An Introduction to Transverse Beam Dynamics

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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

in the horizontal or vertical plane so that it has no torsion. Hence, from the Frenet{Serret

formulae we nd

dr dt db dn

= 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.

individual particle trajectory n

δr t

σ design orbit r (s)

r (s) 0 motion in horizontal plane

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 =

b if the orbit lies in the vertical plane

(

b if the orbit lies in the horizontal plane

e =

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.

ex y t x σ design orbit r (s)

r0 (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 = ;

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

= = 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

+(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

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

F = = e(E + v B ) ; (1.6)

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

(

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

)

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

2 00

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

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

F = e(v B )= (r B ) : (1.9)

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 0

+ y k h y e

y y

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

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

0 0 0 00

x = k h (hB y B ); x

x y t

00 0 0 0

y y = k h + (hB x B );

y x t

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

h hp

00 0

The last equation may b e used to eliminate the term = in the other two.

Expanding the particle momentum in the vicinity of the ideal momentum p , cor-

resp onding to a particle travelling on the design orbit, yields

1 1

= ;

p p (1 + )

0 4

where

p p p

(1.12) =

p p

0 0

is the relative momentum deviation. Hence, the ab ove general equations of motion for a

charged particle in a magnetic eld B may b e rewritten as:

00 0 0 0 1

x (hB y B ) ; x = k h (1 + )

y t x

00 0 0 0 1

y (hB x B ) y = k h +(1+)

x t y

1 e

0 0 0 0 0 0 0 1

(x B y B ) : (1.13) = (k x + k y +2k x +2k y ) (1 + )

y x x y

x y

h hp

Using

v = r ;

it follows that

v v

0 0

jv j = v = jr j = r r ;

0 0

then

0 0

2 02 02

= jr j = h + x + y : (1.14)

On the design orbit (equilibrium orbit) we get

x = x =0

y =y =0

=0:

0 00 0

Consequently h =1; =1; = = 0, and using (1.13) we nd

e e

k = B (0; 0;s) B ;

x y y0

p p

0 0

e e

B (0; 0;s) B : (1.15) k =

x x0 y

p p

0 0

Equivalently,intro ducing the lo cal b ending radii and ,

x y

(s)= ; (1.16)

x;y

k (s)

x;y

we get the b ending eld for the design momentum p

1 e 1 e

= B = B : (1.17)

y 0 x0

p p

x 0 y 0

We adopt the following sign convention: an observer lo oking in the p ositive s-direction

sees a p ositivecharge travelling along the p ositive s-direction de ected to the right (resp. 5

upwards) by a p ositivevertical magnetic eld B > 0 (resp. by a p ositive horizontal

magnetic eld B > 0). x

v bend magnet design orbit

F (s)

Figure 3: Lorentz force (for a p ositivecharged particle).

The term jB j is the b eam rigidity(B and stand for B and or B and ). In

0 0 x0 x y 0 y

more practical units the b eam rigidity reads:

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.

1.2 Linearized equations of motion

0 0

From the general equations of motion (1.13) we retain only linear terms in x; x ;y;y

and . Starting with (1.14) we get

02 02

x y

= h 1+ + h =1+k x+k y

x y

2 2

h h

00 0 0 0 0 0

h = k x + k x + k y + k y :

x y

Hence:

With this approximation and with the rst-order series expansion (1 + ) 1 , the

equations of motion (1.13) in the horizontal and vertical planes b ecome, in the absence

of a tangential magnetic eld (no solenoid eld),

00 2

x = k h (1 )h B (x; y ; s) ;

x y

00 2

y = k h +(1 )h B (x; y ; s) : (1.18)

y x

0 6

For x and y , small deviations from the design orbit, the eld comp onents may b e expanded

in series to the rst order:

@B @B

x x

B (x; y ; s) = B + x + y;

x x0

@x @y

@B @B

y y

B (x; y ; s) = B + y: (1.19) x +

y y 0

@x @

The magnetic eld satis es the Maxwell equations

rB =0 r B =0 ;

from whichwe get, using (1.8),

@B @B @B @B

x y y x

= = : (1.20)

@x @y @x @y

Thus, intro ducing the normalized gradient K , and the skew normalized gradient K

de ned as

! !

e @B e @B

y x

K = K ; (1.21) =

p @x p @x

0 0

x=y=0 x=y=0

the eld comp onents may b e written with (1.15)

B (x; y ; s) = k + K x + K y;

x y 0

B (x; y ; s) = k + K x K y: (1.22)

y x 0

Hence, the transverse equations of motion b ecome

00 2

y ) ; x = k h (1 )h (k + K x K

x x 0

00 2

y = k h (1 )h (k K x K y ) : (1.23)

y y 0

From (1.4) and (1.10) we compute to the rst order in x; y ; ,

h 1+2k x+2k y

x y

k h = k +k u

u u

2 2

uk (1 )h k k +2k

u u u

2 2

u K u; (1 )h K u K u and (1 )h K

0 0

where u stands for x or y . Substituting these approximations into (1.23) and using the

radius instead of the curvature k we obtain the linearized equations of motion

x;y x;y

y = x + K + x K ;

x 7

y K : (1.24) y K x =

The term K intro duces a linear coupling into the equations of motion. If we restrict this

to magnetic elds which do es not intro duce any coupling, wehave to set K = 0 (no skew

linear magnets). Then the equations of motion read:

; x + K + x =

y K : (1.25) y =

The terms K and in the ab ove expressions represent the gradient fo cusing

x;y

and weak sector magnet fo cusing, resp ectively. When the de ection o ccurs only in the

horizontal plane, which is the usual case for synchrotrons, the equation of motion in the

vertical plane simpli es to

y K y =0 : (1.26)

The magnet parameters ;K and K are functions of the s-co ordinate. In practice,

x;y 0

they have zero values in magnet-free sections and assume constantvalues within the mag-

nets. The arrangement of magnets in a b eam transp ort channel or in a circular accelerator

is called a lattice.

1.3 Weak and strong fo cusing

Consider the simple case at a circular accelerator with a b ending magnet which

de ects only in the horizontal plane and whose eld do es not change azimuthally

@B @B

x y

= =0:

@s @s

The design orbit is then a circle of constant radius , with the constant magnetic eld

B on this circle evaluated for the design momentum

y 0

p = eB : (1.27)

0 y0 x

By0

ey e v x s positive particle design orbit (circle) ρ = constant with momentum p ρ x

0 x

Figure 4: Circular design orbit in the horizontal plane. 8

For a particle with the design momentum the equations of motion read:

x + K + x =0;

y K y =0; (1.28)

where K is the constant normalized gradient (1.21). The stability in the horizontal motion

is achieved if the solution is oscillatory, that is:

K + > 0 :

Vertically, the stabilityisachieved if

K < 0 :

Hence, the stabilityisachieved in b oth planes provided

: (1.29) 0 < K <

Using (1.17) and (1.21) the latter inequalities write:

1 1 @B

0 < < : (1.30)

B @x

y 0 x

x=y=0

This condition is called weak- or constant-gradient fo cusing. It allows stable motion in

b oth planes.

The oscillatory solution of (1.28), called b etatron oscillations, is

u(s)= a cos ( Ks ') ; (1.31)

where u stands for x or y; K = + K or K = K , and where a and ' are integration

0 0

constants. The wavelength of the b etatron oscillatio n is

: (1.32) =

The numb er of b etatron oscillatio ns p erformed by particles around a machine circumfer-

ence C is called the tune Q of the circular accelerator (Q stands for Q or Q ):

x y

Q = = K: (1.33)

The weak fo cusing condition (1.29) show that K< , that is:

Q<1 : 9

The b etatron oscillation wavelength is larger than the machine circumference. This

means that the amplitude of the oscillations may b ecome very large as the size of the

machine increases and hence the magnet ap erture may also b e very big. This yields a

practical limit on the size of weak fo cusing accelerators. Historically, the eld index n was

intro duced instead of the normalized gradient K for the weak fo cusing accelerators

x y

n = : (1.34)

B @x

y 0

x=y=0

The weak fo cusing condition (1.30) is then expressed as

Using (1.33) and (1.34) the machine tune may b e written as

1 n Q = n: (1.36) Q =

y x

The combination of b ending and fo cusing forces required for weak fo cusing accelerators

may b e obtained by the magnetic eld shap e called synchrotron magnet or combined dip ole-quadrup ol e magnet.

y ρ x S-pole IIIIII

I Bx By

I I I

I I Fx

B Fy x design F B orbit x y ring centre IIIII IIII III II Bx Fy II

IIIIII N-pole

Figure 5: Synchrotron magnet (p ositive particles approach the reader).

The limitations at weak fo cusing accelerators have b een overcome by the invention

at the strong- or alternating-gradient fo cusing principle. This metho d amounts to splitting

up the machine into an alternate sequence of strongly horizontally fo cusing (n 1or

2 2

K ) and strongly horizontally defo cusing (n 1orK ) magnets. By

0 0

x x

(1.36) this implies that the tunes Q and Q may b ecome arbitrarily large. This means

x y

that the b etatron amplitudes can b e kept small for a given angular de ection as the size

increases, and the magnet ap erture may b e reduced.

x 10 n>>1

n<<Ð1 n<<Ð1 design orbit for p ex 0 n>>1

Figure 6: Strong fo cusing machine.

The b ending and fo cusing forces for strong fo cusing accelerators maybeachieved

either within synchrotron magnets or in a separate b ending magnet, called a dip ole, and

a fo cusing magnet, called a quadrup ole.

The quadrup ole magnet provides fo cusing forces through its normalized gradient,

given by (1.21) and (1.22):

e e

B = K y B = K x: (1.37)

x 0 y 0

p p

0 0

I I

I I N-pole I I S-pole I Fy I I I I I I I I I I I I I I I I I

Fx Fx B

I I

I x I I

I I

I I I

I Fy I I

I S-pole I N-pole

I I

Figure 7: Horizontally fo cusing, F -typ e quadrup ole magnet (p ositive particles approach the

reader).

A quadrup ole that fo cuses in one plane defo cuses in the other plane. The horizontally

defo cusing, D -typ e quadrup ole is obtained by p ermuting the N - and S -p oles of an F -

quadrup ole. 11

In geometrical optics the fo cal length f of a lens is related to the angle of de ection

of the lens by

tan = : f

θ < 0 x focal point s

Figure 8: Principle of fo cusing for light.

It is known that a pair of glass lenses, one fo cusing with fo cal length f > 0, and

the other defo cusing with fo cal length f < 0, separated by a distance d, yields a total

fo cal length f given by

1 1 1 d

= + : (1.38)

f f f f f

F D F D

This lens doublet is fo cusing if f = f . Hence,

D F

d 1

= : (1.39)

f f

The strong fo cusing scheme is based up on the quadrup ole doublet arrangement, which b ecomes a system fo cusing in b oth planes.

fF>0 fD<0

focal point

principal plane d

focal length f

Figure 9: Principle of strong fo cusing for light.

The linear equations of motion (1.25) for strong fo cusing lattices (where the b ending

and fo cusing forces dep end on s)may b e written as

u + K (s)u =0 ; (1.40) 12

where u stands for either the horizontal x or the vertical y co ordinates, and where the

b ending and fo cusing functions are combined in one function (plus sign for x, minus sign

for y ):

K (s)=K (s)+ : (1.41)

(s)

x;y

For synchrotron magnets b oth terms in K (s) are non-zero, while for separated function

magnets (separate dip oles and quadrup oles) either K (s)or (s) is set to zero.

0 x;y

Integrating (1.40) over a short distance, `, where K (s) constant) we nd in the

paraxial approximation, where the defection angle is equal to the slop e of the particle

tra jectory,

00 0 0

u ds = u (`) u (0) = tan

and

K (s)uds Ku` :

Therefore by (1.40)

tan K`u :

By analogy with the expression at the fo cal length of a glass lens, we de ned the fo cal

length of a quadrup ole as

tan = : (1.42)

Hence we get for a thin quadrup ole of length ` [for K (s)= K > 0]

= K `: (1.43)

with f p ositive in the fo cusing plane and negative in the defo cusing plane.

2 LINEAR BEAM OPTICS

2.1 Betatron functions for p erio dic closed lattices

In the case where the b ending magnets of a circular accelerator de ect only in the

horizontal plane, the linear unp erturb ed equations of motion for a particle having the

design momentum p (i.e. = 0) are, according to (1.25),

x =0 x + K +

y K y =0 ; (2.1)

where K (s) and (s) are p erio dic functions of the s-co ordinate due to the orbit b eing

0 x

a closed curve. The p erio d L may b e the accelerator circumference C or the length of a 13

\cell" rep eated N times around the circumference: C = NL. Both the ab ove equations

may b e cast in the form

u + K (s)u =0 ; (2.2)

with

K (s + L)=K(s); (2.3)

where u stands for x or y and where K (s)=K (s)+ (s)or K(s)=K (s).

0 0

Equation (2.2) with p erio dic co ecient K (s) is called Hill's equation. It has a pair

of indep endent stable solutions of the form (see app endix A1)

i(s)

u (s) = w (s)e ;

1 1

i(s)

u (s) = w (s)e ; (2.4)

2 2

such that w (s) and w (s) are p erio dic with p erio d L,

1 2

w (s + L)=w(s) i=1;2 ; (2.5)

i i

and (s) is a function such that

(s + L) (s)=; (2.6)

where is called the characteristic exp onent of Hill's equation de ned by

cos = Tr[M(s+L=s)] ; (2.7)

and is indep endent of the length s. The matrix M (s + L=s) is called the transfer matrix

over one p erio d L [shortly written as M (s) de ned by (see A.9)]

M (s) M (s + L=s)=M(s+L=s )M (s=s ) ;

0 0

in which M (s=s ) is called the transfer matrix b etween the reference p oint s and s, given

0 0

by (see A.6)

C(s) S(s)

M (s=s )= ;

0 0

C (s) S (s)

where C (s) and S (s) are two indep endent solutions of Hill's equation called cosine-like

and sine-line solutions, which satisfy the particular initial conditions (see A.4)

0 0

C (s )=1 C (s )= 0 S(s )=0 S(s )=1:

0 0 0 0

i(s) i(s)

As the functions e and e are already linearly indep endent, we can arbitrarily

make w (s) identical to w (s) so that the pair at indep endent stable solutions (2.4) now

1 2

read:

i(s)

u (s)= w(s)e ; (2.8)

1;2 14

where w (s) is p erio dic with p erio d L. Di erentiating the latter equation twice,

0 0 i(s) 0 i(s)

u (s) = w (s)e i (s)w (s)e

1;2

00 00 0 0 00 0 2 0 0 i(s)

u (s) = [w (s) i (s)w (s) i (s)w (s) (s) w (s) i (s)w (s)]e

1;2

00 0 0 0 2 00 i(s)

= [w (s) 2i (s)w (s) (s) w (s) i (s)w (s)]e ;

i(s)

we obtain by substitution into Hill's equation and cancelling e

00 02 0 0 00

w w + K (s)w i(2 w + w )=0:

Equating real and imaginary parts to zero yields

00 02

w w + K (s)w =0 (2.9)

and

0 0 00

2 w + w =0;

or equivalently

00 0

=0 : (2.10) +

The last equation may b e rewritten as

2 0 0 0

[ln w (s) ] + [ln (s)] =0 ;

which can b e integrated to give

+ln c: ln (s)=ln

w(s)

Hence, cho osing the integration constant c equal to unity

(s)= ; (2.11)

w(s)

which yields on integration

(s)= : (2.12)

w (t)

Substituting (2.11) into (2.9) gives a new di erential equation for w (s):

+ K (s)w =0 : (2.13) w

De ning the so-called b etatron function (s)as

(s)=w (s) ; (2.14) 15

equations (2.12) and (2.13) transform into

(s)= (2.15)

(t)

and

1=2 1=2 3=2

+ K (s) =0 : (2.16)

The latter expression may b e further transformed to give

1 1

00 02 2

+ K (s) =1; (2.17)

2 4

since

1 1

3=2 1=200 00 02

= :

2 4

The function (s) given by (2.15) is called the phase function. Then, the two indep endent

solutions of Hill's equations b ecome

i(s)

(s)e : (2.18) u (s)=

1;2

Every solution of Hill's equation is a linear combination of these two solutions:

q q

i(s) i(s)

u(s)=c (s)e + c (s)e ;

1 2

where c and c are constants, or equivalently

1 2

u(s)=a (s) cos [(s) '] ; (2.19)

with

c c

1 2

a =2 c c tan ' = i :

1 2

c + c

1 2

Any solution (s) that satis es (2.17) together with the phase function (s), whose deriva-

tiveis (s) , can b e used to make (2.19) a real solution of Hill's equation. Such a solution

is a pseudo-harmonic oscillatio n with varying amplitude and frequency, called b etatron

oscillatio n.

