y coupling of the unperturbed antiproton
eigenmodes. Indeed, since the electron response goes in
an orthogonal direction to the original antiproton offset, a
work of the resulting force acting back on the antiproton
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Burov, A. & Lebedev, V. Instabilities of cooled antiproton beam in recycler, article, June 1, 2007; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc901867/m1/4/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.