4.6 Principal trajectories in terms of amplitude and phase function

Transcription

1 4.6 Principal trajectorie in term of amplitude and phae function We denote with C() and S() the coinelike and inelike trajectorie relative to the tart point = : C( ) = S( ) = C( ) = S( ) = Both can be repreented by the form (4.7) of a real trajectory C () > () aco(.(). ) bin(.(). ) with.. ( ) etc. C( ) > > aco(.. ) bin(.. ) > ain(.. ) bco(.. ). From C( ) =, C( ) = we get = a, b = a > > S() can be computed imilarly. The tranformation matrix become > (co,. = in,.) >> in,. C () S () > C () S () > (( = = )co ( == )in ) (co = in ),.,.,.,. >> > (4.4) with,. =.().( ) An intereting pecial cae i the tranformation matrix for one revolution. For implicity we tart at a ymmetry point : >, = Then co F Q > infq C S C S in FQ co FQ > (4.4a) 4.7 Tranformation through a FODO cell Mot high energy accelerator or torage ring have a periodic equence of quadrupole magnet of alternating polarity in the arc, ee Fig. 9. The dipole magnet are put in between. Here we want to dicu the optical propertie of a "FODO" cell diregarding the bending magnet (they are treated a drift pace). The more general cae will be conidered in ection 6. 5

2 Figure 9: FODO cell We work in the "thin-len" approximation, ee ection 3.3d. L/ MF M MD f f The tranformation matrix of the cell i M = M F M O M D M O L L + L f 4f M (4.43) L L L 4 f f f If we compare thi to the Twi repreentation (4.5) of the tranformation matrix over one period we obtain L co tracem 8 f co in (4.44) L in 4 f Thi equation allow one to compute the phae advance per cell from the cell length and the focal length of the quadrupole. Stability require (ee (4.3)): co < i.e. L/ 4f < Stability: f L/4 (4.45) For a phae advance of 9 per cell L f 5

3 Example: HERA proton ring L = 47 m, k (.33 m, l quad (.9 m L f ( 6.4 m, ( 9, > L / in ( 8.7m (in focuing quadrupole, ee Eq. (6.4)) 4 f The limiting cae L = 4f of (4.45) ha a imple interpretation. It i well known from optic that an object at a ditance a=f from a focuing len ha it image at b=f. A regular arrangement of focuing lene with gap L=4f in between provide thu a equence of point-to-point imaging (Fig. 3). Defocuing lene that are inerted at thee image point have no effect at all becaue they are travered on the axi. If however the len ytem i moved further apart (L > 4f), thi i no more true and the divergence of the light or particle beam i increaed by every defocuing len. A another example of the application of FODO cell, Fig. 3 illutrate the evolution of the phae pace ellipe in a very imple circular accelerator coniting of jut one FODO cell. Figure 3: Point-to-point imaging in an arrangement of focuing lene with ditance L = 4f. The defocuing lene have no effect if a point-like object i located exactly on the axi at ditance f from a focuing len. 4.8 Non-periodic beam optic In the previou ection, the Twi parameter =, >, C, have been derived for a periodic, circular accelerator. The condition of periodicity wa eential for the definition of the beta function, ee Eq. (4.7). Quite often, however, a particle beam move only once along a beam tranfer line, but one i nonethele intereted in quantitie like beam envelope and beam divergence. We will now how that the quantitie =, >, C, are ueful in beam tranfer line a well. It will turn out that the main difference i that in tranfer line the beta function i no longer uniquely determined by the tranfer matrix, but alo depend on initial condition which have to be pecified in an adequate way. There i a very imple Gedankenexperiment to how that the Twi parameter are ueful in tranfer line a well: conider a circular accelerator containing the tranfer line a a part of it lattice. Then it i obviou that the optic in the tranfer ection can be decribed in term of Twi parameter. Now one ha to realize that the Twi parameter in the tranfer ection will depend on the complete revolution matrix. Thu, ince there i large arbitrarine in how the tranfer line i complemented, the Twi parameter in the tranfer line are not at all well defined. On the other hand Eq. (4.4) how, that the Twi parameter are perfectly known in the whole tranfer line if only = and > are determined omehow at the entrance of the line. How can thi be done reaonably? 5

