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An Introduction to Transverse Beam Dynamics

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An Introduction to Transverse Beam Dynamics EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN/PS 96{11 (PA) 20 March 1996 AN INTRODUCTION TO TRANSVERSE BEAM DYNAMICS IN ACCELERATORS M. Martini Abstract This text represents an attempt to give a comprehensiveintro duction to the dy- namics of charged particles in accelerator and b eam transp ort 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 b end- ing and fo cusing elds only are solved. This yields the optical parameters of the lattice, useful in strong fo cusing machines, from which the oscillatory motion | the b etatron oscillation | and also the transfer matrices of the lattice elements are expressed. The second part deals with some of the ideas connected with nonlinear- ities and resonances in accelerator physics. The equations of motion are revisited, considering the in uence of nonlinear magnetic elds as pure multip ole magnet elds. A presentation of basic transverse resonance phenomena caused bymulti- p ole p erturbation terms is given, followed by a semi-quantitative discussion of the third-integer resonance. Finally,chromatic e ects in circular accelerators are de- scrib ed and a simple chromaticity correction scheme is given. The b eam dynamics is presented on an intro ductory level, within the framework of di erential equations and using only straightforward p erturbation metho ds, discarding the formulation in terms of Hamiltonian formalism. Lectures given at the Joint Universities Accelerator School, Archamps, 8 January to 22 March 1996 Contents 1 PARTICLE MOTION IN MAGNETIC FIELDS 1 1.1 Co ordinate system 1 1.2 Linearized equations of motion 6 1.3 Weak and strong fo cusing 8 2 LINEAR BEAM OPTICS 13 2.1 Betatron functions for p erio dic closed lattices 13 2.2 Hill's equation with piecewise p erio dic constant co ecients 19 2.3 Emittance and b eam envelop e 21 2.4 Betatron functions for b eam transp ort lattices 24 2.5 Disp ersive p erio dic closed lattices 27 2.6 Disp ersive b eam transp ort lattices 31 3 MULTIPOLE FIELD EXPANSION 35 3.1 General multip ole eld comp onents 35 3.2 Pole pro le 40 4 TRANSVERSE RESONANCES 43 4.1 Nonlinear equations of motion 43 4.2 Description of motion in normalized co ordinates 46 4.3 One-dimensional resonances 51 4.4 Coupling resonances 53 5 THE THIRD-INTEGER RESONANCE 57 5.1 The averaging metho d 57 5.2 The nonlinear sextup ole resonance 60 5.3 Numerical exp eriment: sextup olar kick 68 6 CHROMATICITY 73 6.1 Chromaticity e ect in a closed lattice 73 6.2 The natural chromaticityofaFODO cell 79 6.3 Chromaticity correction 82 APPENDIX A: HILL'S EQUATION 84 A.1 Linear equations 84 A.2 Equations with p erio dic co ecients (Flo quet theory) 86 A.3 Stability of solutions 89 BIBLIOGRAPHY 92 iii 1 PARTICLE MOTION IN MAGNETIC FIELDS 1.1 Co ordinate system The motion of a charged particle in a b eam transp ort channel or in a circular accelerator is governed by the Lorentz force equation F = e(E + v B ) ; (1.1) where E and B are the electric and magnetic elds, v is the particle velo city, and e is the electric charge of the particle. The Lorentz forces are applied as b ending forces to guide the particles along a prede ned ideal path, the design orbit, on which|ideally|all particles should move, and as fo cusing forces to con ne the particles in the vicinityofthe ideal path, from which most particles will unavoidably deviate. The motion of particle b eams under the in uence of these Lorentz forces is called b eam optics. The design orbit can b e describ ed using a xed, right-handed Cartesian reference system. However, using such a reference system it is dicult to express deviations of individual particle tra jectories 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 describ e the ideal b eam path and an individual particle tra jectory.Wehavechosen 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) b e the deviation of the particle tra jectory r (s) from the design orbit r (s). Assume that the design orbit is made of piecewise at curves, which can b e either 0 in the horizontal or vertical plane so that it has no torsion. Hence, from the Frenet{Serret formulae we nd dr dt db dn 0 = t = k (s)n =0 = k (s)t ; (1.2) ds ds ds ds 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 individual particle trajectory n δr t σ design orbit r (s) r (s) 0 motion in horizontal plane s Figure 1: Co ordinate system (n; b; t). 