From (2.15) we compute:

s+L

(s + L) (s)= ;

(t)

and using (2.6) we nd:

s+L

= : (2.20)

(t)

Thus the characteristic exp onent may b e identi ed with the phase advance p er p erio d

or cell (of length L). 16

The oscillation 's lo cal frequency f (s) and wavelength (s) can b e related to the

phase function by

(s)=2f (s)= : (2.21)

(s)

Thus, the b etatron function maybeinterpreted as the lo cal wavelength divided by2

0 1

since (s)= (s) :

(s)=2 (s) : (2.22)

Let the tune Q of a circular accelerator b e de ned as the numb er of b etatron oscillatio ns

executed by particles travelling once around the machine circumference C . Since the lo cal

frequency f (s) denotes the numb er of oscillations p er unit of length, the tune reads:

s+C

Q = f (t)dt : (2.23)

If the accelerator has N cells of p erio d L (i.e., C = NL), then using (2.20) to (2.23), Q

is given by

s+C

N dt 1

Q = = ; (2.24)

2 2 (t)

since (s) is a p erio dic function of p erio d L.

Any solution of Hill's equation can b e describ ed in terms of the b etatron oscillation

(2.19). In particular, the cosine-like solution C (s) and the sine-like solution S (s)maybe

represented by (2.19);

C (s)= a (s) cos [(s) '] ; (2.25)

where a and ' are constants which dep end up on the initial conditions at the reference

p oint s . Setting (s )= and since (s )= 0, C(s )=1, C (s )=0 and

0 0 0 0 0 0

a (s)

C (s)= cos [(s) '] sin [(s) '] ; (2.26)

(s)

we get

a cos ' =1;

cos ' + sin ' =0;

)

from whichwe obtain

; a cos ' =

a sin ' = : (2.27)

0 17

Substituting these two expressions into C (s) and C (s)we nd:

(s)

cos (s) C (s)= sin (s)

( ! !)

0 0 0

1 (s)

0 0

C (s) = cos (s) sin (s) sin (s)+ cos (s)

2 2 2

(s)

0 0 0

1+[ (s) =4] (s)

q q

cos (s) sin (s) : =

2 (s) (s)

0 0

De ning the new variables

(s)

(s)= (2.28)

and (s)=(s)(s )(s), we get

(s)

C (s)= (cos (s)+ sin (s)) ; (2.29)

C (s)= f[ (s) ] cos (s) [1 + (s) ] sin (s)g : (2.30)

0 0

(s)

The sine-like solutions S (s) and S (s) can b e computed similarly. Hence, the transfer

matrix (A.6) b etween s and s reads:

C(s) S(s)

M (s=s )= (2.31)

0 0

C (s) S (s)

1 0

(s)

(cos (s)+ sin (s)) (s) sin (s)

C B

0 0

C B

: =

C B

1+ (s) (s)

0 0

A @

q q

(cos (s) (s) sin (s)) cos (s) sin (s)

(s)

(s) (s)

0 0

From this and using (A.19) the transfer matrix M (s)over one p erio d L may b e written

0 10 1

C(s+L) S(s+L) S (s) S(s)

B CB C

M (s)= ;

@ A@ A

0 0 0

C (s+L) S(s+L) C (s) C(s)

b ecause the determinantofany transfer matrix is equal to unity.Performing the ab ove

matrix multiplication yields

0 1

cos + (s) sin (s) sin

B C

M (s)=B C ; (2.32)

@ A

1+ (s)

sin cos (s) sin

(s) 18

since (s + L)= (s), (s + L)= (s) and (s + L)=(s)+.Intro ducing the new

variable

1+ (s)

(s)= ; (2.33)

(s)

the transfer matrix reads:

0 1

cos + (s) sin (s) sin

B C

M (s)= : (2.34)

@ A

(s) sin cos (s) sin

Wecheck immediately that the determinant of the transfer matrix for one p erio d

is equal to unity and its trace is equal to 2 cos as exp ected. The transfer matrix M (s)

is called the Twiss matrix and the p erio dic functions (s); (s); (s) are called Twiss

parameters. In summary, the solution of Hill's equation and the transfer matrices may

b e expressed in terms of the single function (s), determined by searching the p erio dic

solutions of

1 1

00 02 2

+ K (s) =1:

2 4

2.2 Hill's equation with piecewise p erio dic constant co ecients

The nonlinear di erential equation

1 1

00 02 2

+ K (s) =1 (2.35)

2 4

is completely sp eci ed by the linear optical prop erties of the lattice (fo cusing magnets)

and the condition of p erio dicity. Unfortunately it is not any easier to solve than the

original Hill's equation

u + K (s)u =0 : (2.36)

However, when K (s) is a piecewise constant p erio dic function, explicit determina-

tion of the b etatron function (s)may b e found. Assume K (s) to b e a constant K over a

distance ` between the azimuthal lo cations s and s . There are three cases: K is p ositive,

0 1

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 b e 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:

1) For K>0

0 1

p p

cos K` sin K`

B C

M(s =s )= ; (2.37)

1 0

@ A

p p p

K sin K` cos K`

since b etween s and s

0 1

p p

K (s s ) and S (s)= K (s s ); C (s) = cos sin

0 0

K 19

2) For K<0

q q

1 0

cosh jK j` sinh jK j`

C B

jK j

C B

C ; (2.38) M(s =s )=B

1 0

A @

q q q

jK j sinh jK j` cosh jK j`

3) For K =0

1 `

M(s =s )= ; (2.39)

1 2

0 1

since C (s) = 1 and S (s)=ss between s and s .

0 0 1

The transfer matrix M (s)over one cell (of length L) is then the pro duct of the

individual matrices comp osing the cell. If the cell is made of N elements having transfer

matrices M ;M; ::: M [with M = M (s =s ) for short], we get

1 2 N k k k 1

M (s ) M (s + L=s )= M ::: M M : (2.40)

0 0 0 N 2 1

Let m (s ) the comp onents of M (s ) obtained by (2.40). Equating the twoversions (2.34)

ij 0 0

and (2.40) of M (s ) gives the Twiss parameters at the reference p oint s .We nd:

0 0

m (s )

12 0

(s ) = ;

sin

m (s )

21 0

; (s ) =

sin

m (s ) m (s )

11 0 22 0

(s ) = ;

2 sin

cos = [m (s )+m (s )] : (2.41)

11 0 22 0

Consequently, once the Twiss matrix has b een derived bymultiplication of the individual

transfer matrices in the cell, the Twiss parameters are obtained by (2.41) at any reference

p oint s.

The principle of strong fo cusing is based on the quadrup ole doublet system com-

p osed of one fo cusing quadrup ole and one defo cusing quadrup ole separated by a straight

section of length d.

d fF fD

Figure 10: Quadrup ole doublet. 20

Consider the thin lens approximation which assumes that K ` 1 with K > 0

0 0

and ` ! 0asK `remains constant, where ` is quadrup ole length and K the normalized

0 0

gradient. In that approximation, the transfer matrix (2.37) or (2.38) of a thin quadrup ole

reduces to

! !

1 0

; (2.42) M (`=0) = =

K ` 1

with f = K ` and where the minus sign corresp onds to a horizontally fo cusing quadrup ole

and the plus sign to a horizontally defo cusing quadrup ole.

The transfer matrix M (d=0) of the doublet with fo cal lengths f > 0 and f < 0

db F D

is thus

! ! !

1 d

1 0 1 0

1 d

M (d=0) = ; (2.43) =

1 1

d 1

1 1

0 1

f f

D F

with

1 d d 1 1

+ = = > 0 ; (2.44)

f f f f f f

F D F D

when f = f , with f = K `.

D F 0

As another example, consider a symmetric thin-lens FODO cell comp osed of a hori-

zontally fo cusing quadrup ole (F ) of fo cal length f = f>0, followed by a drift space (O )

of length L, then a horizontally defo cusing quadrup ole (D ) of fo cal length f = jf j < 0

and a drift space (O ) of length L. The transfer matrix through the FODO cell of total

length 2L is then, by (2.39) and (2.42)

! ! !

2 2

L L L

1 2L +

1 0 1 0

1 L 1 L

f f f

M (2L=0) = =

FODO

1 1

L L

1 1

0 1 0 1

f f

where f = K `; K and ` b eing the quadrup ole strength and length.

0 0

2.3 Emittance and b eam envelop e

An invariant can b e found from the solution of Hill's equation:

u(s)=a (s) cos [(s) '] : (2.45)

Computing the derivative

u (s) = (s) cos [(s) '] + sin [(s) '] =

(s)

a (s)

u(s) sin [(s) '] ;

(s)

or equivalently,

(s)u(s)+ (s)u(s)=a (s) sin [(s) '] 21

Squaring and summing the ab ove equations using (2.33) gives

2 0 0 2 2

(s)u(s) +2 (s)u(s)u (s)+ (s)u(s) = a : (2.46)

This is a constant of motion, called the Courant{Snyder invariant. It represents the equa-

0 2

tion of an ellipse in the plane (u; u ), with the area a .

u'(s) α(s) Ða γ(s) particle with "amplitude" a a γ (s) a α(s) Ða β(s) β(s) u(s) a particle with γ(s) "amplitude" b < a

a β(s)

Figure 11: Phase plane ellipses for particles with di erent amplitudes.

It is customary to surround a given fraction at the b eam in the (u; u ) plane, say

95%, at a certain p oint s (for example, the injection p oint) by a phase ellipse describ ed

2 0 0 2

(s)u(s) +2 (s)u(s)u (s)+ (s)u(s) = ; (2.47)

where the parameter is called the b eam emittance:

dudu = : (2.48)

ellipse

All particles inside the phase ellipse will evolve a homothetic invariant ellipse with param-

eters a dictated by the optical prop erties of the lattice (Courant{Snyder invariant).

Thus the phase ellipse will contain the same fraction at the b eam on successive machine

turns: the b eam emittance is conserved (in the absence of acceleration, radiation, and

some collective e ects).

The b etatron oscillatio n for a particle on the phase ellipse reads:

u(s)= (s)cos [(s) ']; (2.49)

where ' is an arbitrary phase constant. The envelop e of the b eam containing the sp eci ed

fraction of particles is de ned by

E (s)= (s) (2.50) 22

and the b eam divergence is de ned by

A(s)= (s): (2.51)

The Twiss parameters (s); (s); (s) determine the shap e and orientation of the ellipse

at azimuthal lo cation s along the lattice. As the particle tra jectory u(s)evolves along

the ring, the ellipse continuously changes its form and orientation but not its area. Every

time the tra jectory covers one cell of length L along s the ellipse is the same, since the

Twiss parameters are p erio dic with p erio d L. Consequently,onevery machine revolution

the particle co ordinates (u; u ) will app ear on the same ellipse;

u(s + kC )=a (s)fcos [(s) '] cos 2kQ sin [(s) '] sin 2kQg ;

since (s + kC ) (s)=2kQ.Thus the particle will app ear cyclically at only n p oints

on the ellipse if the tune is a rational number Q = m=n and the particle tra jectory will

cover the ellipse densely if the tune is an irrational numb er.

u'(s)

turn 2 turn 1 = turn 8 turn 3

turn 4 turn 7 u(s)

turn 6

turn 5

Figure 12: Phase plane motion for one particle after many turns.

This representation of the motion where the tra jectory co ordinates u and u are

picked up at xed azimuthal lo cation s on successive tunes is called `strob oscopic' repre-

0 0

sentation in phase plane (u; u )orPoincare mapping: the series of u ;u(i=1;2:::) 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 p oint s where the initial conditions

of the cosine- and sine-like solutions are given. Setting (s )= , (s )= , (s )= ,

0 0 0 0 0 0

0 0

and u(s )= u , u(s )=u for short, we get

0 0 0

2 02 0 2

= a : + u +2 u u u

0 0 0 0

0 0 0

Furthermore, by de nition of a transfer matrix from s to s,

u(s) C (s) S (s) u

= ;

0 0 0 0

u (s) C (s) S (s) u

0 23

or equivalently by matrix inversion,

u S (s) S (s) u(s)

= ;

0 0 0

u C (s) C (s) u (s)

the Courant{Snyder invariant reads as

0 0 2 0 0 0 0 0 0 2

(S u Su ) +2 (S uSu )(C u + Cu )+ (C u+Cu )

0 0 0

02 0 0 02 2

=(S 2S C +C )u +

0 0 0

0 0 0 0 0

+2(SS +(S C +SC ) C C )uu +

0 0 0

2 2 02 2

+(S 2SC + C )u = a ;

0 0 0

which is the expression of the Courant{Snyder invariantats;

2 0 02 2

(s)u +2 (s)uu + (s)u = a ;

0 0

where wehave set u(s)= u, u(s)=u and S (s)=S, C(s)= C.

Identifying the co ecients of the latter twoinvariants gives

2 2

(s) = C 2SC + S

0 0 0

0 0 0 0

(s) = CC +(S C +SC ) SS

0 0 0

02 0 0 02

(s) = C 2S C + S ;

0 0 0

or in matrix formulation,

1 0 1 1 0 0

2 2

C 2SC S (s)

C B C C B B

0 0 0 0

: (2.52) = CC S C + SC SS (s)

A @ A A @ @

02 0 0 02

C 2S C S (s)

This expression is the transformation rule for phase ellipses through the lattice.

