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

Ball, M.

Brabson, B.

Budnick, J.

Caussyn, D.D.

Chao, A.W.

Collins, J.

Curtis, S.

Derenchuk, V.

Dutt, S.

East, G.

Ellison, M.

Ellison, T.

Friesel, D.

Gabella, W.

Hamilton, B.

Huang, H.

Jones, W.P.

Lamble, W.

Lee, S.Y.

Li, D.

Minty, M.G.

Nagaitsev, S.

Ng, K.Y.

Pei, X.

Rondeau, G.

Sloan, T.

Syphers, M.

Tepikian, S.

Yan, Y.

Zhang, P.L.

English

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TRANSVERSE AND LONGITUDINAL NONLINEAR BEAM DYNAMICS 
M. Ball, B. Brabson, J. Budnick, D.D. Caussyn, J. Collins, S. Curtis, V. Derenchuk, 
G. East, M. Ellison, T. Ellison, D. Friesel, B. Hamilton, H. Huang, W.P. Jones, 
W. Lamble, S.Y. Lee, D. Li, S. Nagaitsev, X. Pei, G. Rondeau, and T. Sloan 
Indiana University Cyclotron Facility, Bloomington, Indiana 47408 
A.W. Chao, S. Dutt, M. Syphers, Y. Yan, and P.L. Zhang 
The SSC Laboratory, 2550 Beckleymeade Avenue, Dallas, Texas 75257-3946 
M.G. Minty 
SLAG, MS26, Box 4349, Stanford, California 94509 
S. Tepikian 
Brookhaven National Laboratory, Upton, New York 11 973 
K.Y. Ng and W. Gabella 
Fermilab, P. 0. Box 500, Batavia, Illinois 6051 0 
In the past year, a great deal of progress has been made in developing the equipment 
and methods required for studying nonlinear beam dynamics, and many interesting results 
have been obtained. These studies track the motion of the beam centroid after collectively 
perturbing the beam. The degree to which the beam centroid motion accurately represents 
the motion of a single particle depends on the emittance of the beam; the smaller the 
emittance of the beam, the more accurate its representation of single particle motion. 
In this respect, the IUCF Cooler Ring provides an ideal laboratory for nonlinear beam 
dynamics experiments. 
In experiments done earlier this year, one dimensional (horizontal) motion of the beam 
in phase space was determined from turn-by-turn measurements of the beam position, 
xl and xz in two capacitive pickup beam position monitors, BPMs, located about 90° 
apart in betatron phase. The conjugate variable of XI, namely p,l, is a function of the 
measured beta functions at the BPMs, the betatron phase advance of the detectors, and 
the measured positions, XI and xz. When the beam is kicked horizontally, the resulting 
motion for a linear machine lies on a circle in the p,l-x1 phase space map. Deviations 
of the phase space map from this behavior provide information about the strength of the 
nonlinearities. 
While the nonlinear terms in the accelerator Hamiltonian are intrinsically weak, par- 
ticle motion can be strongly affected when resonance conditions are encountered. A res- 
onance condition is specified by mu, + nu, = p, where v, and v, are the horizontal and 
vertical betatron tune, and m, n, and p are integers. The order of the nonlinearity is 
related to the values of m and n. For a resonance in one dimension, either m or n is 
zero. In this case, particle oscillation around fixed points in a phase space map can be 
observed and are referred to as resonance islands. In Fig. 1, some recently taken data near 
the fourth order resonance (m = 4 and n = 0) are shown for a series of increasingly large 
horizontal kicks. Here, the motion near the separatrix and around the fixed points is well 
illustrated. These data confirm the work done earlier this past year in which the Cooler 
Figure 1. A transverse phase space map for operation near the fourth order resonance 
(m = 4) where v, = 3.75. 
ring Hamiltonian at v, = 3.75 was determined.' These data will be used for more detailed 
study of the motion within resonance islands, a task made possible by the implementation 
of a method to decouple the vertical and horizontal motion of the beam. 
Our most recent experimental work involves the extension of the studies of transverse 
nonlinear motion from one to two dimensions. Although many numerical studies are avail- 
able, due to their difficulty there have been few attempts to perform 2D beam dynamics 
e ~ ~ e r i m e n t s . ~  However, since beam loss arises essentially from 2D coupling resonances, it is 
especially important to explore these resonances in great detail. To observe the nonlinear 
