Re-evaluation of Spin-Orbit Dynamics of Polarized e+ e- Beams in High Energy Circular Accelerators and Storage Rings: an approach based on a Bloch equation
Klaus Heinemann, Daniel Appelö, Desmond P. Barber, Oleksii Beznosov, James A. Ellison
aa r X i v : . [ phy s i c s . acc - ph ] J a n Re-evaluation of Spin-Orbit Dynamics of Polarized e + e − Beams inHigh Energy Circular Accelerators and Storage Rings: an approachbased on a Bloch equation ∗ Klaus Heinemann Department of Mathematics and Statistics, University of New Mexico,Albuquerque, NM 87131, [email protected]
Daniel Appel¨o Department of Applied Mathematics, University of Colorado Boulder,Boulder, CO 80309-0526, [email protected] P. BarberDeutsches Elektronen-Synchrotron (DESY)Hamburg, 22607, Germanyand:Department of Mathematics and Statistics, University of New MexicoAlbuquerque, NM 87131, [email protected] BeznosovDepartment of Mathematics and Statistics, University of New Mexico,Albuquerque, NM 87131, [email protected] A. EllisonDepartment of Mathematics and Statistics, University of New Mexico,Albuquerque, NM 87131, [email protected] ∗ Based on a talk at IAS, Hong Kong, January 17, 2019. Also available as article:
Int. J. Mod. Phys. , vol. A35, Nos. 15 & 16,2041003, 2020. Moreover available as DESY Report 20-137. Corresponding author. Now at Michigan State University, USA. (email: [email protected]) bstract We give an overview of our current/future analytical and numerical work on the spin polarization inhigh-energy electron storage rings. Our goal is to study the possibility of polarization for the CEPC andFCC-ee. Our work is based on the so-called Bloch equation for the polarization density introduced byDerbenev and Kondratenko in 1975. We also give an outline of the standard approach, the latter beingbased on the Derbenev-Kondratenko formulas.Keywords: electron storage rings, spin-polarized beams, polarization density, FCC, CEPC, stochastic . differential equations, method of averaging. PACS numbers:29.20.db,29.27.Hj,05.10.Gg 2 ontents Introduction
This paper is an update on a talk by K. Heinemann at the IAS Mini-Workshop on Beam Polarization inFuture Colliders on January 17, 2019, in Hong Kong. Our ultimate goal is to examine the possibility ofhigh polarization for CEPC and FCC-ee.We will first briefly review the “standard” approach which is based on the Derbenev-Kondratenko for-mulas. These formulas rely, in part, on plausible assumptions grounded in deep physical intuition. Sothe following question arises: do the Derbenev-Kondratenko formulas tell full story? In fact there is analternative approach based on a Bloch-type equation for the polarization density which we call the Blochequation (BE) and which we believe can deliver more information than the standard approach even if thelatter includes potential correction terms. So we aim to determine the domain of applicability of theDerbenev-Kondratenko formulas and the possibility in theory of polarization at the CEPC and FCC-eeenergies. Of course both approaches focus on the equilibrium polarization and the polarization time. Weuse the name “Bloch” to reflect the analogy with equations for magnetization in condensed matter. Thispaper concentrates on the Bloch approach. The cost of the numerical computations in the Bloch approachis considerable since the polarization density depends on six phase-space variables plus the time variable sothat the numerical solution of the BE, the BE being a system of three PDEs in seven independent variables,is a nontrivial task which