EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH AN INTRODUCTION TO TRANSVERSE BEAM DYNAMICS IN ACCELERATORS. M. Martini. Abstract

Transcription

1 EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN/PS 96{ (PA) March 996 AN INTRODUCTION TO TRANSVERSE BEAM DYNAMICS IN ACCELERATORS M. Martini Abstract This text represents an attempt to give a comprehensive introduction to the dynamics of charged particles in accelerator and beam transport lattices. The rst part treats the basic principles of linear single particle dynamics in the transverse phase plane. The general equations of motion of a charged particle in magnetic elds are derived. Next, the linearized equations of motion in the presence of bending and focusing elds only are solved. This yields the optical parameters of the lattice, useful in strong focusing machines, from which the oscillatory motion the betatron oscillation and also the transfer matrices of the lattice elements are expressed. The second part deals with some of the ideas connected with nonlinearities and resonances in accelerator physics. The equations of motion are revisited, considering the inuence of nonlinear magnetic elds as pure multipole magnet elds. A presentation of basic transverse resonance phenomena caused by multipole perturbation terms is given, followed by a semi-quantitative discussion of the third-integer resonance. Finally, chromatic eects in circular accelerators are described and a simple chromaticity correction scheme is given. The beam dynamics is presented on an introductory level, within the framework of dierential equations and using only straightforward perturbation methods, discarding the formulation in terms of Hamiltonian formalism. Lectures given at the Joint Universities Accelerator School, Archamps, 8 January to March 996

2 Contents PARTICLE MOTION IN MAGNETIC FIELDS. Coordinate system. Linearized equations of motion 6.3 Weak and strong focusing 8 LINEAR BEAM OPTICS 3. Betatron functions for periodic closed lattices 3. Hill's equation with piecewise periodic constant coecients 9.3 Emittance and beam envelope.4 Betatron functions for beam transport lattices 4.5 Dispersive periodic closed lattices 7.6 Dispersive beam transport lattices 3 3 MULTIPOLE FIELD EXPANSION General multipole eld components Pole prole 4 4 TRANSVERSE RESONANCES Nonlinear equations of motion Description of motion in normalized coordinates One-dimensional resonances Coupling resonances 53 5 THE THIRD-INTEGER RESONANCE The averaging method The nonlinear sextupole resonance Numerical experiment: sextupolar kick 68 6 CHROMATICITY Chromaticity eect in a closed lattice The natural chromaticity ofafodo cell Chromaticity correction 8 APPENDIX A: HILL'S EQUATION 84 A. Linear equations 84 A. Equations with periodic coecients (Floquet theory) 86 A.3 Stability of solutions 89 BIBLIOGRAPHY 9 iii

3 PARTICLE MOTION IN MAGNETIC FIELDS. Coordinate system The motion of a charged particle in a beam transport channel or in a circular accelerator is governed by the Lorentz force equation F = e(e + v B) ; (.) where E and B are the electric and magnetic elds, v is the particle velocity, and e is the electric charge of the particle. The Lorentz forces are applied as bending forces to guide the particles along a predened ideal path, the design orbit, on which ideally all particles should move, and as focusing forces to conne the particles in the vicinity of the ideal path, from which most particles will unavoidably deviate. The motion of particle beams under the inuence of these Lorentz forces is called beam optics. The design orbit can be described using a xed, right-handed Cartesian reference system. However, using such a reference system it is dicult to express deviations of individual particle trajectories from the design orbit. Instead, we will use a right-handed orthogonal system (n; b; t) that follows an ideal particle traveling along the design orbit. The variables s and describe the ideal beam path and an individual particle trajectory. We have chosen the convention that n is directed outward if the motion lies in the horizontal plane, and upwards if it lies in the vertical plane. Let r(s) be the deviation of the particle trajectory r(s) from the design orbit r (s). Assume that the design orbit is made of piecewise at curves, which can be either in the horizontal or vertical plane so that it has no torsion. Hence, from the Frenet{Serret formulae we nd dr ds = t dt ds = k(s)n db ds = dn ds = k(s)t ; (.) where k(s) is the curvature, t is the target unit vector, n the normal unit vector and b the binomial vector: b = t n. b n individual particle trajectory δr t σ r (s) design orbit r (s) motion in horizontal plane s Figure : Coordinate system (n; b; t).

4 Unfortunately, n charges discontinuously if the design orbit jumps from the horizontal to the vertical plane, and vice versa. Therefore, we instead introduce the new right-handed coordinate system (e x ; e y ; t): ( n if the orbit lies in the horizontal plane e x = b if the orbit lies in the vertical plane ( b if the orbit lies in the horizontal plane e y = n if the orbit lies in the vertical plane: Thus: de x ds = k x t de y ds = k yt The last equation stands, provided we assume dt ds = k xe x k y e y : (.3) k x (s)k y (s) =; (.4) where k x ;k y are the curvatures in the x-direction and the y-direction. Hence the individual particle trajectory reads: r(x; y; s) =r (s)+xe x (s)+ye y (s); (.5) where (x; y; s) are the particle coordinates in the new reference system. e y σ e x x y r (s) t design orbit r (s) s Figure : Coordinate system (e x ; e y ; t). Consider the length s of the ideal beam path as the independent variable, instead of the time variable t, to express the Lorentz equation (.). Now, from d dt = d dt d d = v ds d d ds = v d ds ;

5 where v = d=dt is the particle velocity and a prime denotes the derivative with respect to s, it follows that dr dt = v dr ds = v r v d r = d dr = v d dt dt dt v r = v v ds r r = v r r d ds ( ) = v r r : Now, from (.3) and (.5), r (s) = r (s)+x e x +y e y +xe x+ye y = t+x e x +y e y +xk x t+yk y t = ( + k x x + k y y)t + x e x + y e y : Similarly, we compute the second derivative: Then, r (s) = (k x x + k x x + k yy + k y y)t + x e x + y e y + x e x + y e y +( + k x x + k y y)t = (k x x + k x x + k yy + k y y)t + x e x + y e y + x k x t + y k y t +( + k x x + k y y)( k x e x k y e y ) r (s) = (k x x + k y y +k xx +k y y )t +[x k x ( + k x x)]e x +[y k y ( + k y y)]e y : The Lorentz force F may be expressed by the change in the particle momentum p as with F = dp dt = e(e + v B) ; (.6) p = mv ; (.7) where m is the particle rest mass and the ratio of the particle energy to its rest energy. Assuming that and v are constants (no particle acceleration), the left-hand side of (.6) can be written as F = m dv dt = m d r v = m r ( dt r = m v (k x x + k y y +k xx +k y y )t+[x k x ( + k x x)]e x +[y k y ( + k y y)]e y [( + k xx + k y y)t + x e x + y e y ] = m v k x x + k y y +k xx +k y y ( + k xx + k y y) t + x k x ( + k x x) x e x + y k y ( + k y y) y e y : ) 3

6 The magnetic eld may be expressed in the (x; y; s) reference system B(x; y; s) =B t (x; y; s)t + B x (x; y; s)e x + B y (x; y; s)e y : (.8) In the absence of an electric eld the Lorentz force equation (.) becomes Using the above results and dening the variable the vector product may be written as F = e(v B) = ev (r B) : (.9) h =+k x x+k y y; (.) r B = (ht + x e x + y e y ) (B t t + B x e x + B y e y ) = (y B t hb y )e x (xb t hb x )e y +(x B y y B x )t: Finally, equating the expression for the rate of change in the particle momentum, the left-hand side of (.6) with (.9) and with (.4) gives since (k x x + k y y +k xx +k y y + y k y h y e y h)t + x k x h x e x = e p [(x B y y B x )t (hb y y B t )e x +(hb x x B t )e y ] ; k x ( + k x x)=k x h and k y ( + k y y)=k y h: Identifying the terms in e x ; e y, and t leads to the equations of motion x x = k x h e p (hb y y B t ); y y = k y h + e p (hb x x B t ); = h (k x x + k y y +k xx +k y y ) e hp (x B y y B x ) : (.) The last equation may be used to eliminate the term = in the other two. Expanding the particle momentum in the vicinity of the ideal momentum p, corresponding to a particle travelling on the design orbit, yields 4 p = p ( + ) ;

7 where = p p p p p (.) is the relative momentum deviation. Hence, the above general equations of motion for a charged particle in a magnetic eld B may be rewritten as: x x = k x h ( + ) e p (hb y y B t ) ; Using y y = k y h +(+) e p (hb x x B t ) = h (k x x + k y y +k xx +k y y ) ( + ) e hp (x B y y B x ) : (.3) v = v r ; it follows that then jvj = v = v jr j = v pr r ; = jr j = q h + x + y : (.4) On the design orbit (equilibrium orbit) we get x = x = y=y = =: Consequently h =; =; = =, and using (.3) we nd k x = e p B y (; ;s) e p B y ; k y = e p B x (; ;s) e p B x : (.5) Equivalently, introducing the local bending radii x and y, x;y (s) = k x;y (s) ; (.6) we get the bending eld for the design momentum p = e B y = e B x : (.7) x p y p We adopt the following sign convention: an observer looking in the positive s-direction sees a positive charge travelling along the positive s-direction deected to the right (resp. 5