2.4 Betatron functions for b eam transp ort lattices

The linear unp erturb ed equation of motion for a particle through an arbitrary b eam

transp ort lattice (non-p erio dic) has a form similar to equation (2.2):

u + K (s)u =0 ; (2.53)

where K (s) is an arbitrary function of s. By analogy with the solution (2.19) of Hill's

equation (2.2) we try a solution of the form

u(s)= a (s) cos [ (s) '] ; (2.54) 24

in which a and ' are integration constants, (s) and (s) are functions of s to b e

determined. Di erentiating this expression twice|writing for short = (s) and =

(s)|

0 0

u = a cos [ '] a sin [ '] ;

00 02 0

a 2

00 0

u = cos [ '] a sin [ ']

3=2

q q

00 02

a sin ( ') a cos [ '] ;

and inserting into (2.53) yields

00 02

2 02 2

+ K (s) cos [ ']

3=2

2 4

0 0 00

( + ) sin ( ')=0 :

Since the cosine and sine terms must cancel separately to make (2.54) true for all (s),

we obtain

02 00

2 02 2

+ K (s)=0; (2.55)

2 4

and

0 0 00 0 0

+ =( ) =0:

The latter equation gives on integration

(s) (s)=c:

Then, cho osing the integration constant c equal to unity,we get

(s)= ; (2.56)

(s)

which after integration gives

(s)= : (2.57)

(t)

Hence, inserting (2.56) into (2.55) we obtain

1 1

00 02 2

+ K (s) =1: (2.58)

2 4

The last two equations are identical to (2.15) and (2.17), derived from Hill's equa-

tion. Therefore, (s) and (s) are also called phase and b etatron functions. However,

unlike that of a p erio dic lattice, the b etatron function of a non-p erio dic lattice is not

uniquely determined by the p erio dicity condition on the cells, but dep ends on the initial 25

conditions at the entrance of the lattice. Similarly to the p erio dic lattice case, we can

de ne the parameters

(s)

(s) = ;

1+ (s)

(s) = ; (2.59)

(s)

and the Courant{Snyder invariantmay b e derived in the same way:

2 0 0 2 2

(s)u(s) +2 (s)u(s)u(s)+ (s)u (s) = a (2.60)

Since the parameters (s), (s), (s) of the non-p erio dic case, and (s); (s), (s)

of the p erio dic case are of similar nature, the star ab ove the letters , , will b e

removed. The derivation of the formulae (2.31) for the general transformation matrix and

(2.52) for the transformation of Twiss parameters pro ceed in the same wayasinthe

p erio dic lattice case.

Designing a b eam transp ort lattice, the Twiss parameters (s ), (s ), (s )atits

0 0 0

entrance s maybechosen such that the phase ellipse

2 0 0 2

(s )u(s ) +2 (s )u(s )u(s )+ (s )u(s ) =

0 0 0 0 0 0 0

closely surrounds a given fraction at the incoming particle b eam distribution in the (u; u )

phase plane. The parameter is the b eam emittance. The description of the distribution

of particles within the b eam is thus reduced to that of a particle travelling along the phase

space ellipse

u(s)= (s) cos [(s) '] :

u'(s0) particle with "amplitude" ε

γ ε (s0) = A(s)

u(s0)

β ε

particle distribution (s0) = E(s)

Figure 13: Particle distribution in phase plane, b eam envelop e and divergence.

The Twiss parameters at the entrance of a transp ort lattice may also b e chosen as those

coming from a join circular machine (at ejection p oint). 26

Another b eam emittance de nition frequently used is

(s)

= ;

(s)

where (s) is the standard deviation of the pro jected b eam distribution onto the u-axis

(b eam pro le).

2.5 Disp ersive p erio dic closed lattices

Particle b eams have a nite disp ersion of momenta ab out the ideal momentum p .

A particle with p = p p will p erform b etatron oscillations ab out a di erent closed

orbit from that of the reference particle. For small deviations in momentum the equation

of motion is, according to (1.25)

u + K (s)u = ; (2.61)

(s)

with

p p p

= = ; (2.62)

p p

0 0

where (s) stands for the lo cal b end radii (s)or (s).

x y

The individual particle deviation from the design orbit can b e regarded as b eing

the sum of two parts:

u(s)=u (s)+u (s) ; (2.63)

where u (s) is the displacement at the closed orbit for the o -momentum particle from

that of the reference particle (with =0); and u (s) is the b etatron oscillation around

this o -momentum orbit. The particular solution u (s) of the inhomogeneous equation

(2.61) is generally re-expressed as

u (s)=D(s); (2.64)

where D (s) is called the disp ersion function, which evidently satis es the equation

D + K (s)D = : (2.65)

(s)

A particular solution of this equation is

Z Z

s s

S (t) C(t)

dt C (s) dt ; (2.66) D (s)= S(s)

(t) (t)

s s

0 0

where C (s) and S (s) are the cosine- and sine-like solutions of the homogeneous Hill's

equation (2.61) (with = 0). The substitution of (2.66) into (2.65) veri es that this is a

valid solution. We compute rst 27

Z Z

s s

C (s) S (s) C (t) S (t)

0 0 0

D (s) = S (s) dt + S (s) C (s) dt C (s)

(t) (s) (t) (s)

s s

0 0

Z Z

s s

C (t) S (t)

0 0

= S (s) dt C (s) dt

(t) (t)

s s

0 0

and

Z Z

s s

C (s) S (s) C (t) S (t)

0 00 0 00 00

dt + S (s) C (s) dt C (s) D (s) = S (s)

(t) (s) (t) (s)

s s

0 0

Z Z

s s

1 C (t) S (t)

00 00

= + S (s) dt C (s) dt ;

(s) (t) (t)

s s

0 0

since the Wronskian is equal to unity (A.7). Hence (2.65) for D reads

Z Z

s s

C (t) S (t)

00 00

[S + K (s)S ] dt [C + K (s)C ] dt =0;

(t) (t)

s s

0 0

where C (s) and S (s) satisfy the homogeneous Hill's equation

00 00

S + K (s)S =0 and C + K (s)C =0:

For a closed lattice wemust nd a disp ersion function which leads to an o -momentum

closed orbit, one for which

0 0

u (s + C )= u (s) u (s+C)=u (s); (2.67)

where wehave considered the machine circumference C = NL as the p erio d, instead of

the cell length L. Other p ossible solutions u (s) are D (s) plus a linear combination of

C (s) and S (s):

u (s) = c C (s)+c S(s)+D (s)

1 2 p

0 0 0 0

u (s) = c C (s)+c S (s)+D (s); (2.68)

1 2

where c and c are constants. The application of the b oundary condition (2.67) to the

1 2

reference p oint s = s , from which C (s) and S (s) are derived, yields

c C (s + C )+c S(s +C)+D (s +C) = c ;

1 0 2 0 p 0 1

0 0 0

(s +C) = c ; (2.69) c C (s +C)+ c S (s + C)+D

0 2 1 0 2 0

since

0 0

C (s )=1 C (s )=0 S(s )=0 S(s )=1

0 0 0 0

and

D (s )=0 D (s )=0:

p 0 0

p 28

In matrix form the system (2.69) reads:

C (s + C ) 1 S (s + C ) c D (s + C )

0 0 1 p 0

= ;

0 0 0

C (s + C ) S (s + C ) 1 c D (s + C )

0 0 2 0

its solution is

! !

S (s + C ) 1 S (s + C ) D (s + C)

0 0 p 0

0 0

D (s + C)

C (s + C ) C (s + C ) 1

0 0 c

= : (2.70)

0 0

[C(s + C) 1][S (s + C ) 1] C (s + C )S (s + C )

0 0 0 0

The denominator of this expression may b e written as

0 0 0

C (s + C )S (s + C ) C (s + C )S (s + C ) [C (s + C )+S (s +C)] + 1

0 0 0 0 0 0

= jM (s )jTr[M(s )] + 1 = 2(1 cos 2Q)=4 sin Q ;

0 0

since

C(s +C) S(s +C)

0 0

M (s )=

0 0

C (s +C) S (s +C)

0 0

is the transfer matrix for one machine revolution, whose determinant is equal to unity,

and the trace is

Tr[M(s )] = 2 cos (s + C ) = 2 cos 2Q ;

0 0

with

s +C

=2Q : (s + C )=

(t)

The solution for the constant c is then

0 0

S (s + C )D (s + C ) [S (s + C ) 1]D (s + C )

0 0 0 p 0

c = : (2.71)

4 sin Q

The numerator may b e expressed as

0 0

S (s + C )D (s + C ) S (s + C )D (s + C )+ D (s + C)

0 0 0 p 0 p 0

s +C

S (t)

0 0

= [S(s + C)C (s + C) S (s + C)C(s + C)] dt

0 0 0 0

(t)

Z Z

s +C s +C

0 0

C (t) S (t)

+S (s + C ) dt C (s + C ) dt

0 0

(t) (t)

s s

0 0

Z Z

s +C s +C

0 0

S (t) C (t)

= [1 C (s + C )] dt + S (s + C ) dt

0 0

(t) (t)

s s

0 0

Furthermore the transfer matrix over a full turn also reads:

cos 2Q + (s ) sin 2Q (s ) sin 2Q

0 0

M (s )= ;

(s ) sin 2Q cos 2Q (s ) sin 2Q

0 0 29

so that

C (s + C ) = cos 2Q + (s ) sin 2Q ;

0 0

S (s + C ) = (s ) sin 2Q :

0 0

The expression (2.31) for the transfer matrix M (t=s ) yields

(t)

C (t) = [cos (t)+ (s ) sin (t)]

(s )

S (t) = (t) (s ) sin (t)

where (t)=(t)(s ). Hence the constant c b ecomes

0 1

s +C

[1 cos 2Q (s ) sin 2Q] (s ) (t) sin (t)dt c =

0 0 1

4 sin Q (t)

s +C

1 (t)

+ (s ) sin 2Q [cos (t)+ (s ) sin (t)]dt

0 0

(t) (s )

0 0

s +C

(s ) 0

0 1

= (t) [(1 cos 2Q) sin (t) + sin 2Q cos (t)] dt

4 sin Q (t)

q q

s +C

(s ) (t)

= [sin Q sin (t) + cos Q cos (t)] dt

2 sin Q (t)

q q

s +C

(s ) (t)

= cos [(t) Q]dt

2 sin Q (t)

At the reference p oint s from (2.64), (2.66) and (2.68) we get

c = u (s )= D(s ):

1 0 0

Since the origin s was chosen arbitrarily,itmay b e replaced by the general co ordinate s,

and the disp ersion function D (s) is found to b e

q q

s+C

(s) (t)

D (s)= cos [(t) (s) Q]dt : (2.72)

2 sin Q (t)

The p erio dic disp ersion function D (s)|also written as (s)|dep ends on all b ending

magnets in the accelerator. Furthermore, to get stable o -momentum orbits, the machine

tune Q must not b e an integer, otherwise resonance e ect will o ccur: u (s) b ecomes

in nite. 30

2.6 Disp ersive b eam transp ort lattices

The disp ersion function D (s) in a b eam transp ort line is the general solution of the

equation

D + K (s)D = ; (2.73)

(s)

where K (s) and (s) are arbitrary functions of s.

Already derived is the solution

Z Z

s s

C(t) S (t)

D (s)=c C(s)+ c S(s)+S(s) dt C (s) dt (2.74)

1 2

(t) (t)

s s

0 0

where c and c are constants and rememb ering that C (s) and S (s) are solutions of the

1 2

equation

D + K (s)D =0:

From the initial conditions at s of the cosine- and sine-like solutions, we obtain

Z Z

s s

C(t) S (t)

D (s)=C(s)D(s )+S(s)D (s )+S(s) dt C (s) dt ; (2.75)

0 0

(t) (t)

s s

0 0

or, in matrix formulation, di erentiating (2.75),

0 Z Z 1

s s

C (t) S (t)

C (s) S (s) S (s) dt C (s) dt

B C

1 0 1 0

(t) (t)

s s

B C

0 0

D (s) D (s )

B C

Z Z

C B B C C B

s s

0 0

C (t) S (t)

= B D (s) : (2.76) C D (s )

A @ A @

0 0 0

C (s) S (s) S (s) dt C (s) dt

B C

(t) (t)

s s

1 1

0 0

@ A

0 0 1

This 3 3 matrix is the extended transfer matrix M (s=s ):

Z 1 0

S (t)

C(t) s

C(s) S(s) S(s) dt C (s) dt

C B

0 (t)

(t)

C B

Z Z

C B

s s

C (t) S (t)

C : (2.77) M (s=s )=B

0 0 0

C (s) S (s) S (s) dt C (s) dt

C B

(t) (t)

s s

0 0

A @

0 0 1

The solution of the equation of motion with a momentum deviation may then b e expressed

0 1 0 1

u(s) u(s )

B C B C

0 0

u (s) = M (s=s ) u (s ) : (2.78)

@ A @ A

0 0

Indeed, using (2.63), (2.64), (2.76) and (A.5) we get 31

0 0

u(s) = C (s)[u (s )+u (s )] + S (s)[u (s )+u (s)]

0 0 0

Z Z

s s

C (t) S (t)

+S (s) dt C (s) dt

(t) (t)

s s

0 0

= C (s)u (s )+S(s)u (s )+C(s)D(s ) +S(s)D (s )

0 0 0 0

Z Z

s s

C(t) S (t)

+S(s) dt C (s) dt = u (s)+D(s)

(t) (t)

s s

0 0

= u (s)+u (s) ;

and, similarly,weverify that

0 0 0

u (s)=u (s)+u (s) :

The 3 3 extended transfer matrix (2.77) is easily computed when the strength K (s) and

the b ending radius (s) are constant functions through the magnet.

Consider a dip ole sector magnet, which is a b ending magnet with entry and exit

p ole faces normal to the incoming and outgoing design orbit, resp ectively.For a sector

magnet: K (s)=1= (b etween s and s over a distance `). The cosine- and sine-like

0 1

solutions are

! !

s s s s

0 0

C (s) = cos and S (s)= sin :

0 0

Hence

! ! !

Z Z

s s

C (t) s s s s t s

0 0 0

S (s) dt = sin cos dt = sin

(t)

s s

0 0

0 0 0

! !

Z Z

s s

S (t) s s t s

0 0

C (s) dt = cos sin dt

(t)

s s

0 0

! ! !

s s s s

0 0

cos 1 = cos

0 0

and then

Z Z

s s

C (t) S (t) s s

S (s) dt C (s) dt = 1 cos

(t) (t)

s s

0 0

By (2.77) the transfer matrix of a sector magnet in the de ecting plane is

0 1

cos sin (1 cos )

0 0

B C

sin cos sin M (s =s )= ; (2.79)

@ A

1 0

0 0 1

where = `= is the b ending angle and ` the arc length of the magnet.

0 32 arc length = s1 Ð s0

π/2 π/2 s0 s1 design orbit (δ = 0)

θ ρ0

Figure 14: Dip ole sector magnet.

The 3 3 transfer matrices for synchrotron magnets (combined dip ole-quadrup ole

magnets) may b e derived similarly.For a strong fo cusing synchrotron sector magnet:

K (s)=K +1= > 0(between s and s over a distance `). In analogy to the dip ole

0 0 1

sector magnet case, replacing 1= by K in the ab ove solutions C (s) and S (s), the

transfer matrix (2.77) reads, after computation of the matrix comp onents m and m

13 23

0 1

sin 1cos

cos

B C

sin

M (s =s )=B K sin cos C (2.80)

1 0

B C

@ A

0 0 1

with = K`, where K is the normalized gradient and ` the magnet length.

For a strong defo cusing synchrotron sector magnet: K (s)=K +1= < 0. The

transfer matrix may b e written as

0 1

sinh cosh 1

cosh

jK j

B C

B q C

B C

sinh

M (s =s )=B jK j sinh cosh C (2.81)

1 0

B C

jK j

@ A

0 0 1

with = jK j`.

