Envelope instability as a source of diffusion

Nuclear Instruments and Methods in Physics Research A 441 (2000) 18}22

Envelope instability as a source of di!usion A. Burov*, S. Nagaitsev Fermi National Accelerator Laboratory P.O. Box 500, Batavia, IL 60510, USA

Abstract Electron cooling increases the phase space density of the cooled particles up to a certain limit. This limit is normally characterized by a strong space charge tune shift, about 0.2}0.3. This tune shift is high enough to bring the cooled beam to the threshold of the envelope instability. The envelope instability can be in a kind of dynamic equilibrium with the cooling, keeping the cooled beam at a threshold with a high level of the coherent noise. This noise acts as a di!usion for the halo particles which puts a limit on the stored current. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 29.20.!c; 29.27.Bd; 29.27.Fh Keywords: Electron cooling; Lifetime; Fokker}Planck equation; Intensity limit

1. Introduction Electron cooling is a conventional method to increase the phase space density of heavy particle beams. However, its possibilities and limits are not completely studied. One of the unclear phenomena is the limit on the cooled beam intensity observed at several cooling facilities [1}5]. The main features of the phenomenon experimentally studied in the IUCF cooler [1] are listed below. f Under the continuous stripping injection of protons in the cooler, the accumulated proton current was limited by a certain value independent on the injected current. The rate of the current increase was seen to be constant until just below

f f f

f * Corresponding author. Tel.: #1-630-840-8852, fax: #1630-840-6311. E-mail address: [email protected] (A. Burov)

f

the limiting current where the rate sharply dropped to zero. The phenomenon can be described as the beam lifetime being a highly nonlinear function of the beam intensity, see Fig. 1. The operation with the bunched proton beam showed that the peak proton current, not the average one, is limited. The beam current decreased smoothly between injection cycles, thus suggesting the beam scraping is not of concern. The size of the cooled beam at the limit current was found to be very small, the rms radius a "0.5 mm, which is deeply inside the electron # beam with the radius a "1.2 cm. The space % charge tune shift in the cooled beam was found to be high and approximately constant during the accumulation, *QK0.2 [6]. The beam halo was signi"cantly increased near the accumulation threshold, see Fig. 2 [7]. Attempts to dilute the transverse emittance of the cooled beam by means of the applied

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A. Burov, S. Nagaitsev / Nuclear Instruments and Methods in Physics Research A 441 (2000) 18}22

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Fig. 1. Accumulated current I as a function of time t during stripping injection with cooling accumulation of 45 MeV protons (IUCF). Solid line corresponds to the measurements, dash line is a "t IJ1!exp(!t/q ) for the lifetime measured independently as q "100 s. 0 0

transverse broad band noise resulted only in the decrease of the accumulated current [7]. f Coherent transverse signals were not observed. For the moment, there is no satisfactory explanation of these facts. It is not clear, why instead of the gradual emittance growth proportionally to the peak current (to keep the tune shift constant), the life time of the protons sharply drops and the accumulation stops. An attempt was made to explain these facts by a microwave electron}proton instability [5]. However, Refs. [9,10] found that the coherent electron}proton interaction introduces a stabilization in the proton beam motion under normal conditions. The main hypothesis of this paper is that it is the envelope instability of the cooled beam which might be a reason for the described phenomenon.

2. Envelope instabilities Due to a periodic modulation of the focusing force, there is a possibility of parametric excitation of coherent second-order (quadrupole) modes, resulting in the envelope instabilities [11,12]. The instability takes place when the space charge of the beam shifts the frequency of one or both quadrupole modes to a parametric resonance with the focusing structure. At the resonance, a ratio of the mode frequency to the structure frequency is a half of integer value, n/2. The coherent quadrupole tune shifts approximately follow the incoherent ones. When the tunes are equal l "l "l, one of the envelope modes x y corresponds to in-phase x}y oscillations and the other to out-of-phase oscillations. Applying the smooth approximation, their tune shifts *l , *l */ 065 are expressed in terms of the incoherent tune shift

SECTION I.

20

A. Burov, S. Nagaitsev / Nuclear Instruments and Methods in Physics Research A 441 (2000) 18}22

Fig. 2. Beam core and halo measurements during the accumulation of bunched beam (IUCF). The upper set of points is 1 of 95% 6 unnormalized emittance, and the lower set is rms emittance. If the transverse distribution were Gaussian, both sets would be equal (see e.g. [8]).