cannot be pursued with traditional approaches like the finite difference method.However we see at least five viable methods:1. Approximating the BE by an effective BE via the Method of Averaging and solving the effectiveBE via spectral phase-space discretization, e.g., a collocation method, plus an implicit-explicit timediscretization.2. Solving the system of stochastic differential equations (SDEs), which underlies the BE, via Monte-Carlospin tracking. See Ref. 6 for the system of SDEs underlying the BE.3. Solving the Fokker-Planck equation, which underlies the BE, via the Gram-Charlier method.4. Solving the BE via a deep learning method.5. Solving the system of SDEs in a way that allows connections with the Derbenev-Kondratenko formulasto be established.We will dwell on Method 1 in this paper. We plan to validate this method by one of the other fourmethods. More details on Method 1 can be found in Ref. 6. The method of averaging we use is discussedin Refs. 7-12. One hope tied to Method 1 is that the effective BE gives analytical insights into the spin-resonance structure of the bunch. Note that Methods 1-4 are independent of the standard approach. Inparticular they do not rely on the invariant spin field. Note also that Methods 1-3 and 5 are based onknowing the system of SDEs, which underlies the BE. For details of this system of SDEs, see the invitedICAP18 paper of Ref. 6. Regarding Method 2 there is a large literature on the numerical solution of SDEs,see Refs. 13, 14 and references in Ref. 15.By neglecting the spin-flip terms and the kinetic-polarization term in the BE one obtains an equationthat we call the Reduced Bloch equation (RBE). The RBE approximation is sufficient for computing theradiative depolarization rate due to stochastic orbital effects and it shares the terms with the BE that arechallenging to discretize. For details on our phase-space discretization and time discretization of the RBE,see Refs. 6,16,17 and 18.We proceed as follows. In Section 2 we sketch the standard approach. In Section 3 we present, for thelaboratory frame, the BE and its restriction, the RBE. In Section 4 we discuss the RBE in the beam frameand in Section 5 we show how, in the beam frame, the effective RBE is obtained via the method of averaging.In Section 6 we describe ongoing and future work. Note that in previous work we sometimes called it the full Bloch equation. Sketching the standard approach based on the Derbenev-Kondratenkoformulas
We define the “time” θ = 2 πs/C where s is the distance around the ring and C is the circumference. Wedenote by y a position in six-dimensional phase space of accelerator coordinates which we call beam-framecoordinates. In particular, following Ref. 19, y is the relative deviation of the energy from the referenceenergy. Then if, f = f ( θ, y ) denotes the normalized 2 π -periodic equilibrium phase-space density at θ and y and ~P loc = ~P loc ( θ, y ) denotes the local polarization vector of the bunch we have Z dy f ( θ, y ) = 1 , Z dy f ( θ, y ) ~P loc ( θ, y ) = ~P ( θ ) , (1)where ~P ( θ ) is the polarization vector of the bunch at θ . For a detailed discussion about ~P loc , see, e.g., Ref.20. Here and in the following we use arrows on three-component column vectors.Central to the standard approach is the invariant spin field (ISF) ˆ n = ˆ n ( θ, y ) defined as a normalizedperiodic solution of the Thomas-BMT-equation in phase space, i.e., ∂ θ ˆ n = L Liou ( θ, y )ˆ n + Ω( θ, y )ˆ n , (2)such that1. (cid:12)(cid:12)(cid:12) ˆ n ( θ, y ) (cid:12)(cid:12)(cid:12) = 