Consider again closed lattices. Like the metho d used for the b etatron oscillations,

the p erio dic disp ersion function may also b e derived applying the matrix formalism rather

than using (2.72). For a ring of circumference C (or one p erio dic cell of length L) comp osed

of N elements having extended 3 3 transfer matrices M ;M ; M [with M =

1 2 N k

M (s =s )], the total transfer matrix over the whole machine (or one machine p erio d)

k k 1

is then obtained bymultiplying the various matrices, yielding

0 1

m (s ) m (s ) m (s )

11 0 12 0 13 0

B C

M (s ) M (s + C=s )= M M M = m (s ) m (s ) m (s ) (2.82)

@ A

0 0 0 N 2 1 21 0 22 0 23 0

0 0 1 33

in which the 2 2 sub-matrix with comp onents (m (s )) is the usual transfer matrix

ij 0 i;j =1;2

over one turn (or one p erio d) for the b etatron oscillations.

The functions D (s) and D (s) b eing p erio dic with p erio d C ,we get from (2.64),

(2.78) in which s is replaced by s + C , and (2.82), the matrix equation

0 1 0 1 0 1

D (s) m (s ) m (s ) m (s ) D (s)

11 0 12 0 13 0

B C B C B C

0 0

D (s) = m (s ) m (s ) m (s D (s) : (2.83)

@ A @ A @ A

21 0 22 0 23 )

1 0 0 1 1

The solution of this equation yields the disp ersion function and its derivativeatthe

starting p oint s

m (s )(1 m (s )) + m (s )m (s )

13 0 22 0 12 0 23 0

D (s ) =

2 m (s ) m (s )

11 0 22 0

m (s )(1 m (s )) + m (s )m (s )

23 0 11 0 21 0 13 0

D (s ) = (2.84)

2 m (s ) m (s )

11 0 22 0

since

m (s )m (s ) m (s )m (s )=1:

11 0 22 0 12 0 21 0

The same calculations can b e p erformed at any azimuthal lo cation s along the machine

circumference, so that the disp ersion function (2.84) can b e 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 p erio dic disp ersion.

As an example, consider a thin-lens FODO lattice with cell length 2L, in which

the drift spaces are replaced by dip ole sector magnets of length L and b ending radius

(assuming that the b ending angle of the dip ole is small: L ). In accordance with

0 0

(2.42), (2.77) and (2.79) the 3 3 transfer matrices of the cell elements read

0 1 1 0

1 0 0 1 L

B C C B

1 L

M = B 0 1 C 1 0 C M = B

QF;D O

B C C B

@ A A @

0 0 1 0 0 1

where the dip ole transfer matrix M in the de ecting plane has b een simpli ed by ex-

panding (2.79) to the rst order in L= .

The extended 3 3 transfer matrix through a FODO cell is obtained from the latter

formulae with M = M M M M

FODO O QD O QF

0 1

2 2 2

L L L L L

1 4+ 2L +

f f f 2 f

B C

L L L L

B C

M = : (2.85)

1+ 4+

FODO

B C

f f 2 f

B C

@ A

0 0 1 34

Hence, by (2.84) the p erio dic disp ersion in the fo cusing or defo cusing thin quadrup ole is

L 4f

1 D = : (2.86)

QF;D

The disp ersion in the defo cusing quadrup ole (minus sign in the formula) is derived by

replacing the fo cal length f by f in (2.85) (i.e. considering a DOFO cell instead, with

M = M M M M ).

DOFO O QF O QD

3 MULTIPOLE FIELD EXPANSION

3.1 General multip ole eld comp onents

Nonlinear magnetic elds will b e considered as pure multip ole magnet comp onents

to b e used in the equations of motion of a charged particle in a transp ort channel or in a

circular accelerator. Togobeyond the linear expression of the magnetic eld B ,we shall

use a multip ole expansion of B in a xed, right-handed, Cartesian co ordinate system

(x; y ; z ), where the z -axis coincides with the s-axis of the co ordinate system moving on

the design orbit. The e ects of curvature will thus b e neglected in the derivation of the

multip ole eld comp onents in the magnets forming a lattice.

Inavacuum environment in the vicinity of the design orbit, the following Maxwell

equations hold:

r B =0 rB =0 (3.1)

b ecause the current density is zero in the region of interest.

The latter equation p ermits the expression of the magnetic eld as the gradientof

a magnetic scalar p otential U (x; y ; z )

B = rU; (3.2)

since for any scalar function U

rrU =0:

This leads to the Laplace equation by the rst equation (3.1)

r rU r U =0: (3.3)

We assume that the eld do es not vary along the z -axis, as is the case for long mag-

nets far from the ends, and that there are only transverse eld comp onents (no solenoid

eld). Then the Laplace equation reads in p olar co ordinates (r;')

@ @U @ U

r =0: (3.4) r +

@r @r @'

Writing the p otential U (r;') as the pro duct of two functions f (r ) and g (')

U (r;')=f(r)g(') (3.5) 35

the Laplace equation transforms into

@ @f @ g

g (')r r + f (r) =0 (3.6)

@r @r @'

or equivalently

d d g r df 1

r = n : (3.7)

f (r ) dr dr g (') d'

Each side of this equation must b e equal to a constant n since the left-hand side dep ends

only on r and the right-hand side on '. Hence, we get

d f df

2 2

r + r n f =0

dr dr

(3.8)

d g

+ n g =0:

The solutions of these equations may b e written as

p r

in'

f (r )= g(')=A e : (3.9)

e n!

The terms (p=e) and n!have b een added for convenience. The constant n is an integer

since the p otential U (r;') is a p erio dic function of ' with p erio d 2 (and also the function

g (')). The constant A is assumed to b e real. Hence, by (3.5) the magnetic scalar p otential

is written

p 1

n in'

U (r;')= A r e : (3.10)

e n!

n=1

Values of n 0have b een discarded to avoid eld singulariti es for r ! 0, and b ecause

n = 0 gives a constant p otential term which do es not contribute to the magnetic eld. In

Cartesian co ordinates the magnetic p otential b ecomes

p 1

U (x; y )= A (x + iy ) : (3.11)

e n!

n=1

Thus, the magnetic p otential may b e decomp osed into indep endentmultip ole terms

U (x; y )= (x+iy ) : (3.12)

e n!

The real and imaginary parts of equation (3.12) are two indep endent solutions of the

Laplace equation (3.3). From

n nk k

(x + iy ) = x (iy )

k !(n k )!

k =0 36

and with for, k 0,

2k k 2k +1 k

i =(1) i =(1) i

it follows that

n=2

n n2m 2m m

Re ((x + iy ) )= x y (1)

(2m)!(n 2m)!

m=0

(n1)=2

n n2m1 2m+1 m

Im ((x + iy ) )= x y (1)

(2m + 1)!(n 2m 1)!

m=0

where the sums extend over the greatest integer less than or equal to n=2 and (n 1)=2,

resp ectively.

and A to the real and imaginary parts of Hence, assigning di erent co ecients A

(3.12), the p otential for the nth order multip ole b ecomes

n=2

2m n2m

p y x

Re [U (x; y )] = (1) A (3.13)

e (n 2m)! (2m)!

m=0

(n1)=2

2m+1 n2m1

p y x

: (3.14) Im [U (x; y )] = (1) A

n n

e (n 2m 1)! (2m + 1)!

m=0

The real and the imaginary part di erentiate b etween two classes of magnet orientation.

The imaginary part has the so-called midplane symmetry, that is

Im [U (x; y )] = Im [U (x; y )] : (3.15)

n n

The magnetic eld comp onents for the nth order multip oles derived from the imaginary

solution (3.14) are, according to (3.2),

B (x; y ) = Im [U (x; y )] =

nx n

(n2)=2

n2m2 2m+1

p x y

= (1) A (3.16)

e (n 2m 2)! (2m + 1)!

m=0

B (x; y ) = Im [U (x; y )] =

ny n

(n1)=2

n2m1 2m

p x y

= (1) A (3.17)

e (n 2m 1)! (2m)!

m=0

n2m1

since (@=@x)x =0 if m=(n1)=2. The midplane symmetry (3.15) yields

B (x; y )= B (x; y )

nx nx 37

and

B (x; y )= B (x; y ) :

ny ny

Thus, in this symmetry there is no horizontal eld in the midplane, B (x; 0) = 0, and a

particle travelling in the horizontal midplane will remain in this plane.

The magnets derived from the imaginary solution of the p otential are called upright

or regular magnets. Rewriting Eq. (3.17) as

(n1)=2

n1 2m n2m1

p x y p x

B (x; y )= A + A (1)

ny n n

e (n1)! e (n 2m 1)! (2m)!

m=1

and di erentiating this expression n 1 times leads to

n1 n1

@ B e e @ B

ny nx

n=2

A = = (1) : (3.18)

n1 n1

p @x p @y

The co ecient A is called the multip ole strength parameter.

The magnets derived from the real solution of the p otential are called rotated or

skew magnets. The magnetic eld comp onents for the nth order skew multip ole derived

from the real solution are

B (x; y ) = Re [U (x; y )] =

nx n

(n1)=2

n2m1 2m

p x y

= A (1) (3.19)

e (n 2m 1)! (2m)!

m=0

Re [U (x; y )] = B (x; y ) =

n ny

n=2

n2m 2m1

p x y

= A (3.20) (1)

e (n 2m)! (2m 1)!

m=0

n2m

since (@=@x)x =0 if m=n=2.

The skew multip ole strength parameters A are derived from (3.19)

n1 n1

@ B e @ B e

ny nx

n=2

A = = (1) : (3.21)

n1 n1

p @x p @y

The skew magnets di er from the regular magnets only by a rotation ab out the z -axis by

an angle = =2n, where n is the order at the multip ole.

n 38

Magnetic multip ole p otentials

1 e 1

x+i y Dip ole U (x; y )=

y x

e 1

2 2

Quadrup ole U (x; y )= K(x y )+iK xy

p 2

e 1 1

3 2 2 3

Sextup ole U (x; y )= S(x 3xy )+i S(3x y y )

p 6 6

e 1 1

4 2 2 4 3 3

Octup ole (x 6x y + y )+i U (x; y )= O O(x y xy )

p 24 6

Here wehave used the notation: A = K; A = S; A = O , and A = K ;A =

2 3 4

2 3

S;A =O. In particular the dip ole strength parameters A and A are given by

4 1

e 1 e 1

A = B = k = = B = k = A :

1 1y x 1x y

p p

x y

Regular multip ole elds

e 1 e

B =0 B = Dip ole

1x 1y

p p

e e

Quadrup ole B = Ky B = Kx

2x 2y

p p

e 1 e

2 2

B = Sxy = B = S(x y ) Sextup ole

3x 3y

p p 2

e 1 e 1

2 3 3 2

Octup ole B = O (3x y y ) B = O (x 3xy )

4x 4y

p 6 p 6 39

Skew multip ole elds

e e 1

Dip ole B = B =0

1x 1y

p p

(=90 )

e e

x y Quadrup ole B = K B = K

2x 2y

p p

( =45 )

e 1 e

2 2

Sextup ole B = S (x y ) B = S xy

3x 3y

p 2 p

( =30 )

e e 1 1

3 2 2 3

Octup ole (x 3xy ) (3x y y ) B = O B = O

4x 4y

p 6 p 6

( =22:5 )

The general magnetic eld expansion including the most commonly used regular

multip ole elements reads

e 1

2 3

B (x; y ) = Ky + Sxy + O(3x y y )+:::

p 6

e 1 1 1

2 2 3 2

B (x; y ) = S(x y )+ O(x 3xy )+::: + Kx +

p 2 6

3.2 Pole pro le

Multip ole elds may b e generated by iron magnets in which the metallic surfaces, in

the limit of in nite magnetic p ermeability, are equip otential surfaces for magnetic elds.

Ignoring the end eld e ects, the p otential of a horizontally de ecting dip ole magnet is

e y

U (x; y )= : (3.22)

An equip otential iron surface is determined by

= constant (3.23)

or y = constant since is constant inside the magnet.

x 40 y SÐpole

g B x g

NÐpole

Figure 15: Dip ole magnet p ole shap e.

For the midplane magnet to b e at y =0,two p ole pro les are needed at y = g ,

where 2g is the gap width.

Similarly, the equip otential iron-surface for a regular quadrup ole magnet is

Kxy = constant : (3.24)

Let R b e the inscrib ed radius at the iron-free region limited by the hyp erb ola. For x = y :

x = R= 2. Then

R = constant : K

Identifying this expression with (3.24) gives the p ole pro le equation

xy = R : (3.25)

Similarly, the equip otential for a skew quadrup ole magnet is

2 2

K (x y ) = constant : (3.26)

For y =0: x=R. Then

K R = constant

The p ole pro le equation is therefore

2 2 2

x y = R : (3.27)

y y S NS R B NNR x x B

Figure 16: Regular and skew quadrup ole magnet p ole shap es. 41

A regular quadrup ole magnet fo cuses in one plane and defo cuses in the other. The

eld pattern is

e e

B = Ky B = Kx :

2x 2y

p p

y B 2x

B 2y

Figure 17: Magnetic led for an horizontally fo cusing quadrup ole (p ositive particles approach

the reader).

The equip otential iron surface for a regular sextup ole magnet is

2 3

S (3x y y ) = constant (3.28)

and for a skew sextup ole magnet

3 2

S (x 3xy ) = constant : (3.29) 42

tNO FILE: g4.eps

Figure 18: Regular and skew sextup ole magnet p ole shap es.

4 TRANSVERSE RESONANCES

4.1 Nonlinear equations of motion

In the absence of a tangential magnetic eld (no solenoid eld) the transverse equa-

tions of a motion (3.13) for a charged particle in a magnetic eld read

00 0 1 0

x x = k h (1 + ) hB

x y

0 1 00 0

y = k h +(1+) y hB (4.1)

y x

where

h =1+k x+k y

x y

and

0 0

2 2 2

= h + x + y : (4.2)

The lo cal curvatures of the design orbit are k , the design momentum is p , and is the

x;y 0

relative momentum deviation of a particle: =(pp )=p .