4 Figure 3: Phae-pace dynamic in a imple circular accelerator coniting of one FODO cell. The two 8 bending magnet are aumed to be located in the drift pace. Their weak focuing contribution i neglected. The periodicity of a, >Ci, reflected by the fact that the phae-pace ellipe i tranformed into itelf after each turn. An individual particle trajectory, however, which tart, for intance, omewhere on the ellipe at the exit of the focuing quadrupole (mall circle), i een to move on the ellipe from turn to turn a determined by the phae angle. Thu, an individual particle trajectory i not periodic, while the envelope of a whole beam i. Figure 3: A particle beam i often reaonably well decribed by a two dimenional Gauian ditribution in phae pace. The line of contant phae-pace denity are then ellipe. Since the phae-pace denity decreae only lowly with amplitude, the phae-pace area containing all particle might be hard to determine (experimentally a well a theoretically). Alo, it i not the quantity relevant for mot of the application. Therefore, the emittance i defined a /F time the phae-pace area containing a certain fraction of the particle (e.g. 9 %). 53

5 Since = and > are related to the beam divergence and beam envelope, a enible approach i to derive the initial condition of = and > from the actual phae pace ditribution of the beam entering the tranfer line. In Fig. 3 we have ketched uch a ditribution. The line of contant phae pace denity are often well fitted by properly parametrized ellipe. (An ellipe i indeed one of the mot imple parametrization one can think of; the main reaon for chooing an ellipe, however, will readily be een.) /F time the area of the ellipe containing a certain fraction of the particle (ay 9 %) i called the beam emittance. The mot general ellipe parametrization in phae pace i given by (y = x or z) Cy + =y y+ >y = a (4.46) Since only three free parameter are needed, we can impoe another condition: >C = = (4.47) With thi normalization of the parameter =, >, C, the parameter a ha the imple meaning that Fa i the ellipe area, thu a = A = emittance (4.48) It i obviou from Eq. ( ) that our beam ellipe parameter =, >, C, a atify the ame equation a the Twi parameter defined for a circular accelerator, ee Eq. (4.8, 4.3, 4.3). In a circular accelerator, however, =, >, C are completely determined by the magnet optic and the condition of periodicity while beam propertie are not at all involved (only A i choen to fit the actual beam ize). In a tranfer line optic, on the other hand, we can chooe >, >, A at the entrance to fit bet the incoming beam. From then on the optic calculation proceed the ame way a in the circular accelerator: Linear optic perform a linear tranformation of phae-pace coordinate, y M( / ) y C S y y y C S y (4.49) The beam ellipe at the entrance i therefore tranformed into another ellipe at : C = > C = > (4.5) y yy y a y yy y a Uing Eq. (4.49) thi can be written in the well known form (ee Eq. (4.4)) > C SC S > = CC SCSC SS = C C S C S C (4.5) Uing det M = it i eay to prove that >C = 54

6 a i.e. F till mean the phae-pace area. By virtue of Eq. (4.5) a A a A i.e. beam emittance i preerved. Compared to the previou ection, there i nothing new with Eq. ( ). It i jut the interpretation of the Twi parameter which make a difference. 5 MOTION OF PARTICLES WITH MOMENTUM DEVIATION The central deign orbit of a circular accelerator i a cloed curve that goe through the center of all quadrupole (auming that the magnet are well aligned). Thi orbit i a poible particle trajectory: particle with nominal momentum p tarting at ome point with zero diplacement and zero lope will move on the deign orbit for an arbitrary number of revolution. However, particle with p = p tarting with non-vanihing initial condition will conduct betatron ocillation about the orbit. Their path around the ring doe not cloe onto itelf ince the Q- value i non-integer. Now we conider a particle with a larger momentum p > p. The deign orbit i no more a poible trajectory for thi particle. Thi i eay to undertand for a weakly focuing machine (or in a homogeneou field): particle with larger momentum need a circle of larger radiu on which they can move indefinitely (ee Fig. 33). Figure 33: Cloed orbit for particle with momentum p p in a weakly (a) and trongly (b) focuing circular accelerator. If they do not tart on exactly that circle, they will perform betatron ocillation about thi new, larger circle which therefore i the reference orbit for the particle with the given momentum deviation dp = p p. Since it radial ditance x(,dp/p ) from the deign orbit i proportional to 55