8 upwards) by a positive vertical magnetic eld B y magnetic eld B x > ). > (resp. by a positive horizontal B bend magnet v design orbit s F ρ(s) Figure 3: Lorentz force (for a positive charged particle). The term jb j is the beam rigidity (B and stand for B x and x or B y more practical units the beam rigidity reads: and y ). In B (tesla m) =3:3356 p (GeV=c) =3:3356 E (GeV) ; where E is the particle total energy, p the particle momentum, and = v=c.. Linearized equations of motion From the general equations of motion (.3) we retain only linear terms in x; x ;y;y and. Starting with (.4) we get s Hence: = h + x h +y h h=+k xx+k y y h = k x x + k xx + k y y + k yy : h With this approximation and with the rst-order series expansion ( + ), the equations of motion (.3) in the horizontal and vertical planes become, in the absence of a tangential magnetic eld (no solenoid eld), x = k x h ( )h e p B y (x; y; s) ; y = k y h +( )h e p B x (x; y; s) : (.8) 6

9 For x and y, small deviations from the design orbit, the eld components may be expanded in series to the rst order: B x (x; y; s) = B x x y; The magnetic eld satises the Maxwell equations from which we get, using (.8), B y (x; y; s) = B y x y y: (.9) rb= rb y : (.) Thus, introducing the normalized gradient K, and the skew normalized gradient K dened as K = y K = x ; (.) x=y= the eld components may be written with (.5) Hence, the transverse equations of motion become x=y= e p B x (x; y; s) = k y + K x + K y; e p B y (x; y; s) = k x + K x K y: (.) x = k x h ( )h (k x + K x K y) ; y = k y h ( )h (k y K x K y) : (.3) From (.4) and (.) we compute to the rst order in x; y;, h +k x x+k y y k u h=k u +k u u ( )h k u k u +k u u k u ( )h K u K u and ( )h K u K u; where u stands for x or y. Substituting these approximations into (.3) and using the radius x;y instead of the curvature k x;y we obtain the linearized equations of motion x + K + x x K y = x ; 7

10 y K y K x = : (.4) y y The term K introduces a linear coupling into the equations of motion. If we restrict this to magnetic elds which does not introduce any coupling, we have to set K = (no skew linear magnets). Then the equations of motion read: The terms K and x;y x + K + x y K y x = x ; y = y : (.5) in the above expressions represent the gradient focusing and weak sector magnet focusing, respectively. When the deection occurs only in the horizontal plane, which is the usual case for synchrotrons, the equation of motion in the vertical plane simplies to y K y =: (.6) The magnet parameters x;y ; K and K are functions of the s-coordinate. In practice, they have zero values in magnet-free sections and assume constant values within the magnets. The arrangement of magnets in a beam transport channel or in a circular accelerator is called a lattice..3 Weak and strong focusing Consider the simple case at a circular accelerator with a bending magnet which deects only in the horizontal plane and whose eld does not change =: The design orbit is then a circle of constant radius x, with the constant magnetic eld B y on this circle evaluated for the design momentum p = eb y x : (.7) B y positive particle with momentum p e x e y ρ x v s design orbit (circle) ρ x = constant Figure 4: Circular design orbit in the horizontal plane. 8

11 For a particle with the design momentum the equations of motion read: x + K + x x =; y K y =; (.8) where K is the constant normalized gradient (.). The stability in the horizontal motion is achieved if the solution is oscillatory, that is: K + x > : Vertically, the stability is achieved if K < : Hence, the stability is achieved in both planes provided < K < x : (.9) Using (.7) and (.) the latter inequalities write: < B x=y= < x : (.3) This condition is called weak- or constant-gradient focusing. It allows stable motion in both planes. The oscillatory solution of (.8), called betatron oscillations, is u(s) =a cos ( p Ks ') ; (.3) where u stands for x or y; K = x +K or K = K, and where a and ' are integration constants. The wavelength of the betatron oscillation is = p K : (.3) The number of betatron oscillations performed by particles around a machine circumference C is called the tune Q of the circular accelerator (Q stands for Q x or Q y ): Q = C = xp K: (.33) The weak focusing condition (.9) show that K<x, that is: Q<: 9

12 IIIIII The betatron oscillation wavelength is larger than the machine circumference. This means that the amplitude of the oscillations may become very large as the size x of the machine increases and hence the magnet aperture may also be very big. This yields a practical limit on the size of weak focusing accelerators. Historically, the eld index n was introduced instead of the normalized gradient K for the weak focusing accelerators n = x B x=y= The weak focusing condition (.3) is then expressed as Using (.33) and (.34) the machine tune may be written as : (.34) <n<: (.35) Q x = p n Q y = p n: (.36) The combination of bending and focusing forces required for weak focusing accelerators may be obtained by the magnetic eld shape called synchrotron magnet or combined dipole-quadrupole magnet. ρ x y S-pole IIIIIIIIIIIIIIII B x F x B y ring centre design orbit IIIIII B IIIIIIIIIIIIIIII N-pole F x B x F y B y F y x Figure 5: Synchrotron magnet (positive particles approach the reader). The limitations at weak focusing accelerators have been overcome by the invention at the strong- or alternating-gradient focusing principle. This method amounts to splitting up the machine into an alternate sequence of strongly horizontally focusing (n or K x ) and strongly horizontally defocusing (n ork x ) magnets. By (.36) this implies that the tunes Q x and Q y may become arbitrarily large. This means that the betatron amplitudes can be kept small for a given angular deection as the size x increases, and the magnet aperture may be reduced.

13 n>> n<< n<< design orbit for p e x n>> s Figure 6: Strong focusing machine. The bending and focusing forces for strong focusing accelerators may be achieved either within synchrotron magnets or in a separate bending magnet, called a dipole, and a focusing magnet, called a quadrupole. The quadrupole magnet provides focusing forces through its normalized gradient, given by (.) and (.): e p B x = K y e p B y = K x: (.37) y N-pole I I I I I I I I I I I I I F y S-pole I I I I I I I I I I I I I S-pole F x I I I I I I I I I I I I I F x B F y N-pole I I I I I I I I I I I I I x Figure 7: Horizontally focusing, F -type quadrupole magnet (positive particles approach the reader). A quadrupole that focuses in one plane defocuses in the other plane. The horizontally defocusing, D-type quadrupole is obtained by permuting the N- and S-poles of an F - quadrupole.

14 In geometrical optics the focal length f of a lens is related to the angle of deection of the lens by tan = x f : x θ < focal point s f Figure 8: Principle of focusing for light. It is known that a pair of glass lenses, one focusing with focal length f F >, and the other defocusing with focal length f D <, separated by a distance d, yields a total focal length f given by f = + d : (.38) f F f D f F f D This lens doublet is focusing if f D = f F. Hence, f = d f F : (.39) The strong focusing scheme is based upon the quadrupole doublet arrangement, which becomes a system focusing in both planes. f F > f D < focal point s principal plane d focal length f Figure 9: Principle of strong focusing for light. The linear equations of motion (.5) for strong focusing lattices (where the bending and focusing forces depend on s) may be written as u + K(s)u =; (.4)

15 where u stands for either the horizontal x or the vertical y coordinates, and where the bending and focusing functions are combined in one function (plus sign for x, minus sign for y): K(s) =K (s)+ x;y(s) : (.4) For synchrotron magnets both terms in K(s) are non-zero, while for separated function magnets (separate dipoles and quadrupoles) either K (s) or x;y (s) is set to zero. Integrating (.4) over a short distance, `, where K(s) constant) we nd in the paraxial approximation, where the defection angle is equal to the slope of the particle trajectory, and Therefore by (.4) Z ` u ds = u (`) Z ` u () = tan K(s)uds Ku` : tan K`u : By analogy with the expression at the focal length of a glass lens, we dened the focal length of a quadrupole as tan = u f : (.4) Hence we get for a thin quadrupole of length ` [for K(s) =K >] f = K `: (.43) with f positive in the focusing plane and negative in the defocusing plane. LINEAR BEAM OPTICS. Betatron functions for periodic closed lattices In the case where the bending magnets of a circular accelerator deect only in the horizontal plane, the linear unperturbed equations of motion for a particle having the design momentum p (i.e. = ) are, according to (.5), x + K + x = x y K y =; (.) where K (s) and x (s) are periodic functions of the s-coordinate due to the orbit being a closed curve. The period L may be the accelerator circumference C or the length of a 3

