Beam Polarization

1. Beam Polarization e+/- p SPEAR HERA SLC LEP MIT/Bates PETRA Tristan past and present polarized beam facilities: ZGS AGS IUCF RHIC + many lower energy facilities HERA-P TeV-P (?) possible future polarized beam facilities: NLC JLC TESLA CLIC outline: • Thomas-BMT equation • Spinor algebra • Equation of motion for spin • Periodic solution to the eom • Depolarizing resonances • Polarization preservation in storage rings • Siberian snakes • Partial Siberian snakes • Resonance strength • Summary

2. Thomas-Bargmann-Michel-Telegdi (Thomas-BMT) equation (1959) Uhlenbeck and Goudsmit (1926): gyromagnetic ratio magnetic moment angular momentum, “spin”, |s|=hbar/2 (e+/-,p) a=(g-2)/2 = anomolous part of the electron magnetic moment G=(g-2)/2= anomolous part of the proton magnetic moment a = 0.0011596 (e-) G = 1.7928 (p) x = 0.001166 (μ) The “spin” (e.g. angular momentum) of the particle interacts with the external electromagnetic fields through influence on its magnetic moment. The equation of motion in an external magnetic field B, in the rest frame of the particle is orthogonal fields precess the spin see e.g. J.D. Jackson “Classical electro- dynamics” angular precession frequency: It is convenient to normalize s and use S with the normalization |S|=1

3. Thomas-BMT equation (modern form) In the laboratory frame, the spin precession of a relativistic particle is given by the Thomas-BMT equation (derived from a Lorentz transformation of the electro-magnetic fields including relativistic time dilation): assuming that the particle velocity is along the direction of external electric fields and that there are no significant trans-verse electric fields; i.e. vE=0 the spin precession due to B depends on the beam energy (=E/m); the higher the beam energy, the more the spin precession the spin precession due to B|| is energy-independent Beam polarization the beam polarization is defined as the ensemble average over the spin vectors S of the particles within the bunch: N = number of particles per bunch The polarization is the quantity that is measurable (e.g. by measuring a scattering asymmetry of a fixed targets in a proton accelerator):

4. B. Spinor algebra using SU(2) can transport the (31) spin vector or, equivalently, the (21) spin wave function The transformation between the two representations is given by with the Pauli matrices defined† by y †this is a cyclic permutation of the “standard” Pauli matrix definitions which conforms with the axes defi- nitions prefered by the high-energy physics community v s x Example: spinor representations for vertical polarization given: let then b=0 a=1

5. C. Spin equation of motion (reference: Courant and Ruth) In 1980, Courant and Ruth expressed the magnetic fields of the Thomas-BMT equation in terms of the particle coordinates and reexpressed the equation of motion for the spin in terms of a (complicated) Hamiltonian. In doing so a simple expression resulted: where θ is the orbital angle: θ local bending radius In the absence of depolarizing resonances, H has a simple form where κ=G for protons and κ=a for electrons/positrons Courant and Ruth introduced another (now conventional) form for the EOM.It is assumed that H is time-independent and that there are no perturbing fields. Then H can be reexpressed as a linear combination of the 3 components of the Pauli vector: =|| is the amplitude of the precession frequency=[(g-2)/2] Pauli matrices =<x, z, y> unit vector aligned with  The solution to the eom is with After expanding the exponential, using the algebra of the -matrices, the solution for the spinor is

6. D. Periodic solution to the spin equation of motion M=M1M2…Mn express the spin matrix M as the product of n precession matrices: M0(+2)=M0() the one-turn spin map M0 for the closed orbit is periodic: for the single element precession the spin, we had with  giving the precession frequency and  the precession angle for the one-turn-map, since M0 is unitary, it may also be expressed as > y n0=“stable spin direction” > n0 P s 0=/2=“spin tune” with =(g-2)/2 x > n0 = stable stable spin direction (axis which returns to same place in every turn around the ring) 0 = spin tune (number of times the spin precesses about n0 in one turn around the ring) >

7. Periodic solution to the spin equation of motion, cont’d The one-turn-map is given, after algebra, by (previous solution) please remove subscript 2 in Eq. 10.24 stable spin direction spin tune Expanding the Pauli matrices, the solution is given equivalently by So, if the Hamiltonian is time-independent (e.g. the influence of spin resonances may be neglected – as can be made the case with most low energy accelerators), the spin tune and the stable spin direction may be easily evaluated. The spin tune is given by determining M (multiplying all rotation matrices) and taking the trace of the spin-OTM: > y n0 > y For the stable spin direction n0, it is convenient to parametrize n0 using directional cosines s s with normalization x x

8. Example: spin tune and stable spin direction for a planar ring with perfect alignment the one-turn spin map for a ring with only vertical dipole fields is with expanding the exponential, i.e. the spin tune is derived from the trace of the OTM: The orientation of the stable spin direction is found by equating components of the OTM. Recall, > or, from above, y n0 y s s x x