7 dp/p, it i practical to divide by the momentum deviation and define the cloed diperion orbit D() by the equation dp dp x (, ) D () p p In a machine with weak focuing, D() i of coure a contant. In trongly focuing machine uch an orbit exit, too, but it look a lot more complicated. The focuing quadrupole do not permit the particle to deviate too much from the central orbit and bend the trajectory toward the central orbit wherea the defocuing quadrupole bend it away. It i then eay to undertand that the cloed diperion orbit ha it maximum deviation from the deign orbit in the center of the focuing quadrupole and it minimum deviation in the defocuing quadrupole. Thi i ketched in Fig. 33 b. In the following we want to derive the mathematical expreion for the cloed diperion orbit. 5. Cloed orbit for, p A particle with,p = p p atifie the inhomogeneou Hill equation for the horizontal motion, p x K() x (5.) H p The total deviation of the particle from the reference orbit of the machine can be written a x() = x D () + x > () (5.), p Here xd () D() decribe the deviation of the cloed orbit for off-momentum particle p with a fixed,p from the reference orbit; x > () decribe the betatron ocillation around thi cloed diperion orbit. D() i the "periodic diperion". It atifie jut the ame differential equation a the diperion trajectory in ection 3.: D" + K() D= H() (5.3) But now we impoe periodic boundary condition D(+C) = D(), D(+C) = D() C = N L i the circumference of the machine. The periodic diperion i often denoted a D(). In general, alo the equation for the vertical coordinate z contain a nonvanihing right hand ide, e.g. due to field and alignment error in the magnet. We therefore conider more generally the equation 56

8 y" + K() y = F() (5.5) K() and /H() have the period L but in general F() ha only the period C = NL, e.g. if we have one magnet error. We look for a periodic olution of the inhomogeneou equation (5.5). The general olution i y() = a C() + b S() + u() (5.6) u() i a pecial olution of the inhomogeneou equation. It i given by (cf. (3.)) u () S () FtCtdt () () C () FtStdt () () (5.7) The reference point = i arbitrary; C() and S() are the coinelike and inelike trajectorie referred to thi point. Our goal i to find a periodic olution of the type (5.6) Y(+C) = Y(), Y(+C) = Y() (5.8) Since i arbitrary we can evaluate the condition (5.8) at = to compute the unknown contant a and b. ac + b + u = ac + b + u ac + b + u = ac + b + u Here c = C( ) =, c = C( ) = c = C( + C), c = C( + C) u =, u =, u = u( + C) etc. We obtain u ( ) u a ( c )( ) c The denominator i ( cc) ( c ) det M " trace "! M 4in ( F Q) cofq C ( C) S ( C) Here M = C ( C) S( C) The numerator i i the tranformation matrix for one revolution. 57

9 ( ) FC c FS FC c FS C abbreviation: FS F() t S() t dt. ( ) " "! det M Num c c FS c FS FC (ee Eq.(4.4)) c = C( + C) = cofq += infq = > infq C. co( F ) = in( F ) > > ( ) ( )in,.( ) Num Q Q t F t t dt C in( ) ( ) ( ) co ( ) in ( ) > F Q > t F t,. t =,. t dt C in( ) ( ) ( )co ( ) > F Q > t F t. t. FQ dt Now a = Y( ). The point wa choen arbitrarily. Therefore we get for the cloed trajectory > () C Y( ) > ( t) F( t)co ( t) ( ) FQdt inf Q.. > () > ( tft ) ( )co. ( t). ( ) FQdt inf Q (5.9) Note that in the firt equation t > i required, i.e. the integration ha to tart at, while in the econd one thi retriction doe not apply., p The cloed periodic diperion function i obtained for Ft () : H() t p () () t D () > > D() co () t () FQdt in FQ H( t).. (5.) Thee equation exhibit an eential intability of a circular accelerator: a finite diperion exit only if the number of betatron ocillation per revolution Q i different from an integer. Conider an integer Q and look at a certain dipole. A particle with,p will receive a different kick angle than the reference particle with p = p. Since Q i integer, thee angular deviation add up coherently from turn to turn, and oon the particle hit the vacuum chamber. While Eq. (5.9) and (5.) are very ueful for undertanding the overall ocillatory behaviour of cloed orbit in the preence of dipole error or momentum error, numerical calculation will be, p eaier if matrix formalim i applied. Thi i provided by Eq. (3.), ince xd () D() i a p poible particle trajectory (we ue now D, D for the diperion trajectory part of the tranfer matrix and D, D for the periodic diperion). 58