*l (see e.g. Ref. [8]): *l "*l; *l "1.5*l. */ 065

(1)

The structure period of a storage ring is, generally, its total circumference. Taking this into account, it can be concluded that a threshold of the envelope instability in the storage ring is reached when the space charge shifts one of the envelope tunes to a half-integer. It means that the single particle tune shift is limited by a value about 0.5/1.5+0.3. This value decreases if a di!erence between the tune and the nearest half integer below it goes down. In principle, the envelope instabilities can be avoided for any space charge level, if a storage ring is constructed from identical or almost identical cells. Such a storage ring has to obey the limit *l(N /4 which would be valid for any space # charge if the number of cells N is high enough, # N '4l. Due to the ring curvature, there is a min#

imal periodicity to satisfy this requirement. A storage ring of such a kind was suggested for experiments aimed to reach a crystalline state of a stored ion beam [13], the minimal periodicity was found to be N "8. # To avoid the envelope instability, booster synchrotrons are designed with high periodicity. But even if there are no systematic half-integer resonances near the working point, the resonances related to imperfections of the periodicity can limit the tune shift. The numerous simulations show that if the beam envelope is unstable initially, the coherent oscillations start to grow so that after a few structure periods the beam emittance is diluted enough to be in the stable area. The situation is di!erent if the beam is under cooling. Cooling would not allow the beam just permanently stay in the stable area: it would continue to shrink the emittance and lead the beam again to the instability threshold.

A. Burov, S. Nagaitsev / Nuclear Instruments and Methods in Physics Research A 441 (2000) 18}22

3. Di4usion by envelope oscillations Thus, the electron cooling and the envelope instability can be considered to be in a dynamic equilibrium keeping the cooled beam at the threshold of the instability. Behavior of such systems is typically rather complicated, turbulent, with time correlations restricted by several periods. The electric "eld of the oscillating beam core acts on the outer, hot protons. Mainly, the proton transverse energy is changed when it passes near the oscillating beam, at impact parameters less or about equal to the beam radius a . For a proton # with an amplitude a
21

f The longitudinal correlations of the envelope oscillations extend not much further than the core radius. Due to the longitudinal velocity of the proton, every pass it sees the core at various longitudinal coordinates where the "elds are not correlated. f Finally, this assumption of random interactions can be modi"ed: it could be assumed that the correlation time is by a certain factor f higher 3 than the average time between consecutive interactions. However, the calculations show that the resulting dependence on this factor is very weak, it enters as f 2@7. 3 Thus, the interactions of the remote proton with the oscillating beam core are considered as independent inelastic scatterings causing a di!usion of the proton energy. The proton angle changes due to the simultaneous action of this di!usion with the coe$cient 4o2 r2a3c2 D"(*h)2/*t" 1 1 # . u a5b2c6 "

(3)

The electron cooling rate is given by 4pn r r ¸ gc % % 1 C j" h3b3c5

(4)

q8K(a /a)u~1 " # is a time of the interaction, a8 /a K1 is the relative # # amplitude of the beam size oscillations responsible for the considered interaction. During the interaction time q8, the core envelope changes by a factor of Ku q8 which gives the perturbation of the proton " energy by this coherent oscillation Ju q82. " The main hypothesis of the present model is that this energy change of the hot proton can be considered as random. In other words, it is assumed that the oscillating "elds of the beam seen by the proton when it passes the core, are completely unrelated to the "elds seen on the previous pass. This assumption seems acceptable due to the following factors:

where n is the electron beam density, r , r are the % % 1 electron and proton classical radii, ¸ is the C Coulomb logarithm and g is the cooling length related to the ring circumference, h"au /(bc) is " the proton angle amplitude. Evolution of the proton distribution f (h) under an in#uence of the cooling and di!usion is described by the Fokker}Plank equation:

f A correlation time for oscillations of a system near its instability threshold is typically not more than several periods of the oscillations.

with f K1/h , where h "a u /(bc) is the angle 0 # # # " amplitude of the core particles. Assuming the aperture limit h"h "a u /(bc), the lifetime q follows ! . "

A

B

Rf R D Rf " jhf# . Rt Rh 2 Rh

(5)

This determines an equilibrium distribution:

A P

f (h)"f exp !2 0

hj(h@)h@ dh@ D(h@) 0

B

(6)

SECTION I.

