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

Alexeeva, L.V.

Anferov, V.A.

Baiod, R.

Caussyn, D.D.

Chu, C.M.

Crandell, D.A.

Gladycheva, S.E.

Hu, S-Q.

Krisch, A.D.

Lee, S.Y.

Ohmori, C.

Phelps, R.A.

Rinckel, T.

Russell, A.

Sato, H.

Schwandt, P.

Sperisen, F.

Stephenson, E.J.

Varzar, S.M.

von Przewoski, B.

Wong, V.K.

English

IUScholarWorks

ACCELERATOR PHYSICS 
FIRST DEMONSTRATION OF A HIGHER ORDER DEPOLARIZING RESONANCE 
L.V. ~lexeeva,(') V.A. ~nferov,(') C.M. ~ h u , ( ~ )  D.D. caussyn,(~) 
D.A. Crandell, S.E. Gladycheva, S-Q. Hu, A.D. Krisch, R.A. Phelps, 
S.M. ~arzar,(") and V.K. Wong 
Randall  Laboratory of Physics,  Universi ty  of Michigan,  Ann Arbor, Michigan 481  09  
S.Y. Lee, T. Rinckel, P. Schwandt, F. Sperisen, E.J. Stephenson, and B. von Przewoski 
Indiana Universi ty  Cyclotron Facility, Bloomington,  Indiana 4 74 0 8  
R. Baiod and A. Russell, 
Fermilab, Batavia,  Illinois 60510 
C. Ohmori, 
Ins t i tu te  for Nuclear S tudy ,  Univers i ty  of Tokyo,  Tanashi ,  Tokyo 188, Japan 
E. Sato 
KEK, Nat ional  Laboratory for High Energy  Physics,  1-1  Oho, Tsukuba-shi ,  
Ibaraki-ken 505, Japan 
The spin of a polarized proton in a storage ring will precess around the vertical ring 
fields with a frequency f ,- = fc  Gy , where fc  is the circulation frequency, G is the anomolous 
magnetic moment of the proton, and y = Elmp.  The spin tune Gy is the number of spin 
rotations during each turn around the ring. Protons also experience non-vertical magnetic 
fields that can kick the spin away. from the vertical direction. At certain energies, the 
protons see these non-vertical fields at a frequency that is synchronized with f,; these 
small kicks then can add coherently, causing a depolarizing resonance which destroys the 
polarization. 
We have extensively studied the first-order-imperfection1 and intrinsic2 depolarizing 
resonances in the IUCF Cooler Ring. Higher order depolarizing resonances can also exist 
in a ring because of coupling between the x and y motion, nonlinear fields, and synchrotron 
motion. A depolarizing resonance can occur when 
where j ,  k ,  m, and n are integers, v, and v, are respectively the vertical and horizontal 
betatron tunes, and vSyn, is the synchrotron tune. With a Siberian Snake present, the 
spin tune v, is not equal to Gy; thus we have written the resonance condition in Eq. 1 in 
terms of the spin tune us, which depends on the snake strength. 
Recently we observed two new depolarizing resonances in the Cooler Ring. Coupling 
due to the solenoids in the Cooler Ring can mix the vertical and horizontal betatron 
0.4 / 20% Snake @ Figure 1 .  The transverse polarization Pt is plotted against the horizont a1 betatron tune v,. The expected us = 6 - v, reso- 
: v,=4.710 nance position of 3.861 with a 20% snake 
0.2 180 MeV is shown. 
* 
3.76 3.80 3.84 3.88 3.92 
vx 
motion. This will give a vertical betatron frequency component f,v, that can cause a 
depolarizing resonance when v, = j f v,. We observed evidence for the 6 - v, horizontal 
intrinsic depolarizing resonance in a November 1994 run at 160 MeV with a 20% snake; 
Fig. 1 shows the transverse polarization Pt = plotted against the horizontal 
betatron tune v, with a fixed vertical betatron tune of vy = 4.710. With a 20% snake, 
the spin tune was then 2.139 at 160 MeV. The expected resonance position should be 
v, = 6 - us = 3.861; this agrees quite well with the data in Fig. 1. 
We also found an even narrower resonance at 160 MeV with a 20% snake while varying 
the vertical betatron tune v, with the horizontal tune v, fixed at 3.700. The data are shown 
in the two plots in Fig. 2. The lower plot shows a wide dip centered around vy = 4.87, 
which is the 7 - v, resonance. There is also a narrow dip at vy = 4.841. We thought 
that this dip might- be the second-order v, = 1 + vy - v, resonance; its calculated position 
seems to agree with the data. To test this hypothesis, we shifted v, to 3.640 and then 
repeated the v, scan. As indicated in the upper graph of Fig. 2, the narrow resonance 
moved by Av, = 0.059 f 0.001 to v, = 4.782; this shift agrees almost exactly with its 
predicted movement, which is equal to Av, = 3.700 - 3.640 = 0.060. Therefore, this 
narrow dip must be the us = 1 + v, - v, second-order intrinsic depolarizing resonance. 
This agreement in Fig. 2 appears to be the first experimental proof for the existence of a 
second-order depolarizing resonance. 
These studies show clearly that higher-order depolarizing resonances can exist in the 
IUCF Cooler Ring with a 20% snake. These resonances should be considered for future 
polarized beam running. 
This research was supported by grants from the U.S. Department of Energy and the 
U. S. National Science Foundation. 
(a) Also at: Moscow State University, Moscow, Russia. 
(b) Now at: National Central University, Chungli, Taiwan. 
(c) Now at: Florida State University, Tallahassee, Florida. 
'C'  I 71 f I I 
Figure 2. The transverse polarization Pt 
is plotted against the vertical betatron 
tune vy , for two values of the horizon- 
tal betatron tune v,. The expected v, = 
1 + vy = v, resonance position based on 
the spin tune with a 20% snake is also 
shown in each plot. 
1. Krisch, A.D. et al., Phys. Rev. Lett. 63, 1137 (1989). 
2. Goodman, J.E. et aL, Phys. Rev. Lett. 64, 2779 (1990). 


First Demonstration of a Higher Order Depolarizing Resonance

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First Demonstration of a Higher Order Depolarizing Resonance

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