1 Unfortunately, n charges discontinuously if the design orbit jumps from the horizontal to the vertical plane, and vice versa. Therefore, we instead intro duce the new right-handed co ordinate system (e ; e ; t): x y ( 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 de dt x y = k t = k t = k e k e : (1.3) x y x x y y ds ds ds The last equation stands, provided we assume k (s)k (s)=0; (1.4) x y where k ;k are the curvatures in the x-direction and the y -direction. Hence the individual x y particle tra jectory reads: r (x; y ; s)=r (s)+xe (s)+ye (s) ; (1.5) 0 x y where (x; y ; s) are the particle co ordinates in the new reference system. ey ex y t x σ design orbit r (s) r0 (s) s Figure 2: Co ordinate system (e ; e ; t). x y Consider the length s of the ideal b eam path as the indep endentvariable, instead of the time variable t, to express the Lorentz equation (1.1). Now, from d d d d ds d v = = v = ; 0 dt dt d d ds ds 2 where v = d =dt is the particle velo city and a prime denotes the derivative with resp ect to s, it follows that dr dr v v 0 = = r v 0 0 dt ds ! ! 2 2 0 2 00 d v d r d dr v r v 00 0 = = = v r r 2 0 0 02 0 02 dt dt dt ds ! ! 0 00 2 2 1 r d v v 00 00 02 0 r r = ( ) = r : 02 02 02 0 2 ds Now, from (1.3) and (1.5), 0 0 0 0 0 0 r (s) = r (s)+xe +y e +xe +ye x y 0 x y 0 0 = t+x e +y e +xk t + yk t x y x y 0 0 = (1 + k x + k y )t + x e + y e : x y x y Similarly,we compute the second derivative: 00 0 0 0 0 00 00 0 0 0 0 r (s) = (k x + k x + k Y + k y )t + x e + y e + x e + y e x y x y x y x y 0 +(1 + k x + k y )t x y 0 0 0 0 00 00 0 0 = (k x + k x + k y + k y )t + x e + y e + x k t + y k t x y x y x y x y +(1 + k x + k y )(k e k e ) x y x x y y Then, 00 0 0 0 0 r (s) = (k x + k y +2k x +2k y )t x y x y 00 00 +[x k (1 + k x)]e +[y k (1 + k y )]e : x x x y y y The Lorentz force F may b e expressed by the change in the particle momentum p as dp F = = e(E + v B ) ; (1.6) dt with p = m v ; (1.7) 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 (1.6) can b e written as ! 2 2 00 dv d r v 00 0 F = m = m = m r r 2 02 0 dt dt ( 2 v 0 0 00 0 0 = m y +2k x +2k y )t+[x k (1 + k x)]e x + k (k x y x x x y x 02 ) 00 00 0 0 +[y k (1 + k y )]e [(1 + k x + k y )t + x e + y e ] y y y x y x y 0 ! 2 00 v 0 0 0 0 = m (1 + k x + k y ) t k x + k y +2k x +2k y x y x y x y 02 0 ! ! 00 00 00 0 00 0 + x k (1 + k x) x e + y k (1 + k y ) y e : x x x y y y 0 0 3 The magnetic eld may b e expressed in the (x; y ; s) reference system B (x; y ; s)= B (x; y ; s)t + B (x; y ; s)e + B (x; y ; s)e : (1.8) t x x y y In the absence of an electric eld the Lorentz force equation (1.1) b ecomes ev 0 F = e(v B )= (r B ) : (1.9) 0 Using the ab ove results and de ning the variable h =1+k x+k y; (1.10) x y the vector pro duct may b e written as 0 0 0 r B = (ht + x e + y e ) (B t + B e + B e ) x y t x x y y 0 0 0 = (y B hB )e (xB hB )e +(xB y B )t : t y x t x y y x Finally, equating the expression for the rate of change in the particle momentum, the left-hand side of (1.6) with (1.9) and with (1.4) gives ! 00 00 0 0 0 0 00 0 (k x + k y +2k x +2k y e h)t + x k h x x y x x x y 0 0 ! 00 00 0 + y k h y e y y 0 e 0 0 0 0 0 = [(x B y B )t (hB y B )e +(hB x B )e ] ; y x y t x x t y p since k (1 + k x)= k h and k (1 + k y )= k h: x x x y y y Identifying the terms in e ; e , and t leads to the equations of motion x y 00 e 0 0 0 00 x = k h (hB y B ); x x y t 0 p 00 e 00 0 0 0 y y = k h + (hB x B ); y x t 0 p 00 1 e 0 0 0 0 0 0 0 = (k x + k y +2k x +2k y ) (x B y B ) : (1.11) x y y x x y 0 h hp 00 0 The last equation may b e used to eliminate the term = in the other two.
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