1,2. ˆ n ( θ + 2 π, y ) = ˆ n ( θ, y ),and where L Liou denotes the Hamiltonian part of the Fokker-Planck operator L y FP , the latter being introducedin Section 3 below. For some of our work on the ISF see Refs. 21 and 22. The unit vector of the ISF on theclosed orbit is denoted by ˆ n ( θ ) and it is easily obtained as an eigenvector of the one-turn spin-transportmap on the closed orbit. There are many methods for computing the ISF but none are trivial (for arecent technique see Ref. 23). In fact the existence, in general, of the invariant spin field is a mathematicalissue which is only partially resolved, see, e.g., Ref. 21. The standard approach assumes that a function P DK = P DK ( θ ) exists such that ~P loc ( θ, y ) ≈ P DK ( θ )ˆ n ( θ, y ) . (3)Thus, by (1) and (3), ~P ( θ ) = Z dy f ( θ, y ) ~P loc ( θ, y ) ≈ P DK ( θ ) Z dy f ( θ, y )ˆ n ( θ, y ) . (4)The approximation (3) leads to P DK ( θ ) = P DK ( ∞ )(1 − e − θ/τ DK ) + P DK (0) e − θ/τ DK , (5)where τ DK and P DK ( ∞ ) are given by the Derbenev-Kondratenko formulas P DK ( ∞ ) := τ − τ − , (6) τ − := 5 √ r e γ ~ m C π Z π dθ D | R | [1 −
29 (ˆ n · ˆ β ) + 1118 (cid:12)(cid:12)(cid:12) ∂ y ˆ n (cid:12)(cid:12)(cid:12) ] E θ , (7) τ − := r e γ ~ m C π Z π dθ D | R | ˆ b · [ˆ n − ∂ y ˆ n ] E θ , (8)with • D · · · E θ ≡ R dy f ( θ, y ) · · · ˆ b = ˆ b ( θ, y ) ≡ normalized magnetic field, ˆ β = ˆ β ( θ, y ) ≡ normalized velocity vector, γ ≡ Lorentz factorof the reference particle, R ( θ, y ) ≡ radius of curvature in the external magnetic field, r e ≡ classicalelectron radius, m ≡ rest mass of electrons or positrons.By (4) and for large θ ~P ( θ ) ≈ P DK ( ∞ ) Z dy f ( θ, y )ˆ n ( θ, y ) , (9)where P DK ( ∞ ) is given by (6) and where the rhs of (9) is the approximate equilibrium polarization vector.Note that the latter is 2 π -periodic in θ since f ( θ, y ) and ˆ n ( θ, y ) are 2 π -periodic in θ . Defining τ − dep := 5 √ r e γ ~ m C π Z π dθ D | R | (cid:12)(cid:12)(cid:12) ∂ y ˆ n (cid:12)(cid:12)(cid:12) E θ , (10)we can write (7) as τ − = τ − dep + 5 √ r e γ ~ m C π Z π dθ D | R | [1 −
29 (ˆ n · ˆ β ) ] E θ . (11)For details on (6), (7), (8), (10) and (11) see, e.g., Refs. 24 and 19.We now briefly characterize the various terms in the Derbenev-Kondratenko formulas. First, τ − dep isthe radiative depolarization rate. Secondly, the term r e γ ~ m C π R π dθ D | R | ˆ b · ˆ n E θ in τ − and the term √ r e γ ~ m C π R π dθ D | R | E θ in τ − cover the Sokolov-Ternov effect. Lastly, the term − r e γ ~ m C π R π dθ D | R | ˆ b · [ ∂ y ˆ n ] E θ in τ − covers the kinetic polarization effect and the term in τ − which is proportional to 2 / P DK ( ∞ ) via the Derbenev-Kondratenko formulas. Allthree approaches use (6) but they differ in how τ − and τ − are computed.(i) Compute τ − via (8) and τ − via (7) by computing f and ˆ n as accurately as needed.(ii) Approximate τ − by neglecting the usually-small kinetic polarization term in (8) and by approximatingthe remaining term in (8) by replacing ˆ n by ˆ n . Compute τ − via (11) where τ − dep is not computed via(10) but via Monte-Carlo spin tracking and where the remaining terms in (11) are approximated byusing the ˆ n -axis. (iii) Compute τ − via (8) and τ − via (7) by linear approximation in orbit and spin variables via theso-called SLIM formalism. Approach (ii) is the most practiced while approach (i) is only feasible if one can compute f and ˆ n as accuratelyas needed (which is not easy!). Approach (iii), which was historically the first, is very simple and is oftenused