coupling resonance, we studied the nonlinear coupling resonance at v,o - 2vZo = -6. In 
Fig. 2, some data for the horizontal and vertical motion near this resonance are shown. 
The growth in oscillation amplitude in one dimension with a corresponding decrease in 
amplitude in the other dimension is a characteristic of the nonlinear difference resonance. 
With further analysis of these data, the contribution of this resonance to the Hamiltonian 
will also be determined. 
Turn number 
0 500 1000 1500 2000 
Turn number 
Figure 2. The x and z positions from single BPMs near a third order difference coupling 
resonance at v, w 3.70, v, x 4.85. 
The existence of chaotic solutions and transition to chaos in parametric resonant dy- 
namics system are of interest to many disciplines of physics. One such system is the syn- 
chrotron motion3 discovered in 1945 by McMillan and Veksler. For a circular accelerator, 
the equation of phase oscillation is given by 
where w, is the small amplitude synchrotron angular frequency; is the synchronous 
phase angle, wo is the angular frequency, and 6 = Aplp and 4 = -he are conjugate 
variables with 0 being the orbital angle and h the harmonic number. 
Recently, we have tracked the longitudinal motion of a single particle and used it to 
study synchrotron motion for the first time. The beam was kicked longitudinally by phase 
shifting the rf. The resulting beam-centroid displacement in momentum was tracked by 
digitizing and recording the horizontal transverse position of the beam at a high dispersion 
location; the beam-centroid displacement in phase was tracked by digitizing the phase of 
the signal from a longitudinal pickup relative to the rf. The momentum deviation, Aplp, 
is related to the off momentum closed orbit, Ax,,, by Ax,, = D,Ap/p, where D, is the 
horizontal dispersion function. 
To study the effect of a driven motion, the phase of the rf was modulated producing a 
modulation of the synchronous phase. Fig. 3 shows that the numerical simulations of this 
system agree well with the measured responses at small driven amplitudes, a. When the 
modulation amplitude is larger than about 5.7O, the numerical simulations predict that the 
the particles can jump outside the rf bucket and can be recaptured into the same bucket 
( h  = 1 rf system). 
In conclusion, our experimental studies of nonlinear beam dynamics have produced 
interesting results from studying both transverse and longitudinal motion. The ground 
work for possible six dimensional phase space tracking in nonlinear beam dynamics experi- 
ments has been layed. Further experimental studies using these techniques may contribute 
to a better understanding of the bunch diffusion process caused by phase and amplitude 
noises. Another avenue of experimental exploration, now available, is the effect of ampli- 
tude modulation. Amplitude modulation can create islands within the dominant rf bucket. 
Such manipulations may offer a possible superslow extraction for fixed target experiments 
at S S C energies. Also, for highly damped parametric resonance physical systems , 4 9 5  many 
interesting phenomena, such as strange attractors, period doubling, etc., can be observed. 
Using a rf phase feedback loop, highly damped parametric resonances can also be studied 
in a synchrotron. 
1. S.Y. Lee, et al., Phys. Rev. Lett. 67, 3768 (1991). 
2. See for example the Fifth Advanced Beam Dynamics Workshop. 
3. E.M. MacMillan, Phys. Rev., 68, 143 (1945); V.I. Veksler, Compt. Rend. Acad. Sci. 
U.R.S.S., 43, 329 (1944); 44, 365 (1944). 
4. M. Octavio, Phys. Rev. B29, 1231 (1984). 
5. D. D'Humieres, M.R. Beasley, B.A. Huberman and Libchaber, Phys. Rev. A26, 3483 
(1982). 
x Response T, (mS) 
0 Response A,/A, 
x 5.3 deg. Phase Modulation 
X 
I I I I  I l l 1  I I I I  1 1 1 I  I I I I  I I I I  
400 450 500 553 600 650 700 
Modulation Frequencji Hz 
Figure 3. The measured amplitude and period of amplitude modulation as a function of 
modulation frequency for various phase modulation amplitudes, a = 0.88', 1 .go, 5 . 3 O .  


Transverse and Longitudinal Nonlinear Beam Dynamics

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Transverse and Longitudinal Nonlinear Beam Dynamics

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