Krisch, A D

Mane, S R

Roser, T

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Particle Accelerators, 1988, Vol. 23, pp. 73-77Reprints available directly from the publisherPhotocopying permitted by license only© 1988 Gordon and Breach Science Publishers, Inc.Printed in the United States of AmericaHORIZONTAL FIELDS IN A RING WITHONE SIBERIAN SNAKE AND A VERTICALLYPOLARIZED BEAMA. D. KRISCH, S. R. MANE,t and T. ROSERRandall Laboratory of Physics, The University of Michigan Ann Arbor, MI48109(Received June 8, 1987)We show that a single Type 1 Siberian snake, such as a solenoid, is able to eliminate thedepolarization caused by certain harmonics of horizontal fields when a beam is vertically polarized.However, the corresponding depolarization due to the orthogonal set of harmonics cannot beeliminated by a single Type 1 snake. The two orthogonal sets of horizontal magnetic fields arerespectively in phase with radial and longitudinal fields at the point in the ring opposite the snake.1. INTRODUCTIONAn important problem in accelerating polarized proton beams to high energies isthe large number of depolarizing spin resonances that must be crossed. At theSSC, there will be about 36,000 depolarizing resonances; thus it appearsimpractical to use the jumping and correction techniques developed at the ZGS1and AGS.2 A more promising method of overcoming so many depolarizingresonances is to use Siberian snakes.3 Siberian snakes would, in principle,eliminate the effects of all depolarizing resonances by making the spin tune Voindependent of energy; specifically Vo would be 1/2. Therefore Siberian snakeshave generated considerable theoretical interest, but, during the last 12 years, noexperimental test of the snake concept has yet been performed.The original analysis3 and most subsequent theoretical work4 on snakesassumes that the polarization vector P is periodic around the ring, i.e.,pee + 2n) = pee), (1)where e is the ring azimuth angle. For a ring with one snake, this periodicity canonly occur when P is horizontal and can therefore· precess in the ring bendingmagnets. However for some experiments vertical polarization is more desirable.By using pairs of snakes, it may be possible to keep P always vertical, whilechanging direction between up and down in traversing each snake. Thistwo-snake scheme seems to be the most desirable solution for the SSC.In this paper we study a ring with only one snake and a vertically polarizedbeam, so that P alternates between up and down upon consecutive turns. We arestudying this case because we wish to consider possible near-term experimentaltests of the snake conceptS,6 with P vertical. For various practical reasons one cant Present Address: Fermilab, P.O. Box 500, Batavia, IL 60510.7374 A. D. KRISCH, S. R. MANE AND T. ROSERuse only a single Type 1 snake; thus we must ask if a single snake and a vertical Pwill allow valid tests of the Siberian snake concept. In this paper we shall studythe effect of a single Type 1 snake upon vertical spin polarization. We find thatthe effect is very different depending upon whether the horizontal magnetic fieldsare in phase with a radial or longitudinal field at the position opposite the snake.2. BASIC MODELWe begin by defining a model of a horizontal storage ring. The ring contains onesolenoid, which rotates a particle's spin by 1800 around the longitudinal axis. Thissolenoid is equivalent to a Type 1 Siberian snake. In the absence of a snake, thespin of a particle on the closed orbit will precess around the vertical axis, y, witha frequency 2Jlvo radians per circumference. It is easily verified that the spin tuneVo is equal to Gy, where y is the particle energy in units of rest mass andG == (g - 2)/2, where g is proportional to the particle's magnetic moment.Some details will not be discussed in this paper. We will ignore the rotation andfocusing of the particles' betatron orbits caused by the solenoid spin rotator.These effects can be compensated by a number of quadrupole magnets,6 whichwill also affect the ring structure. Moreover, it may be difficult to ramp thesolenoid magnetic field to exactly match the ramping of the beam energy; thus thesolenoid spin rotation angle may not be exactly 1800 •We will use the coordinate system with the orbit frame Oxyz. As shown in Fig.1, 0 is the location of a particle on the closed orbit, i is pointing radially inward,y is vertical, and z is longitudinal. We can solve the Thomas-BMT equation7 forspin motion in this frame, without the snake and without dep~larizing resonances.The resulting equation is ds-= Qy'" Xs (2)dB 'where ~ Q dB = 2Jlvo is the total spin rotation caused by the vertical ringmagnetic field in one turn around the orbit. Equation (2) has three linearlytYZ I_-POINT OPPOSITE- SNAKE I foSNAKE -_ flo!Ilof"~r' ..... I ~lo 0/IXnot1---..