for ballparking P DK ( ∞ ). Since the inception of the Derbenev-Kondratenko formulas correction termsto the rhs of (10) have been suspected. See Refs. 4, 28 as well as Z. Duan’s contribution to this workshop.These correction terms, associated with so-called resonance crossing, in turn associated with large energyspread, are not as well understood as the rhs of (10), partly because of their peculiar form. Nevertheless,careful observation of spin motion during the Monte-Carlo tracking in approach (ii), might provide a way toinvestigate their existence and form. Prominent Monte-Carlo spin tracking codes are SLICKTRACK by D.P. Barber, SITROS by J. Kewisch, Zgoubi by F.Meot, PTC/FPP by E. Forest, and Bmad by D. Sagan. This approach provides a useful first impression avoiding thecomputation of f and ˆ n . For more details on this approach see Ref. 19. Monte-Carlo tracking can also be extended beyondintegrable orbital motion to include, as just one example, beam-beam forces. Note that Monte-Carlo tracking just gives anestimate of τ − dep but it does not provide an explanation. Nevertheless, insights into sources of depolarization can be obtainedby switching off terms in the Thomas-BMT equation. In principal such diagnoses can also be applied in approach (i). Suchinvestigations can the systematized under the heading of “spin matching”. The Bloch equation and the Reduced Bloch equation in thelaboratory frame
In the previous section we used the beam frame and we will do so later. However the BE was first presentedin Ref. 3 for the laboratory frame and in that frame it also has its simplest form. In this section we focuson the laboratory frame.In a semiclassical probabilistic description of an electron or positron bunch the spin-orbit dynamics isdescribed by the spin- / Wigner function ρ (also called the Stratonovich function ) written as ρ ( t, z ) = 12 [ f lab ( t, z ) I × + ~σ · ~η lab ( t, z )] , (12)with z = ( ~r, ~p ) where ~r and ~p are the position and momentum vectors of the phase space and t is the time, andwhere f lab is the phase-space density of particles normalized by R dzf lab ( t, z ) = 1, ~η lab is the polarizationdensity of the bunch and ~σ is the vector of the three Pauli matrices. As explained in Ref. 20, ~η lab isproportional to the spin angular momentum density. In fact it is given by ~η lab ( t, z ) = f lab ( t, z ) ~P loc,lab ( t, z )where ~P loc,lab is the local polarization vector. Thus f lab = T r [ ρ ] and ~η lab = T r [ ρ~σ ]. The polarization vector ~P lab ( t ) of the bunch is ~P lab ( t ) = R dz~η lab ( t, z ) = R dzf lab ( t, z ) ~P loc,lab ( t, z ).Then, by neglecting collective effects and after several other approximations, the phase-space densityevolves according to Ref. 3 via ∂ t f lab = L labF P ( t, z ) f lab . (13)Using the units as in Ref. 3 the Fokker-Planck operator L labF P is defined by L labF P ( t, z ) := L labLiou ( t, z ) + ~F rad ( t, z ) + ~Q rad ( t, z ) + 12 X i,j =1 ∂ p i ∂ p j E ij ( t, z ) , (14)where L labLiou ( t, z ) := − ∂ ~r · mγ ( ~p ) ~p − ∂ ~p · [ e ~E ( t, ~r ) + emγ ( ~p ) ( ~p × ~B ( t, ~r ))] , (15) ~F rad ( t, z ) := − e m γ ( ~p ) | ~p × ~B ( t, ~r ) | ~p , (16) ~Q rad ( t, z ) := 5548 √ X j =1 ∂ [ λ ( t, z ) ~pp j ] ∂p j , (17) E ij ( t, z ) := 5524 √ λ ( t, z ) p i p j , λ ( t, z ) := ~ | e | m γ ( ~p ) | ~p × ~B ( t, ~r ) | , (18) γ ( ~p ) := 1 m p | ~p | + m , (19)and with e being the electric charge of the electron or positron and ~E and ~B being the external electric andmagnetic fields.The Fokker-Planck operator L