10 , p, p D D p p C S D, p, p D C S D D p p, p, p p p or D C S D D D C S D D Thi mean: D and D are eigenvector component of the revolution matrix M for the eigenvalue. (We leave the proof that the 3 3 matrix M alway ha an eigenvalue a an exercie.) Explicitly, the lat equation implie D ( S) DSD ( S) DSD C S 4in FQ CD ( C) D D 4in FQ Again it i een that integer Q value are to be avoided. Once D and D have been determined at one point, the value at any other point are calculated by a imple matrix multiplication: D D D M( / ) D Momentum compaction A particle with,p/p > travel on a revolution a longer ditance than the reference particle (,p = ). Conider a particle moving on the cloed diperion trajectory., p xd () D() p The circumference for thi particle i 59

11 C = xd () d H C +,C, C, p = # C p = D ( ) d C H() # The quantity = i the relative change in orbit length divided by the relative momentum deviation. It i called "momentum compaction factor", which i a rather mileading notion. A rough etimate in term of the horizontal Q-value i given by = Q x 5. Diperion in tranfer line In tranfer line, the diperion function i alo derived from Eq. (5.6) and (5.7), but periodicity cannot be aumed anymore. Intead, initial condition are derived from the beam propertie at the entrance of the tranfer line. If here a ignificant correlation exit between the phae pace coordinate of the particle and their repective momenta, appropriate initial condition pi i i p p p p i i i D x ; D = x ; where < > i denote averaging over the whole enemble of particle at the entrance. The diperion a a function of the poition i then D() = D C() + D S() + S() C(t) dt C() H() t i S(t) dt (5.3) H() t If no uch correlation i known at the entrance, one uually chooe D = D =, ee Eq. (3.). A tranfer line i called non-diperive if Ct () St () D = D =, dt H() t dt H() t It i intructive to conider alo the generalization of the momentum compaction factor. The relative change in orbit length per relative momentum deviation i given by, L/ L Dt ( ) = (, ) dt, p/ p L with L = H( t) dt Finally we conider the relative change in time of flight per relative momentum deviation: L K, t/ t p, K = (, ) = (, ), p/ p t, p c C 6

12 Thi quantity i ometime denoted by D(, ). If D(, ) =, the tranfer line i called iochronou, becaue then the time of flight doe not depend on momentum. For ultrarelativitic particle ( ), there i no difference between = and D. C A more trict definition of a tranfer line to be iochronou would include not only the contribution of off-energy particle to the time of flight but alo the contribution of betatronocillation. The condition that the time of flight i independent of energy and initial condition require, in addition to D Ct () St (), alo dt, dt H() t H() t. In torage ring, the quantity C define the tranition energy C, i. e. the energy = for which Dbecome zero and longitudinal focuing i lot. tr mc tr 5.3 Influence of field error In thi ection we want to touch very briefly the effect of a dipole or quadrupole error. a) Dipole error If B z = B +,B or B x =,B, we have an additional, -dependent Lorentz force. Thi lead again to an equation of the type (5.5). The periodic cloed orbit i given by (5.9) with e Ft (), Bt (). In order to have bounded motion the Q value mut be non integer, Q n. We p ee that even for particle with reference momentum p an integer Q value i forbidden, ince mall field error are alway preent. The main ource of orbit error in accelerator i the diplacement of quadrupole from the deign orbit. Correction dipole are needed to correct the orbit. Eq. (5.9) tell u that their effect i larget if they are placed at point with a large beta function. So a horizontally deflecting correction dipole hould be placed cloe to a horizontally focuing quadrupole and a vertically deflecting correction dipole cloe to a vertically focuing quadrupole. b) Quadrupole error Let K () be the deign quadrupole trength,,k() the error. For a particle with,p = we have y" + (K () +,K()) y = Call M the tranformation matrix for a revolution in the unditurbed machine M = I co + J in = FQ Suppoe the error occur only at = over a hort length d and let M denote the tranformation matrix for the diturbed machine. 6

