RIGID-ROTOR VLASOV EQUILIBRIUM FOR AN INTENSE CHARGED-PARTICLE BEAM PROPAGATING THROUGH A PERIODIC SOLENOIDAL MAGNETIC FIELD Chiping Chen and Renato Pakte Plama Science and Fuion Cente Maachuett Intitute of Technology Camidge, Maachuett 0139 Ronald C. Davidon Plama Phyic Laoatoy Pinceton Univeity Pinceton, New Jeey 08543 ABSTRACT A new igid-oto Vlaov equiliium i otained fo an intene, axiymmetic chagedpaticle eam with unifom denity in the adial diection popagating though a peiodic olenoidal focuing field. The eam envelope equation i deived, and example of peiodically focued igid-oto Vlaov equiliia ae peented. Statitical popetie and poile application of the peent eam equiliium ae alo dicued. PACS Nume: 9.7.-a, 41.75.-i, 41.85.-p
A fundamental undetanding of the kinetic equiliium and taility popetie of an intene chaged-paticle eam in peiodic electic and magnetic field i impotant to the development of advanced paticle acceleato and advanced coheent adiation ouce fo a wide ange of application [1-3]. Until thi pape, the Kapchinkij-Vladimikij (KV) equiliium [4] ha een the only known colliionle (Vlaov) equiliium fo continuou intene chaged-paticle eam popagating though eithe an altenating-gadient quadupole magnetic focuing field [4,5] o a peiodic olenoidal focuing field [5,6]. Studie of the KV eam equiliium and it taility popetie [1-7] have contiuted ignificantly to the phyic of intene chaged-paticle eam. A lage ody of liteatue exit on the Vlaov equiliium and taility popetie of otating nonneutal chaged-paticle eam popagating paallel to a unifom olenoidal focuing field B 0 e [8], whee B0 = cont., dating ack to the oiginal wok of Davidon and Kall [9]. In the peent pape, it i hown that thee exit a igid-oto Vlaov equiliium fo an intene chaged-paticle eam with unifom denity in the adial diection popagating though a peiodic olenoidal focuing field. In the peent analyi, the eam i aumed to have a unifom denity pofile in the adial diection, and a igidoto angula flow velocity in addition to a contant axial velocity β c. A pecial limiting cae, the peent analyi include oth the familia KV equiliium fo an intene eam popagating though a peiodic olenoidal focuing field [6,8], and the familia unifomdenity igid-oto Vlaov equiliium in a unifom olenoidal field [10]. The eam envelope equation i deived and ued to detemine the axial dependence of the oute eam adiu. Statitical popetie and poile application of the peent eam equiliium ae alo dicued. We conide a thin, continuou, axiymmetic ( / θ = 0 ), intene chaged-paticle eam popagating with contant axial velocity β ce though an applied peiodic olenoidal focuing field. The applied olenoidal focuing field inide the thin eam can e appoximated y ext B, = B e B e, (1) / ( ) ( ) ( ) whee = i the axial coodinate, = ( x + y ) 1/ i the adial ditance fom the eam
axi, B ( + S) = B ( ) i the axial magnetic field, S i the fundamental peiodicity length of the focuing field, and pime denote deivative with epect to. Hee, S >> i aumed, whee i the chaacteitic adiu of the oute eam envelope. To detemine the elf-electic and elf-magnetic field of the eam elf-conitently, we aume that the denity pofile of the eam i unifom, i.e., (, ) n N / π ( ) ( ), 0 <, = 0, > ( ), whee ( ) = ( + S) i the equiliium eam adiu, and N = π dn (, ) = cont. i the nume of paticle pe unit axial length. In the paaxial appoximation, the Budke paamete of the eam i aumed to e mall compaed with unity, i.e., q N / mc << 1, and the tanvee kinetic enegy of a eam paticle i aumed to e mall compaed with it axial kinetic enegy. Hee, c i the peed of light in vacuo, and q and m ae the paticle chage and et ma, epectively. Fom the equiliium Maxwell equation, we find that the elf-electic and elf-magnetic field, E e and B e, ae given y 1 qn E (, ) = β Bθ (, ) = ( ) in the eam inteio ( 0 < ). It i convenient to expe the elf field in tem of the cala and vecto potential defined fo 0 < ( ) y (, qn ) = β 1 A (, ) = ( ), (4) Φ whee A (, ) = A (, ) e, E (, ) = Φ /, and ( ) θ θ 0 () (3) Bθ, = A /. Futhemoe, we chooe the vecto potential fo the applied peiodic olenoidal field to e ext ext ext A, / B e B, = A,. ( ) = ( ) ( ) with ( ) ( ) θ It can e hown that the tanvee motion fo an individual paticle in the comined ext B, B, e E, e, ae decied y the nomalied pependicula field, ( ) + θ ( ) θ and ( ) Hamiltonian H $ = H / γ β mc [6] 3
Hee, ( x P x ) 1 K ( ) [ ( )] [ ( )] ( ) ( x y ) x y x y $,, $, $, $ $ H x y P P P y P x. = + κ + κ + (5), $ and ( y P y ), $ ae canonical conjugate pai, κ ( ) = qb ( ) / γ β mc i the nomalied Lamo fequency, K = q N / γ 3 β mc i the elf-field peveance [1], γ ( 1 β) = 1/ $ momentum P ( P$, $ x Py ) i the elativitic ma facto, and the nomalied tanvee canonical = i elated to the tanvee mechanical momentum p y $ P = mc p + qa c. It i ueful to intoduce the canonical tanfomation 1 ext ( γ β ) ( / ) fom the Cateian canonical vaiale ( x y Px Py ) ( X P P ),Y,, in the Lamo fame defined y [6] X Y x = w X coφ + wy in φ,,, $, $ to the new canonical vaiale y = w X inφ + wy co φ, 1 1 ( ) φ ( ) P$ = w P + w X co + w P + w Y in φ, x X Y (6) 1 1 ( ) φ ( ) P$ = w P + w X in + w P + w Y co φ. y X Y In Eq. (6), φ( ) = d κ ( ) i the accumulated phae of otation of the Lamo fame of 0 efeence elative to the laoatoy fame, the peiodic function w( ) = w( + S) olve the diffeential equation ( ) + K w ( ) ( ) ( ) w = κ 3 1, (7) w ( ) and pime denote deivative with epect to. The canonical tanfomation in Eq. (6) can e otained y ucceive application of the geneating function ~ ( ) = ( ) + ( in + co ) F x, y; P ~ x, P ~ ~ y, x co φ y in φ Px x φ y φ Py, (8) ( ) ( ) ( X Y X Y )( ) ~ F ~ x, ~ y ; P, P, = ~ xp + ~ yp / w + w / w ~ x + ~ y. (9) It follow fom Eq. (8) that ~ x = x coφ y inφ and ~ y = x inφ + y coφ. The Hamilton equation fo the pependicula motion in the Lamo fame can e expeed a 4
whee X ( X ) H P X = =, P w ( ) H X P = =, X w ( ) =,, and the new Hamiltonian i defined y =,Y, P ( P P ) X Y H ( X Y P P 1 ) ( ) (,, X Y P P ) X, Y, = + + X + Y. w (10) (11) Becaue A = X + P, A = Y + P, and the canonical angula momentum X X Y Y PΘ = XPY YPX ae exact ingle-paticle contant of the motion fo the Hamiltonian in Eq. (5), we define a poile choice of Vlaov equiliium ditiution function y [ X Y Y X T ] N f( X, Y, PX, PY ) = X + Y + P + P ( XP YP ) ( ), π ε δ ω ω ε 1 (1) T whee df / d = 0 = f /, ε T = cont. > 0 i an effective emittance, δ( x ) i the Diac δ -function, and the otation paamete ω = cont. i allowed to e in the ange 1 < ω < 1 fo adially confined equiliia. A hown elow, Eq. (1) i conitent with the aumed denity pofile in Eq. (). While f i defined in tem of a δ -function, it hould povide a vey good deciption of a well-matched eam equiliium in expeimental application. It i eadily hown that the eam equiliium decied y the ditiution function f in Eq. (1) ha the following tatitical popetie. Fit, the eam ha the unifom-denity pofile ( ) ( ) n, = w fdpx dpy pecied y Eq. (), povided ( ) = ε 1/ w( ). In T othe wod, the oute equiliium adiu of the eam ( ) = ( + S) oey the familia envelope equation [1,6] d ( ) ( ) ( ) d K εt + κ = ( ) ( ) 3 0. (13) Second, in dimenional unit, the aveage (macocopic) tanvee velocity of the eam equiliium decied y Eq. (1) i given in the Lamo fame y ~ ~ ( ) ~ ~ ~ V (, ) [ n (, ) w ( )] v f dp dp ( ) = X Y = 1 βce + Ω eθ, (14) ( ) 5
