Generalized Kapchinskij-Vladimirskij Distribution and Beam Matrix for Phase-Space Manipulations of High-Intensity Beams
Moses Chung, Hong Qin, Ronald C. Davidson, Lars Groening, Chen Xiao
aa r X i v : . [ phy s i c s . acc - ph ] O c t Generalized Kapchinskij-Vladimirskij Distribution and BeamMatrix for Phase-Space Manipulations of High-Intensity Beams
Moses Chung, ∗ Hong Qin,
2, 3
Ronald C. Davidson, † Lars Groening, and Chen Xiao Department of Physics, Ulsan National Instituteof Science and Technology, Ulsan 689-798, Korea Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 Department of Modern Physics, University of Scienceand Technology of China, Hefei, Anhui 230026, China GSI Helmholtzzentrum f ¨ u r Schwerionenforschung GmbH,Planckstrasse 1, D-64291 Darmstadt, Germany (Dated: October 5, 2018) Abstract
In an uncoupled linear lattice system, the Kapchinskij-Vladimirskij (KV) distribution, formu-lated on the basis of the single-particle Courant-Snyder (CS) invariants, has served as a funda-mental theoretical basis for the analyses of the equilibrium, stability, and transport properties ofhigh-intensity beams for the past several decades. Recent applications of high-intensity beams,however, require beam phase-space manipulations by intentionally introducing strong coupling. Inthis Letter, we report the full generalization of the KV model by including all of the linear (bothexternal and space-charge) coupling forces, beam energy variations, and arbitrary emittance par-tition, which all form essential elements for phase-space manipulations. The new generalized KVmodel yields spatially uniform density profiles and corresponding linear self-field forces as desired.The corresponding matrix envelope equations and beam matrix for the generalized KV model pro-vide important new theoretical tools for the detailed design and analysis of high-intensity beammanipulations, for which previous theoretical models are not easily applicable.
PACS numbers: 29.27.Bd, 41.85.Ct uncoupled linear lattice system.One of the recent areas of investigation by the beam physics community, however, is tomanipulate the beam phase-space by intentionally introducing strong coupling. The round-to-flat beam transformation [2–6] and the transverse-to-longitudinal emittance exchange[7–10] have been investigated for electron injectors. The generation of flat hadron beamshas recently drawn significant attention in the context of optimizing emittance budgets inheavy ion synchrotrons [11, 12], and improving space-charge and beam-beam luminositylimitations in colliders [13]. For muon ionization cooling, special arrangements of solenoidalmangets are employed to achieve six-dimensional emittance reduction [14, 15].Various attempts have been made to extend the uncoupled CS theory to the case ofgeneral linear coupled systems [16–19]. However, due to the lack of a proper CS invari-ant for the coupled dynamics, previous analyses did not retain the elegant mathematicalstructure present in the original CS theory. Recently, Qin et al. [20, 21] have identifiedthe generalized CS invariant for the linear coupled systems including both solenoidal andskew-quadrupole magnets, and variation of beam energy along the reference orbit. Forphase-space manipulations, solenoidal and skew-quadrupole magnets are frequently used toprovide strong coupling, as mentioned previously. Moreover, the relativistic mass increasemay be important when there is a rapid acceleration of low-energy beams.For some of the beam manipulations, space-charge effects are non-negligible as well;hence in those cases, we require a further generalization that incorporates space-chargeeffects into linear coupling lattices. In the original CS theory, space-charge effects wereconsidered by means of the Kapchinskij-Vladimirskij (KV) distribution [22]. For an in-tense beam propagating trough an alternating-gradient lattice, the KV distribution is theonly known exact solution to the nonlinear Vlasov-Maxwell equations [23, 24], and it gen-erates linear space-charge forces consistent with the CS theory. Through the concept ofrms-equivalent beams [24–26], the KV beam model remains the most important basic de-sign tool for high-intensity beam transport, even in the presence of nonlinear space-chargecontributions. Several generalizations have been proposed for the KV model in order thatit can be applied to coupled systems as well [27–32]. However, none of them incorporatesthe solenoids and skew-quadrupoles simultaneously with a proper CS invariant.In this Letter, we report the first complete generalization of the KV model for the gen-2ral linear coupled system, so that the model describes all of the important processes fortransverse phase-space manipulations of high-intensity beams. Due to the existence of thegeneralized CS invariant, the KV model developed here provides a self-consistent solutionto the nonlinear Vlasov-Maxwell equations for high-intensity beams in coupled lattices, andleads to a matrix version of the envelope equation with an elegant Hamiltonian structure.We emphasize that space-charge effects during emittance manipulation, illustrated by anumerical example in this Letter, is one area that previous KV models could not address.First, we consider a transverse Hamiltonian in general linear focusing lattice of the form H ⊥ = 12 z T A c ( s ) z , A c ( s ) = κ RR T m − . (1)Here, z = ( x, y, p x , p y ) T denotes the transverse canonical coordinates, s is the path lengththat plays the role of a time-like variable, and κ and m − are 2 × × R is not symmetric in general. The canonical momenta arenormalized by a fixed reference momentum p = γ m b β c . Based on the generalized CStheory developed in Refs. [20, 21], we obtain the solution for the coupled dynamics governedby the Hamiltonian (1) in the form of a linear map z ( s ) = M ( s ) z , where z is the initialcondition and M is the transfer matrix defined by M ( s ) = Q − P − P Q = W V W − T P T W − − V T W T , (2)where subscript “0” denotes initial conditions at s = 0, P T = P − and P is a symplecticrotation, and P is set equal to the unit matrix I without loss of generality. Here, the 2 × W and V are defined by W = w T and V = m (cid:16) dw T ds − R T w T (cid:17) . Furthermore, the2 × w is obtained by solving the matrix envelope equation given by [20, 21] dds (cid:18) dwds m − wRm (cid:19) + dwds mR T + w ( κ − RmR T ) − (cid:0) w T wmw T (cid:1) − = 0 . (3)We note that the second-order matrix differential equation (3) can be expressed in terms oftwo first-order equations, i.e., W ′ = m − V + R T W, V ′ = − κW − RV + (cid:0) W T mW W T (cid:1) − . (4)The variable V can be considered to be the matrix associated with the envelope momentum[33]. We also note that Eq. (4) has similar Hamiltonian structure to the single particle3quations of motion except for the term (cid:0) W T mW W T (cid:1) − [see Eq. (12) for comparison andRef. [34] for a more detailed discussion].The 4 × P has the following form [20, 21] P = C o − S i S i C o . (5)Here, C o and S i are the 2 × C ′ o = − S i (cid:0) W T mW (cid:1) − and S ′ i =+ C o (cid:0) W T mW (cid:1) − , where the term (cid:0) W T mW (cid:1) − represents the phase advance rate. Fromthe symplecticity of P , we note that S i C To = C o S Ti and S i S Ti + C o C To = I . The generalizedCS invariant of the Hamiltonian (1) is given by I ξ = z T Q T P T ξP Q z , where ξ is a constant4 × ξ matrix acquires a mean-ing associated with emittance when the beam distribution is defined in terms of the CSinvariant I ξ [see, for example, Eq. (13)]. The two symplectic eigenvalues of ξ are