This research was sponsored by the National Science Foundation Grant NSF PHY-931478

Anderson, D.

Ball, M.

Derenchuk, V.

East, G.

Ellison, M.

Ellison, T.

Hamilton, B.

Nagaitsev, S.

Schwandt, P.

English

IUScholarWorks

1. Jacques Bosser, CERN (private communication: FAX'S 14 and 15 January 1993); a 
maximum of 3 mA had been stored in the LEAR ring. 
2. Gerard Tranquille, CERN (private communication, E-mail 15 March 1994); a record 
of 5 mA had been achieved. 
3. Dag Reistad, CELSIUS (private telephone communication 11 March 1993). 
4. Dag Reist ad, CELSIUS (private communication, E-mail 26 November 1992). 
5. S. Nagaitsev, T. Ellison, M. Ellison, D. Anderson, in the Proceedings of t h e  1993 
Workshop  o n  B e a m  Cooling, October 3-8, 1993, Montreux, Switzerland. 
6. T. Ellison, S. Nagaitsev, M. Ball, D. Caussyn, M. Ellison, and B. Hamilton, Phys. 
Rev. Lett. 70, 790 (1993). 
7. M. Ball, D.D. Caussyn, T. Ellison, B. Hamilton, and S.Y. Lee, IUCF Sci. and Tech. 
Rep. May 1990 - April 1991, p. 139. 
8. D. Anderson, M. Ball, V. Derenchuk, G. East, M. Ellison, T. Ellison, B. Hamilton, S. 
Nagaitsev, P. Schwandt, in this report. 
THE INVESTIGATION OF SPACE CHARGE DOMINATED 
BEAMS IN A SYNCHROTRON 
S. Nagaitsev, D. Anderson, M. Ball, V. Derenchuk, G. East, 
T. Ellison, M. Ellison, B. Hamilton, and P. Schwandt 
I nd iana  Univers i ty  Cyc lo t ron  Facili ty,  B looming ton ,  I nd iana  47408 
Knowledge of the longitudinal momentum spread of an electron-cooled proton beam 
is important for experiments which rely upon the unique properties of this beam. Since 
the RF cavity produces a conservative force, it cannot change the beam longitudinal phase 
density. Consequently, for a knownRF voltage, the beam time spread normally provides a 
direct measurement of the beam momentum spread. An electron-cooling system,' however, 
can reduce the ion beam emittance to extremely small values. In this regime the ion 
beam, interacting with its surrounding, generates longitudinal self-fields which significantly 
modify, and in fact to first order determine, the bunch shape.2 It can be shown that for 
the ion beams below transition, the most significant part of this interaction comes from 
the electrostatic repulsion between the ions within the bunch. Since in all cases of interest 
the bunch lengths are much greater than the radius of the surrounding vacuum chamber, 
one can treat the beam as a thin thread and reduce the electrostatic problem to one 
dimension. The Vlasov technique can then be used to find the self-consistent longitudinal 
particle distribution f ~ n c t i o n . ~  
For electron-cooled ion beams, however, one needs to modify the Vlasov equation to 
include both damping and diff~sion.~ Although the cooling force is generally a complex 
function of proton m ~ m e n t u m , ~  it can be approximated well as a linear function of relative 
proton momentum, 5, for 5 smaller than the longitudinal relative momentum spread of the 
electron beam. The latter is determined by longitudinal-longitudinal intrabeam scattering, 
the electron beam space charge depression, and by the high-voltage power supply ripple; 
the resulting relative momentum spread is typically > 1 x Consequently the linear 
approximation is valid for a < 1 x where a is a relative rms proton momentum 
spread. The modified Vlasov equation can then be written in Fokker-Planck form: 
where $ = !P(t, s ,  6) is the longitudinal distribution function, t is time, s is the longitudinal 
coordinate, X is the cooling rate and d is the diffusion rate. For a stationary distribution, 
the time dependence of vanishes and the solution of Eq. (1) for the particles experiencing 
synchrotron oscillations in the presence of space-charge force can be written as:6 
where a2 = d/2X and p(s) is the longitudinal linear charge density given by the expression: 
R2 
p(s)e-ap(s) = p(o)e-.p(o) [-w (1 - cos (%))I 
The value of p(0) must be chosen so that p(s) is normalized to qN,  where q is the ion 
charge and N is a total number of particles per bunch. The constants a and a, are given 
by the expressions: 
where y, ,B are the usual relativistic parameters, Zo = 377R, g is a geometrical factor (= 
ln(pipe radiuslbeam radius) + 1 /2 for a cylindrical perfectly-conductive vacuum pipe and 
a round uniform density beam), M is the particle mass, Vrf is the RF voltage, h is the 
harmonic number, 1) is the phase slip factor (x -0.86 for the IUCF Cooler ring), fo  is 
revolution frequency of the synchronous particle, and R is the radius of the synchrotron 
storage ring (x 13.8 m for the IUCF Cooler ring). 
Measurements of the longitudinal bunch profile and longitudinal bunch oscillation 
frequencies of an electron-cooled 45-MeV proton beam were performed in the IUCF cooler 