16 \cell" repeated N times around the circumference: C = NL. Both the above equations may be cast in the form u + K(s)u =; (.) with K(s + L) =K(s); (.3) where u stands for x or y and where K(s) =K (s)+ x (s)ork(s)= K (s). Equation (.) with periodic coecient K(s) is called Hill's equation. It has a pair of independent stable solutions of the form (see appendix A) u (s) = w (s)e i(s) ; such that w (s) and w (s) are periodic with period L, and (s) is a function such that u (s) = w (s)e i(s) ; (.4) w i (s + L) =w i (s) i=;; (.5) (s + L) (s) =; (.6) where is called the characteristic exponent of Hill's equation dened by cos = Tr[M(s+L=s)] ; (.7) and is independent of the length s. The matrix M(s + L=s) is called the transfer matrix over one period L [shortly written as M(s) dened by (see A.9)] M(s) M(s + L=s) =M(s+L=s )M(s=s ) ; in which M(s=s ) is called the transfer matrix between the reference point s and s, given by (see A.6) C(s) S(s) M(s=s )= C (s) S ; (s) where C(s) and S(s) are two independent solutions of Hill's equation called cosine-like and sine-line solutions, which satisfy the particular initial conditions (see A.4) C(s )= C (s )= S(s )= S (s )=: As the functions e i(s) and e i(s) are already linearly independent, we can arbitrarily make w (s) identical to w (s) so that the pair at independent stable solutions (.4) now read: u ; (s) =w(s)e i(s) ; (.8) 4

17 where w(s) is periodic with period L. Dierentiating the latter equation twice, u ;(s) = w (s)e i(s) i (s)w(s)e i(s) u ; = [w (s) i (s)w (s) i (s)w(s) (s) w(s) i (s)w (s)]e i(s) = [w (s) i (s)w (s) (s) w(s) i (s)w(s)]e i(s) ; we obtain by substitution into Hill's equation and cancelling e i(s) w w + K(s)w i( w + w)=: Equating real and imaginary parts to zero yields w w + K(s)w = (.9) and w + w =; or equivalently w w The last equation may be rewritten as + =: (.) [ln w(s) ] + [ln (s)] =; which can be integrated to give ln (s) =ln +ln c: w(s) Hence, choosing the integration constant c equal to unity which yields on integration (s) = w(s) ; (.) (s) = Z s s dt w(t) : (.) Substituting (.) into (.9) gives a new dierential equation for w(s): Dening the so-called betatron function (s) as w + K(s)w =: (.3) 3 w (s)=w (s); (.4) 5

18 equations (.) and (.3) transform into (s) = Z s s dt (t) (.5) and The latter expression may be further transformed to give since = + K(s) = 3= =: (.6) 4 + K(s) =; (.7) 3= = = 4 : The function (s) given by (.5) is called the phase function. Then, the two independent solutions of Hill's equations become q u ; (s) = (s)e i(s) : (.8) Every solution of Hill's equation is a linear combination of these two solutions: q u(s) =c (s)e i(s) + c q(s)e i(s) ; where c and c are constants, or equivalently with u(s) =a a = p c c q (s) cos [(s) '] ; (.9) c c tan ' = i c + c Any solution (s) that satises (.7) together with the phase function (s), whose derivative is(s), can be used to make (.9) a real solution of Hill's equation. Such a solution is a pseudo-harmonic oscillation with varying amplitude and frequency, called betatron oscillation. From (.5) we compute: and using (.6) we nd: (s + L) (s) = = Z s+l s Z s+l s dt (t) ; : dt (t) : (.) Thus the characteristic exponent may be identied with the phase advance per period or cell (of length L). 6

19 The oscillation's local frequency f(s) and wavelength (s) can be related to the phase function by (s) =f(s) = (s) : (.) Thus, the betatron function may be interpreted as the local wavelength divided by since (s) =(s) : (s)=(s) : (.) Let the tune Q of a circular accelerator be dened as the number of betatron oscillations executed by particles travelling once around the machine circumference C. Since the local frequency f(s) denotes the number of oscillations per unit of length, the tune reads: Q = Z s+c s f(t)dt : (.3) If the accelerator has N cells of period L (i.e., C = NL), then using (.) to (.3), Q is given by Q = N = Z s+c dt s (t) ; (.4) since (s) is a periodic function of period L. Any solution of Hill's equation can be described in terms of the betatron oscillation (.9). In particular, the cosine-like solution C(s) and the sine-like solution S(s) maybe represented by (.9); q C(s) =a (s) cos [(s) '] ; (.5) where a and ' are constants which depend upon the initial conditions at the reference point s. Setting (s )= and since (s )=,C(s )=,C (s )=and C (s)= q a (s) (s) cos [(s) '] sin [(s) '] ; (.6) we get from which we obtain q a cos ' =; a q ) cos ' + sin ' =; a cos ' = p ; a sin ' = p : (.7) 7

20 Substituting these two expressions into C(s) and C (s) we nd: s (s) C(s) = cos (s) sin (s) C (s) = ( q (s) (s) = (s) q (s) cos (s) cos (s) sin (s) +[ (s) =4] q(s) sin (s) : sin (s)+ cos (s) ) Dening the new variables (s) = and (s) =(s) (s )(s), we get (s) (.8) C (s) = C(s) = s (s) (cos (s)+ sin (s)) ; (.9) q(s) f[(s) ] cos (s) [ + (s) ] sin (s)g : (.3) The sine-like solutions S(s) and S (s) can be computed similarly. Hence, the transfer matrix (A.6) between s and s reads: C(s) S(s) M(s=s )= C (s) S (.3) (s) = s (s) (cos (s)+ sin (s)) (s) cos (s) q(s) +(s) q(s) sin (s) s q (s) sin (s) (cos (s) (s) C (s) sin (s)) A : From this and using (A.9) the transfer matrix M(s) over one period L may be written as C(s+L) S(s+L) S M(s) = C A B (s) C A ; C (s+l) S (s+l) C (s) C(s) because the determinant ofany transfer matrix is equal to unity. Performing the above matrix multiplication yields 8 M(s) = cos + (s) sin +(s) (s) (s) sin sin cos (s) sin C A ; (.3)

21 since (s + L) = (s), (s + L) = (s) and (s + L) = (s) +. Introducing the new variable (s) = +(s) ; (.33) (s) the transfer matrix reads: M(s) = cos + (s) sin (s) sin (s) sin cos (s) sin C A : (.34) We check immediately that the determinant of the transfer matrix for one period is equal to unity and its trace is equal to cos as expected. The transfer matrix M(s) is called the Twiss matrix and the periodic functions (s); (s); (s) are called Twiss parameters. In summary, the solution of Hill's equation and the transfer matrices may be expressed in terms of the single function (s), determined by searching the periodic solutions of 4 + K(s) =:. Hill's equation with piecewise periodic constant coecients The nonlinear dierential equation 4 + K(s) = (.35) is completely specied by the linear optical properties of the lattice (focusing magnets) and the condition of periodicity. Unfortunately it is not any easier to solve than the original Hill's equation u + K(s)u =: (.36) However, when K(s) is a piecewise constant periodic function, explicit determination of the betatron function (s) may be found. Assume K(s) to be a constant K over a distance ` between the azimuthal locations s and s. There are three cases: K is positive, K is negative, and K is equal to zero. For the rst two cases, Hill's equation reduces to the simple harmonic oscillator equation. The solutions may be expressed in terms of the functions C(s) and S(s) (see A.6) satisfying the initial conditions (A.4) at s. In terms of transfer matrices we get: ) For K> M(s =s )= since between s and s cos p K` p K sin p K` p K sin p K` cos p K` C(s) = cos p K(s s ) and S(s) = p K sin p K(s s ); C A ; (.37) 9

22 ) For K< M(s =s )= q cosh jkj` q q jkj sinh jkj` q p sinh jkj` jkj cosh q jkj` C A ; (.38) 3) For K = ` M(s =s )= ; (.39) since C(s) = and S(s) =s s between s and s. The transfer matrix M(s) over one cell (of length L) is then the product of the individual matrices composing the cell. If the cell is made of N elements having transfer matrices M ;M ; ::: M N [with M k = M(s k =s k ) for short], we get M(s ) M(s + L=s )=M N ::: M M : (.4) Let m ij (s ) the components of M(s ) obtained by (.4). Equating the two versions (.34) and (.4) of M(s ) gives the Twiss parameters at the reference point s.we nd: (s ) = m (s ) sin ; (s ) = m (s ) sin ; (s ) = m (s ) m (s ) sin ; cos = [m (s )+m (s )] : (.4) Consequently, once the Twiss matrix has been derived by multiplication of the individual transfer matrices in the cell, the Twiss parameters are obtained by (.4) at any reference point s. The principle of strong focusing is based on the quadrupole doublet system composed of one focusing quadrupole and one defocusing quadrupole separated by a straight section of length d. f F d f D s Figure : Quadrupole doublet.