labF P whose explicit form is taken from Ref. 3 is a linear second-order partialdifferential operator and, with some additional approximations, is commonly used for electron synchrotronsand storage rings, see Ref. 29 and Section 2.5.4 in Ref. 19. As usual, since it is minuscule compared toall other forces, the Stern-Gerlach effect from the spin onto the orbit is neglected in (13). The polarizationdensity ~η lab evolves via eq. 2 in Ref. 3, i.e., via that which we call the Bloch equation, namely ∂ t ~η lab = L lab FP ( t, z ) ~η lab + M ( t, z ) ~η lab − [1 + ∂ ~p · ~p ] λ ( t, z ) 1 mγ ( ~p ) ~p × ~a ( t, z ) | ~a ( t, z ) | f lab ( t, z ) , (20)7here M ( t, z ) := Ω lab ( t, z ) − λ ( t, z ) 5 √
38 [ I × − m γ ( ~p ) ~p~p T ] , (21) ~a ( t, z ) := em γ ( ~p ) ( ~p × ~B ( t, ~r )) . (22)The BE was derived in Ref. 3 from the semiclassical approximation of quantum electrodynamics and it isa generalization, to the whole phase space, of the Baier-Katkov-Strakhovenko equation which just describesthe evolution of polarization along a single deterministic trajectory. Note also that, while the BE wasnew in 1975, the orbital Fokker-Planck equation (13) was already known thanks to research of the 1950s,e.g., Schwinger’s paper on quantum corrections to synchrotron radiation. The skew-symmetric matrixΩ lab ( t, z ) takes into account the Thomas-BMT spin-precession effect. Thus in the laboratory frame theThomas-BMT-equation (2) reads as ∂ t ˆ n lab = L labLiou ( t, z )ˆ n lab + Ω lab ( t, z )ˆ n lab . (23)The quantum aspect of (13) and (20) is embodied in the factor ~ in λ ( t, z ). For example ~Q rad is aquantum correction to the classical radiation reaction force ~F rad . The terms − λ ( t, z ) √ ~η lab and − λ ( t, z ) mγ ( ~p ) ~p × ~a ( t,z ) | ~a ( t,z ) | f lab ( t, z ) take into account spin flips due to synchrotron radiation and encapsulate theSokolov-Ternov effect. The term λ ( t, z ) √
38 29 m γ ( ~p ) ~p~p T ~η lab encapsulates the Baier-Katkov correction, andthe term ∂ ~p · ~p λ ( t, z ) mγ ( ~p ) ~p × ~a ( t,z ) | ~a ( t,z ) | f lab ( t, z ) encapsulates the kinetic-polarization effect. The only terms in(20) which couple the three components of ~η lab are the Thomas-BMT term and the Baier-Katkov correctionterm.As mentioned above, there exists a system of SDEs underlying (20) (for details, see Ref. 6). In particular, f lab and ~η lab are related to a spin-orbit density P lab = P lab ( t, z, ~s ) via f lab ( t, z ) = Z R d~s P lab ( t, z, ~s ) , (24) ~η lab ( t, z ) = Z R d~s ~s P lab ( t, z, ~s ) , (25)where P lab satisfies the Fokker-Planck equation corresponding to the system of SDEs in Ref. 6. These SDEscan be used as the basis for a Monte-Carlo spin tracking algorithm, i.e., for Method 2 mentioned in Section1 above. This would extend the standard Monte-Carlo spin tracking algorithms, which we mentioned inSection 2 above, by taking into account all physical effects described by (20), i.e., the Sokolov-Ternov effect,the Baier-Katkov correction, the kinetic-polarization effect and, of course, spin diffusion.If we ignore the spin-flip terms and the kinetic-polarization term in the BE then (20) simplifies to theRBE ∂ t ~η lab = L labF P ( t, z ) ~η lab + Ω lab ( t, z ) ~η lab . (26)The RBE models spin diffusion due to the effect of the stochastic orbital motion on the spin and thus containsthose terms of the BE which are related to the radiative depolarization rate τ − dep . This effect is clearly seenin the SDEs (see, e.g., (28) and (29)). In the beam frame, i.e., in the accelerator