directlyconnected to the eigen-emittances of the beam [35].By using s as an independent coordinate, and treating | p x − q b A x /p | , | p y − q b A y /p | ≪ p and | q b φ sc | ≪ γ b m b c , we can express the transverse Hamiltonian (normalized by p ) tosecond order in the transverse momenta as [19, 23] H ⊥ = 12 p b /p "(cid:18) p x − q b A x p (cid:19) + (cid:18) p y − q b A y p (cid:19) + (cid:18) γ b (cid:19) q b φ sc β b cp − q b A exts p , (6)where we have used the fact that the longitudinal vector potential is composed of bothexternal ( A exts ) and self-field ( A scs ) contributions, and the self-field potentials φ sc and A scs arerelated approximately by A scs = β b φ sc /c . Also, it is assumed that the reference trajectory isa straight line, that the longitudinal motion is independent of the transverse motion, andthat there is no external electric focusing. Furthermore, p b ( s ) = γ b m b β b c , γ b ( s ), and β b ( s )are regarded as prescribed functions of s set by the acceleration schedule of the beamline[24]. Hence, for a combination of the quadrupole, skew-quadrupole, and solenoidal fields,we obtain the following matrices for the Hamiltonian (1) κ ext ( s ) = κ q + (cid:16) β γ β b γ b (cid:17) Ω L κ sq κ sq − κ q + (cid:16) β γ β b γ b (cid:17) Ω L , (7) R ( s ) = − (cid:16) β γ β b γ b (cid:17) Ω L (cid:16) β γ β b γ b (cid:17) Ω L , m − ( s ) = (cid:16) β γ β b γ b (cid:17) (cid:16) β γ β b γ b (cid:17) . (8)4ere, κ q = q b B ′ q ( s ) /p , κ sq = q b B ′ sq ( s ) /p , and Ω L = q b B s ( s ) / p .For low energy (i.e., β b ≪
1) beams, we note that the longitudinal acceleration acts todamp particle oscillations more rapidly [24]. In such cases, so-called reduced coordinatesare often introduced to avoid the complication due to the acceleration [36]. Since the newgeneralized KV model has been formulated in terms of the canonical momenta with therelativistic mass increase already included, an additional transformation to the reducedcoordinates is unnecessary.Since the focusing matrix κ used in the generalized CS theory is an arbitrary 2 × − κ x = − κ ext x − κ sc x = − κ ext x − (cid:18) β γ β b γ b (cid:19) ∇ ψ, (9)where x = ( x, y ) T , and κ ext is constructed from the external lattices. The normalized self-field potential is defined by ψ = q b φ sc /γ β cp . In this coupled linear focusing system, ψ ( x , s ) and the beam distribution function f ( x , p , s ) evolve according to ∂f∂s + x ′ · ∂f∂ x + (cid:20) − κ ext x − (cid:18) β γ β b γ b (cid:19) ∇ ψ − R p (cid:21) · ∂f∂ p = 0 , (10) ∇ ψ = − πK b N b Z f dp x dp y = − πK b N b n. (11)Here, x ′ = ( x ′ , y ′ ) T is the normalized transverse velocity, and p = ( p x , p y ) T is the normalizedcanonical momentum defined from the Hamiltonian equations of motion ( d z /ds = J A c z ,where J is the unit symplectic matrix) as x ′ = m − p + R T x , p ′ = − κ x − R p . (12)The self-field perveance is defined by K b = (1 / πǫ )(2 N b q b /γ β cp ) in SI units, and theline density N b = R f dxdydp x dp y is assumed to be constant. Based on the analysis in Ref.[32], we consider the following distribution function f = N b p | ξ | π δ ( I ξ − . (13)which is a solution of the Vlasov equation (i.e., df /ds = 0 because I ξ is a constant of motion),and generates the coupled linear space-charge force (i.e., R f dp x dp y is spatially uniform inthe beam interior).Using the Cholesky decomposition method, the momentum integral in Eq. (11) can becarried out in a straightforward manner. First, we decompose Q T P T ξP Q in terms of a lower5riangular matrix L according to Q T P T ξP Q = L T L , and then introduce new coordinates Z = ( X, Y, P X , P Y ) T = L z defined by X = ( X, Y ) T = ¯ D T/ W − x and P = ( P X , P Y ) T =( D − / B T W − − D T/ V T ) x + D T/ W T p . We note that, similar