0 0 43

The equations of motion will b e expanded to the second order in ;x;y, and their

derivatives. Di erentiating Eq. (4.2) gives, dividing the result by ,

00 0 0 00 0 00

hh + x x + y y

0 2

which, after some sorting may b e expressed as

d 1

0 2 2 2 2 02 02

h (k x + k y x y ) (4.3)

x y

2 ds

where wehave assumed a piecewise at design orbit, so that

k (s)k (s)=0:

x y

Then

0 2 0 0 0 0 0 0

x k x + k x y + k xx + k yx :

x y

Furthermore it will b e assumed that

u 1 u = ; (4.4)

2 0 2

where u stands for x or y , so that u and uu = may b e neglected since

k =

where (s) is the lo cal b ending radius. Therefore, to the second order we nd

00 00

0 0

x = y 0 :

0 0

Expressing the general magnetic eld in terms of multip ole comp onents, the equations of

motion (4.1) b ecome to the second-order expansion

00 2 0 2 2

x = k h (1 + ) h k + K x + S (x y ) K y S xy

x x 0 0

0 0

00 2 0 2 2

y = k h +(1 + ) h k +K y +S xy + K x + S (x y ) (4.5)

y y 0 0

0 0

The index zero in K ;K ;S and S means that the quadrup ole and sextup ole strengths

0 0

are evaluated at the design momentum p .We compute the following quantities to the

second order:

2 2 2 2 2

h = 1+2k x+2k y +k x +k y

x y

x y 44

0 0

2 2

x y x y

0 2 2 2

h = h 1+ + h 1+ + h ;

2 2

h h 2 2

0 0

2 2

since x and y are neglected due to (4.4). Furthermore

2 0 2 2 2 3

(1 + ) hk k (1 + )+(2k 2k +k x)x

x x

x x x

Similarly

2 0

(1 + ) hK x (1+2k x+2k y )K x

0 x y 0

2 0

(1 + ) hK y (1 + 2k x +2k y )K y

x y

0 0

1 1

2 0 2 2 2 2

(1 + ) h S (x y ) S (x y )

0 0

2 2

(1 + )S xy S xy

0 0

Thus the equations of motion (4.5) b ecome

00 2 2 2

x + (K + k )x K y = k k +(K +2k )x

0 x x 0

x x

2 2 2 2

S (x y ) 2K k xy (2K + k )k x

0 0 y 0 x

K y +2K k y +(S +2K k )xy (4.6)

y x

0 0 0 0

and

00 2 2 2

y (K k )y K x = k k (K 2k )y +

0 y y 0

y y

2 2

+ (2K k )k y +(S +2K k )xy

0 y 0 0 x

2 2 2

+2K k x +K k xy + S (x y ) : (4.7) K

x y

0 0 0 0

We shall further assume that

K S K S ; (4.8)

0 x;y 0 x;y

0 0

which is generally valid in large synchrotrons. Hence, some terms in (4.6) and (4.7) may

b e neglected since

2 2 2 2

K k xy S (x y ) K k xy S (x y )

0 y 0 y

0 0

K k xy S xy K k xy S xy

0 y 0 x

0 0

2 2

K k u K u K k u K u

0 u 0 u

0 0

3 2 2

k u k u

u u

where u denotes x or y . Then the equations of motion reduce to the form

! !

1 2

y + x + K + x = K + K + x

0 0

2 2

x x

x x 45

2 2

S (x y ) K y + S xy (4.9)

0 0

! !

2 1

K y K y = K x + y +

0 0

2 2

y y

2 2

+ S xy K x + S (x y ) : (4.10)

0 0

If we consider particles with the design momentum p , the equations of motion expanded

up to sextup ole terms read

1 1

00 2 2

x + K + x = K y S (x y )+S xy (4.11)

0 0

1 1

00 2 2

y K y = K x + S xy + S (x y ) : (4.12)

0 0

When the design orbit lies only in the horizontal plane, the vertical equation of motion

simpli es to

00 2 2

y K y = K x + S xy + S (x y ) : (4.13)

0 0

Rememb er that the magnet parameters ; K ; K ; S and S are generally

x;y 0 0

0 0

functions of the s-co ordinate. In practice they are piecewise constant functions along the

lattice.

4.2 Description of motion in normalized co ordinates

All terms on the right-hand sides of equations (4.9) and (4.10) will b e treated as

small p erturbations. The left-hand side of these equations yields the linear unp erturb ed

equations of motion 46

x + K + x = 0

y K y = 0 : (4.14)

For closed lattices these equations are Hill's equations with p erio dic co ecients. They can

b e written in the compact form, writing u for either x or y ,

u + K (s)u =0 (4.15)

where K (s) is a p erio dic function with p erio d L equal to the length of a machine cell

K (s + L)=K(s) (4.16)

with

: K (s)=K (s)+

(s)

x;y

Perturbation terms in the equations of motion may lead to unstable b eam motion,

called resonances, when the p erturbating eld acts in synchronism with the particle os-

cillations. A multip ole of nth order is said to generate resonances of order n. Resonances

b elow the third order (i.e. due to dip ole and quadrup ole eld errors for instance) are called

linear resonances. The nonlinear resonances are those of third order and ab ove.

By an appropriate transformation of variables we can express the equations of

motions (4.9) and (4.10) in the form of a p erturb ed harmonic oscillator with constant

frequency.To this end weintro duce the normalized co ordinates (; ), called Flo quet's

co ordinates, though the transformation (u; s) ! (; )

u 1 dt

= = (4.17)

Q (t)

(s)

where Q is the machine tune

s+C

1 dt

(4.18) Q =

2 (t)

C is the ring circumference: C = NL, where N is the numb er of p erio dic cells.

Since u = u( ; )we compute

@u @u 1

0 0 1=2 0 0 0

u = + : + =

1=2

@ @ 2

0 0 0 0

Similarly, since u = u ( ; ;;)we nd 47

0 0 0 0

@u @u @u @u

00 0 00 0 00

u = + + + =

0 0

@ @ @ @

0 0

1 1

0 0 00 0 1=2 00

= + + + + =

3=2 1=2 1=2 1=2

4 2 2 2

2 0 00

1=2 00 0

: = + +

1=2 1=2 3=2

2 2

Hence, with

d d d d 1

= =

ds ds d Q d

and

! !

2 2 0 0

d d d d d d d 1 1 d 1

= = + = + ;

2 2 2 2 2 2

ds ds Q d Q d Q ds d Q d Q d

it follows that

! ! !

0 00 2

1 1 1

1=2 0 00 1=2

_ + + _ + u =

2 2 2 1=2 3=2

Q Q Q 2 2

in whichapoint denotes the derivative with resp ect to . Inserting u into the Hill's

equation (4.15) yields the unp erturb ed equation of motion expressed in normalized co or-

dinates

1 1 1

3=2 1=2 00 1=2 3=2 2

=0: + +2K(s)

Q 2 2

At this stage weintro duce a general p erturbation term p(x; y ; s) in the right-hand

side of the latter equation [such a term may b e the right-hand side of equation (4.9) or

2 3=2

(4.10)]. Multiplying the resulting equation by Q yields

1 1

2 2 3=2 00 2 2

+ Q = Q p(x; y ; s) : (4.19) + K (s)

2 4

It is known that the b etatron function (s) satis es the di erential equation

1 1

00 2 2

+ K (s) =1 : (4.20)

2 4

Consequently, the p erturb ed Hill's equation is transformed into

2 2 3=2

+ Q = Q p(; ) (4.21)

assuming a p erturbation of the form p(u; s) (one degree of freedom). When the p erturba-

tion vanishes, equation (4.21) reduces to a harmonic oscillator with frequency Q, whose

solution is

()= acos (() ') (4.22) 48

with

()=Q = : (4.23)

(t)

Di erentiating (4.22) with resp ect to the variable gives

= a sin [() ']

and then

2 2

+ = a : (4.24)

The particle tra jectory in the phase plane (; )isthus a circle of radius equal to the

amplitude a of the oscillation . The phase () advances by2 every b etatron oscillation

(i.e. the tra jectory describ es one full phase circle) or by2Q every machine revolution.

dη dµ

a µ

Figure 19: Phase circle in normalized co ordinates.

In terms of normalized co ordinates, the equations of motion (4.11) and (4.12) for

an on-momentum particle (with = 0) reads

S S

0 0

2 2 3=2 1=2 2 3=2 2 2 1=2 5=2

+ Q = Q K S + (4.25)

0 0

x x x y x x y x

2 2

1 1

2 2 1=2 3=2 3=2 2 5=2 2 1=2 2

S S + Q = Q K + + S (4.26)

x 0

0 0 0

y y x y y y x y

2 2

in which is the normalized co ordinate for the vertical plane

y x

q q

= (4.27) =

(s)

where (s) is the horizontal or vertical b etatron function, and since is the indep endent

x;y

variable in the di erential equations (4.25) and (4.26), it can b e de ned as 49

1 dt

= (4.28)

Q (t)

x;y x;y

; where Q is the horizontal or vertical machine tune. The multip ole strengths K ;K

x;y 0

S;S , and the b etatron functions are p erio dic functions of with p erio d 2 .

0 x;y

Nowwe consider the case where the general p erturbation term p(x; y ; s) represents

the magnetic eld error B , that is the eld not considered in setting up the design

x;y

orbit of the machine. Equivalently,B is the eld error with resp ect to the ideal lattice,

x;y

even if such a lattice includes nonlinear magnets like sextup oles. Expanding B into

x;y

multip oles up to the third order, in which the strength parameters are replaced by their

variations: B for the dip ole eld error, K and K for the quadrup ole gradient errors,

x ;y

0 0

S and S for the sextup ole errors, we obtain

e e 1

2 2

B = B + K y + K x + S xy + S(x y )

x x

p p 2

0 0

e 1 e

2 2

S (x y ) S B = B + K x K y + xy :

y y

p p 2

0 0

Hence, using (4.21), (4.27) and (4.28) we get the equations of motion in normalized co or-

dinates

2 2 3=2 2 3=2 1=2

+ Q = Q B + K K +

x x x 0 x x y

S S

2 3=2 2 2 1=2 5=2

S +

x x y x

2 2

2 2 3=2 2 1=2 3=2

+ Q = Q B + K K +

y y y 0 y x y

1 1

1=2 2 3=2 2 5=2 2

S + S S : (4.29) +

x y y y

2 2

We can rewrite the last two equations considering only the nth order multip ole term in

the horizontal plane and the nth order multip ole term in the vertical plane. We get

2 n1 r 1

+ Q = p ()

nr x

2 n1 r 1

+ Q = p () : (4.30)

nr y

y 50

Perturbation terms

n1 r 1 n1 n1

Order p () p ()

nr x nr y

n r

e e

2 3=2 2 3=2

1 1 Q B Q B

y x

x x 0 y y 0

p p

0 0

2 3=2 1=2 2 2

1 2 Q K Q K

x x y y y

2 2 2 1=2 3=2

2 1 Q K Q K

x x y x y

2 2 1=2 2 1=2 2

2 2 Q S Q S

x x y y x y

1 1

2 3=2 2 2 5=2 2

1 3 Q S Q S

x x y y

2 2

1 1

2 5=2 2 2 3=2 2

3 1 Q S Q S

x x y y

2 2

4.3 One-dimensional resonances

Considering only the nth order multip ole horizontal p erturbation term, the equation

of motion (4.30) in one dimension may b e written [with r = 1 and Q stands for Q and

p () for p ()] as

n n1x

2 n1

+ Q = p () : (4.31)

Expanding the p erturbation into Fourier series yields

p ()= p^ (m)e (4.32)

n n

m=1

wherep ^ (m) is the Fourier co ecient of the expansion

p^ (m)= p ()e d : (4.33)

n n

The unp erturb ed oscillation , solution of the inhomogeneous equation (4.31), may b e writ-

ten as

iQ iQ

()=ae + be (4.34)

where a and b are constants of integration.

Assuming the p erturbation is small, we can insert ()into the right-hand side of

the equation of motion (4.31). We nd

2 im iQ iQ n1

+ Q = p^ (m)e (ae + be ) :

m=1 51

Using the binomial expansion, we get

(n 1)!

n1 iQ iQ n1 nk 1 i(nk 1)Q k ik Q

() = (ae + be ) = a e b e

k !(n k 1)!

k =0

(n 1)!

nk 1 k i(n2k 1)Q

a b e : =

k !(n k 1)!

k =0

Performing the change of variable p =2kn+ 1, the ab ove expression may b e rewritten

n1 ipQ

() = c(p)e

p=n+1

with

h i

n+p1 np1

(n1)! 1

n+p1

2 2

c(p)= b : 1+(1) a

n+p1 np1

! !

2 2

Hence (4.31) reads

1 n1

X X

2 i(mpQ)

+ Q = p^ (m)c(p)e : (4.35)

m=1

p=(n1)

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 comp onent of the p erturbation

2 i(mpQ)

+ Q =^p (m)c(p)e : (4.36)

The solution of this equation is the sum of the solution of the homogeneous equation and

a particular solution of the form

i(mpQ)

= Ae : (4.37)

The constant A is determined byintro ducing (4.37) into (4.36)

i h

i(mpQ) i(mpQ) 2 2

Ae =^p (m)c(p)e (m pQ) + Q

from which A is found and

p^ (m)c(p)

i(mpQ)

()= e : (4.38)

[Q(p1) m][Q(p +1) m]

Thus the motion b ecomes unstable when the tune satis es the resonance conditions

m =(p1)Q with jpjn1 (4.39)

or equivalently

jmj =(jpj1)Q (4.40) 52

where jpj + 1 is called the order of resonance and m the order of the p erturbation Fourier

harmonic.

The resonance conditions apply only for index p such that the co ecient c(p) is non-

zero. Thus for the third order resonance n = 3 driven by sextup ole elds, we compute for

p = 2; 1; 0; 1; 2:

2 2

c(2) = a c(1)=0 c(0)=2ab c(1)=0 c(2) = 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 (4.41)

where k is any p ositiveinteger.

4.4 Coupling resonances

The two-dimensional equations of motion with only the nth order multip ole hori-

zontal and r th order vertical multip ole p erturbation are given by (4.30)

2 n1 r 1

+ Q = p ()

nr x

2 n1 r 1

+ Q = p () :

nr y

Again expanding the p erturbation in Fourier series yields

p () = p^ (m)e

nr x nr x

m=1

p () = p^ (m)e : (4.42)

nr y nr y

m=1

Furthermore, the solutions of the unp erturb ed equations are

iQ iQ

x y

() = a e + b e

0 x x

(4.43)

iQ iQ

y y

() = a e + b e

0 y y

in which a and b are constants at integration. Hence

x;y x;y

i`Q n1

c (`)e () =

x 0

`=n+1

r 1

r 1 ipQ

() = c (p)e

0 y

p=r +1 53

with

h i

n+`1 n`1

1 (n 1)!

n`1

2 2

b c (`) = 1+(1) a

x x

n+`1 n`1

! !

2 2

h i

r p1

r +`1

1 (r 1)!

r p1

2 2

c (p) = b : 1+(1) a

y y

r p1 r +p1

! !

2 2

In the same way as in the uncoupled treatment of resonances, we obtain, after substitu-

tion of the unp erturb ed oscillations into the right-hand side of the p erturb ed di erential

equations (4.30),

X X X

2 i(m`Q pQ )

x y

+ Q = p^ (m)c (`)c (p)e

nr x x y

m p

X X X

2 i(m`Q pQ )

x y

+ Q = p^ (m)c (`)c (p)e : (4.44)

nr y x y

m p

Then considering a single comp onent of the p erturbation, particular solutions of

these equations may b e derived. We obtain

p^ (m)c (`)c (p)

nr x x y

i(m`Q pQ )

x y

e () =

[Q (` 1) + pQ m][Q (` +1)+pQ m]

x y x y

p^ (m)c (k )c (q )

nr y x y

i(mkQ qQ )

x y

() = e :

[Q (q 1) + kQ m][Q (q +1)+kQ m]

y x y x

(4.45)

The resonance conditions are then

m =(`1)Q + pQ with j`jn1 and jpjr1;

x y

m=(q1)Q + kQ with jk jn1 and jq jr1 : (4.46)

y x

For example, we consider the resonances driven by a regular sextup ole for which the

equations of motion are

2 2 2

+ Q = p () + p ()

31x 13x

+ Q = p () : (4.47)

22y

The resonance conditions are given in the following table. 54

Order Horizontal motion Vertical motion

n r ` p m =(`1)Q + pQ m =(p1)Q + `Q

x y y x

2 2 1 1 m =

2Q Q

y x

1 1 m =

2Q Q

y x

2Q + Q

y x

1 1 m =

2Q + Q

y x

1 1 m =

Q 2Q

x y

1 3 0 2 m =

Q 2Q

x y

0 0 m =

Q +2Q

x y

0 2 m =

Q +2Q

x y

3 1 2 0 m =

0 0 m =

2 0 m =

Discarding the redundant conditions, we are left with

jmj = Q jmj =2Q Q jmj=3Q :

x y x x

Having considering a skew sextup ole instead, the resonance conditions would have b een

jmj = Q jmj =2Q Q jmj=3Q :

y x y y

Thus, the general resonance conditions in two degrees of freedom may b e cast into the

form

MQ + NQ = P; (4.48)

x y 55

where M; N; and P are integers, P b eing non-negative, and jM j + jN j is the order of

the resonance, and P is the order of the p erturbation harmonic. Plotting the resonance

lines (4.47) for di erentvalues of M; N and P in the (Q ;Q ) plane yield the so called

x y resonance or tune diagram.