coordinates y of Section 2, the RBE (26) becomes ∂ θ ~η = L y FP ( θ, y ) ~η + Ω( θ, y ) ~η . (27)Because the coefficients of L y FP are θ -dependent, the RBE (27) is numerically and analytically quite complex.So we first approximate it by treating the synchrotron radiation as a perturbation. Then, in order to solve8t numerically to determine the long-time behavior that we need, we address the system of SDEs underlying(27) and apply the refined averaging technique presented in Ref. 32 (see also 7), for the orbital dynamics,and extend it to include spin. The averaged SDEs are then used to construct an approximate RBE whichwe call the effective RBE.The system of SDEs underlying (27) reads as dYdθ = ( A ( θ ) + ǫ R δA ( θ )) Y + √ ǫ R p ω ( θ ) e ξ ( θ ) , (28) d~Sdθ = [Ω ( θ ) + ǫ S C ( θ, Y )] ~S , (29)where the orbital dynamics has been linearized in Y and Ω = Ω + ǫ S C has been linearized in Y so that C ( θ, Y ) = X j =1 C j ( θ ) Y j . (30)Also, A ( θ ) is a Hamiltonian matrix representing the nonradiative part of the orbital dynamics and Y hasbeen scaled so that ǫ R is the size of the orbital effect of the synchrotron radiation. Thus ǫ R δA ( θ ) representsthe orbital damping effects due to synchrotron radiation and the cavities, √ ǫ R ξ ( θ ) represents the associatedquantum fluctuations, ξ is the white noise process and e := (0 , , , , , T . In the spin equation (29), Ω is the closed-orbit contribution to Ω so that ǫ S C ( θ, Y ) is what remains and C ( θ, Y ) is chosen O (1). Hence ǫ S estimates the size of Ω − Ω . Both Ω ( θ ) and C ( θ, Y ) are, of course, skew-symmetric 3 × ǫ R and ǫ S are small in some appropriate sense.Eqs. (28) and (29) can be obtained by transforming the system of SDEs in Ref. 6 from the laboratoryframe to the beam frame. However, since in this section we only deal with the RBE (not with the BE),(28) and (29) can also be found in older expositions on spin in high-energy electron storage rings, e.g., Ref.34. Note that these expositions make approximations as for example with the linearity of (28) in Y and thelinearity of C ( θ, Y ) in Y .With (28) and (29) the evolution equation for the spin-orbit joint probability density P = P ( θ, y, ~s ) isthe following spin-orbit Fokker-Planck equation ∂ θ P = L y FP ( θ, y ) P − ∂ ~s · (cid:18) Ω( θ, y ) ~s (cid:19) P ! , (31)where L y FP is the orbital Fokker-Planck operator. The phase-space density f and the polarization density ~η corresponding to P are defined by f ( θ, y ) = Z R d~s P ( θ, y, ~s ) , ~η ( θ, y ) = Z R d~s ~s P ( θ, y, ~s ) , (32)which are the beam-frame analogs of (24) and (25). The local polarization vector ~P loc from Section 2 aboveis related to f and ~η by ~η ( θ, y ) = f ( θ, y ) ~P loc ( θ, y ) . (33)The RBE (27) follows from (31) by differentiating (32) w.r.t. θ and by using the Fokker-Planck equation for P . This proves that (28) and (29) is the system of SDEs which underlie the RBE (27). For (27), see alsoRef. 20. The effective RBE is, by definition, an approximation of the RBE (27) obtained by approximating the systemof SDEs (28) and (29) using the method of averaging, see Refs. 7-12. We call the system of SDEs underlying We denote the random dependent variables like Y in (28) by capital letters to distinguish them from independent variableslike y in (27). ǫ := ǫ S = ǫ R is small.To apply the method of averaging to (28) and (29) we must transform them to a standard form foraveraging, i.e., we must transform the variables Y, ~S to slowly varying variables. We do this by using afundamental