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A. Burov, S. Nagaitsev / Nuclear Instruments and Methods in Physics Research A 441 (2000) 18}22

4. Avoidance of the overcooling

from Eqs. (5) and (6):

A P

B

h! j(h)hdh j h q~1K ! ! exp !2 , j "j(h ). (7) ! ! h D(h) # 0 These lifetime estimations imply that the relaxation time of the injected particles is shorter than their lifetime. This condition can be expressed as JD (2j )(h !h (8) * * ! * with h as an initial angle of the injected particles, * D "D(h ) and j "j(h ). * * * * Lifetime (7) contains current-dependent terms in the exponent, so it is a highly nonlinear function of the beam intensity. Assuming a "nite accumulation time q "d lnN/dt, 4 and taking into account dependencies (3) and (4) the intensity limitation follows: j h2 ! ! "¸ , ¸ "ln(j q a /a ), D "D(h ). (9) ! ! ! 4 . # ! ! 2D ! Then, the proton beam linear density o and its 1 radius a determine the space charge tune shift # o r R2 *l" 1 1 (10) 2lb2c3a2 # where R is the storage ring average radius. The threshold condition (9) can be resolved for the proton beam peak current I "o ebc and presen1 1 ted in the following form:

A

B A

B

nn g¸ r 2@7 2l*lb2 3@7 % C % . (11) r R2 2¸ r 1 ! 1 The obtained threshold (11) can be calculated for the parameters of the IUCF cooler. The aperture logarithm ¸ (9) calculated with a "1.5 cm, ! . a "0.5 mm, j "0.3 s~1, q "18 s, comes out # ! 4 ¸ "6. Substituting I "o ebc"400 mA, a " ! % % % 1.2 cm, b"0.3, ¸ "3 (only magnetized e}p C collisions contribute), g"0.03, l"4, *l"0.25, R"15 m, it gives I "6 mA, in agreement with 1 the observations [1]. Assuming the same aperture limitations and electron current, similar result (K4}6 mA) is obtained for the proton threshold current in CELSIUS [2] and the Kr34` threshold current in SIS [4], which is reasonably close to the observed values [5,3].

I "ecbc11@7a8@7 . 1

A slight misalignment t between the electron and proton beams could increase the threshold. In this case the protons with smaller angles cannot be accumulated because the corresponding phase space area is unstable for the single-particle motion (the so-called monochromatic instability [14]). The angle provided in both transverse planes should be larger than the threshold beam angles h but small# er than the angle at the injection h : * h (t (h . (12) # x,y * The larger is the misalignment, the higher is both the threshold current and the temperature of the accumulated protons. An improvement with restrictions similar to Eq. (12) could be reached by means of an electron cooler with a hollow cathode [1,15].

References [1] T. Ellison et al., Cooled Beam Intensity Limits in the IUCF Cooler, Proc. of Workshop on Beam Cooling and Related Topics, Montreux, CERN 94-03, October 1993, p. 377. [2] D. Reistad et al., Measurements of Electron Cooling and &Electron Heating' at CELSIUS, Proc. of Workshop on Beam Cooling and Related Topics, Montreux, CERN 9403, October 1993, p. 183. [3] V. Parkhomchuk, Nucl. Instr. and Meth. A 441 (2000) 9. [4] M. Steck et al., Commissioning of the Electron Cooling Device in SIS, Proceedings of the EPAC'98, Stockholm, p. 550. [5] V. Parkhomchuk, Limitation of Ion Beam Intensity in Electron Cooling Systems, Proceedings of the HEACC'98, Dubna, Russia. [6] S. Nagaitsev et al., Nucl. Instr. and Meth. A 391 (1997) 32. [7] S. Nagaitsev, Ph.D. Thesis, Indiana University, 1995. [8] M. Reiser, Theory and Design of Charged Particle Beams, Wiley, 1994. [9] A. Burov, Part. Accel. 57 (1997) 131. [10] D. Pestrikov, Nucl. Instr. and Meth. A 412 (1998) 283. [11] I. Hofmann, L.J. Laslett, L. Smith, I. Haber, Part. Accel. 13 (1983) 145. [12] J. Struckmeier, M. Reiser, Part. Accel. 14 (1984) 227. [13] L. Tecchio et al., CRYSTAL Ring, Feasibility Study, LNL-INFN Rep., Legnaro, Italy, 1996. [14] Ya. S. Derbener, A.N. Skrinsky, Part. Acc. 8 (1997) 1. [15] A. Sharapa, A. Shemyakin, Nucl. Instr. and Meth. A 336 (1993) 6.