13 We get M = mm M Here m i the matrix of the ection of length d in the unditurbed cae, m the matrix in the diturbed cae m K ( ) d m K ( ), K( ) d Simple matrix algebra yield mm, thu, uing Eq. (4.5) :, K( ) d = > M co + in, Kd a, Kd C >, Kd - = co = tracem = co >,Kd in If the perturbation i ditributed around the ring, we get in,(co ) co co > ( ), Kd ( ) Now, for mall,, co co(,) co co, in in, co,in, thu,,(co ) co co, in or, Q = F Fin A gradient error therefore lead to a hift in the Q value, Q > () () 4F, K d (5.4) The Q hift i proportional to both the magnitude of the gradient error and the beta function at the location of the error. A gradient error change the beta function itelf. Without proof we tate the reult > (),>() = ßt () in( FQ),K(t) co (.(t).() FQ) dt (5.5) Again the change of the beta function i proportional to the magnitude of the perturbation and the amplitude of the beta function itelf at the point of the perturbation. Gradient error in the interaction region quadrupole are therefore mot dangerou. The econd important reult i that,> remain only finite if Q i different from a half-integer number (in(fq) in the denominator). The obervation that dipole error lead to integer reonance, quadrupole error to reonance at half-integer Q value indicate that extupole field excite reonance at third-integer Q value. Thi i in fact the cae but not the ubject of the preent introductory coure. Moreover, one ha Reonance are dicued in detail in E. Wilon' lecture at thi and the previou accelerator chool (ee e.g. the proceeding of the 984 accelerator chool, CERN 85-9). 6

14 in general a coupling between horizontal and vertical betatron ocillation due to extupole or mialignment of magnet (kew quadrupole field, ee the table at the end of ection f) etc. Thi lead to a more general condition mq x + nq z l (5.6) (m, n, l integer number; m, n mall) The working point (Q x, Q z) ha to be choen in a reaonable ditance from the reonance line (Fig. 34) Figure 34: Reonance diagram up to fourth order. Some of the reonance line have been identified explicitly. 5.4 Chromaticity Particle with,p are focued differently in the quadrupole. Thi lead to a hift of the Q value. From eg K p we get dk eg, p, p, K, p K dp p p p Formally thi i the ame a a gradient error. We can therefore ue (5.4) to calculate the Q hift 63

15 Q p p () () 4 K d,,, > N F p p N > () K ()d 4F #% N i called the chromaticity of the machine. For a linear magnet lattice it i alway negative. The main contribution to the chromaticity come from quadrupole which are trongly excited and where the > function i large (e.g. interaction region quadrupole). In big accelerator the chromaticity ariing from the linear lattice (alo called the "linear" or "natural" chromaticity) i a large quantity (e.g. N 6 in the HERA torage ring). Then, the "tune" pread due to the finite momentum band become o large that ome part of the beam unavoidably hit dangerou reonance line. In the HERA cae, for intance, a beam with a relative momentum pread of only 3 would cover a tune range of.! For thi reaon, and in order to avoid the o-called "head-tail" intability, one ha to compenate the chromaticity. Thi can be achieved with extupole. The extupole magnet have to be placed at location where the cloed diperion orbit D() i nonzero, (Fig. 35). Figure 35: Diperion trajectory in a extupole magnet Conider a particle with,p moving without betatron ocillation on the cloed diperion trajectory., p xd () D() p Now put a extupole magnet at a place with D(). The extupole field at x = x D in the horizontal plane z = i B z = gx (5.8) 64