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Mitglied der Helmholtz-Gemeinschaft | Yurij Senichev ELECTROSTATIC LATTICE for srEDM with ALTERNATING SPIN ABERRATION 24. Mai 2017 “Tomas-Bargmann, Michel,Telegdi” equation The spin is a quantum value, but in the classical physics representation the “spin” means an expectation value of a quantum mechanical spin operator: dS S dt 1 e G B 2 G E E B 1 m 2 g 2 G , 2 d e / 4mc 24. Mai 2017 Folie 2 SE The EDM search methods in Storage Ring: 1. Resonant method with initial spin orientation in ring S║B; S={0,Sy ,0} and B={0,By,0} 2. “Magic” method with initial spin orientation in ring S║p; S┴E; S={0,0,Sz} and E={Ex,0,0} 24. Mai 2017 Folie 3 In resonant method* the spin frequency is parameterized : 1 e G B 2 G E E B 1 m 2 S p z x i ( f rf kfrev )t By y Ex RF field By using RF flipper E ~ e . f rf In case of parametric resonance when f k G; k 0,1,2,... rev we shall observe the resonant build up: h 2 2 s2 Erf lrf B y Lcir ~ EDM signal Tf -peiod of fundamental oscillation hn cos 2 s n s G Advantage: the method can be realized in COSY ring Disadvantage: the high requirement to stability of frf 24. Mai 2017 *A.Lehrach, Sz max S z (n) (2 e s h s ) sin 2 s n Te - period of envelope B.Lorentz, W.Morse, N.Nikolaev and F.Rathmann Folie 4 “Magic” method for purely electrostatic ring In the “magic” method the beam is injected in the electrostatic ring with the spin directed along momentum S║p and S┴E; S={0,0,Sz} and E={Ex,0,0} at “magic” energy : 1 m2 ag 1 G 0 dS S dt 1 e G B 2 G E E B m 2 1 g 2 G , 2 External fields 24. Mai 2017 EDM Folie 5 “Magic” method in purely electrostatic ring In purely electrostatic ring the spin of particle with “magic energy” rotates with the same angular frequency as the momentum and it tilts up in the YZ plane due to the EDM with angular rate e dS E S dt 2m S y p ARC2 z y x By=0 electrostatic field E x S EDM p ARC1 24. Mai 2017 Folie 6 Spin tune aberration in purely electrostatic ring In reality the beam has energy spread γmag±Δγ and all particles move in different external field. Therefore the spin tune has the aberrations dependent on energy γ and trajectory r(t) of particles. dS e 1 G n E S 2 dt m0c 1 vs energy vs field distribution At “magic” energy it is no precession of spin. For no “magic” energy γmag±Δγ 0 ( , r ) Spin tune aberration 24. Mai 2017 Folie 7 Spin Coherence Time is time when RMS spin orientation of the bunch particles reaches one radian (YS,BNL), and it has to be > 1000sec. During SCT each particle performs ~109 turns in the storage ring moving with different trajectories through the optics elements. At such conditions the spin-rotation aberrations associated with various types of space and the time dependent nonlinearities start to play a crucial role. 24. Mai 2017 S y p z y x S t p Folie 8 Spin tune aberration due to energy spread Longitudinal component dS x eL E orb 2x d 2mc 1 G Sz 2 1 eL E dS z orb 2x d 2mc 1 Sx G 2 1 1 s G 2m0c 2 2 1 eLorbE x d 2dt / Trev 2 2 1 p 1 3 p 2 G 2G G ... 2 p p 1 24. Mai 2017 Folie 9 RF cavity as first step to increase SCT RF off: 2 d S z e E x Lcir p 2G Sz 0 2 2 p d 2m0 c 2 for Δp/p=10-4 SCT=6300 turns, which is ~ 1 msec. RF on: p / p p / p max cos z Idea of using the RF cavity was expressed long time ago by many authors, for instance [ A.P. Lysenko et al., Part.Accel. 18, 215 (1986) ]. 2 p d S z e E x Lcir 2G cos z S z 0 2 2 d p max 2m0c 2 The spin swing in a rapidly oscillating field with RF frequency and it is bounded within a very narrow angle ~10-6 dependent on max ~ s / z 2 24. Mai 2017 Folie 10 RF on: Second order approach of spin tune versus Δp/p However, in the second approach versus momentum the average tune spin is not zero 2 2 eE x L p p 1 3 2 1 3 2 G p 2 cir sz 2G cos z G cos z 2 2 2 2 p p 2 p 2m0 c 2 m c max max 0 e E x Lcir At (Δp/p)max=10-4 and an axial particle the number of turns for SCT is ~6 107 turns, that is ~180 sec. spin drift term spin oscillation 24. Mai 2017 max The code COSY infinity simulation Folie 11 Off-axial particle: Longitudinal-transverse coupling in electrostatic storage ring The electrical deflector has the central field symmetry: E n r where angular momentum M mv r . const Due to this fact the total energy can be represented as the function of the coordinates : 2 W ( r, r) m 2 M r r 2 2 2mr The radial motion is one dimensional motion in the field with the effective potential, and the equilibrium radius: Req M n 24. Mai 2017 Folie 12 Off-axial particle: influence on spin motion The particle with different momentum oscillates with respect to different energy levels: COSY infinity tracking results for initial coordinates x=0, y=0 and x=3mm, y=0. Thus, RF cavity will not be able to reduce the oscillation of the spin for off-axial particles, since: e E x Lcir p sz G 2 p m0 c 24. Mai 2017 Folie 13 Equilibrium energy level modulation as method to increase SCT Following physical logic the only solution to increase SCT is the modulation of the energy level itself relative of the magic level. For this purpose, we have increase coupling between longitudinal and transverse motion that is, approach frequencies as close as possible to each other. In result we have got SCT=400 sec COSY infinity tracking results for initial coordinates x=0, y=0 and x=3mm, y=0. 