3Qx = 3k + 1 2Qx = 2k + 1 3Qx = 3k + 2 k + 1 Qx + 2Qy = 3k + 2

2Qx + QyQx = 3k + + Qy1 = 2k + 1 Ð Qx = k + 1 2Qy

Qy = 0 Ð Qy = k + 1 Ð Qx 2Qx k + 2 3Qy = 3k + 2 3

k + 1 2Qy = 2k + 1 2

k + 1 3Qy = 3k + 1 3 2Qx + Qy = 3k + 2

Qy = k Ð Qx + 2Qy = 3k + 1

2Qx Ð Qx = k 2Qy

Qx k + 1 k k + 1 k + 1 k + 2

3 2 3

Figure 20: Resonance diagram in a unit square for regular and skew multip ole elds up to the

3rd order.

If the integers M and N are p ositive, the resonance is called a sum resonance and

can lead to a loss of b eam. If M and N are of opp osite sign, the resonance is called

a coupling resonance or a di erence resonance and do es not cause a loss of b eam, but

rather leads to a b eating b etween the horizontal and the vertical oscillatio ns. It can b e

shown that the \strength" of the resonances decrease with increasing order, so that only

low-order resonances need to b e 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 b e unstable, dep ending

on the oscillation amplitude. When the resonance is linear (b elow the third order), the

resonance width is called a stopband b ecause the entire b eam b ecomes unstable if the

op erating p oint Q ;Q reaches this region of tune values. The largest oscillation amplitude

x y

which is still stable in the presence of nonlinearities is called the dynamic ap erture.

The following gures are tune diagrams in a unit square around the origin showing

the resonance lines driven by regular multip ole elds up to the 12th order. 56

Figure 21: Resonance diagrams for regular multip ole elds up to 3rd and 6th orders.

Figure 22: Resonance diagrams for regular multip ole elds up to 9th and 12th orders.

5 THE THIRD-INTEGER RESONANCE

5.1 The averaging metho d

Consider the equation of motion in Flo quet co ordinates for an nth order multip ole

p erturbation

2 n1

+ Q = p () (5.1)

where the p erturbation term p () is p erio dic in , with p erio d 2 .

Assume that the tune Q is close to an nth order resonance

Q Q = : (5.2)

in which m is a harmonic of the p erturbation p ().

n 57

De ning the tune deviation Q between Q and Q as

Q = Q Q

the equation of motion can then b e written as

2 n1

+ Q = p () 2Q Q (5.3)

n r

since for small tune deviation

2 2

Q Q +2Q Q :

The averaging metho d of Krilov and Bogoliub ov assumes that the solution of (5.3) is

nearly p erio dic in the variable .Writing this solution as

()= a() cos [Q + '()] (5.4)

with slowly-varying amplitude a() and phase '()over a \cycle" 2 ,we imp ose the

condition that _ () should have the form

_ ()=a()Q sin [Q + '()] : (5.5)

r r

By di erentiating (5.4), this assumption leads to the additional condition

a_ () cos () a()_'() sin ()= 0 ; (5.6)

where ()= Q +'(). Di erentiating now_() yields

= Q a cos aQ_ sin Q a '_ cos :

r r

Substituting and into (5.3) gives

n1 n1

Q a_ sin Q a'_ cos = p ()a cos 2Q Qa cos :

r r n r

Adding this expression, multiplied by cos , to equation (5.6) multiplied by Q sin , one

obtains

p () d'

n2 n 2

= a cos +2Q cos (5.7)

d Q

p () da

n1 n1

= a cos sin +2a Q cos sin ; (5.8)

d Q

where the latter equation is derived from (5.6), using' _ given in (5.7).

The p erio dic p erturbation p ()may b e expanded in a trigonometric Fourier series

p^ (0)

(k) sin k] (5.9) [^p (k) cos k +^p p ()= +

n n

k =1 58

with

p () cos k d p^ (k ) =

n n

p^ (k) = p () sin k d

wherep ^ (k ) andp ^ (k ) are the Fourier co ecients of the series. Considering only the single

Fourier comp onent^p (m) which drives the resonance Q = m=n, the di erential equations

n r

for a and ' may b e rewritten as

da n

n1 n1

= p^ (m)a cos n( ') cos sin +

d m

+2aQcos sin (5.10)

d' n

n2 n 2

= p^ (m)a cos n( ') cos +2Q cos (5.11)

d m

using

' n

= ( ') : =

Q m

As a() and '() are slowly varying functions of over a \cycle" 2 ,_aand' _ may

b e treated as p erio dic in with p erio d 2 and thus expanded in a Fourier series

^ ^

a_ ( )= a_ (k )e + a_ (0) (5.12)

k=1

k 6=0

^ ^

'_ (k )= '_ (k )e + '_ (0) (5.13)

k=1

k 6=0

with

a_( )e d (5.14) a_ (k )=

'_ (k )= '_( )e d : (5.15)

^ ^

Exp onential Fourier series have b een considered, and the co ecients a_ (0) and '_ (0)

have b een explicitly written for convenience. The principle of the averaging metho d con-

sists in replacinga _ ( ) and' _ ( )by their average parts ha_ i and h'_ i, which is equivalent

to assuming thata _ and' _ are not in uenced by small rapid oscillatio ns, that is

a_ = ha_ i + small rapidly oscillating terms ha_i

'_ = h'_i+ small rapidly oscillating terms h'_i: 59

Hence, taking into account (5.12) to (5.15) we obtain

* +

da da da 1

= a_ (0) = d (5.16)

d d 2 d

* +

d' d' d' 1

= '_ (0) = d (5.17)

d d 2 d

or equivalently, using (5.10) and (5.11)

da n

n1 n1

p^ (m)a cos n( ') cos sin d (5.18)

d 2m

d' n

n2 n

Q p^ (m)a cos n( ') cos d : (5.19)

d 2m

In summary, the solution of the nonlinear equation of motion (5.1) when the tune

is close to a resonance Q m=n is

()= a() cos [Q + '()]

where a() and '() are given in the rst approximation by the averaged di erential

equations (5.18) and (5.19).

5.2 The nonlinear sextup ole resonance

The third-integer resonance is driven by sextup olar elds

Q = :

The equation of motion for the mth harmonic of the p erturbation p ()is

2 2

+Q 1+2 =^p (m) cos m : (5.20)

For Q close to Q the solution of the equation of motion is given by (5.4)

()= a() cos + '() : (5.21)

The averaged di erential equations for a() and '() are, after integration of (5.18) and

(5.19) for n =3.

da 3

= p^ (m)a sin 3' (5.22)

d 8m

d' 3

= Q p^ (m)a cos 3' (5.23)

d 8m 60

where the following trigonometric formulae have b een used

cos 3( ') = cos 3 cos 3' + sin 3 sin 3'

cos 3 = 3 cos + 4 cos

sin 3 = 3 sin 4 sin

and

(2m)!(2n)!2

2m 2n

sin cos d =

2(m+n)

m!n!(m + n)!2

m n

sin cos d = 0 if m and=or n odd :

Eliminating the variable from (5.22) and (5.23) yields

d' 3^p (m)a cos 3' 8mQ

= : (5.24)

da 3^p (m)a sin 3'

By means of the change of variable

z = a cos 3'

the latter equation transforms into

# "

a 4m Q

da + dz =0 ; z

p^ (m) 2

which can b e integrated to give

" #

1=2

4m Q

ln a +ln z =ln A;

p^ (m)

where A is a constant dep ending on the initial conditions. Transforming back to the

original variable ' we nd

" #

4m Q

cos 3' a = A: (5.25)

p^ (m)a

The solutions () and _ ()may b e represented on the phase plane with Cartesian axes

; _ .Asincreases, [ (); _ ()] traces out a phase path. A more appropriate phase plane

can b e chosen with axes ; p , with

3 m

p () _ ()= a() sin + '() : (5.26)

m 3

The phase plane co ordinates may b e rewritten, using the trigonometric formulae for the

sum of the angles

m m

= X cos + Y sin (5.27)

3 3 61

m m

p = X sin + Y cos (5.28)

3 3

with the new phase plane variables X ();Y() de ned by

X ()=a() cos '() (5.29)

Y ()=a() sin '() : (5.30)

The link b etween the variables (; p ) and (X; Y ) is a rotation of an angle m=3. Thus

it is sucient to study the representative phase plane with Cartesian co ordinates (X; Y )

or p olar co ordinates (a; '). In particular, for every multiple of three machine revolutions

(i.e. for =6k, with k =0;1;2:::), one has

(6k)= X(6k) and p (6k)= Y(6k) :

Y Phase path pη φ) η a (

ϕ (φ) m φ 3 X

Figure 23: Co ordinate systems for the (; p ) and (X; Y ) phase planes.

Fixed p oints in phase plane are p oints for which

dX dY

= =0 ; (5.31)

d d

^ ^ ^ ^

that is X ()=X and Y ()= Y where X and Y are constants. Fixed p oints are equilib-

rium p oints (stable or unstable). It follows from this that

da d'

= =0 (5.32)

d d

b ecause

_ _

a_ = X cos ' Y sin '

_ _

'_ = (X sin ' + Y cos ') :

a 62

Thus, according to (5.22) and (5.23), xed p oints are solutions of the equations

sin 3' =0

p^ (m)a cos 3' = Q

which gives

3^' =2k and 3^' =(2k+1) k =0;1;2

2 4 5

'^ =0; ; and '^ = ;; : (5.33)

3 3 3 3

The rst three xed p oints give cos 3' ^ = 1, the other three give cos 3' ^ = 1. It follows

that

^a = Q (5.34)

3^p (m)

where the sign + is chosen when Q > 0 and vice versa, since the amplitudea ^ cannot b e

negative [assumingp ^ (m) p ositive]. One more xed p ointisgiven considering the trivial

case

^ ^

a^ =0 (i:e:X = Y =0) :

Inserting (5.34) into the general solution (5.25) of the averaged equations for a and '

yields

3 a^

cos 3' a = A: (5.35)

2 a

Expressed with the variables X and Y this equation b ecomes, using (5.29) and (5.30),

3 2 2 2

X 3XY ^a(X + Y )=A (5.36)

or, solving the quadratic equation,

3 2

2X 3^aX 2A

Y = (5.37)

6X 3^a

with

3 3 3 3 2

a cos 3' = a (4 cos ' 3 cos )=4X 3a X

2 2 2

a =X +Y :

The constant A dep ends on the initial conditions X (0) and Y (0). In particular, the con-

ditions (5.33) and (5.34) for xed p oints give

a^ a^ a^ ^a

^ ^

X =^a; ; and X = ; a^;

2 2 2 2

p p p p

3 3 3 3

^ ^

Y =0; a;^ and Y = a;^ 0; a:^

2 2 2 2 63

^ ^

Replacing X and Y by X and Y in (5.36), or b etter a and ' by^aand' ^ in (5.35), one

nds

A =

where the minus sign is chosen when Q > 0 and vice-versa.

From now on, we shall assume that Q > 0. Hence, for the ab ove xed p oint

conditions, equation (5.36) reduces to

3 a^

3 2 2 2

X 3XY a^(X + Y )=

2 2

whichmay b e factorized to give

! ! !

X a^ X a^ a^

p p

Y Y + X + =0 : (5.38)

3 3

This yields a family of three straight lines, called separatrices. These three separatrices

a^ 3

de ne a triangular area in the phase plane. The three xed p oints (a ^; 0); ; a^ , and

2 2

a^ 3

; a^ are at the intersections of the separatrices.

2 2

Y Separatrix

Separatrix â 2π ϕ = ˆ 3 X Ð â 0 2 â δQ > 0

Separatrix

Figure 24: Phase plane separatrices near the third-integer resonance.

Plotting equation (5.37) for di erentvalues of the constant A gives the phase plane

p ortrait of the equation of motion (5.20). 64

Figure 25: Phase p ortraits near the third-integer resonance fora ^ = 2 (left) and ^a =1=10 (right).

The phase path description in phase plane (X; Y ) is equivalent to the \strob oscopic"

representation of paths in phase plane (; p ) in which a sequence of p oints are plotted

at interval 2k(k =0;1;2:::) corresp onding to every machine turn at xed azimuthal

lo cation. The separatrices de ne a b oundary b etween stable motion (b ounded oscillatio ns)

and unstable motion (expanding oscillations). The xed p oints are equilibrium p oints

(stable or unstable) of p erio d 6 corresp onding to three machine turns.

If we had considered a negative tune variation (Q < 0) rather than a p ositive one

p p

^a 3 3 a^

(Q > 0), the three xed p oints would have b een lo cated at (a^; 0); ; ^a ; ; ^a ,

2 2 2 2

yielding a reversed phase p ortrait with resp ect to the Y -axis.

When the tune shift Q is reduced to zero, the phase plane triangle shrinks to a

p oint. The stable area disapp ears and the separatrix crosses the origin.

The stability ab out the xed p oints may b e examined by expanding a asa ^ +a

and ' as' ^ +' in (5.22) and (5.23). Keeping only the linear terms in a and ',we

obtain

d 3 6

(â +a) = p^ (m)â sin 3' ^ p^ (m)â sin 3' â

3 3

d 8m 8m

9 9

2 2

p^ (m)^a cos 3' ^ ' = p^ (m)^a '

3 3

8m 8m

since cos 3' ^ = 1. Therefore

da 9

= p^ (m)^a ':

d 8m

Similarly we compute to the rst order

d' 3

= p^ (m)a:

d 8m 65

These two equations maybecombined to give

d a

k a =0 (5.39)

d '

k ' =0 (5.40)

with

3 3

k = p^ (m)^a:

Solutions for a and ' near xed p oints are of the form

k k

a = A e + A e

1 2

k k

' = B e + B e ;

1 2

where A ;A ;B ;B are constant, dep ending up on the initial conditions. These solutions

1 2 1 2

show that the motion may either converge to or diverge from the xed p oint, dep ending

up on the initial conditions. This kind of xed p oint is said to b e hyp erb olic. Hyp erb olic

xed p oints are unstable b ecause any p oint near to them will eventually moveaway from

them.

For the trivial xed p oint^a() = 0 one gets

da d'

=0 and = Q ;

d d

which gives

a = C and ' = Q + C ;

1 2

where C and C are constants. The motion describ es circles around the origin, and the

1 2

xed p oint a = 0 is said to b e elliptic. Elliptic xed p oints are stable b ecause any p oint

near to them remains always in the vicinity of them.