~ = V ~, / = ε ω β c /. Note in Eq. (14) that the aimuthal flow whee ( ) ( ) ( ) Ω θ T velocity V ( ) ~ θ, i popotional to the aveage (nomalied) canonical angula momentum in the Lamo fame, ( ) ( ) ( ) θ 1 [ ] ( ) Θ X Y εtω P $, = n, w P f dp dp = /. In the laoatoy fame, the -dependent aveage tanvee flow velocity and angula otation velocity of the eam equiliium decied y Eq. (1) can e expeed a V (, ) = [ n (, ) w ( ) ] v fdpx dpy = 1 βce + Ω ( ) eθ, (15) ( ) ( ) ε β c qb ( ) ( ) = ω, (16) ( ) γ mc Ω T epectively. Becaue the eam otate macocopically a a igid ody at a ate that i a peiodic function of the axial popagation ditance, we efe to the Vlaov equiliium decied y Eq. (1) a a peiodically focued igid-oto Vlaov equiliium. A a thid tatitical popety, the eam equiliium decied y Eq. (1) ha the effective tanvee tempeatue pofile (in dimenional unit) whee m 1 γ T (, ) = [ n (, ) w ( ) ] ( v V ) f dp X dp Y = T ( 0, ) 1, ( ) (17) mγ βc εt T ( 0, ) = ( 1 ω). ( ) Note fom Eq. (18) that the poduct T (, ) ( ) i a coneved quantity ( d d ) 0 (18) / = 0 a the eam i axially modulated. A a fouth popety, the m emittance of the eam equiliium decied y Eq. (1) i given in the Lamo fame y ( ) 1 ( ) ε ~ ~ ~~ = ~ ~ ~~ x x xx y y yy 1 = ε /, T 4 (19) / / whee tatitical aveage ae defined in the uual manne y L = N 1 ( L) f dxdydp dp. Note that the definition of ε T in Eq. (19) include X Y diected tanvee motion a well a motion elative to the mean. Defining the themal emittance of the eam y 6
it i eadily veified that ε [ ] ( β c) x ( v V ) 1 = 4 th x x ( 1 ) ε = ω ε = th T ( ) ( ) 1/, (0) T 0,. (1) mγ β c It follow fom Eq. (1) that ε = ε + ω ε, whee the ω T th T tem coepond to the aveage aimuthal motion in the Lamo fame. Making ue of Eq. (16) and (1), the envelope equation (13) can alo e expeed a d ( ) ( ) [ ( ) ( )] ( ) d Ω K εth Ω + Ωc = β c ( ) ( ) 3 0, () whee ( ) = qb ( ) / γ mc i the elativitic cycloton fequency including it ign. Ω c The peiodically focued igid-oto Vlaov equiliium ha two limiting cae which ae well known. It ecove the familia KV eam equiliium [4-6] y etting the otation paamete ω = 0. It alo ecove the familia contant-adiu, unifom-denity igidoto Vlaov equiliium [10] y taking the limit of a unifom magnetic field with B ( ) = B0 = cont. We now illutate with example of peiodically focued igid-oto Vlaov equiliia in a peiodic olenoidal focuing channel with tep-function lattice. Figue 1 how plot of the nomalied axial magnetic field (olid cuve), eam adiu (dahed cuve), and aveage angula velocity in the laoatoy fame (dotted cuve) defined in Eq. (16) veu the axial popagation ditance fo a igid-oto Vlaov equiliium in a peiodic olenoidal focuing channel defined y the ideal peiodic tep-function κ ( ) = κ 0 = cont., η / / S < η /, 0, η / / S < 1 η /. Hee, η i the o-called filling facto. In Fig. 1, the eam adiu and aveage angula velocity ae detemined uing Eq. (13) and (16), epectively, fo the choice of ytem paamete coeponding to: S vaiale, κ ( ), ( ), and ( ) and S (3) κ 0 = 316., η = 0., SK / ε T = 10, and ω = 0. 9 ; the Ω ae caled y the multiplie S 1, S, ( ) /, ε T S 1 / β c, epectively. The vacuum and pace-chage-depeed phae advance of the 7
paticle etaton ocillation aveaged ove one lattice peiod ae evaluated to e σ v = ε d / ( ) = 86. 6 o and σ = ε d / ( ) = 1. 8 o, epectively. Hee, T S 0 0 ( ) = ( + S) i the oute equiliium eam adiu when K = 0. 0 0 S T 0 To illutate the influence of otation on the peiodically focued igid-oto Vlaov KV equiliium, we plot the elative equiliium eam adiu ( 0) / ( 0) at = 0 veu the otation paamete ω in Fig., a otained y olving the envelope equation () numeically fo the cae of the tep-function lattice defined in Eq. (3). Hee, KV ( 0 ) i the oute eam adiu fo the KV equiliium ( ) ytem paamete coepond to: S ω = 0 at = 0, S, S,L. The choice of κ 0 = 316., η = 0., and SK / ε th = 10, whee ε th i the themal emittance. It i evident in Fig. that the equiliium eam adiu i a minimum when ω = 0 (tonget magnetic