to the original KV model,the distribution function f in Eq. (13) represents the trajectories of all particles lying on thesurface of the 4D hyper-ellipsoid, Z T Z = X + Y + P X + P Y = 1 [26]. Here, the square-rootof a symmetric and positive definite matrix D is defined by D / D T/ = D . The ¯ D matrixis known as the Schur complement of D , and it has the following definitions and properties. P T ξP = A BB T D = ¯ D / BD − T/ D / ¯ D T/ D − / B T D T/ , (14)where ¯ D = A − BD − B T = ¯ D T and | P T ξP | = | ξ | = | ¯ D || D | .The Jacobians of the linear coordinate transformations are given by dXdY = | ¯ D T/ W − | dxdy , and dP X dP Y = | D T/ W T | dp x dp y . Then, it can be readily shown thatthe number density n ( x, y, s ) of the beam particles is given by n ( x, y, s ) = Z f dp x dp y = N b | ¯ D T/ W − | π , ≤ X T X < , , < X T X , (15)where R n ( x, y, s ) dxdy = N b is the line density. From Eq. (15), we note that n is spa-tially uniform and a function only of s . The boundary of the beam is determined from X T X = x T ( W − T ¯ DW − ) x = 1, which is a tilted ellipse in ( x, y ) space with area equal to π | ¯ D T/ W − | − . The transverse dimensions of the tilted ellipse, a and b , are determined bythe two eigenvalues (1 /a , /b ) of the matrix W − T ¯ DW − . Therefore, the coupled linearspace-charge force coefficient κ sc can be expressed as κ sc = − (cid:18) β γ β b γ b (cid:19) K b a + b G /a
00 1 /b G − . (16)Here, G is the matrix constructed by the two normalized eigenvectors v and v of the matrix W − T ¯ DW − as G = ( v , v ). Note that G is a rotation matrix, i.e., G − = G T . Whenthe space-charge force term κ sc is substituted back into Eq. (9), the envelope equations(4) become a set of closed nonlinear matrix equations for the envelope matrix W and itsassociated envelope momentum matrix V .To demonstrate the exact connection between Q T P T ξP Q and the beam matrix, we intro-duce the geometric factor g [23] and the symmetric matrix Σ defined by Q T P T ξP Q = g Σ − .6e will show that there exits a real number g which makes Σ equal to the beam matrix (cid:10) zz T (cid:11) , in which h· · · i denotes statistical average over the distribution function f . Sincethe matrix Σ is real and symmetric, we consider the eigenvalue equation for Σ given byΣ u i = λ i u i . We can then make use of the orthonormality of the eigenvectors [37] to express z = P j =1 y j u j , where y j = u Tj z . It then follows that (cid:10) zz T (cid:11) = p | ξ | π X i =1 u i u Ti Z δ ( g X k =1 y k λ k − y i d y . (17)Here, we have used the fact that the above integral vanishes by symmetry unless y i = y j .After some straightforward algebra, the above integral yields ( λ i /g ) hQ j =1 p λ j /g i ( π / (cid:10) zz T (cid:11) = g P i =1 u i u Ti λ i = g Σ. Therefore, if g = 1 /
4, then Σ = (cid:10) zz T (cid:11) = Q − P − εP − T Q − T , where the emittance matrix is defined by ε = ξ − . We notethat the transverse rms emittance is ǫ ⊥ = p | Σ | = p | ε | . This is the natural generalizationof the original KV model, in which the total (or 100%) emittance is 4 times larger than therms emittance for each transverse phase-space.Once the initial beam matrix Σ is prescribed, the beam matrix at an arbitrary position s can be calculated in terms of the transfer matrix M as Σ( s ) = M Σ M T . In principle,the transfer matrix M is independent of the choice of the parametrization because M issolely determined by the equations of motion. Therefore, the envelope equations (4) can besolved for arbitrary choices of the initial conditions ( W, V ) . Furthermore, for the case ofnegligible space-charge, the envelope equations (4) become independent of the initial beammatrix Σ = Q − εQ − T as well. On the other hand, for the case of intense space-charge, thebeam envelopes