Anferov, V A

Phelps, R A

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Particle Accelerators, Vol. 59, pp. 169-174Reprints available directly from the publisherPhotocopying permitted by license only© 1998 OPA (Overseas Publishers Association) N.V.Published by license underthe Gordon and Breach SciencePublishers imprint.Printed in India.SYNCHROTRON DEPOLARIZINGRESONANCES IN THE PRESENCE OF ASOLENOIDAL SIBERIAN SNAKE*RICHARD A. PHELPS+ and VLADIMIR A. ANFEROVtRandall Laboratory of Physics, University of Michigan, Ann Arbor,MI48109-1120, USA(Received 19 November 1997; In final form 29 January 1998)Particle's energy variation arising from synchrotron oscillations modulates the spin tune,which causes synchrotron sidebands around depolarizing resonances. Studying the spintune modulation in a ring with a single solenoidal Siberian snake, we found that for a givenenergy it is possible to tune the strength of the snake solenoid to eliminate spin the tunemodulation, thus eliminating the synchrotron sideband resonances.Keywords: Spin dynamics; Siberian snake; Depolarizing resonancesPolarized proton beams encounter many depolarizing resonances dur-ing acceleration to high energies. Most of the depolarizing resonancesare caused by small horizontal magnetic fields! which act on theparticles traveling around the ring. In addition, many of these reso-nances are found to have synchrotron sideband resonances, which aremainly caused by the energy modulated spin tune.2 The Siberiansnake technique3 was proposed to universally cancel all depolarizingresonances. Each full Siberian snake rotates the proton's spin by 1800around some horizontal axis and has no effect on the beam motion.While full Siberian snakes work best at high energies, one still needs to*This research was supported by grants from the U.S. Department of Energy and theU.S. National Science Foundation.t Corresponding author. E-mail: anferov@spinbud.physics.lsa.umich.edu.:j: Present address: IBM, Essex Junction, VT 05452.169170 R.A. PHELPS AND V.A. ANFEROVcorrect a few depolarizing resonances that occur at low energies. In thispaper we will study how a solenoidal partial Siberian snake could beused to cancel the synchrotron sideband resonances.In an accelerator without Siberian snakes, the spin tune v s , which isthe number of spin revolutions around the spin precession axis in oneturn around the ring, is energy dependent(1)where G == (g - 2)/2 is the gyromagnetic anomaly (G == 1.792847 forprotons) and ry is the Lorentz energy factor. During acceleration of apolarized proton beam, the energy dependent spin tune crossesmany depolarizing resonances, which ultimately destroy the beampolarization.In synchrotrons, the energy of each particle is modulated due tosynchrotron oscillations,ry == ryo + ~ry cos(Vsyn(1), (2)where ~ry is an amplitude of the energy modulation, vsyn is the syn-chrotron tune and (1 == 21ft/, is time in units of the revolution period, ofthe ring. The synchrotron energy modulation causes spin tune mod-ulation, which in turn leads to a spectrum of synchrotron sidebandsaround the depolarizing resonances.In a ring with a single Siberian snake, the spin tune V s can beexpressed in terms of Gry and the snake strength s:cos(1fVs ) == cos(1fGry) cos(s1f/2). (3)For a full snake (s == 1) the spin tune becomes a half-integer at allenergies, which eliminates the problem of crossing many depolarizingresonances during the acceleration. In the presence of a full Siberiansnake the spin tune is energy independent only if the snake strength isenergy independent. This is certainly true for dipole snakes at highenergy. At low energies, dipoles cause large orbit excursions, making adipole snake impractical. Low energy snakes are usually solenoids. Thesnake strength of a solenoid is given by:s == e(1 + G) JBdl,pc(4)SYNCHROTRON DEPOLARIZING RESONANCES 171where p is the proton's momentum. For a constant field in a solenoid thesnake's strength is proportional to p-l; particles with slightly differentenergies will see different snake strengths, and therefore will have dif-ferent spin tunes. If the energy has synchrotron modulation, so does thesnake strength and the spin tune, even for a "full" snake (average s == 1).Using the approach discussed by Montague,4 it is easy to show thatthe energy modulation of the spin tune naturally leads to synchrotronsideband resonances. Assuming a localized horizontal spin kick ofstrength EO at a frequency Vy, the resonance strength can be computed asthe spin perturbation averaged over many turns (N ---+ (0):(5)Without energy modulation, the sum in Eq. (5) would be equal to Nwhenever G10 - V r is equal to an integer. The energy modulation term inthe exponent can be expanded in series using,00eixsin(y) == L Jm(x)e-imy ,m=-oo(6)where Jm is the mth Bessel function of the first kind. The resonancestrength equation then becomes,00 (G~ ) 1 N .E = L EOJm~ N L e'27Tn(G'Yo- II,-mIlSyn ).m=-oo syn n=l(7)Thus, along with the main resonance at G10 == k + Vy, the spin tunemodulation creates synchrotron sideband resonances at G10 ==k +Vr +mvsyn, where k and m are integers. The strength of each syn-chrotron sideband resonance is(8)This analysis can be easily repeated for the case of a single partial orfull Siberian snake in a ring. One finds that the strength of the mthsynchrotron sideband is then equal to cOJm(~vs/vsyn), where ~vsis the spin tune modulation amplitude. To find the strength of the172 R.A. PHELPS AND V.A. ANFEROVsynchrotron sideband resonances one only needs to compute theamplitude of the spin tune modulation in the presence of a Siberiansnake.Using Eq. (3), the spin tune modulation amplitude ~vs can bewritten:(9)The second term takes into account the energy dependence of thesnake strength, which can be obtained from Eq. (4). Substituting thederivatives of Eq. (3), the amplitude of the spin tune modulationbecomes:,(j.vs = Sin~:vs) [G sin(1fG/,) cos(1fs/2) - ; (/'2~ 1) cos(1fG)') sin(1fS/2)] .