23 Consider the thin lens approximation which assumes that p K ` with K > and ` ask `remains constant, where ` is quadrupole length and K the normalized gradient. In that approximation, the transfer matrix (.37) or (.38) of a thin quadrupole reduces to M(`=) = K ` f D d = f ; (.4) with f = K ` and where the minus sign corresponds to a horizontally focusing quadrupole and the plus sign to a horizontally defocusing quadrupole. The transfer matrix M db (d=) of the doublet with focal lengths f F > and f D < is thus M db (d=) = = d f F d ; (.43) with f = + f F f D f F d = d f F f D ff d f f D > ; (.44) when f D = f F, with f F = K `. As another example, consider a symmetric thin-lens FODO cell composed of a horizontally focusing quadrupole (F ) of focal length f F = f>, followed by a drift space (O) of length L, then a horizontally defocusing quadrupole (D) of focal length f D = jfj < and a drift space (O) of length L. The transfer matrix through the FODO cell of total length L is then, by (.39) and (.4) L M FODO (L=) = f L f = L f where f = K `; K and ` being the quadrupole strength and length. L f L f L + L f + L f.3 Emittance and beam envelope An invariant can be found from the solution of Hill's equation: q u(s) =a (s) cos [(s) '] : (.45) Computing the derivative u (s) = a q(s) (s) cos [(s) '] + sin [(s) '] = (s) (s) u(s) a q(s) sin [(s) '] ; or equivalently, q (s)u(s)+(s)u (s)=a (s) sin [(s) ']

24 Squaring and summing the above equations using (.33) gives (s)u(s) +(s)u(s)u (s)+(s)u (s) =a : (.46) This is a constant of motion, called the Courant{Snyder invariant. It represents the equation of an ellipse in the plane (u; u ), with the area a. a γ (s) a β(s) particle with "amplitude" b < a u'(s) α(s) a γ(s) a γ(s) a β(s) particle with "amplitude" a α(s) a β(s) u(s) Figure : Phase plane ellipses for particles with dierent amplitudes. It is customary to surround a given fraction at the beam in the (u; u ) plane, say 95%, at a certain point s (for example, the injection point) by a phase ellipse described by (s)u(s) +(s)u(s)u (s)+(s)u (s) =; (.47) where the parameter is called the beam emittance: ZZ dudu = : (.48) ellipse All particles inside the phase ellipse will evolve a homothetic invariant ellipse with parameters a dictated by the optical properties of the lattice (Courant{Snyder invariant). Thus the phase ellipse will contain the same fraction at the beam on successive machine turns: the beam emittance is conserved (in the absence of acceleration, radiation, and some collective eects). The betatron oscillation for a particle on the phase ellipse reads: q u(s) = (s)cos [(s) ']; (.49) where ' is an arbitrary phase constant. The envelope of the beam containing the specied fraction of particles is dened by q E(s) = (s) (.5)

25 and the beam divergence is dened by A(s) = q (s): (.5) The Twiss parameters (s);(s);(s) determine the shape and orientation of the ellipse at azimuthal location s along the lattice. As the particle trajectory u(s) evolves along the ring, the ellipse continuously changes its form and orientation but not its area. Every time the trajectory covers one cell of length L along s the ellipse is the same, since the Twiss parameters are periodic with period L. Consequently, onevery machine revolution the particle coordinates (u; u ) will appear on the same ellipse; u(s + kc) =a q (s)fcos [(s) '] cos kq sin [(s) '] sin kqg ; since (s + kc) (s) =kq. Thus the particle will appear cyclically at only n points on the ellipse if the tune is a rational number Q = m=n and the particle trajectory will cover the ellipse densely if the tune is an irrational number. u'(s) turn 3 turn turn = turn 8 turn 4 turn 7 u(s) turn 5 turn 6 Figure : Phase plane motion for one particle after many turns. This representation of the motion where the trajectory coordinates u and u are picked up at xed azimuthal location s on successive tunes is called `stroboscopic' representation in phase plane (u; u )orpoincare mapping: the series of u i ;u i(i=;:::) dots is a mapping of the (u; u ) plane onto itself. The Courant{Snyder invariant enables us to determine how the Twiss parameters transform through the lattice. Consider the reference point s where the initial conditions of the cosine- and sine-like solutions are given. Setting (s )=,(s )=,(s )=, and u(s )=u,u (s )=u for short, we get u + u u + u =a : Furthermore, by denition of a transfer matrix from s to s, u(s) C(s) S(s) u u = (s) C (s) S (s) u ; 3

26 or equivalently by matrix inversion, u S = (s) C (s) u S(s) C(s) u(s) u (s) ; the Courant{Snyder invariant reads as (S u Su ) + (S u Su )( C u + Cu )+ ( C u+cu ) =(S S C +C )u + +( SS +(S C+SC ) C C )uu + +(S SC + C )u = a ; which is the expression of the Courant{Snyder invariant at s; (s)u +(s)uu + (s)u = a ; where we have set u(s) =u,u (s)=u and S(s) =S,C(s)=C. Identifying the coecients of the latter two invariants gives (s) = C SC + S (s) = CC +(S C+SC ) SS (s) = C S C + S ; or in matrix formulation, (s) C (s) A = (s) C SC S CC S C + SC SS C S C S C A C A : (.5) This expression is the transformation rule for phase ellipses through the lattice..4 Betatron functions for beam transport lattices The linear unperturbed equation of motion for a particle through an arbitrary beam transport lattice (non-periodic) has a form similar to equation (.): u + K(s)u =; (.53) where K(s) is an arbitrary function of s. By analogy with the solution (.9) of Hill's equation (.) we try a solution of the form 4 u(s) =a q (s) cos [ (s) '] ; (.54)

27 in which a and ' are integration constants, (s) and (s) are functions of s to be determined. Dierentiating this expression twice writing for short = (s) and = (s) q u = a p cos [ '] a sin [ '] ; u = a 4 and inserting into (.53) yields a 3= cos [ '] a q 3= sin [ '] q q a sin ( ') a cos [ '] ; + K(s) cos [ '] 4 a p ( + ) sin ( ')=: Since the cosine and sine terms must cancel separately to make (.54) true for all (s), we obtain + K(s) =; (.55) 4 and + =( ) =: The latter equation gives on integration (s) (s) =c: Then, choosing the integration constant c equal to unity, we get which after integration gives (s) = (s) ; (.56) (s) = Hence, inserting (.56) into (.55) we obtain Z s s dt (t) : (.57) 4 + K(s) =: (.58) The last two equations are identical to (.5) and (.7), derived from Hill's equation. Therefore, (s) and (s) are also called phase and betatron functions. However, unlike that of a periodic lattice, the betatron function of a non-periodic lattice is not uniquely determined by the periodicity condition on the cells, but depends on the initial 5

28 conditions at the entrance of the lattice. Similarly to the periodic lattice case, we can dene the parameters (s) = (s) (s) = + (s) (s) ; ; (.59) and the Courant{Snyder invariant may be derived in the same way: (s)u(s) + (s)u(s)u (s)+ (s)u (s) =a (.6) Since the parameters (s), (s), (s) of the non-periodic case, and (s); (s), (s) of the periodic case are of similar nature, the star above the letters,, will be removed. The derivation of the formulae (.3) for the general transformation matrix and (.5) for the transformation of Twiss parameters proceed in the same way as in the periodic lattice case. Designing a beam transport lattice, the Twiss parameters (s ), (s ), (s )atits entrance s may bechosen such that the phase ellipse (s )u(s ) +(s )u(s )u (s )+(s )u (s ) = closely surrounds a given fraction at the incoming particle beam distribution in the (u; u ) phase plane. The parameter is the beam emittance. The description of the distribution of particles within the beam is thus reduced to that of a particle travelling along the phase space ellipse q u(s) = (s) cos [(s) '] : u'(s ) particle with "amplitude" ε γ(s )ε = A(s) u(s ) particle distribution β(s )ε = E(s) Figure 3: Particle distribution in phase plane, beam envelope and divergence. The Twiss parameters at the entrance of a transport lattice may also be chosen as those coming from a join circular machine (at ejection point). 6

29 Another beam emittance denition frequently used is = u(s) (s) ; where u (s) is the standard deviation of the projected beam distribution onto the u-axis (beam prole)..5 Dispersive periodic closed lattices Particle beams have a nite dispersion of momenta about the ideal momentum p. A particle with p = p p will perform betatron oscillations about a dierent closed orbit from that of the reference particle. For small deviations in momentum the equation of motion is, according to (.5) with u + K(s)u = = p p p (s) ; (.6) = p p ; (.6) where (s) stands for the local bend radii x (s) or y (s). The individual particle deviation from the design orbit can be regarded as being the sum of two parts: u(s) =u (s)+u (s); (.63) where u (s) is the displacement at the closed orbit for the o-momentum particle from that of the reference particle (with =);and u (s) is the betatron oscillation around this o-momentum orbit. The particular solution u (s) of the inhomogeneous equation (.6) is generally re-expressed as u (s) =D(s); (.64) where D(s) is called the dispersion function, which evidently satises the equation A particular solution of this equation is D + K(s)D = (s) : (.65) D p (s) =S(s) Z s s C(t) (t) dt C(s) Z s s S(t) dt ; (.66) (t) where C(s) and S(s) are the cosine- and sine-like solutions of the homogeneous Hill's equation (.6) (with = ). The substitution of (.66) into (.65) veries that this is a valid solution. We compute rst 7