focuing), and inceae apidly a ω 1. Thi i ecaue the centifugal foce aociated with the eam otation i defocuing, theey eulting in a lage eam adiu. Finally, we point out poile application of the peiodically focued igid-oto Vlaov equiliium peented in thi aticle. Fit, thi equiliium in pinciple allow fo pecie matching of an intene eam into a peiodic olenoidal focuing channel in tem of detailed phae-pace ditiution. Fo example, the value of otation paamete ω can e adjuted expeimentally [11] y paing the eam though a tep in B ( ) efoe the eam ente a peiodic olenoidal focuing field. Second, the equiliium allow fo tudie of taility popetie of a cla of intene eam with equiliium ditiution othe than the KV ditiution. To ummaie, a igid-oto Vlaov equiliium ha een otained fo an intene chaged-paticle eam popagating though a peiodic olenoidal focuing field. In the peent analyi, the eam ha a unifom denity pofile in the adial diection, and a igidoto angula flow velocity in addition to a contant axial velocity. A pecial limiting cae, the peent analyi include oth the Kapchinkij-Vladimikij (KV) eam equiliium (when ω = 0) fo an intene eam popagating though a peiodic olenoidal 8
focuing field, and the unifom-denity, igid-oto Vlaov equiliium fo a eam popagating in a unifom magnetic field when B ( ) = B0 = cont. The eam envelope equation wa deived, and example of peiodically focued igid-oto Vlaov eam equiliia wee peented. Statitical popetie and poile application of the peent equiliium wee alo dicued. Study of the taility popetie of the peent equiliium i an impotant aea fo futue invetigation. 9
ACKNOWLEDGMENTS Thi eeach wa uppoted y Depatment of Enegy Gant No. DE-FG0-95ER- 40919, Contact No. DE-AC0-76-CHO-3073, and Ai Foce Office of Scientific Reeach Gant No. F4960-94-1-0374. The Reeach y R. Pakte wa alo uppoted y CAPES, Bail. 10
REFERENCES 1. R. C. Davidon, Phyic of Nonneutal Plama (Addion-Weley, Reading, Maachuett, 1990).. M. Reie, Theoy and Deign of Chaged-Paticle Beam (Wiley & Son, Inc., New Yok, 1994). 3. Space Chage Dominated Beam and Application of High Bightne Beam, edited y S. Y. Lee, Ameican Intitute of Phyic Confeence Poceeding 377 (1996). 4. I. M. Kapchinkij and V.V. Vladimikij, Poceeding of the Intenational Confeence on High Enegy Acceleato (CERN, Geneva, 1959), p. 74. 5. I. Hofmann, L. J. Lalett, L. Smith, and I. Hae, Paticle Acceleato 13, 145 (1983). 6. C. Chen and R. C. Davidon, Phy. Rev. E49, 5679 (1995). 7. R. L. Gluckten, W.-H. Cheng, and H. Ye, Phy. Rev. Lett. 75, 835 (1995). 8. See, fo example, Chapte 4 and 9 of Ref. 1. 9. R. C. Davidon and N. A. Kall, Phy. Fluid 13, 1543 (1970). 10. See p. 99 of Ref. 1. 11. A. J. Thei, R. A. Mahaffey, and A. W. Tivelpiece, Phy. Rev. Lett. 35, 1436 (1975). 11
FIGURE CAPTIONS Fig. 1 Plot of the nomalied axial magnetic field (olid cuve), eam adiu (dahed cuve), and aveage angula velocity (dotted cuve) veu the axial popagation ditance fo a peiodically focued igid-oto Vlaov equiliium in an applied magnetic field decied y the peiodic tep-function lattice in Eq. (3). Hee, the choice of ytem paamete coepond to: S SK / ε T = 10, and ω = 0. 9. The vaiale, ( ) caled y the multiplie S 1, S, ( ε T S) 1 κ 0 = 316., η = 0., κ, ( ), and ( ) ae Ω /, and S / β c, epectively. KV Fig. Plot of the elative equiliium eam adiu ( 0) / ( 0) at = 0 veu the otation paamete ω a otained fom Eq. () fo the tep-function lattice defined in Eq. (3). Hee, KV ( 0 ) i the oute eam adiu fo the KV equiliium ( ω = 0 ). The choice of ytem paamete coepond to: S κ 0 = 316., η = 0., and SK / ε th = 10, whee ε th i the themal emittance. 1
4 η κ,, Ω 0 - Field Envelope Angula Velocity -4 0.0 0.5 1.0 1.5.0 Figue 1 Chen, Phy. Rev. Lett. 13
1.0 (0) / KV (0) 1.15 1.10 1.05 1.00-1.0-0.5 0.0 0.5 1.0 ω Figue Chen, Phy. Rev. Lett. 14