9. E. Depolarizing resonances depolarizing resonances occur whenever the spin tune is harmonically related (“beats”) with any of the natural oscillation frequencies of the particle motion: q,r,s,t,and u are integers m=t+uP, where P is the superperiodicity betatron tunes synchrotron tune resonance order: |m|+|q|+|r|+|s| Types of depolarizing resonances 0=t+uP imperfection resonances due to magnet imperfections, dipole rotations, and vertical quadruple misalignments these are in practice usually the most significant for existing accelerators with polarized beams 0=(t+uP)+rQyintrinsic resonances due to gradient errors 0=(t+uP)+sQssynchrotron sideband resonances due to coupling between longitudinal and transverse motion these resonances become increasingly important at higher beam energies 0=(t+uP)+qQx+rQy(higher-order) betatron coupling resonances

10. Example: SLC collider arc 1 mile total length, E=45.6 GeV (a~103), 23 achromats 108° phase advance per cell Simulated particle and spin motion in the SLC arc (courtesy P. Emma, 1999) orbit with initial offset error of 500 m longitudinal polarization vertical polarization in practice, vertical “spin bumps” were used to properly orient the spin (longitudinally) at the interaction point

11. Intermediate summary equation of motion (Eq. 10.13) solution (Eq. 10.17) y periodic solution > (Eq. 10.24) n0 > P s n0 = stable stable spin direction (axis which returns to same place in every turn around the ring) 0 = spin tune (number of times the spin precesses about n0 in one turn around the ring) > x spacing of (strong) imperfections resonances: electrons: 0=a=E/0.411 [GeV] protons: 0=G=E/0.523 [GeV] (as will be shown) resonance strength (i.e. the Fourier harmonic of the off-diagonal elements of H which couple the up and down components of ) (Eq. 10.49) linear in the particle energy depends on the vertical displacement

12. F. Polarization preservation in storage rings 1. Injection > align the beam polarization of the injected beam Pinj with the stable spin direction n0 injected polarization stable spin direction component of polarization surviving injection polarization that one would measure > > using directional cosines (’s for Pinj and ’s for n0), project Pinj onto n0: the measured polarization is given by projection onto the plane of interest:

13. 2. Harmonic correction (Petra, Tristan, AGS, HERA, LEP,…) concept: correct those orbital harmonics close to 0 n is the harmonic of interest orbital angle Fourier harmonics example: correction of imperfection resonances using pulsed dipoles at the AGS during proton ramp to 16.5 GeV (courtesy A. Krisch, 1999) pulsed dipole currents main dipole current (~E)

14. Example: lepton beam polarization at 27.5 GeV measured at HERA after correction of the strength of the nearest imperfection resonance (courtesy the HERMES experiment, 2002)

15. HERA-II HERA-I spin rotators at fixed-target experiment spin rotators at all experiments particular concerns for the colliding-beam experiments (H1 and ZEUS): solenoidal fields not locally compensated (beam trajectories not perfectly parallel to solenoid axis) increased lepton beam emittance coupling (for matched IP beam sizes) effect of beam-beam interaction on lepton beam polarization complicated spin-matched optics closed-orbit control and harmonic spin matching no validation of theory by experiment (2003 data)

16. 3. Adiabatic spin flip Froissart-Stora formula (for describing spin transport through a single, isolated resonance): final polarization resonance strength initial polarization “ramp rate” limiting behavior +1 if  is small and/or if  is large -1 if  is large and/or if  is small Pfinal/Pinitial = Example: spin flipping of a vertically polarized beam (courtesy A. Krisch, 1999) t~1/ t=10 ms t=30 ms Vsol~

17. 4. Tune jump From the Froissart-Stora equation, If the resonance is crossed quickly ( large), then the polarization will be preserved. Intrinisic resonances may therefore be crossed by rapidly pulsing a quadrupole at the appropriate time. 0 energy ramp integer + Qy example: correction of intrinsic resonances using pulsed quad- rupoles at the AGS integer resonances integer - Qy G pulsed dipole currents pulsed quad- rupole currents rapid traversal of resonance main dipole current (~E)

18. recap: method particle type energy application fixed ramped ramped minimize || of nearby resonances (same) – empirical correction as function of beam energy, E maximize ||/2 at nearby resonance as function of E harmonic orbit correction lepton proton proton harmonic orbit correction tune jump For high energy polarized protons, the above methods were anticipated to be of limited applicability (empirically determined corrections are time consuming to develop and dependent on the closed orbit; adiabatic spin flip harder as ||increases). The solution, proposed in 1976, was first tested over a decade later and proved effective.

