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