24. Mai 2017 Folie 14 Spin tune aberration vs momentum and axial deviation Assuming violation “magic” condition for non-reference particle the spin tune aberration is defined: Lorb E x 1 e s 2 G Lorb E x 1 2 2mc 1 L E orb x with 2 2 1 p 1 3 p ... 2 G 2G G 2 p p 1 2 Lorb p p ... 1 2 Lorb p p E x Ex 24. Mai 2017 2 x x k1 k2 ... R R Folie 15 Spin tune aberration vs momentum and axial deviation Grouping the coefficients of powers up to second order, we obtain an equation having a parabolic form : 2 eLorbE xG x p x p 2 F1 k1, k2 , s F , k , k , 2 2 1 1 2 R p R p 2m0c with coefficients having a parabolic dependence on axial deviation 2 x 1 3 2 x 1 3 2 x 5 2 1 F2 1, k1, k2 , k2 k1 21 2 2 2 R R R x x x F1 k1, k2 , k1 k2 R R R 24. Mai 2017 2 Folie 16 x Spin tune aberration s vs p p and Convex surface, or concave surface depends on the sign of x F1, 2 2 x p x p F2 1 , k1 , k 2 , 2 F1 k1 , k 2 , R p R p + 24. Mai 2017 Folie 17 Two steps to minimize the spin aberrations - The lattice with a compensation of the mutual influence of deflector parameters k1 , k2 and lattice parameters 1 - The lattice must provide the alternate change of the spin aberration surface from concave to convex and vice versa 24. Mai 2017 Folie 18 Electrostatic lattice consisting of electrostatic deflectors and electrostatic quadrupoles Figure: Twiss functions of electrostatic ring for ring and one cell 24. Mai 2017 Folie 19 Method realization Question is how to customize the required k1 , k2 ? 24. Mai 2017 Folie 20 Alternating spin aberration method The ring is equipped with two types of deflector with k1 ; k 2 k av k and k1 ; k 2 k av k changing from one deflector to another. - In such optics is easier to achieve minimum spin aberration Raising the field strength between the plates in even deflectors and reducing in the odd deflectors it effectively adjusts the required coefficients k1 and k2. It allows to adjust the spin of aberration to minimum. 24. Mai 2017 Folie 21 Alternating spin aberration method Another possibility is the creation of the required potential distribution due to potential changes in stripline deposited on the surface of the ceramic plates. 1 n or 1 n Such plates may be the additional corrective elements placed on the sides of the main deflector 24. Mai 2017 Folie 22 The limit capabilities of alternating spin aberration method The spread due to final Δp/p it is impossible to remove completely using the correct k1 and k2 only. Nevertheless the total spread of spin deflection angle does not exceed ±0.5 rad after billion turns, which one corresponds to a SCT about 5000 seconds. 24. Mai 2017 Folie 23 Tracking results: We used differential algebra methods realized in COSY Infinity and integrating program with symplectic Runge-Kutta methods. 1. Cylindrical deflector: after 106 turns Sxrms≈0.002 that is SCT~500 sec 2. Alternating k2 deflector: after 106 turns Sxrms≈0.0002 that is SCT~5000 sec 24. Mai 2017 Folie 24 Conclusion: -we have studied the behavior of spin aberration in the structure and developed techniques to minimize it; -one of the most effective methods is the alternating spin aberration; -the analytical model allows finding the general solution of the retention of aberrations within the values allowed SCT to have about 5000 seconds confirmed by COSY-Infinity. Nearest future plan: -3D shape deflector -spin-orbital tracking in 3 D deflector -including B field 24. Mai 2017 Folie 25 First step We first investigate the structure with deflector having a purely cylindrical electrodes: k1 1, k2 1 Figure: Maximum spin deflection angle after billion turns versus Δp/p at initial horizontal deviation x=-2 mm, 0 mm and 2 mm (a) and versus x at Δp/p=10-4 , 0, -10-4 (b) p/p 10 -4 x, mm s / N F p / p x p / p 0.35 x 2 2 24. Mai 2017 Folie 26 First step The maximum flatness of spin aberration surface is reached by choice of parameters of deflector k1 , k2 and α1 momentum compaction factor. After optimization: k1 0.94, k2 0.96 s / N F 0.012 x 2 Figure: Maximum spin deflection angle after billion turns versus x deviation at Δp/p=2·10-4 , 0, -2·10-4 24. Mai 2017 Folie 27 Second step: Alternating spin aberration The second step is to alternately change the deflector parameters and thereby alternating the rotation of the spin. In mathematical terms, this means minimizing all the factors F0 , F1, F2 and by averaging them in time s / N F 0.004 x 2 Figure: Maximum spin deflection angle after 109 turns versus x deviation at Δp/p=0 (a) and ±2·10-4 , 0 (b) 24. Mai 2017 Folie 28