(10)Note that with no snake (s == 0), the spin tune modulation amplitudereduces to ~vs == G~"Y, and for a fun solenoidal snake (s == 1),(11 )which is nonzero except when G"Y is a half-integer. The depolarizingresonances that occur even in the presence of a Siberian snake (socalled "snake resonances"s) could then have synchrotron sidebands.The first-order sidebands (m == ±1) would have strength CI ';::5 co~vs/(2vsyn). The sidebands of a "snake resonance" are strongest when G"Y isclose to an integer; their maximum strength is 2Gryf32 times smallerthan the strength of the synchrotron sidebands without a snake.Equation (10) can be used to predict the snake strength required forzero spin tune modulation at a given energy; if a depolarizing resonanceoccurs at this energy, its synchrotron sidebands would then vanish.Setting the right side of Eq. (10) to zero gives an implicit equation forcanceling the spin tune modulation, which can be solved numerically.The predicted snake strength is plotted in Figure 1 against fractionalGry for the integer part of Gry from 2 to 6, or an energy range of0.1-2.7 GeV. Note that for half-integer Gry, a full snake is required toSYNCHROTRON DEPOLARIZING RESONANCES 1732.0,..q.....,J 1.50.0~Q)H.....,Jrn 1.0Q)~coQrn 0.50.00 .0 0.2 0.4 0.6Fractional Gy0.8 1.0FIGURE 1 The snake strength calculated for cancelling the spin tune modulation isplotted as a function of fractional G~. The curves are for the integer part of G~ of 2through 6. The sharpness of the curves increases with the integer part of G~.cancel the spin tune modulation. Also note that for the upper half ofeach fractional range, one must use a snake strength greater than 1; thiscould limit practical application of this method. Why cannot a snake'sspin precession of ¢ > 1r be replaced with 21r - ¢, which gives the samespin tune? Although the spin tune would be the same for these twosnake's spin precessions, the energy modulation of 21r - ¢ spin preces-sion is in opposite phase to the modulation of ¢; the spin tune mod-ulation due to the direct energy effect would no longer cancel themodulating snake spin precession. Finally, note that the profile of s as afunction of fractional G1 is increasingly sharp for larger integer values(higher energies); tuning the snake for values near integer G1 would bemore difficult at higher energies.Canceling the spin tune modulation could have application in theacceleration of polarized beams. One could arrange for a particularsnake strength at some point in an energy ramp where it is suspected thatstrong synchrotron sidebands are causing polarization losses. Thismight also be useful for intrinsic resonance jumping in low energy174 R.A. PHELPS AND V.A. ANFEROVaccelerators, where spin tune modulation limits the efficiency ofthe tunejump method. 6 It should also be noted that nowhere in the derivationofEq. (10) is it necessary to assume that ~'Y be a modulation. The sameapproach could be used to stabilize the spin tune against random energyfluctuations. Besides the energy modulation of the spin tune, synchro-tron resonances could be caused by vertical synchrotron oscillationsdue to a non-zero vertical dispersion function. 7 The synchrotrondepolarizing resonances caused by this effect cannot be compensated bysolenoidal snake; however, their strength is expected to be significantlyweaker than the one caused by the energy modulated spin tune.2In summary, it has been shown that in the presence of a solenoidalSiberian snake depolarizing resonances can have synchrotron side-bands. The spin tune modulation which causes the synchrotron side-bands can be eliminated by adjusting the solenoidal snake strength. Thistechnique may be helpful in accelerating polarized protons in thelow and medium energy machines such as the Brookhaven AGS, theFermilab Booster, and DESY3 at DESY.References[1] M. Froissart and R. Stora, Nucl. Instrum. and Methods 7, 297-305 (1960); Englishtranslation ANL-TRANS-570, (1968).[2] S.Y. Lee and M. Berglund, Phys. Rev. E 54, 807 (1996).[3] Ya.S. Derbenev and A.M. Kondratenko, Sov. Phys. Doklady 20, 562 (1978); Part.Accel. 8, 115 (1978).[4] B.W. Montague, Physics Reports 113(1), 1-96 (1984).[5] S.Y. Lee and S. Tepikian, Phys. Rev. Lett. 56, 1635 (1986); R.A. Phelps et al., Phys.Rev. Lett. 78, 2772 (1997).[6] S. Hiramatsu et aI., AlP Conf. Proc. 187, 1436 (1988).[7] J. Buon and J.-P. Koutchouk, CERN Preprint SL/94-80 (AP).

Synchrotron Depolarizing Resonances in the Presence of A solenoidal siberian Snake

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