30 and Z s Dp(s) = S (s) s = S (s) Z s s C(t) dt + S(s)C(s) (t) (s) C(t) Z s (t) dt C (s) Z s D (s) = C(t) p S (s) s (t) dt + S (s) C(s) (s) = (s) + S (s) Z s s C(t) (t) dt s Z s C S(t) (s) s (t) dt S(t) (t) dt C (s) Z s C S(t) (s) s (t) dt Z s S(t) (t) dt ; s C(s)S(s) (s) C (s) S(s) (s) since the Wronskian is equal to unity (A.7). Hence (.65) for D p reads [S + K(s)S] Z s s C(t) (t) dt [C + K(s)C] Z s s S(t) dt =; (t) where C(s) and S(s) satisfy the homogeneous Hill's equation S + K(s)S = and C + K(s)C =: For a closed lattice we must nd a dispersion function which leads to an o-momentum closed orbit, one for which u (s + C) =u (s) u (s+c)=u (s); (.67) where we have considered the machine circumference C = NL as the period, instead of the cell length L. Other possible solutions u (s) are D p (s) plus a linear combination of C(s) and S(s): u (s) = c C(s)+c S(s)+D p (s) u (s) = c C (s)+c S (s)+d p(s); (.68) where c and c are constants. The application of the boundary condition (.67) to the reference point s = s, from which C(s) and S(s) are derived, yields c C(s + C)+c S(s +C)+D p (s +C)=c ; c C (s +C)+c S (s +C)+D p (s +C)=c ; (.69) since and C(s )= C (s )= S(s )= S (s )= D p (s )= D p(s )=: 8

31 In matrix form the system (.69) reads: C(s + C) S(s + C) C (s + C) S (s + C) c = c Dp (s + C) D p (s ; + C) its solution is c = c S (s + C) S(s + C) C (s + C) C(s + C) Dp (s +C) D p(s +C) [C(s +C) ][S (s + C) ] C : (.7) (s + C)S(s + C) The denominator of this expression may be written as since C(s + C)S (s + C) C (s + C)S(s + C) [C(s + C)+S (s +C)] + = jm(s )j Tr[M(s )] + = ( cos Q) =4 sin Q ; M(s )= C(s +C) S(s +C) C (s +C) S (s +C) is the transfer matrix for one machine revolution, whose determinant is equal to unity, and the trace is Tr [M(s )] = cos (s + C) = cos Q ; with (s + C) = The solution for the constant c is then Z s +C s dt (t) =Q : c = S(s + C)Dp(s + C) [S (s + C) ]D p (s + C) : (.7) 4 sin Q The numerator may be expressed as S(s + C)D p(s + C) S (s + C)D p (s + C)+D p (s +C) Z = [S(s +C)C (s +C) S s +C S(t) (s +C)C(s +C)] s (t) dt Z s +C C(t) +S(s + C) s (t) dt C(s + C) Z s +C Z s +C S(t) = [ C(s + C)] s (t) dt + S(s + C) Furthermore the transfer matrix over a full turn also reads: s Z s +C s S(t) (t) dt C(t) (t) dt M(s )= cos Q + (s ) sin Q (s ) sin Q (s ) sin Q cos Q (s ) sin Q ; 9

32 so that C(s + C) = cos Q + (s ) sin Q ; S(s + C) = (s ) sin Q : The expression (.3) for the transfer matrix M(t=s ) yields where (t) =(t) c = = = = C(t) = S(t) = vu ut (t) (s ) [cos (t)+(s ) sin (t)] q (t)(s ) sin (t) (s ). Hence the constant c becomes 4 sin Q [ cos Q (s ) sin Q] Z s +C +(s ) sin Q q (s ) 4 sin Q q (s ) sin Q q (s ) sin Q Z s +C s Z s +C s Z s +C s (t) s (t) q (t) vu q (t) [( Z s +C q (s )(t) sin (t)dt s (t) t (t) (s ) [cos (t)+(s ) sin (t)]dt cos Q) sin (t) + sin Qcos (t)] dt [sin Qsin (t) + cos Qcos (t)] dt (t) q (t) cos [(t) Q]dt (t) At the reference point s from (.64), (.66) and (.68) we get c = u (s )=D(s ): Since the origin s was chosen arbitrarily, itmay be replaced by the general coordinate s, and the dispersion function D(s) is found to be q (s) D(s) = sin Q Z s+c s q (t) (t) cos [(t) (s) Q]dt : (.7) The periodic dispersion function D(s) also written as (s) depends on all bending magnets in the accelerator. Furthermore, to get stable o-momentum orbits, the machine tune Q must not be an integer, otherwise resonance eect will occur: u (s) becomes innite. 3

33 .6 Dispersive beam transport lattices The dispersion function D(s) in a beam transport line is the general solution of the equation D + K(s)D = (s) ; (.73) where K(s) and (s) are arbitrary functions of s. Already derived is the solution D(s) =c C(s)+c S(s)+S(s) Z s s C(t) (t) dt C(s) Z s s S(t) dt (.74) (t) where c and c are constants and remembering that C(s) and S(s) are solutions of the equation D + K(s)D =: From the initial conditions at s of the cosine- and sine-like solutions, we obtain D(s) =C(s)D(s )+S(s)D (s )+S(s) Z s s C(t) (t) dt C(s) Z s s S(t) dt ; (.75) (t) or, in matrix formulation, dierentiating (.75), D(s) D C (s) A = C(s) S(s) S(s) C (s) S (s) S (s) Z s s Z s s C(t) (t) dt C(t) (t) dt Z s C(s) S(t) (t) dt C (s) s Z s s S(t) (t) dt C A D(s ) D C (s ) A : (.76) This 3 3 matrix is the extended transfer matrix M(s=s ): M(s=s )= C(s) S(s) S(s) R s C(t) s dt (t) C (s) S (s) S (s) Z s s C(t) (t) dt Z s C(s) S(t) (t) dt C (s) s Z s s S(t) (t) dt C A : (.77) The solution of the equation of motion with a momentum deviation may then be expressed as u(s) u(s u (s) C A = M(s=s ) B u (s ) C A : (.78) Indeed, using (.63), (.64), (.76) and (A.5) we get 3

34 u(s) = C(s)[u (s )+u (s )] + S(s)[u (s )+u (s)] +S(s) Z s s C(t) (t) dt C(s) Z s s S(t) (t) dt = C(s)u (s )+S(s)u (s )+C(s)D(s )+S(s)D (s ) +S(s) Z s s C(t) (t) dt C(s) Z s s S(t) (t) dt = u (s)+d(s) = u (s)+u (s); and, similarly, we verify that u (s) =u (s)+u (s): The 3 3 extended transfer matrix (.77) is easily computed when the strength K(s) and the bending radius (s) are constant functions through the magnet. Consider a dipole sector magnet, which is a bending magnet with entry and exit pole faces normal to the incoming and outgoing design orbit, respectively. For a sector magnet: K(s) == (between s and s over a distance `). The cosine- and sine-like solutions are Hence and then C(s) = cos s s Z s C(t) S(s) s (t) dt = sin s s C(s) Z s s S(t) (t) dt = cos s s = cos Z s C(t) S(s) s (t) dt C(s) Z s and Z s Z s s cos s s s S(s) = sin s s t s s s dt = sin sin t s dt s cos s s S(t) (t) dt = cos s s By (.77) the transfer matrix of a sector magnet in the deecting plane is M(s =s )= cos sin ( cos ) sin cos sin where = `= is the bending angle and ` the arc length of the magnet. 3 C : A ; (.79)

35 arc length = s s π/ s s π/ design orbit (δ = ) θ ρ Figure 4: Dipole sector magnet. The 3 3 transfer matrices for synchrotron magnets (combined dipole-quadrupole magnets) may be derived similarly. For a strong focusing synchrotron sector magnet: K(s) =K += > (between s and s over a distance `). In analogy to the dipole sector magnet case, replacing = by p K in the above solutions C(s) and S(s), the transfer matrix (.77) reads, after computation of the matrix components m 3 and m 3 M(s =s )= cos sin p K cos K p K sin cos sin p K with = p K`, where K is the normalized gradient and ` the magnet length. For a strong defocusing synchrotron sector magnet: K(s) =K += transfer matrix may be written as M(s =s )= C A sinh p cosh jkj jkj cosh q jkj sinh cosh sinh p jkj C A (.8) <. The (.8) q with = jkj`. Consider again closed lattices. Like the method used for the betatron oscillations, the periodic dispersion function may also be derived applying the matrix formalism rather than using (.7). For a ring of circumference C (or one periodic cell of length L) composed of N elements having extended 3 3 transfer matrices M ;M ; M N [with M k = M(s k =s k )], the total transfer matrix over the whole machine (or one machine period) is then obtained by multiplying the various matrices, yielding M(s ) M(s + C=s )=M N M M = m (s ) m (s ) m 3 (s ) m (s ) m (s ) m 3 (s ) C A (.8) 33

