SuperB, the 2nd-generation B-Factory with a luminosity of 10{sup 36}/cm{sup 2}/s proposed for LNF, is being designed from the start to be capable of providing a spin-polarized electron beam in the low-energy ring (LER) with longitudinal polarization at the interaction point. Due to the high luminosity at moderate beam current the beam lifetime is short (a few minutes), and a polarized injector will be used. Spin rotators have been designed and the equilibrium polarization evaluated. It will be shown that an average polarization of about 70% can be expected

/INFN, Rome

/Novosibirsk, IYF

/SLAC

Barber, Desmond

Biagini, Maria

Koop, Ivan

Nikitin, Sergei

Nosochkov, Yuri

Raimondi, Pantaleo

Sinyatkin, Sergey

Sullivan, Michael

Tech., /Cockcroft Inst. Accel. Sci.

Wienands, Ulrich

Wittmer, Walter

UNT Digital Library

POLARIZATION IN SUPERB∗U. Wienands† , Yu. Nosochkov, M.K. Sullivan, W. Wittmer, SLAC, Menlo Park, CA 94025, USAD.P. Barber, DESY, Hamburg, F.R. Germany and Cockroft Institute, Warrington, UKM.E. Biagini, P. Raimondi, LNF, Frascati, ItalyI. Koop, S. Nikitin, S. Sinyatkin, BINP, Novosibirsk, RUAbstractSuperB, the 2nd-generation B-Factory with a luminosityof 1036/cm2/s proposed for LNF, is being designed fromthe start to be capable of providing a spin-polarized elec-tron beam in the low-energy ring (LER) with longitudi-nal polarization at the interaction point.[1] Due to the highluminosity at moderate beam current the beam lifetime isshort (a few minutes), and a polarized injector will be used.Spin rotators have been designed and the equilibrium po-larization evaluated. It will be shown that an average polar-ization of about 70% can be expected.INTRODUCTIONRelativistic electrons circulating in the guide field of astorage ring emit synchrotron radiation and a tiny fractionof the photons can cause spin flip from “up” along an initialdirection to “down”, and vice versa. However, the up-to-down and down-to-up rates differ, with the result that thebeam can become spin polarized. In a perfectly aligned flatring with no solenoids the polarization is anti-parallel tothe vertical guide field, reaching a maximum polarization,Pst, of 85√2= 92.4%. This, the Sokolov-Ternov (S-T) po-larizing process[2], is very slow on the time scale of otherdynamical phenomena occurring in storage rings, and theinverse time constant for the exponential build up in a uni-form dipole field is[2]:τ−1st =5√38reγ5h̄me|ρ3|where re is the classical electron radius, γ, the Lorentz fac-tor, ρ, the bending radius in the guide field, and the othersymbols have their usual meanings.In a simplified picture the majority of the photons in thesynchrotron radiation do not cause spin flip but tend insteadto randomize the orbital motion in the magnetic fields dueto the presence of dispersion. Then the spin-orbit couplingembodied in the Thomas-BMT equation can cause spin dif-fusion, i.e. depolarization. The equilibrium polarizationis then less than 92.4% and will depend on the relativestrengths of the polarization and depolarization processes.Analytical estimates of the attainable equilibrium polar-ization are best based on the Derbenev-Kondratenko (D-K)∗Work supported by US DOE under contract DE-AC03-76SF00515† uli@slac.stanford.eduformalism.[3, 4] The equilibrium polarization is given byPdk =85√3∮ds〈1|ρ3| b̂ ·(n̂− ∂n̂∂δ)〉s∮ds〈1|ρ3|(1− 29 (n̂ · ŝ)2 + 1118(∂n̂∂δ)2)〉where <>s denotes the average over phase space at az-imuth s, ŝ is the direction of motion, and b̂, the magneticfield direction. Furthermore, n̂ is a unit 3-vector satisfyingthe Thomas-BMT equation along particle trajectories andis 1-turn periodic. The ensemble of n̂-vectors for a beam iscalled the invariant spin field. On the closed orbit, i.e. atthe origin in phase space, n̂ becomes n̂0.