The third-integer resonance may b e used for resonant extraction. The principle con-

sists in approaching the third integer resonance byvarying the strengths of quadrup oles.

Hence, as the triangle area shrinks, the particles in a b eam are moved out along the three

arms of the separatrices jumping from one arm to the next each turn. When the displace-

ment of a particle reaches the values X , it jumps the extraction septum. The septum

width is equal to the growth in the X -direction in three turns.

s 66 Y

Turn 1

Stable area Septum ∆ s X Xs

Turn 2 Turn 3

δQ > 0 Kick from septum

Figure 26: Resonant extraction scheme in phase plane.

Finally,we shall estimate the third-integer resonance width. Supp ose that the tune

Q is initially set far from the resonance Q = m=3 so that the b eam emittance may

b e represented by a phase ellipse entirely lo cated inside the (; p ) phase plane triangle

de ned by the tune di erence Q = Q Q . The phase ellipse in the (u; u ) phase plane,

given by the Courant{Snyder invariant

2 0 2

(s)u +2 (s)uu + (s)u = (5.41)

transforms to a phase circle in the phase planes (; p ) and (X; Y )

2 2 2 2

+ p = X + Y = (5.42)

by (5.27), (5.28), and (4.17), and b ecause

u = (p ) :

1=2

Let Q slowly approache Q , then the triangle area shrinks and the phase circle

gradually distorts into a triangular shap e, while its area is kept conserved. We de ne

the resonance width Q as b eing twice the tune di erence Q which yields the triangle

area equal to .

Y Distorted phase 3â 2 ellipse: area πε

X Ð â â 2 Triangle of area 3 3 â2

Figure 27: Distorted b eam phase ellipse within the stable phase plane region de ned by the

separatrices near the third-integer resonance. 67

By (5.34), the width of the third-integer resonance may b e written as

1=2

( 3)

Q 2jQj = p^ (m) : (5.43)

5.3 Numerical exp eriment: sextup olar kick

We shall study more quantitatively the motion of an on-momentum particle in a

circular accelerator sub ject to a p erio dic sextup olar p erturbation. The sextup ole eld

is assumed to b e a \p oint-like" p erturbation at lo cation s (i.e. the particle p osition is

assumed not to vary as the particle traverses the eld). Thus, at each turn the lo cal

magnetic eld gives a \kick" to the particle, de ecting it from its unp erturb ed tra jectory.

This metho d of description is referred to as a \kick" mo del.

Written in the normalized co ordinate (; ) the one-dimensional equation of motion

is, according to (4.31),

2 2

+ Q = A () ( 2m) (5.44)

m=0

where Q is the machine tune, () is the Dirac distribution, and A is the strength of

the impulse. The right-hand side of (5.44) corresp onds to the p erio dic p erturbation term

p () in (4.31), with p erio d 2 . The kick strength A may b e written as

5=2

A = Q S ` (5.45)

in which is the b etatron function at s (i.e. at = 0), the factor S ` is the integrated

0 0 0

sextup ole strength (in thin-lens approximation the sextup ole length ` ! 0 while keeping

constant the pro duct of the sextup ole strength S by `). Between kicks the right-hand

side of (5.44) is zero, and the unp erturb ed solution is

()=a cos (Q + ' ) for 2 (n 1) <2n (5.46)

n n

where a and ' are integration constants, and

n n

_ ()=a Qsin (Q + ' ) (5.47)

n n

where n =1; 2; ::: .

For eachvalue of n,at =2n, the angle _ changes discontinuously while the

p osition is not a ected since the p erturbation is p oint-like. The increase of _ at the

passage of the sextup olar kick on the nth machine turn is obtained byintegrating (5.44)

from 2n to 2n + , b eing an arbitrarily small numb er, and then taking the limit

! 0 68

" #

Z Z

2n+ 2n+

2 2

_ = ()d = Q () A () ( 2m) d

2n 2n

m=0

2 2

= A (2n) A ;

where denotes the limit of () for ! 2n. De ning the new phase plane variable

p ()as

p ()= _();

the discontinuous change in \angle" p ()reads

p p (2n)= : (5.48)

According to (5.46) wehave for n +1

()= a cos (Q + ' ) for 2n<2(n+1)

n+1 n+1

p ()=a sin (Q + ' ) ; (5.49)

n+1 n+1

where a and ' are other integration constants. In particular

n+1 n+1

= a cos (2nQ + ' )

n n n

p = a sin (2nQ + ' ) ; (5.50)

n n n

where p denotes the left-limit of p () for ! 2n from the left [also written as

p (2n)].

Similarly, the right-limits of () and p () for ! 2n from the right, written

(2n+) and p (2n+), read

(2n+) = a cos (2nQ + ' )

n+1 n+1

p (2n+) = a sin (2nQ + ' ) : (5.51)

n+1 n+1

Hence, since () do es not change b etween 2n and 2n + (with ! 0) while p ()

changes according to (5.48), wemust have

(2n+) =

p (2n+) = p +p (5.52)

n n 69

that is

a cos (2nQ + ' ) = a cos (2nQ + ' )

n+1 n+1 n n

a sin (2nQ + ' ) = a sin (2nQ + ' )+ : (5.53)

n+1 n+1 n n

Multiplying the rst equation (5.53) by cos 2nQ and the second by sin 2nQ, and then

summing the results, we nd, using the formulae

cos (2nQ + ' ) = cos 2nQ cos ' sin 2nQ sin '

n n n

sin (2nQ + ' ) = sin 2nQ cos ' + cos 2nQ sin ' (5.54)

n n n

the relationship b etween the co ecients in the nth interval and those in the (n + 1)th

interval

a cos ' = a cos ' sin 2nQ : (5.55)

n+1 n+1 n n

Similarly,we can derive

a sin ' = a sin ' cos 2nQ : (5.56)

n+1 n+1 n n

Now, by equations (5.49), we can write for ! 2 (n +1)

= a cos [2 (n +1)Q+' ]

n+1 n+1 n+1

p = a sin [2 (n +1)Q+' ] : (5.57)

n+1 n+1 n+1

Expanding (5.57) using (5.54), and inserting (5.55) and (5.56) into the result yields after

some sorting

= cos 2Q + p + sin 2Q

n+1 n n

cos 2Q : (5.58) p = sin 2Q + p +

n+1 n n

These equations give the particle p osition and \angle" just b efore the phase 2 (n +1)

(i.e. just b efore the passage of the sextup olar eld on turn n + 1) in terms of p osition

and angle just b efore the phase 2n. Equations (5.58) form a quadratic mapping in phase

plane (; p ), called Henon mapping. Mappings with di erent initial conditions ( ;p )

0 0

have b een computed and plotted in Figs. 28{29 for various values of Q (with A=Q = 1).

This yields a strob oscopic representation of phase-space tra jectories on every machine

turn at s (i.e. at the phase values of =0; 2; 4 ; :::).

0 70

Figure 28: Phase plane plots caused by a single thin sextup ole for Q =0:324 (close to 1/3) and

for Q =0:320 (close to 1/3).

Figure 29: Phase plane plots caused by a single thin sextup ole for Q =0:252 (close to 1/4) and

for Q =0:211 (close to 1/5). 71

Tune Characteristics of the motion

Q =0:324 Regular (i.e. predictable) b ounded motion at small `amplitudes'

close to (within the stable triangle).

Divergent (unstable) regular motion at larger amplitudes

(outward tra jectory near the outermost contour).

Q =0:320 Regular b ounded (stable) motion at small amplitudes.

close to

Three regular resonance islands at larger amplitudes

(the tra jectory breaks into a chain of three closed curves).

Eight secondary regular resonance islands are visible

within each of the three primary islands.

Chaotic (i.e. unpredictable) b ounded motion at larger amplitudes.

Q =0:252 Regular b ounded motion at small amplitude.

close to

Four regular resonance islands at larger amplitudes,

surrounded by regular tra jectories.

Chaotic b ounded and unb ounded motion at still larger amplitudes.

Q =0:211 Regular b ounded motion at small amplitudes.

close to

Five regular resonance islands at larger amplitudes,

surrounded by regular tra jectories. A chain of 16 smaller regular

islands is visible at larger amplitudes.

Chaotic motion at still larger amplitudes.

The analytic approach for studying the third-integer resonance tra jectories by p er-

turbation techniques (e.g. the averaging metho d) is useful when the fractional part of the

machine tune is close to 1/3. Indeed, the triangular phase-plane b oundary b etween stable

and unstable motions as well as the separatrices may b e determined. However, unlike the

kick mo del, the large amplitude description of the motion is incorrect: islands are not

revealed and divergent tra jectories, which might return, do not.

In general, the equations of motion in the presence of nonlinear elds are untractable 72

for any but the simplest situations. An alternative approach to study phase plane tra-

jectories and determine the regions that are stable consists to p erform numerical particle

\tracking" exp eriments using computer programs. Tracking consists to simulate particle

motion in circular accelerators in the presence of nonlinear elds. Test particles with initial

phase plane co ordinates at a xed azimuthal lo cation along the accelerator are tracked

for many turns through the lattice using various numerical techniques. The particle co-

ordinates at the end of one turn may b e viewed as a nonlinear mapping of the initial

conditions, that sp eci es the motion for one turn. The resulting strob oscopic phase plane

plot of circulating particles for many turns (obtained by iterative mapping) thus reveals

the largest surviving oscillatio n amplitudes (that enclose a central stable phase plane

region), which then de ne the dynamic ap erture. A simple technique used in tracking

programs consists to describ e the mapping by means of the kick mo del exp osed ab ove:

any nonlinear magnet is treated in the \p oint-like" approximation, the motion in all other

elements of the lattice is assumed to b e linear. Alternative metho ds of description of the

mapping are available (e.g. the \higher-order" matrix metho d based on Taylor series ex-

pansion, the canonical integration metho d whichnumerically integrate the equations of

motion, and the Lie algebraic formalism).

6 CHROMATICITY

6.1 Chromaticity e ect in a closed lattice

The fo cal prop erties of lattice elements dep end up on the momentum deviation, since

the equations of motion of an o -momentum particle are, from (4.9) and (4.10), without

skew magnets and sextup oles, and in a zero-curvature region

x + K (1 )x = 0

y K (1 )y = 0 (6.1)

where is the relative momentum deviation from the design momentum p : =p=p .

0 0

This momentum dep endence of the fo cusing, which, in turn, causes tune changes,

is called chromatic e ect. The variation of the tune Q with the momentum is called the

chromaticity and is de ned as (Q stands for the horizontal or the vertical tune)

Q = : (6.2)

p=p

The relativechromaticity is de ned as

Q=Q

= : (6.3)

p=p

0 73

The control of the chromaticity is imp ortant for two reasons:

1) Toavoid shifting the b eam on resonances due to tune changes induced bychromatic

e ects.

2) To prevent some transverse instabilities (head{tail instabilities).

Focusing quadrupole

δ ≥ 0, δ < 0 δ < 0

δ = 0 δ > 0 s

δ ≥ 0, δ < 0 f (δ < 0) f (δ = 0)

f (δ > 0)

Figure 30: Schematic representation of chromatic e ect in a disp ersion-free region.

Focusing quadrupole

δ > 0 δ < 0

δ = 0 s δ > 0 δ < 0 f (δ < 0)

f (δ > 0)

Figure 31: Schematic representation of chromatic e ect in a region with disp ersion.

Particles with di erent momenta are spread where there is disp ersion. Higher-

momentum particles are fo cused less than particles with the design momentum p ,lower-

momentum particles are fo cused more. Indeed, if f ( ) and K ( ) are the fo cal length and

normalized gradient, at momentum p = p (1 + ), of a quadrup ole of length `, with

@B e @B e

y y

K = K ( )=

p @x p @x

0 74

we nd

1 K `

= : (6.4)

f ( ) 1+

Therefore, once the particles are spread by momentum in a region with disp ersion, we

can apply fo cusing corrections dep ending on the momentum using a sextup ole magnet.

y y Particles with ∆ ∆p/p < 0 N p/p > 0 S ∆p/p < 0 SS 0 NN∆ p/p0 > 0 B F x x F B NN SS

s S s N

e ∂2 By e ∂2 By S = > 0 S = Ð < 0 0 0 2

p0 ∂x2 p0 ∂x

Figure 32: Sextup ole elds and forces in a region with disp ersion (p ositive particles approach

the reader).

Here, the sextup ole is fo cusing for the higher-momentum particles and defo cusing

for the lower-momentum particles. Hence, it can b e used to correct the chromatic fo cusing errors in a region with non-zero disp ersion.

Focusing quadrupole δ > 0

Sextupole with S0 > 0

δ = 0 s

δ < 0

Figure 33: Schematic representation of chromaticity correction with a sextup ole added b eside

a fo cusing quadrup ole. 75 Defocusing quadrupole

δ > 0

Sextupole with S < 0 δ = 0 0 s

δ < 0

Figure 34: Schematic representation of chromaticity correction with a sextup ole added b eside

a defo cusing quadrup ole.

Let us consider the equations of motion with quadratic terms in ;x, and y .We

get, using (4.9) and (4.10) with zero curvature, and considering regular magnets only

2 2 00

S (x y ) x + K x = K x

0 0 0

y K y = K y + S xy : (6.5)

0 0 0

Substituting the total co ordinates for the o -momentum particle, assuming no disp ersion

in the vertical plane,

x = x + D

y = y (6.6)

where x and y are the horizontal and vertical b etatron oscillations , the equations of

motion (6.5) b ecome

00 00 2 2 2

x + K x +(D + K D ) = K x S (x y )+K D

0 0 x 0 0 0 x

2 2

S x D S D

0 x 0

y K y = K y + S x y + S D y :

0 0 0 0 x

Discarding the terms which do not dep end on the b etatron motion, b ecause they

do not contribute to the chromatic tune shift, we are left with

00 2 2

x + K x = K x S D x S (x y )

0 0 0 x 0

K y = K y + S D y + S x y : (6.7) y

0 0 0 x 0

2 2

Ignoring the non-chromatic terms of second order (i.e. terms in x ;y and x y ) yields

x + K x = (K S D )x

0 0 0 x

y K y = (K S D )y : (6.8)

0 0 0 x

Intro ducing the normalized co ordinates (4.27) and (4.28)

y x

p q

= =

dt 1

Q (t)

x;y 0 x;y

transforms the equations (6.8) into

2 2 2

+ Q = Q [K () S ()D ]

0 0 x

x x x

2 2 2

+ Q = Q [K () S ()D ] (6.9)

0 0 x

y y y

in which K ;S ; and D are p erio dic functions of with p erio d 2 . These equations

0 0 x;y x

may b e written in the form:

+ Q = Q p ()

x x

+ Q = Q p () (6.10)

y y

with

p ()= Q [K () S ()D ]: (6.11)

x;y x;y 0 0 x

x;y

Let us solve the p erturb ed equation of motion for . Expanding p ()inFourier series

yields

p ()= p^ (m)e (6.12)

x x

m=1

in which thep ^ (m) are the Fourier co ecients given by

p^ (m)= p ()e d : (6.13)

x x

From this we obtain

# "

im 2

p^ (m)e =0: (6.14) + Q 1

m=1

which is a di erential equation with p erio dic co ecients of the Hill typ e.