--------~---7I fa 9,,>./10 Yt' /x1/Z AFIGURE 1 Horizontal ring with a Siberian snake at (J = 1'C and the rotating frame {fo, no, 10}, ~hichis defined at the point opposite the snake «(J = 0) to be fo=x (radial), no = y (normal), and 10 = Z(longitudinal). Typical orientations of the rotating frame also shown just before and just after passingthrough the snake at the end of the first turn.HORIZONTAL FIELDS WITH SIBERIAN SNAKE 75(3)dsde = e(8) x s.independent solutions, such as {fo, no, 1o}, which is a right-handed orthonormaltriad that precesses along with the spinning particle. If we choose fio to pointalong the vertical (normal) direction V. then 10 and fo clearly precess in thehorizontal plane with spin tune frequency Vo = Gy. We will define the frame bychoosing the initial conditions, fo=x (radial) and 10 = z (longitudinal) at 8 = 0,which is the point opposite the snake. The snake produces a spin rotation of 1800around z at 8 = Jr as shown in Fig. 1. With these initial conditions, it can beshown that 10 points along the stable polarization direction at every point arounda ring with a Type 1 Siberian snake at (J = n.Notice that the snake rotates both the spin and the {fo, no, to} frame by 1800about the longitudinal direction. Thus, on the second turn around the ring(n < e< 3n), fo and no will both point opposite to their direction on the firstturn. The third turn will be identical to the first, 'and the fourth turn will beidentical to the second. Thus the Siberian snake changes the ring's 'spinperiodicity from 2n to 4n, both by making the spin tune 1/2 and by making theperiodicity of the {fo, fio, 1o} frame 4Jr. Let us denote the horizontal imperfectionfields by e(8). Then in the rotating frame {fo, no, 1o}, the Thomas-BMT equationhas the simple formIn this frame the spin will precess around the horizontal fields.3. INSTABILITIESIf we assume that the spin is initially vertical, we can study the spin evolutioncaused by the horizontal imperfection field(4)On the first turn around the ring, E(8) has components in the rotating frame{fo, no, 1o} given bye(e) == (E:, 0, E}) = [Ex cos (Gye) - Ez sin (Gy8), 0, Ex sin (Gy8) + Ez cos (Gy8)],(5)while on the second turn e(8) is given bye(8)==(E;,0, E;)=(-E~,O, EI). (6)We will now calculate the spin evolution in the {fo, no, 1o} frame as a particlemakes two turns around the ring. Assume that the spin is initially vertical (s =y)at 8 = -Jr. For the first turn around the ring we then have, using Eq. (3) andkeeping only first-order terms in e,s(8 = Jr) = (0,1,0) +IJt ddS d8-Jt 8= (0, 1,0) + [f:1< E}(O) dO, 0, - f:1< E~(O) dO]' (7)76 A. D. KRISCH, S. R. MANE AND T. ROSERThe spin motion in the second turn around the ring is given byj3Jf dss(0=3n)=s(0=n)+ 7< dOdO= [J:7< E}(0) dO, 1, - J:7< E~(0) dO ]+ [J:7< E}(O) dO, 0, J:7< E~(O) dO ]= [2J:7< E}(O) dO, 1, 0J. (8)In general, successive pairs of turns will tilt the spin increasingly away from thevertical direction. However, it is important to notice that only the integral of theE[ component of the imperfection field,contributes to this depolarization, whereas the integral of the Er component,is cancelled by successive turns. Note that E[ is in phase with a longitudinal field atthe position opposite the snake while Er is in phase with a radial field at the same(J = 0 position opposite the snake. A similar analysis can be done for a singleType 2 snake, which rotates the spin by 1800 around the radial direction x ratherthan the longitudinal direction z. The effects are then reversed: E r causesdepolarization, whereas E[ does not.Let us now evaluate the integral in Eq. (9) for the simple case of a localizedimperfection E = (ax + f3Z)D(fJ - fJo) at fJ = fJo, where a and f3 are constants. Weobtain(11)Depending on the exact value of GyfJo, the magnitude of E[ is typically in therange IE[12 = !(a2+ f32). Thus the tilt of the spin away from the vertical increasesby approximately [2(a 2 + f32)]112 radians every two turns.4. CONCLUSIONIn a ring with a vertically polarized beam and a single Siberian snake,depolarization will in general occur if the imperfections are distributed randomlyaround the ring. However, it may be possible to maintain vertical polarization ifE[, which is the integral of the horizontal fields in phase with a longitudinal fieldHORIZONTAL FIELDS WITH SIBERIAN SNAKE 77at (J = 0, can be made to vanish. A Type 1 snake could then eliminate thedepolarization due to the remaining horizontal fields in phase with a radial field atthe (J =0 position opposite the snake.This work was supported by a research grant from the U.S. Department ofEnergy. We would like to thank E. D. Courant, S. Y. Lee, R. E. Pollock, L. G.Ratner, L. C. Teng, and K. M. Terwilliger for their comments.REFERENCES1. T. Khoe et aI., Particle Accelerators 6, 213 (1975).2. A.D. Krisch, in High Energy Spin Physics (Marseille, 1984), J. Phys. (Paris), 46 C2, 51 (1985).3. Ya.S. Derbenev and A.M. Kondratenko, Proc. 10th Int. Con! on High Energy Accel., Protvino,USSR, 1977; Ya.S. Derbenev et aI., Particle Accelerators 8, 115 (1978).4. B.W. Montague, Phys. Reports 113, No.1 (1984), and references therein.5. L.G. Ratner, AlP Conf. Proc. 145, 100 (1986).6. A.D. Krisch, S.R. Mane, T. Roser, K.M. Terwilliger, H.O. Meyer, R.E. Pollock, F. Sperisen,E.J. Stephenson, E.D. Courant, S.Y. Lee, and L.G. Ratner, Experimental Test of the SiberianSnake Concept, Proposal to Indiana University Cyclotron Facility, April 13, 1987 (CE-05).7. L.H. Thomas, Phi/os. Mag. 3, 1 (1927); V. Bargmann, L. Michel, and V. Telegdi, Phys. Rev. Lett.2, 435 (1959).

Horizontal Fields In A Ring With One Siberian Snake And A vertically polarized Beam

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