[5] In conventionalsituations in electron rings, < n̂ >s is very nearly alignedwith n̂0(s). In the presence of radiative depolarization, therate given above must be replaced byτ−1dk =5√38reγ5h̄me1C∮ds〈1− 29 (n̂ · ŝ)2 + 1118(∂n̂∂δ)2|ρ (s) |3〉.the quantity d2 =(∂n̂∂δ)2is a key parameter in evaluatingthe expected polarization. Large values of d2 cause lowequilibrium polarization Pens,dk(s) and small time con-stants τdk, thus reducing the polarization attainable. d2 canbecome very large at the spin-orbit resonances where thespin tune—which is γG for a flat ring—equals an integeror a harmonic of the machine tunes.[5] Here,G = (g−2)/2is the gyromagnetic anomaly, G ≈ 0.00116 for electrons.SUPERBQuantitative evaluation of τst for SuperB gives about5. . . 7 hours for either ring, which is too long to be useful asa polarization mechanism. Therefore SuperB will achievepolarized beams by injecting polarized electrons into theLER. We chose the LER rather than the HER because thespin rotators (see below) employ solenoids. These rotatorsare more compact than dipole-based spin rotators, but thesolenoids need to be scaled in strength with energy thusmaking LER spin rotators more compact than spin rotatorsin the HER would be.In SuperB at high luminosity the beam lifetime willbe only 3. . . 5 minutes and continuous injection (“trickle-charge”) operation is a key component of the proposal. Byinjecting at a high rate with a polarized beam one can over-ride the depolarization in the ring as long as the depolariza-tion time constant is not too small. The equilibrium polar-ization under continuous injection is given byP = Piτdkτdk + τb+ Pdkτbτdk + τb,SLAC-PUB-15100Table 1: Parameters of Spin RotatorParameter Value CommentDesign Energy 4.18 GeVSpin rotation of solenoids 90◦ one sideSolenoid field integral 4*10.94 Tm 4 solenoidsSolenoid field 2.39 TTot. length of solenoid 23.07 m incl. de-section couplingSpin rotation of dipoles 270◦ one sideBending of dipoles 28.4◦ one sidewhere τb is the beam lifetime in the ring and Pi, the polar-ization of the injected beam. As long as τb < τdk, the firstterm dominates (for high Pi).Spin RotatorsIn the arcs of the ring, the vector n̂0, which gives the di-rection of the polarization, must be close to vertical to min-imize depolarization. In order to obtain longitudinal polar-ization at the IP, a net rotation of n̂0 by 90◦ about the radialaxis is required. A rotation of 90◦ in a solenoid followed bya spin rotation of 90◦ in the horizontal plane provides therequired net rotation about the radial axis without any ver-tical bending—thus avoiding vertical emittance growth—and is adopted for SuperB. After the IP, n̂0, and with it thepolarization, has to be restored to vertical by a second spinrotator. Two geometries are possible: an antisymmetric ge-ometry where the dipoles and solenoids after the IP havepolarities opposite to those before the IP and a symmetricgeometry, where the polarities are all the same. The twosolutions have significantly different properties[6]:• With the antisymmetric geometry and perfect align-ment, n̂0 is vertical in the arcs at all energies; the ro-tators are to lowest order spin matched in energy.• With the symmetric geometry, n̂0 is vertical in thearcs and longitudinal at the IP at just one energy andthe whole interaction region will normally not be spinmatched in energy.For SuperB at high luminosity, the LER beam lifetime isabout 3. . . 5 min. Under these conditions it turns out that asymmetric spin-rotator scheme is feasible and can achieve70% polarization or better. For the solenoids to avoidcoupling the betatron planes, we employ the decouplingscheme worked out by Zholents and Litvinenko[7], whichinserts quadrupoles in between two half-solenoids to invertone of the planes, thus decoupling each full spin-rotatorsolenoid.LER Spin Rotator LayoutFig. 