36 in which the sub-matrix with components (m ij (s )) i;j=; is the usual transfer matrix over one turn (or one period) for the betatron oscillations. The functions D(s) and D (s) being periodic with period C, we get from (.64), (.78) in which s is replaced by s + C, and (.8), the matrix equation D(s) D (s) C A = m (s ) m (s ) m 3 (s ) m (s ) m (s ) m 3 (s ) C A D(s) D (s) C A : (.83) The solution of this equation yields the dispersion function and its derivative at the starting point s D(s ) = m 3(s )( m (s )) + m (s )m 3 (s ) m (s ) m (s ) D (s ) = m 3(s )( m (s )) + m (s )m 3 (s ) m (s ) m (s ) (.84) since m (s )m (s ) m (s )m (s )=: The same calculations can be performed at any azimuthal location s along the machine circumference, so that the dispersion function (.84) can be obtained everywhere in the lattice. The matrix approach is useful when the strength K(s) and the radius of curvature (s) are piecewise constant functions over the magnets, allowing an explicit determination of the periodic dispersion. As an example, consider a thin-lens FODO lattice with cell length L, in which the drift spaces are replaced by dipole sector magnets of length L and bending radius (assuming that the bending angle of the dipole is small: L ). In accordance with (.4), (.77) and (.79) the 3 3 transfer matrices of the cell elements read M QF;D = f C A M O = L L L where the dipole transfer matrix M O in the deecting plane has been simplied by expanding (.79) to the rst order in L=. The extended 3 3 transfer matrix through a FODO cell is obtained from the latter C A formulae with M FODO = M O M QD M O M QF M FODO = L f L f L f L + L f + L f L 4+ L f L 4+ L f C A : (.85) 34

37 Hence, by (.84) the periodic dispersion in the focusing or defocusing thin quadrupole is D QF;D = 4f L : (.86) 4f The dispersion in the defocusing quadrupole (minus sign in the formula) is derived by replacing the focal length f by f in (.85) (i.e. considering a DOFO cell instead, with M DOFO = M O M QF M O M QD ). 3 MULTIPOLE FIELD EXPANSION 3. General multipole eld components Nonlinear magnetic elds will be considered as pure multipole magnet components to be used in the equations of motion of a charged particle in a transport channel or in a circular accelerator. To go beyond the linear expression of the magnetic eld B, we shall use a multipole expansion of B in a xed, right-handed, Cartesian coordinate system (x; y; z), where the z-axis coincides with the s-axis of the coordinate system moving on the design orbit. The eects of curvature will thus be neglected in the derivation of the multipole eld components in the magnets forming a lattice. In a vacuum environment in the vicinity of the design orbit, the following Maxwell equations hold: rb = rb= (3.) because the current density is zero in the region of interest. The latter equation permits the expression of the magnetic eld as the gradient of a magnetic scalar potential U(x; y; z) since for any scalar function U B = ru ; (3.) rru =: This leads to the Laplace equation by the rst equation (3.) rru r U=: (3.3) We assume that the eld does not vary along the z-axis, as is the case for long magnets far from the ends, and that there are only transverse eld components (no solenoid eld). Then the Laplace equation reads in polar coordinates (r; @r U =: Writing the potential U(r; ') as the product of two functions f(r) and g(') U(r; ')=f(r)g(') (3.5) 35

38 the Laplace equation transforms @r + g = or equivalently r f(r) d dr r df = dr g(') d g d' n : (3.7) Each side of this equation must be equal to a constant n since the left-hand side depends only on r and the right-hand side on '. Hence, we get r d f dr + r df dr n f = d g d' + n g =: (3.8) The solutions of these equations may be written as f(r) = p e r n n g(')=a n e in' : (3.9) The terms ( p=e) and n have been added for convenience. The constant n is an integer since the potential U(r; ') is a periodic function of ' with period (and also the function g(')). The constant A n is assumed to be real. Hence, by (3.5) the magnetic scalar potential is written U(r; ')= p X e n A nr n e in' : (3.) n= Values of n have been discarded to avoid eld singularities for r, and because n = gives a constant potential term which does not contribute to the magnetic eld. In Cartesian coordinates the magnetic potential becomes U(x; y) = p e X n= n A n(x + iy) n : (3.) Thus, the magnetic potential may be decomposed into independent multipole terms U n (x; y) = p e A n n (x+iy)n : (3.) The real and imaginary parts of equation (3.) are two independent solutions of the Laplace equation (3.3). From 36 (x + iy) n = nx k= n k(n k) xn k (iy) k

39 and with for, k, it follows that i k =( ) k i k+ =( ) k i Im ((x + iy) n )= Re ((x + iy) n )= X (n )= m= n= X m= n (m)(n m) xn m y m ( ) m ( ) m n (m + )(n m ) xn m y m+ where the sums extend over the greatest integer less than or equal to n= and (n )=, respectively. Hence, assigning dierent coecients A n and A n to the real and imaginary parts of (3.), the potential for the nth order multipole becomes Re [U n (x; y)] = p e n= X m= ( ) m x n m A n (n m) y m (m) (3.3) Im [U n (x; y)] = p e X (n )= m= ( ) m A n x n m (n m ) y m+ (m + ) : (3.4) The real and the imaginary part dierentiate between two classes of magnet orientation. The imaginary part has the so-called midplane symmetry, that is Im [U n (x; y)] = Im [U n (x; y)] : (3.5) The magnetic eld components for the nth order multipoles derived from the imaginary solution (3.4) are, according to (3.), B nx (x; y) = = Im [U n(x; y)] = X (n )= m= ( ) m A n x n m (n m ) y m+ (m + ) (3.6) B ny (x; y) = = Im [U n(x; y)] = X (n )= m= ( ) m A n x n m (n m ) y m (m) (3.7) since (@=@x)x n m =ifm=(n )=. The midplane symmetry (3.5) yields B nx (x; y)= B nx (x; y) 37

40 and B ny (x; y) =B ny (x; y) : Thus, in this symmetry there is no horizontal eld in the midplane, B nx (x; ) =, and a particle travelling in the horizontal midplane will remain in this plane. The magnets derived from the imaginary solution of the potential are called upright or regular magnets. Rewriting Eq. (3.7) as x n B ny (x; y) = p e A n (n ) + p e A n X (n )= m= ( ) m x n m (n m ) y m (m) and dierentiating this expression n times leads to A n = n B ny = e n p B nx )n= : n The coecient A n is called the multipole strength parameter. The magnets derived from the real solution of the potential are called rotated or skew magnets. The magnetic eld components for the nth order skew multipole derived from the real solution are B nx (x; y) = = p e Re [U n(x; y)] = X (n )= m= ( ) m x n m (n m ) y m (m) (3.9) B ny (x; y) = = p e Re [U n(x; y)] = n= X m= m m xn ( ) (n m) y m (m ) (3.) since (@=@x)x n m =ifm=n=. The skew multipole strength parameters A n are derived from (3.9) A n = e n B nx = n p B ny )n= : n The skew magnets dier from the regular magnets only by a rotation about the z-axis by an angle n = =n, where n is the order at the multipole. 38

41 Magnetic multipole potentials Dipole e p U (x; y) = y x+i x y Quadrupole e p U (x; y) = K(x y )+ikxy Sextupole Octupole e p U 3(x; y) = 6 S(x3 3xy )+i 6 S(3x y y 3 ) e p U 4(x; y) = 4 O(x4 6x y + y 4 )+i 6 O(x3 y xy 3 ) Here we have used the notation: A = K; A 3 = S; A 4 = O, and A = K; A 3 = S;A 4 =O. In particular the dipole strength parameters A and A are given by A = e p B y = k x = x A = e p B x = k y = y : Regular multipole elds Dipole Quadrupole Sextupole Octupole e p B x = e p B x = Ky e p B 3x = Sxy e p B 4x = 6 O(3x y y 3 ) e p B y= x e p B y = Kx e p = B 3y = S(x y ) e p B 4y = 6 O(x3 3xy ) 39