19. G. Siberian snakes (Derbenev and Kondratenko, 1976) (P denotes a polarimeter) P concept: make the spin tune 0 independent of energy (and equal to some non-resonant value) B  snake,  example: a type-I Siberian snake (rotation of spin around longitudinal axis per turn) A one-turn spin matrix snake B A expanding M and taking the trace gives =0 (no snake)  s=G as before = (full snake)  s=n/2 with n odd independent of the beam energy with a spin tune of ½, the depolarizing resonance condition can never be satisfied

20. Types of Siberian snakes Type I = about longitudinal axis Type II = about radial axis Type III = about vertical axis Design of Siberian snakes longitudinal snake depends linearly on  best suited for low energy beams transverse snake independent of  so fixed-field magnets may be used. However, a dipole produces an orbital deflection angle of /G which is large at low beam energies therefore best suited for high energy beams

21. Example: type-I (= about longitudinal axis), transverse snake (courtesy A. Chao, 1999) magnet orientation: H = horizontal dipole V = vertical dipole vertical orbit excursion optically transparent horizontal orbit excursion orbital angle  chosen for a total spin precession of /2 > > > > > > > original notation ( ,x)( ,-x)( ,y)( ,-x)(,-y)( ,x)( ,y) spin precession axis (in direction of field) spin precession angle

22. Example: polarization preservation near an imperfection resonance using a Siberian snake (spin precession) snake on snake off snake on snake off depolarization polarization maintained at all beam energies Result: all high-energy polarized proton facilities plan to or do use Siberian snakes

23. H. Partial Siberian snakes (T. Roser) again dependence of the spin tune on G for various strengths of partial snake: location of intrinsic resonance with Qy=0.2 (=0) imperfection resonance location of intrinsic resonance with Qy=0.2 (=) only a few % snake is needed to avoid strong imperfection resonances larger partial snakes can be used to avoid intrinsic resonances (over narrow energy range)

24. Polarized beams at RHIC

25. I. Resonance strength,  The spin equation of motion was solved previously disregarding the influence of depolarizing resonances EOM with Courant and Ruth gave the general form of H, where t and r are complicated functions of the particle coordinates While we defer here the extension of their work (see text), the definition of resonance strength warrants mention. Due to the periodic nature of a circular accelerator, the coupling term may be expanded in terms of the Fourier components; i.e. where  is the particle orbital angle, ±res,k=k for imperfections resonances, ±res,k=k±Qy for first order intrinsic resonances, etc., and k is the resonance strength given by the Fourier amplitude for the case of an imperfection resonance,  is given approximately by summing over the radial error fields encountered by a particle in one turn: optics programs (e.g. DEPOL) exist to calculate  given the magnetic optics

26. J. Summary electrons and protons possess a magnetic moment proportional to the spin angular momentum, or polarization: magnetic fields orthogonal to the polarization change the orientation of the polarization the Thomas-BMT equation shows this explicitly in the rest frame of the particle polarization transport can be equivalently described in terms of the spin wave function, or spinors, given in terms of the Pauli matrices in terms of spinors, the equation of motion (Courant and Ruth) has a simple form with solution the periodic solution is 0 is the spin tune > n0 is the stable spin direction

27. depolarizing resonances result when the spin frequency is harmonically related to any natural oscillation frequency of the beam the resonance strengths can be evaluated (for widely spaced resonances): which shows that the resonance strength increases with increasing energy polarization preservation includes matching the polarization onto the stable spin direction at injection other preservation methods include: adiabatic spin flip (ala Froissart and Stora) betatron tune jump (AGS) harmonic correction (AGS, LEP, HERA,…) Siberian snakes (Derbenev and Kondratenko) Siberian snakes force 0=1/2 (full snake) so the resonance condition is never satisfied at any energy snake designs generally are optically transparent. The choice of solenoidal or dipole snakes depends on the beam energy partial Siberian snakes (Roser) are useful for curing selected resonances (AGS)

28. Exercises (for Wednesday, June 30): 1. Derive the spinor representations for a) horizontal, b) vertical, and c) longitudinal polarization. Recall the transformation: y s particle trajectory (along s) where the Pauli matrices are given by x 2. The spinor rotation matrix for precession by an angle  about the j-axis (j=x,y,s) is given by exp(ij/2). Assuming an initial polarization that is purely a) horizontal, b) vertical, c) longitudinal, determine the final polarization after precession about the longitudinal (s) axis by an angle  and sketch the final orientation in each of these cases. Note: exp(ij) = cos()+ijsin(). 3. Consider a hexagonally-shaped accelerator with a type-I Siberian snake. a) with a proton kinetic energy of 108.4 MeV, what is G? b) show that at the location of the snake the stable spin direction is in the longitudinal direction (i.e. parallel to the nominal particle velocity) c) draw the orientation of the stable spin direction in each of the 6 seg- ments of the ring d) assuming that the beam is fully polarized, what are the amplitudes of the components of the transverse beam polarization measured at the location P? snake P beam direction