evolve under the influence of the beam matrix Σ, because the space-chargefocusing coefficient κ sc depends on Σ. Hence, in this case, it is important to ensure thatthe generalized CS parametrization generates the initial beam matrix Σ correctly. Thiscan be achieved by requiring ε = ξ − = 4 Q Σ Q T . When Q is calculated for specifiedinitial conditions ( W, V ) , the ten free parameters in ε (or ξ − ) are determined accordingly.Note that when different initial conditions ( W, V ) are used, the emittance matrix ε itself iscalculated differently; however, it generates the same Σ , M , and eigen-emittances.Based on Refs. [2, 11], we specify the initial beam matrix of a cylindrically symmetric7eam in the following formΣ = σ κ σ σ − κ σ − κ σ σ ′ + κ σ κ σ σ ′ + κ σ , (18)where σ = h x i = h y i , and σ ′ = h x ′ i = h y ′ i . It can be shown that the two eigen-emittances are given by ǫ , = ǫ eff ± L , where L = κ σ and ǫ eff = p ( σσ ′ ) + L . The initialbeam matrix Σ in the form of Eq. (18) can be obtained, either by generating an electronbeam inside a solenoid as in the round-to-flat beam (RTFB) transformation experiment[3], or by stripping an ion beam inside a solenoid as in the emittance transfer experiment(EMTEX) [11, 12]. For the RTFB transformation experiment, κ is given by κ = B s Bρ ) c ,where B s and ( Bρ ) c are the solenoidal magnetic field and beam rigidity at the cathode,respectively. For the EMTEX experiment, κ = h ( Bρ ) in ( Bρ ) out − i B s Bρ ) in , where ( Bρ ) in and( Bρ ) out are the beam rigidity before and after the stripping foil, respectively. To remove thecorrelation in Σ , a beam-line constructed by three skew-quadrupoles is often used [2, 11].As a numerical example, we consider an initial beam matrix with parameters of the EM-TEX in Ref. [11]. The focusing coefficients of the skew-quadrupoles are kept fixed at thevalues used to decouple the beam produced by a solenoidal field of 1.0 T, with the condi-tions of zero space-charge and zero acceleration. Figures 1(a) and 1(b) indicate that thedecoupling processes are not sensitive to the solenoidal field strength B s , particulary when B s . .
5. This tendency has been investigated in detail in Refs. [12, 40]. Therefore, for agiven skew-quadrupole triplet setting, one can obtain arbitrary emittance ratios by simplychanging the single parameter B s . Figures 1(c) and 1(d) show the effects of the space-chargeforces on the decoupling processes. If the normalized beam intensity K b S/ ǫ ⊥ (in which S is the axial periodicity length or the characteristic length of the beamline) is greater thanabout 1.0 (i.e., the space-charge force becomes comparable to or greater than the emit-tance contribution), the deviations of the rms emittances from the eigen-emittances becomesignificant and increase continuously. Conventional multi-particle tracking simulations in-cluding space-charge effects show a good agreement ( <
7% of relative errors in projectedrms eimttances) with the present KV model. Figures 1(e) and 1(f) show the effects of thebeam energy variation on the decoupling processes. The rms emittances deviate from the8 ç ç ç ç ç ç ç ç ç çá á á á á á á á á á áç ç ç ç ç ç ç ç ç ç çá á á á á á á á á á á H a L Theory ç Ε á Ε y ç Ε á Ε x Solenoidal magnetic field, B s @ T D E m itt a n ce s @ mmm r a d D æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à àæ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à H b L Simulation æ Ε à Ε y æ Ε à Ε x Solenoidal magnetic field, B s @ T D E m itt a n ce s @ mmm r a d D ç ç ç ç ç ç ç ç ç ç çá á á á á á á á á á áç ç ç ç ç ç ç ç ç ç çá á á á á á á á á á á H c L Theory ç Ε á Ε y ç Ε á Ε x Normalized beam intensity, K b S (cid:144) Ε ¦ E m itt a n ce s @ mmm r a d D æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à