42 Skew multipole elds Dipole (=9 ) Quadrupole ( =45 ) Sextupole ( =3 ) Octupole ( =:5 ) e p B x = e y p B y = e p B x = Kx e p B 3x = S(x y ) e p B 4x = 6 O(x3 3xy ) e p B y = e p B 3y = Ky Sxy e p B 4y = 6 O(3x y y 3 ) The general magnetic eld expansion including the most commonly used regular multipole elements reads e p B x(x; y) = Ky + Sxy + 6 O(3x y y 3 )+::: e p B y(x; y) = x + Kx + S(x y )+ 6 O(x3 3xy )+::: 3. Pole prole Multipole elds may be generated by iron magnets in which the metallic surfaces, in the limit of innite magnetic permeability, are equipotential surfaces for magnetic elds. Ignoring the end eld eects, the potential of a horizontally deecting dipole magnet is e p U (x; y) = y x : (3.) An equipotential iron surface is determined by y x = constant (3.3) or y = constant since x is constant inside the magnet. 4

43 y S pole g g B x N pole Figure 5: Dipole magnet pole shape. For the midplane magnet to be at y =,two pole proles are needed at y = g, where g is the gap width. Similarly, the equipotential iron-surface for a regular quadrupole magnet is Kxy = constant : (3.4) Let R be the inscribed radius at the iron-free region limited by the hyperbola. For x = y: x = R= p. Then K R = constant : Identifying this expression with (3.4) gives the pole prole equation Similarly, the equipotential for a skew quadrupole magnet is xy = R : (3.5) K(x y ) = constant : (3.6) For y =:x=r. Then KR = constant The pole prole equation is therefore x y = R : (3.7) N y B R S x N S B y R N x S N S Figure 6: Regular and skew quadrupole magnet pole shapes. 4

44 A regular quadrupole magnet focuses in one plane and defocuses in the other. The eld pattern is e p B x = Ky e p B y = Kx : y B x B y x Figure 7: Magnetic led for an horizontally focusing quadrupole (positive particles approach the reader). The equipotential iron surface for a regular sextupole magnet is S(3x y y 3 ) = constant (3.8) and for a skew sextupole magnet 4 S(x 3 3xy ) = constant : (3.9)

45 tno FILE: g4.eps Figure 8: Regular and skew sextupole magnet pole shapes. 4 TRANSVERSE RESONANCES 4. Nonlinear equations of motion In the absence of a tangential magnetic eld (no solenoid eld) the transverse equations of a motion (3.3) for a charged particle in a magnetic eld read where and x x = k x h ( + ) e hb y p y y = k y h +(+) e hb x (4.) p h =+k x x+k y y = q h + x + y : (4.) The local curvatures of the design orbit are k x;y, the design momentum is p, and is the relative momentum deviation of a particle: =(p p )=p. 43

46 The equations of motion will be expanded to the second order in ; x;y, and their derivatives. Dierentiating Eq. (4.) gives, dividing the result by, = hh + x x + y y which, after some sorting may be expressed as h d ds (k x x + k y y x y ) (4.3) where we have assumed a piecewise at design orbit, so that Then k x (s)k y (s) =: x k x x + k y x y + k x xx + k y yx : Furthermore it will be assumed that u u u = k u ; (4.4) where u stands for x or y, so that u and uu = u may be neglected since k u = u u where u (s) is the local bending radius. Therefore, to the second order we nd x = y : Expressing the general magnetic eld in terms of multipole components, the equations of motion (4.) become to the second-order expansion x = k x h ( + ) h k x + K x + S (x y ) K y S xy y = k y h +( + ) h k y +K y+s xy + K x + S (x y ) (4.5) The index zero in K ;K ;S and S means that the quadrupole and sextupole strengths are evaluated at the design momentum p.we compute the following quantities to the second order: h = +k x x+k y y+k x x +k y y 44

47 s h = h + x h +y h h + x +y since x and y are neglected due to (4.4). Furthermore Similarly h ; ( + ) hk x k x ( + )+(k x k x +k3 x x)x ( + ) hk x (+k x x+k y y )K x ( + ) hk y ( + k x x +k y y )K y ( + ) h S (x y ) S (x y ) ( + )S xy S xy Thus the equations of motion (4.5) become and We shall further assume that x + (K + k x)x K y = k x k x +(K +k x)x (K + k x)k x x S (x y ) K k y xy K y +K k y y +(S +K k x )xy (4.6) y (K k y)y K x = k y k y (K k y)y + + (K k y )k yy +(S +K k x )xy K +K k x x +K k y xy + S (x y ) : (4.7) K x;y S K x;y S ; (4.8) which is generally valid in large synchrotrons. Hence, some terms in (4.6) and (4.7) may be neglected since K k y xy S (x y ) K k y xy S (x y ) K k y xy S xy K k x xy S xy K k u u K u K k u u K u ku 3u ku u where u denotes x or y. Then the equations of motion reduce to the form x + K + x = K y + x + K + x x x x 45

48 S (x y ) K y + S xy (4.9) y K y = K x + y K y + y y y + S xy K x + S (x y ) : (4.) If we consider particles with the design momentum p, the equations of motion expanded up to sextupole terms read x + K + x = K y x S (x y )+S xy (4.) y K y = K x + S xy + y S (x y ) : (4.) When the design orbit lies only in the horizontal plane, the vertical equation of motion simplies to y K y = K x + S xy + S (x y ) : (4.3) Remember that the magnet parameters x;y ; K ; K ; S and S are generally functions of the s-coordinate. In practice they are piecewise constant functions along the lattice. 4. Description of motion in normalized coordinates All terms on the right-hand sides of equations (4.9) and (4.) will be treated as small perturbations. The left-hand side of these equations yields the linear unperturbed equations of motion 46

49 x + K + x y K y x = y = : (4.4) For closed lattices these equations are Hill's equations with periodic coecients. They can be written in the compact form, writing u for either x or y, u + K(s)u = (4.5) where K(s) is a periodic function with period L equal to the length of a machine cell with K(s + L) =K(s) (4.6) K(s) =K (s)+ x;y(s) : Perturbation terms in the equations of motion may lead to unstable beam motion, called resonances, when the perturbating eld acts in synchronism with the particle oscillations. A multipole of nth order is said to generate resonances of order n. Resonances below the third order (i.e. due to dipole and quadrupole eld errors for instance) are called linear resonances. The nonlinear resonances are those of third order and above. By an appropriate transformation of variables we can express the equations of motions (4.9) and (4.) in the form of a perturbed harmonic oscillator with constant frequency. To this end we introduce the normalized coordinates (; ), called Floquet's coordinates, though the transformation (u; s) (; ) = q u (s) = Q Z s s dt (t) (4.7) where Q is the machine tune Q = Z s+c s dt (t) C is the ring circumference: C = NL, where N is the number of periodic cells. Since u = u(;)we compute (4.8) Similarly, since u = u (; ;; )we nd @ = = + = : 47

50 @ = = 4 3= + + = = + = + = = = = + = + : = 3= Hence, with and d ds = d ds d Q d = d ds = d d ds d = d Qd d Q d + d Q ds d d = d Q d + d Q d ; it follows that u = = Q _ + Q + = Q _ + = 3= in which apoint denotes the derivative with respect to. Inserting u into the Hill's equation (4.5) yields the unperturbed equation of motion expressed in normalized coordinates Q 3= + = +K(s) = 3= =: At this stage we introduce a general perturbation term p(x; y; s) in the right-hand side of the latter equation [such a term may be the right-hand side of equation (4.9) or (4.)]. Multiplying the resulting equation by Q 3= yields + Q + 4 K(s) = Q 3= p(x; y; s) : (4.9) It is known that the betatron function (s) satises the dierential equation + 4 K(s) =: (4.) Consequently, the perturbed Hill's equation is transformed into + Q = Q 3= p(; ) (4.) assuming a perturbation of the form p(u; s) (one degree of freedom). When the perturbation vanishes, equation (4.) reduces to a harmonic oscillator with frequency Q, whose solution is () =acos (() ') (4.) 48

51 with () =Q = Dierentiating (4.) with respect to the variable gives Z s s dt (t) : (4.3) d d = a sin [() '] and then + d = a : (4.4) d The particle trajectory in the phase plane (; )isthus a circle of radius equal to the amplitude a of the oscillation. The phase () advances by every betatron oscillation (i.e. the trajectory describes one full phase circle) or by Q every machine revolution. dη dµ a µ η Figure 9: Phase circle in normalized coordinates. In terms of normalized coordinates, the equations of motion (4.) and (4.) for an on-momentum particle (with = ) reads + Q x = Q x 3= x = y K + 5= x S 3= x S y x = y S + Q y = Q y = x 3= y K + x 3= y S 5= y S + = x y S in which is the normalized coordinate for the vertical plane = x q x (s) y = q y (s) (4.5) (4.6) (4.7) where x;y (s) is the horizontal or vertical betatron function, and since is the independent variable in the dierential equations (4.5) and (4.6), it can be dened as 49