solution matrix X of the unperturbed ǫ = 0 part of (28), i.e., X ′ = A ( θ ) X , (34)and a fundamental solution matrix Φ of the unperturbed ǫ = 0 part of (29), i.e.,Φ ′ = Ω ( θ )Φ . (35)We thus transform Y and ~S into the slowly varying U and ~T via Y ( θ ) = X ( θ ) U ( θ ) , ~S ( θ ) = Φ( θ ) ~T ( θ ) . (36)Hence (28) and (29) are transformed to U ′ = ǫ D ( θ ) U + √ ǫ p ω ( θ ) X − ( θ ) e ξ ( θ ) , (37) ~T ′ = ǫ D ( θ, U ) ~T , (38)where D and D are defined by D ( θ ) := X − ( θ ) δA ( θ ) X ( θ ) , (39) D ( θ, U ) := Φ − ( θ ) C ( θ, X ( θ ) U )Φ( θ ) . (40)Of course, (37) and (38) carry the same information as (28) and (29). Now, applying the method of averagingto (37) and (38), we obtain the following effective system of SDEs V ′ = ǫ ¯ D V + √ ǫ B ( ξ , ..., ξ k ) T , (41) ~T ′ a = ǫ ¯ D ( V ) ~T a , (42)where the bar denotes θ -averaging, i.e., the operation lim T →∞ (1 /T ) R T dθ · · · . Moreover ξ , ..., ξ k are statis-tically independent versions of the white noise process and B is a 6 × k matrix which satisfies BB T = ¯ E with k = rank ( ¯ E ) and where ¯ E is the θ -average of E ( θ ) = ω ( θ ) X − ( θ ) e e T X − T ( θ ) . (43)For physically reasonable A and Ω the fundamental matrices X and Φ are quasiperiodic functions whence D , D ( · , U ) and E are quasiperiodic functions so that their θ averages ¯ D , ¯ D ( V ) and ¯ E exist.Our derivation of (41) from (37) is discussed in some detail in Ref. 6. We are close to showing that U = V + O ( ǫ ) on θ -intervals of length O (1 /ǫ ) and it seems likely that this error is valid for 0 ≤ θ < ∞ ,because of the radiation damping. This is a refinement of Ref. 32 and assumes a non-resonance condition.Since the sample paths of U are continuous and U is slowly varying it seems likely that ~T a is a goodapproximation to ~T and we are working on the error analysis. Spin-orbit resonances will be an importantfocus in the construction of ¯ D ( V ) from (40) which contains both the orbital frequencies in X and the spinprecession frequency in Φ.Since, by definition, the effective system of SDEs underly the effective RBE, the latter can be obtainedfrom the former in the same way as we obtained (27) from (28) and (29) (recall the discussion after (32)).Thus the evolution equation for the spin-orbit probability density P V = P V ( θ, v , ~t ) is the following Fokker-Planck equation: ∂ θ P V = L V FP (v) P V − ǫ∂ ~t · (cid:18) ¯ D (v) ~t (cid:19) P V ! , (44)10here L V FP (v) = − ǫ X j =1 ∂ v j ( ¯ D v) j + ǫ X i,j =1 ¯ E ij ∂ v i ∂ v j . (45)The polarization density ~η V corresponding to P V is defined by ~η V ( θ, v) = Z R d~t ~t P V ( θ, v , ~t ) , (46)so that, by (44), the effective RBE is ∂ θ ~η V = L V FP (v) ~η V + ǫ ¯ D (v) ~η V . (47)This then is the focus of our approach in Method 1. For more details on this section, see Refs. 6, 17 and 18. • Further development of Bloch-equation approach (numerical and theoretical), i.e., of Method 1 andwith a realistic lattice. • Development of validation methods, i.e., Methods 2-4. Note that Method 2 is an extension of thestandard Monte-Carlo spin tracking algorithms and for that matter we will study Refs. 13, 14 and 15. • Investigating the connection between the Bloch-equation approach and the standard approach basedon the Derbenev-Kondratenko formulas, and studying the potential for correction terms to τ − byusing the RBE. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office ofHigh Energy Physics, under Award Number DE-SC0018008.
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