ring as a function of the RF voltage and beam current. The bunch shape was measured 
using a high bandwidth longitudinal electrostatic pick-up electrode and a 10-bit full scale, 
1 GSample/s digitizing scope. The data were transferred to a PC computer for offline 
analysis. This analysis consisted of fitting the experimental longitudinal bunch density to 
the theoretical prediction (Eq. 3) using two parameters: a and g. A comparison between 
the predicted and measured bunch shapes is presented in Fig. 1: the dashed line "2" is the 
-300 -200 -100 0 100 200 300 
Time, (nsec) 
Figure 1. Measured (solid, 2) and theoretical (dashed, 2) linear density. Cosine (1) and 
Gaussian (3) fits are also presented. Ip = 400 pA,Vrf = 10 V. 
best theoretical fit (g = 2.1 and o = 3.5 x the solid line "1" is a truncated cosine 
fit (g = 2.4, o = 0) and the solid line "3" a Gaussian fit (g = 0, o = 8.8 x loy5). 
The area under the linear density curve (Fig. 1, curve 2) was used to determine the 
proton beam current. There is a 5% normalization error due to the uncertainty in the 
capacitance of the longitudinal pick-up electrode. 
The RF cavity voltage was measured both with an RF probe capacitively coupled to 
the cavity, and by measuring the frequency, us,  of a 10"-amplitude dipole synchrotron os- 
cillation at low beam currents. These oscillations were excited by nonadiabaticaly shifting 
the phase of the cavity voltage by 10" in less than 40 ps. 
The fitted values for g and o can be used to calculate the ratio of the beam rest 
frame kinetic and electrostatic potential energy spreads, or alternately, the ratio Ve /Vrf 
where Veff is the effective RF voltage amplitude inside the bunch due to the presence of 
space-charge. This ratio is plotted in Fig. 2 as a function of beam current. 
A measurement of the bunch length ( "quadrupole" ) oscillation frequency, w, , can 
also be used to determines the degree by which the space charge force within the bunch 
compensates the external RF focusing force. For short bunches, where the linear RF force 
and parabolic bunch distribution is a good approximation to the sinusoidal RF waveform 
and truncated cosine bunch distribution, there exists a simple analytic expression7 relating 
the ratio ws/wq to the ratio Ve /Vrf: 
In the absence of space charge, Veff = Vrf and wq = 2ws, which simply corresponds to 
the rotation of the bunch in the longitudinal phase space; in the case of a zero momentum 
spread beam, Vef = 0, and wq = &us. 
The amplitude of the RF voltage was incremented by a fixed value (z 6 V) to excite 
the bunch length oscillations. The frequency ofythe bunch length oscillations was measured 
0.0 
10 -* 10 -I 1 
Proton current, ( m ~ )  
Figure 2. Ratio of V, to VTf derived by fitting the measured bunch profiles using Eq. 3. 
1. f 
10 -= 10 -' 1 
Proton current, (mA) 
Figure 3. Ratio of quadrupole frequency to synchrotron frequency vs. beam current. 
Dashed line shows fi value. VTf = 10 V. 
0.0 
10 -2 10 -' 1 
Proton current, (mA) 
Figure 4.  Ratio of effective RF voltage to applied RF voltage derived from the ratio 
wq/ws. VTf = 10 v. 
by monitoring the power in one of the higher revolution frequency harmonics as a function 
of time. The harmonic number, typically 20-40, was chosen to be sensitive to the bunch 
length changes. 
The measured ratio wq/w,, shown in Fig. 3, was much closer to f i  than to 2, indicating 
the high degree of space charge compensation. The ratio Veff/Vrf calculated from the 
bunch oscillation frequencies using Eq. 6 is plotted in Fig. 4 as a function of beam current. 
We conclude that the monitoring of the bunch modes has the potential to be used on- 
line in the future experiments which require knowledge of the momentum spread. We shall 
also continue the measurements of the longitudinal properties of the beam as a function 
of beam energy and internal target thickness. 
1. G.I. Budker, Sov. At. Energy 22, 346 (1967). 
2. Timothy J.P. Ellison, Sergei S. Nagaitsev, Mark S. Ball, David D. Caussyn, Michael 
J. Ellison and Brett J .  Hamilton, Phys. Rev. Lett. 70, 790 (1993). 
3. David Neuffer, IEEE Trans. Nucl. Sci. 26, 3031 (1979). 
4. J. Hai'ssinski, Nuovo Cimento 18B, 72 (1973). 
5. Tim Ellison, Ph.D. thesis, Indiana University (1990) p. 83. 
6. Alexander Wu Chao, Phys i c s  of Col lect ive  B e a m  Instabi l i t ies  in High  Energy  Acceler- 
a tors  (John Wiley & Sons, Inc., 1993) p. 287. 
7. S. Nagaitsev, T. Ellison, M. Ellison, D. Anderson, in the Proc. of t h e  1993 W o r k s h o p  
o n  B e a m  Cooling, October 3-8, 1993, Montreux, Switzerland. 


The Investigation of Space Charge Dominated Beams in a Synchrotron

https://scholarworks.iu.edu/dspace/bitstream/2022/529/1/iucf1993-1994_91-95.pdf

The Investigation of Space Charge Dominated Beams in a Synchrotron

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