1 shows the IR of the LER with spin rotators. Therotator parameters are given in Table 1. Note that the dipolesection rotates the spin by 270◦ instead of 90◦; this wasdone in order to integrate the rotators with the local chro-maticity correction needed in the IR. With 90◦ dipole angle,the total bending would have been too small and the disper-sion insufficient for effective chromaticity correction.Spin DynamicsIn order to evaluate the rate of depolarization we used thecode SLICKTRACK. This code is an extension of the codeSLICK which can perform analytic first-order thick-lensevaluations using the SLIM[8] formalism. The extensioncomprises a Monte-Carlo spin-orbit tracking algorithm forsimulating full 3-d spin-orbit motion in the presence of syn-chrotron radiation.The following results are based on the MAD latticemodel of the LER to which accelerating cavities have beenadded. A limited set of misalignments (in the arcs only)was implemented. Orbit correction was done in SLICK-TRACK using a reduced set of correctors. SLICKTRACKcalculates beam emittances and the values obtained areclose to the design values in the horizontal plane, whilelarger than the design in the vertical plane (due to limitedattempts at orbit correction). The energy spread and syn-chrotron tune are close to the design values. The differ-ences w.r.t. the MAD parameters are explained by the dif-ferent treatment of RBENDs in MAD vs SLICKTRACK(MAD takes the length as hardware length and then cal-culates the orbit length, whereas SLICKTRACK takes thelength as orbit length) as well as the different misalignmentand orbit-correction setups.Fig. 2 shows the (de-)polarization time τdk vs ring en-ergy (which is 0.441·γG [GeV]). The behavior seen mainlyreflects the variation in d2 in the arcs as the angle of tilt ofn̂0 from the vertical in the arcs varies with the ring energyas a result of using the symmetric rotator. Moreover sincethe interaction regions are not spin transparent, d2 is notsmall even when n̂0 is vertical in the arcs. Consequentlyτdk is always much smaller than the Sokolov-Ternov time.The maximum equilibrium polarization (without continu-Figure 2: Depolarization time vs ring energy in the LER.Figure 1: Interaction Region layout with spin rotators. The arrows indicate the orientation of the n̂ 0 axis.Figure 3: Average polarization in the LER under continu-ous injection and at 3.5 min. beam lifetime.ous injection) under these conditions is about 10%.For the above parameters we can evaluate the expecteddegree of polarization under continuous injection. Fig. 3shows the result for 90% polarization at injection and abeam lifetime of 3.5 min (i.e. at full luminosity). Thereis a significant band in energy where the polarization is ex-pected to exceed 70%. We also evaluated the deviation ofn̂0 from the longitudinal direction at the IP vs beam energy,shown in Fig. 4. Given that the longitudinal component ofthe polarization scales with the cosine of this angle, there isa wide plateau where effectively Plong = P . This depen-dence is important in assessing any systematic effect for theprecision polarimetry.SUMMARYThe results obtained so far give confidence that a polar-ization in excess of 70% at high luminosity can be achievedin the SuperB LER with the injection of polarized beams.Further studies will focus on the assessment of the needto improve the spin matching[5] of the interaction regionincluding the rotator solenoid sections, the effect of im-perfections including residual detector solenoid fields, andspin-tracking to include the higher order and beam-beameffects. The preservation of the polarization during the in-jection process will also be studied.|n 0,s|^Figure 4: Tilt of the n̂0 axis against the beam direction atthe IP. The longitudinal component is plotted.REFERENCES[1] M.E. Biagini et al., this conference.[2] A.A. Sokolov and I.M. Ternov, Sov. Phys. Dokl. 8(12),(1964) 1203.[3] Ya.S. Derbenev, A.M. Kondratenko, Sov. Phys. JETP. 37, 968(1973).[4] S.R. Mane, Phys. Rev. A36, 105 (1987).[5] D.P. Barber, G. Ripken, in Handbook of Accel. Phys.,A.W. Chao, M. Tigner, eds., World Scientific, 3rd edition(2006).[6] D.P. Barber et al. Part. Acc. 17 (1985) 243.[7] A.A. Zholents, V.N. Litvinenko, DESY Report L—Trans 289(1984).[8] A.W. Chao, Nucl. Inst. Meth. 180 (1981), 29.

Polarization in SuperB

Polarization in SuperB

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