àæ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à H d L Simulation æ Ε à Ε y æ Ε à Ε x Normalized beam intensity, K b S (cid:144) Ε ¦ E m itt a n ce s @ mmm r a d D ç ç ç ç ç ç ç ç ç ç çá á á á á á á á á á áç ç ç ç ç ç ç ç ç ç çá á á á á á á á á á á H e L Theory ç Ε á Ε y ç Ε á Ε x @ MV D E m itt a n ce s @ mmm r a d D æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à àæ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à H f L Simulation æ Ε à Ε y æ Ε à Ε x @ MV D E m itt a n ce s @ mmm r a d D FIG. 1: Plots of the eigen-emittances (solid lines with circles) and projected rms emittances(dashed lines with squares) at the exit of the skew-quadrupole triplet. Frames (a), (c), and (e)represent the results of the present KV model analyzed by MATHEMATICA [38], and frames (b),(d), and (f) represent the results of the multi-particle tracking simulations using TRACK code [39].Frames (a) and (b) correspond to the cases with K b = 0 and zero acceleration; frames (c) and (d)to the cases with B s = 1 T and zero acceleration; frames (e) and (f) to the cases with B s = 1 Tand K b = 0, respectively. eigen-emittances when an RF voltage is applied to the acceleration gap located between thesolenoid and the skew-quadrupole triplet.In summary, we have fully generalized the KV model by including all the linear couplingelements, so that it provides a new advanced theoretical tool for the design and analysis ofcomplex beamlines with strong coupling. In the numerical example summarized in Fig. 1,we have demonstrated the usefulness and effectiveness of the new generalized KV model inunderstanding phase-space manipulations of high-intensity beams, for which previous KVmodels are inapplicable.This research was supported by the National Research Foundation of Korea (NRF-2015R1D1A1A01061074). This work was also supported by the U.S. Department of Energy9rant No. DE-AC02-09CH11466. ∗ Electronic address: [email protected] † Deceased.[1] E. D. Courant and H. S. Snyder, Annals of Physics , 1 (1958).[2] K.-J. Kim, Phys. Rev. ST Accel. Beams , 104002 (2003).[3] Y.-E. Sun, P. Piot, K.-J. Kim, N. Barov, S. Lidia, J. Santucci, R. Tikhoplav, and J. Wenner-berg, Phys. Rev. ST Accel. Beams , 123501 (2004).[4] A. Burov, S. Nagaitsev, and Y. Derbenev, Phys. Rev. E , 016503 (2002).[5] P. Piot, Y. E. Sun, and K. J. Kim, Phys. Rev. ST Accel. Beams , 031001 (2006).[6] D. Stratakis, Nucl. Instrum. Methods in Phys. Res., Sect A , 6 (2016).[7] M. Cornacchia and P. Emma, Phys. Rev. ST Accel. Beams , 084001 (2002).[8] J. Ruan, A. S. Johnson, A. H. Lumpkin, R. Thurman-Keup, H. Edwards, R. P. Fliller, T. W.Koeth, and Y.-E. Sun, Phys. Rev. Lett. , 244801 (2011).[9] P. Emma, Z.Huang, K.-J. Kim, and P. Piot, Phys. Rev. ST Accel. Beams , 100702 (2006).[10] K.-J. Kim, in Proceedings of the Particle Accelerator Conference, Albuquerque, NM (2007), p.775.[11] C. Xiao, O. K. Kester, L. Groening, H. Leibrock, M. Maier, and P. Rottl¨ a nder, Phys. Rev.ST Accel. Beams , 044201 (2013).[12] L. Groening, M. Maier, C. Xiao, L. Dahl, P. Gerhard, O. K. Kester, S. Mickat, H. Vormann,M. Vossberg, and M. Chung, Phys. Rev. Lett. , 264802 (2014).[13] A. Burov, Phys. Rev. ST Accel. Beams , 061002 (2013).[14] Y. Derbenev and R. P. Johnson, Phys. Rev. ST Accel. Beams , 041002 (2005).[15] D. Stratakis and R. B. Palmer, Phys. Rev. ST Accel. Beams , 031003 (2015).[16] L. C. Teng, Fermi National Accelerator Laboratory Report FN-229 (1971).[17] G. Ripken, Deutsches Elektronen-Synchrotron Internal Report R1-70/04 (1970).[18] A. V. Lebedev and S. A. Bogacz, JINST , P10010 (2010).[19] A. Wolski, Beam Dynamics in High Energy Particle Accelerators (Imperial College Press,London, 2014).[20] H. Qin, R. C. Davidson, M. Chung, and J. W. Burby, Phys. Rev. Lett. , 104801 (2013).
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