52 = Z s dt Q x;y x;y (t) s (4.8) where Q x;y is the horizontal or vertical machine tune. The multipole strengths K ; K ; S ;S, and the betatron functions x;y are periodic functions of with period. Now we consider the case where the general perturbation term p(x; y; s) represents the magnetic eld error B x;y, that is the eld not considered in setting up the design orbit of the machine. Equivalently, B x;y is the eld error with respect to the ideal lattice, even if such a lattice includes nonlinear magnets like sextupoles. Expanding B x;y into multipoles up to the third order, in which the strength parameters are replaced by their variations: B x ;y for the dipole eld error, K and K for the quadrupole gradient errors, S and S for the sextupole errors, we obtain e B x = e B x + K y + K x + S xy + p p S(x y ) e p B y = e p B y + K x K y + S(x y ) S xy : Hence, using (4.), (4.7) and (4.8) we get the equations of motion in normalized coordinates + Q x = Q x 3= x e B y + xk 3= p S + 5= x 3= x S y x = y K + x = y S + Q y = Q y 3= y e K = x 3= y K + B x + y p + = x y S + x 3= y S 5= y S : (4.9) We can rewrite the last two equations considering only the nth order multipole term in the horizontal plane and the nth order multipole term in the vertical plane. We get + Q x = p nrx() n r + Q y = p nry() n r : (4.3) 5

53 Perturbation terms Order p nrx () n r p nry () n n n r Q x 3= x e p B y Q y 3= y p B x Q x 3= x = y K Q y y K Q x x K Q y = Q x x = y S Q y = x e x 3= y y 3 Q x 3= x y S Q y 5= y 3 Q x 5= x S Q y xy 3= K S S S 4.3 One-dimensional resonances Considering only the nth order multipole horizontal perturbation term, the equation of motion (4.3) in one dimension may be written [with r = and Q stands for Q x and p n () for p nx ()] as + Q = p n () n : (4.3) Expanding the perturbation into Fourier series yields p n () = X m= where ^p n (m) is the Fourier coecient of the expansion ^p n (m) = Z ^p n (m)e im (4.3) p n ()e im d : (4.33) The unperturbed oscillation, solution of the inhomogeneous equation (4.3), may be written as () =ae iq + be iq (4.34) where a and b are constants of integration. Assuming the perturbation is small, we can insert () into the right-hand side of the equation of motion (4.3). We nd + Q = X m= ^p n (m)e im (ae iq + be iq ) n : 5

54 Using the binomial expansion, we get () n = (ae iq + be iq ) n = = nx k= nx k= (n ) k(n k ) an k b k e i(n k )Q : Performing the change of variable p =k as with Hence (4.3) reads c(p) = n+p () n = (n ) + Q = n p X (n ) k(n k ) an k e i(n k )Q b k e ikq n+, the above expression may be rewritten nx p= n+ c(p)e ipq h +( ) n+p i a n p b n+p : nx m= p= (n ) ^p n (m)c(p)e i(m pq) : (4.35) Whenever a term on the right-hand side of equation (4.35) has the same frequency as the frequency Q of the harmonic oscillator, the motion gets in resonance. To see this, we consider a single Fourier component of the perturbation + Q =^p n (m)c(p)e i(m pq) : (4.36) The solution of this equation is the sum of the solution of the homogeneous equation and a particular solution of the form = Ae i(m pq) : (4.37) The constant A is determined by introducing (4.37) into (4.36) h i (m pq) + Q Ae i(m pq) i(m pq) =^p n (m)c(p)e from which A is found and () = ^p n (m)c(p) [Q(p ) m][q(p +) m] ei(m pq) : (4.38) Thus the motion becomes unstable when the tune satises the resonance conditions m =(p )Q with jpj n (4.39) or equivalently jmj =(jpj)q (4.4) 5

55 where jpj + is called the order of resonance and m the order of the perturbation Fourier harmonic. The resonance conditions apply only for index p such that the coecient c(p) is nonzero. Thus for the third order resonance n = 3 driven by sextupole elds, we compute for p = ; ; ; ; : c( ) = a c( )= c()=ab c()= c() = b : The resonance conditions are then m = 3Q: (third-order resonance) and m = Q: (integer resonance). Hence the Q values driving third integer resonances are Q = k 3 (4.4) where k is any positive integer. 4.4 Coupling resonances The two-dimensional equations of motion with only the nth order multipole horizontal and rth order vertical multipole perturbation are given by (4.3) + Q x = p nrx() n r + Q y = p nry() n r : Again expanding the perturbation in Fourier series yields p nrx () = p nry () = X X m= m= ^p nrx (m)e im Furthermore, the solutions of the unperturbed equations are ^p nry (m)e im : (4.4) () = a x e iqx + b x e iqy (4.43) () = a y e iqy + b y e iqy in which a x;y and b x;y are constants at integration. Hence () n = () r = nx `= n+ rx p= r+ c x (`)e i`qx c y (p)e ipqy 53

56 with c x (`) = c y (p) = n+` r+p (n ) n ` (r ) r p h +( ) n ` i a n ` h +( ) r p i a r p x b n+` x y b r+` y : In the same way as in the uncoupled treatment of resonances, we obtain, after substitution of the unperturbed oscillations into the right-hand side of the perturbed dierential equations (4.3), + Q x = X m + Q y = X m X X ` p X X ` p i(m `Qx pqy) ^p nrx (m)c x (`)c y (p)e ^p nry (m)c x (`)c y (p)e i(m `Qx pqy) : (4.44) Then considering a single component of the perturbation, particular solutions of these equations may be derived. We obtain () = () = ^p nrx (m)c x (`)c y (p) `Qx pqy) ei(m [Q x (` ) + pq y m][q x (`+)+pq y m] ^p nry (m)c x (k)c y (q) [Q y (q ) + kq x m][q y (q +)+kq x m] ei(m kqx qqy) : (4.45) The resonance conditions are then m =(`)Q x + pq y with j`j n and jpj r ; m=(q)q y + kq x with jkj n and jqj r : (4.46) For example, we consider the resonances driven by a regular sextupole for which the equations of motion are + Q x = p 3x() + p 3x () + Q y = p y() : (4.47) The resonance conditions are given in the following table. 54

57 Order Horizontal motion Vertical motion n r ` p m =(`)Q x + pq y m =(p)q y + `Q x m = m = Q y Q y Q x Q x Q x Q x m = Qy + Q x Q x m = Qy + Q x Q x Qx Q 3 m = y Q x Q y m = Qx Qx m = Qx +Q y Q x +Q y 3 m= Qx m= 3Q x Qx Qx m= 3Qx Q x Discarding the redundant conditions, we are left with jmj = Q x jmj =Q y Q x jmj=3q x : Having considering a skew sextupole instead, the resonance conditions would have been jmj = Q y jmj =Q x Q y jmj=3q y : Thus, the general resonance conditions in two degrees of freedom may be cast into the form MQ x + NQ y = P; (4.48) 55

58 where M; N; and P are integers, P being non-negative, and jmj + jnj is the order of the resonance, and P is the order of the perturbation harmonic. Plotting the resonance lines (4.47) for dierent values of M; N and P in the (Q x ;Q y ) plane yield the so called resonance or tune diagram. k + k + 3 Qy Qx + Qy = 3k + Qx + Qy = k + Qx + Qy = 3k + 3Qx = 3k + Qx = k + 3Qx = 3k + Qy Qx = k + Qx Qy = Qx Qy = k + 3Qy = 3k + k + Qy = k + k + 3 Qx + Qy = 3k + 3Qy = 3k + Qx Qy = k Qy Qx = k k k + 3 k + k + 3 Qx + Qy = 3k + k + Qx Figure : Resonance diagram in a unit square for regular and skew multipole elds up to the 3rd order. If the integers M and N are positive, the resonance is called a sum resonance and can lead to a loss of beam. If M and N are of opposite sign, the resonance is called a coupling resonance or a dierence resonance and does not cause a loss of beam, but rather leads to a beating between the horizontal and the vertical oscillations. It can be shown that the \strength" of the resonances decrease with increasing order, so that only low-order resonances need to be avoided. Around every resonance line in the resonance diagram, there is a band with some thickness, called the resonance width, in which the motion may be unstable, depending on the oscillation amplitude. When the resonance is linear (below the third order), the resonance width is called a stopband because the entire beam becomes unstable if the operating point Q x ;Q y reaches this region of tune values. The largest oscillation amplitude which is still stable in the presence of nonlinearities is called the dynamic aperture. The following gures are tune diagrams in a unit square around the origin showing the resonance lines driven by regular multipole elds up to the th order. 56

59 Figure : Resonance diagrams for regular multipole elds up to 3rd and 6th orders. Figure : Resonance diagrams for regular multipole elds up to 9th and th orders. 5 THE THIRD-INTEGER RESONANCE 5. The averaging method Consider the equation of motion in Floquet coordinates for an nth order multipole perturbation + Q = p n () n (5.) where the perturbation term p n () is periodic in, with period. Assume that the tune Q is close to an nth order resonance in which m is a harmonic of the perturbation p n (). Q Q r = m n : (5.) 57