Furthermore, since

X X

im im

p^ (m)e p^ (m)e =^p (0) +

x x x

m=1

m6=0 77

equation (6.14) may b e written as

+ Q () =0 (6.15)

where Q () has a static and an oscillatory part

# "

p^ (0)

im 2 2

Q p^ (m)e : (6.16) Q ()=Q 1

x x

m6=0

~ ~

The oscillatory part of the tune Q averages to zero over one p erio d 2 . Replacing Q ()

x x

by its average part hQ i (see the averaging metho d) gives the static tune shift due to

chromatic e ect. Then equation (6.15) transforms into

+[Q Q p^ (0)] =0 ; (6.17)

x x

whichmay b e rewritten as

+(Q +Q ) =0; (6.18)

x x

where Q is the chromatic tune shift. It follows that for a small tune shift

2 2 2

Q Q p^ (0)=(Q +Q ) Q +2Q Q :

x x x x x x

x x

Hence, from (6.11) and (6.13) we obtain the tune shift as

Q = p^ (0) =

x x

= Q ()[K () S ()D ()]d: (6.19)

x 0 0 x

Returning to the original variables x and s we nd

s +C

Q = (s)[K (s) S (s)D (s)]ds (6.20)

x x 0 0 x

where C is the machine circumference. Similarly,we compute

s +C

(s)[K (s) S (s)D (s)]ds : (6.21) Q =

y 0 0 x y

Intro ducing these equations into the de nition (6.3) of the chromaticity yields, replacing

s by s

s+C

= (t)[K (t) S (t)D (t)]dt

x x 0 0 x

s+C

= (t)[K (t) S (t)D (t)]dt : (6.22)

y y 0 0 x

y 78

The contribution to chromaticity arising from pure quadrup ole elements (and also pure

dip oles) is called the natural chromaticity.Thus, the natural chromaticity of a lattice is

given by

s+C

(t)K (t)dt =

x 0 x

s+C

= + (t)K (t)dt : (6.23)

y y 0

6.2 The natural chromaticityofaFODO cell

As an example we compute the natural chromaticity of a synchrotron formed only

of thin-lens FODO cells of length 2L.

Cell length 2L

QF QD QF s

L L

Horizontal plane

Figure 35: FODO cell (QF : fo cusing quadrup ole, QD : defo cusing quadrup ole.)

The transfer matrix of a thin-lens FODO cell has already b een derived to b e

2 2

L L L

1 2L +

f f f

M = (6.24)

FODO

L L

f f

where f = K `>0;K and ` are the quadrup ole strength and length. The FODO

0 0

transfer matrix may b e identi ed with the general transfer matrix through any section

(from s to s )

1 2 79

2 3

(cos + sin ) sin

1 1 2

6 7

M (s =s )= (6.25)

2 1

4 5

1 2 1 2

p p

cos sin (cos sin )

1 2 1 2

where = (s ) (s ) is the phase advance p er cell. However, the lattice b eing

2 1

comp osed only of FODO cells, p erio dic conditions yield = = and = =

1 2 1 2

. Hence, over one FODO cell the latter expression reduces to

1 0

QF QF

cos + sin sin

x x x

x x

C B

(6.26) M (s =s )=M =

A @

2 1 FODO

QF QF

sin cos sin

x x x

x x

where

= :

Identifying the matrix comp onents, we obtain

L 2L L

QF QF

2L + = sin or = 1+

x x

f sin 2f

2 2

L 1 L

= 2 cos or = (1 cos ) = sin : Tr(M )=2

x x FODO

2 2

f 4f 2 2

Hence

2 1

= sin (6.27)

f L 2

and

= 1 + sin : (6.28)

sin 2

Here, denotes the b etatron function of a horizontally thin fo cusing quadrup ole, for

QF 1 QD

which f = jf j > 0, with f = K `.We can derive the b etatron function of a

horizontally thin defo cusing quadrup ole, for which f = jf j < 0, by starting the cell

by a defo cusing quadrup ole instead of a fo cusing one. In that case we obtain the transfer

matrix of a DOFO cell, in which the matrix comp onents are those of a FODO cell, where f

is replaced by f . Hence, by identi cation with the general transfer matrix with p erio dic

QD QD

conditions = = and = = ,we nd

1 2 1 2

x x

: (6.29) 1 sin

sin 2

From these results, the natural relativechromaticityofaFODO cell may b e computed as 80

s+C

= (t)K (t)dt =

x x 0

Z Z

QF QF QD QD

K ds + K ds = =

x x

1 1

QF QF QD QD x^

= ( K ` + K `)= = +

x x

QF QD

4Q 4Q f f

x x

QF QD

x x

= ;

4Q f

where, according to the thin-lens approximation

QF ;D QF ;D

K ds = K ` = :

QF ;D

Intro ducing and the fo cal length f in the expression (6.23) yields, using the trigono-

metric formula,

x x

sin sin = 2 cos

2 2

the natural chromaticity

= ; (6.30) tan

Q 2

where is the horizontal phase advance for the FODO cell. If the full lattice is made

of N similar FODO cells, the natural chromaticity of the machine b ecomes

= : (6.31) tan

Q 2

Furthermore, since

Q = [ (s + C ) (s )] ;

x x 0 x 0

where C is the machine circumference, we can write

Q = :

Thus, from (6.31)

= tan : (6.32)

QF QD

To compute the natural vertical chromaticitywe use the fact that = (with

y x

QF QD QF QD

f = jf j > 0), and = (with f = jf j < 0). Therefore

y x

QF QD QF QD

QF QD

1 1 1

y y y y

x x

= + = = :

QF QD

4Q f f 4Q f 4Q f

y y y 81

Similarly, with

Q = ;

where is the vertical phase advance for the FODO cell, we nd

= tan : (6.33)

Thus, the natural chromaticities of a synchrotron made up of FODO cells are negative.

For instance, if = =2we obtain = = 4= . More generally the natural

x;y x y

0 0

chromaticities are always negative since for higher-momentum particles (i.e. >0) the

fo cusing is less e ective, and then the tune is reduced (i.e. Q< 0).

6.3 Chromaticity correction

The chromaticity equations suggest the insertion of sextup oles close to each quadrup ole,

where the disp ersion function is non-zero, in order to correct the chromaticity.Thus, for

the chromaticitytovanish, the sextup ole strength S would b e, from (6.22)

K `

0 Q

S ` = (6.34)

0 s

where ` ;` are the quadrup ole and sextup ole lengths, K is the quadrup ole strength,

Q S 0

and D is the disp ersion function value at the sextup ole lo cation.

Unfortunately, such lo calized connections are seldom feasible. A standard wayof

adjusting b oth the horizontal and the vertical chromaticities is to use families of sextup oles

with mo derate strength, distributed around the ring. Rewriting the equations for the

chromaticityas

s+C

= + (t)S (t)D (t)dt

x x x 0 x

s+C

= (t)S (t)D (t)dt (6.35)

y y y 0 x

and using the thin-lens approximation, we obtain

S D ` = +

x 0 x s x x

i i i i 0

i=1

= S D ` (6.36)

y y y 0 s s

0 i i i i

i=1

in which the sextup oles are lo cated at s , their strengths and lengths b eing S and ` ,

i 0 s

i i

and N is the total numb er of sextup oles. If the chromaticities have to b e adjusted to the

values and , the sextup ole strengths are obtained by solving the linear system of

x y

equations 82

S D ` = 4Q

x 0 x s x x

i i i i

i=1

S D ` = 4Q (6.37)

y 0 x s y y

i i i i

i=1

with = . Assuming there are only two sextup oles in the ring, the linear

x;y x ;y x;y

0 0

system of equations reduces to

( S D + S D )` = 4Q

x 0 x x 0 x s x x

1 1 1 2 2 2

( S D + S D )` = 4Q ; (6.38)

y 0 x y 0 x s y y

1 1 1 2 2 2

where wehave assumed that the two sextup oles have the same lengths ` . Solving these

two equations yields

4 Q + Q

y x x x y y

2 2

S =

` D

s x x y x y

1 1 2 2 1

4 Q + Q

y x x x y y

1 1

S = : (6.39)

` D

s x x y x y

2 1 2 2 1

The sextup oles have to b e placed where the disp ersion function is high and the b etatron

functions well separated to minimize the sextup ole strengths (e.g. at one sex-

x y

tup ole and at the other sextup ole). Furthermore, the strengths S and S are

x y 0 0

1 2

opp osite in sign (i.e. S S < 0). In the case of a two-family sextup ole with N sextup oles

0 0 1

1 2

of strength S and N sextup oles of strength S , the terms in (6.37) may b e group ed in

0 2 0

1 2

two sums and solved for S and S . The \two-family sextup ole" chromaticity correction

0 0

1 2

scheme yields sextup ole strengths lower than those based on the \two sextup oles in a

ring" scheme. 83

APPENDIX A:

HILL'S EQUATION

A.1 Linear equations

Consider a linear di erential equation of the form, called Hill's equation

u + K (s)u =0 ; (A.1)

where K (s) is a p erio dic function with p erio d L,

K (s + L)=K(s): (A.2)

A prime denotes the derivative with resp ect to the variable s.For any second-order di er-

ential equation there exist two indep endent solutions, C (s) and S (s), called a fundamental

set of solutions and such that every solution u(s) is a linear combination of these two:

u(s)= c C(s)+ c S(s) ; (A.3)

1 2

where c and c are arbitrary constants. However, there is only one solution u(s) that

1 2

0 0

meets the initial conditions u(s )=u and u (s )=u at s . Assume that a fundamental

0 0 0 0

set of solutions C (s) and S (s)|cosine-like and sine-like solutions|of (A.1), whether

0 0

K (s) is p erio dic or not, have b een found satisfying the initial conditions

C (s )=1 C (s )=0

0 0 0

S (s )=0 S (s )=1: (A.4)

0 0 0

Then, the solution u(s) of Hill's equation with initial conditions u and u at s can b e

0 0

expressed as a linear combination of C (s) and S (s),

0 0

u(s) = u C (s)+u S (s) ;

0 0 0

0 0 0 0

u(s) = u C (s)+u S (s) ;

0 0 0

or equivalently,

u(s) C (s) S (s) u(s)

0 0 0

= : (A.5)

0 0 0 0

u (s) C (s) S (s) u(s)

0 0

The ab ove matrix is called a transformation matrix or transfer matrix from s to s, written

C (s) S (s)

0 0

(A.6) M (s=s )=

0 0

C (s) S (s)

0 0

Equations (A.2) to (A.5) hold whether K (s) is p erio dic or not. The name cosine- and sine-

like solutions of C (s) and S (s) come from the case where K (s)= K is a p ositive constant

0 0

p p p

for which a fundamental set of solutions is C (s) = cos Ks and S (s) = sin Ks= K;

0 0 84

0 0

with C (0)=1;C(0) = 0, and S (0)=0;S(0) = 1. The determinant of the transfer

0 0

matrix M (s=s ) is called the Wronskian. Di erentiation of the Wronskian yields

d d

0 0 00 00

jM (s=s )j = [C (s)S (s) C (s)S (s)] = C (s)S (s) C (s)S (s) :

0 0 0 0 0

0 0 0

ds ds

Since b oth C (s) and S (s) are solutions of Hill's equation, we nd

0 0

jM (s=s )j =0

and thus the Wronskian is a constant. Consequently, since by (A.4) jM (s =S )j =1,we

0 0

get the general result, true whether K (s) is p erio dic or not:

C (s) S (s)

0 0

=1: (A.7) jM (s=s )j =

0 0

C (s) S (s)

0 0

The transfer matrix from s 6= s to s, written as M (s=s ), is built with the cosine- and

1 0 1

sine-like functions C (s) and S (s), which are generally di erent from C (s) and S (s).

1 1 0 0

The conditions (A.4) are not generally satis ed by C (s) and S (s). However,

1 1

C (s )=1 and C (s )=0

1 1 1

S (s )=0 and S (s )=1:

1 1 1

0 0 0

Using (A.5) in which u(s);u(s) are replaced by C (s);C(s)| S (s);S(s), resp ectively|

0 0

0 0 0

and u(s );u(s) are replaced by C (s );C(s)|S (s );S(s), resp ectively|we obtain

0 0 0 1 1 0 1 1

0 0

with C (s) and S (s) the fundamental set of solutions

1 1

C (s) S (s) C (s) S (s) C(s) S(s)

0 0 1 1 0 1 0 1

= ;

0 0 0 0 0 0

C (s) S (s) C (s) S (s) C(s) S(s)

1 1

0 0 1 1 0 0

where wehave combined two matrix equations in one. By the de nition of a transfer

matrix this result may b e written as

M (s=s )= M(s=s )M (s =s ) ; (A.8)

0 1 1 0

with

C (s) S (s)

1 1

M (s=s )= :

0 0

C (s) S (s)

1 1

From now on the subscript in C (s) and S (s), or in C (s) and C (s), will b e omitted, since

0 0 1 2

the reference p oint s already app ears in the notation M (s=s ). When K (s) is p erio dic

0 0

the transfer matrix M (s + L=s)over one p erio d L will b e written as M (s). By (A.8),

M (s) M (s + L=s)=M(s+L=s )M (s=s ) : (A.9)

0 0 85

A.2 Equations with p erio dic co ecients (Flo quet theory)

If K (s) is p erio dic of p erio d L, and since C (s) and S (s) are a fundamental set of

solutions of Hill's equation, C (s + L) and S (s + L) are also a fundamental set of solutions

of the same equation, b ecause

2 2

d C (s + L) d C (s + L)

+ K (s)C (s + L) = + K (s + L)C (s + L)

2 2

ds ds

d C (t)

= + K (t)C (t)=0

with the change of variables t = s + L.

Hence wemay write alternatively the cosine- and sine-like solutions C (s + L) and

S (s + L) as a linear combination of C (s) and S (s) using (A.3):

C (s + L) = a C (s)+ a S (s);

11 12

S (s + L) = a C (s)+ a S (s) ; (A.10)

21 22

where a are the comp onents of a constant matrix A. Similarly,any general solution of

Hill's equation is a linear combination of the fundamental set C (s) and S (s)

u(s)= c C(s)+ c S(s) : (A.11)

1 2

The solution u(s) is not necessarily p erio dic, although K (s) is p erio dic. However, let us

try to nd a solution u(s) with the prop erty

u(s + L)=u(s) : (A.12)

Using (A.10) to (A.12) we obtain

u(s + L) = c C (s + L)+c S(s+L)

1 2

= c (a C (s)+a S (s)) + c (a C (s)+a S (s))

1 11 12 2 21 22

and

u(s + L)=[c C(s)+ c S(s)] :

1 2

Comparison of the last two equations gives

[(a )c + a c ] C (s)+ [a c +(a )c ] S (s)=0:

11 1 21 2 12 1 22 2

Since C (s) and S (s) are linearly indep endent, the co ecients of the last equation must

vanish, yielding the system

(a )c + a c = 0

11 1 21 2

a c +(a )c = 0 :

12 1 22 2

The condition for nonvanishing solution c ;c is

1 2

a a

11 21

= jA I j =0; (A.13)

a a

12 22 86

or equivalently,

Tr(A)+jAj=0: (A.14)

This expression is called the characteristic equation. Now, wewant to express (A.14) in

terms of the transfer matrix, rather than in terms of the matrix A with co ecients a .

Rewriting (A.10) in matrix formulation

C (s + L) C (s)

= A

S (s + L) S (s)

and transp osing this expression gives

[C (s + L) S (s + L)] = [C (s) S (s)]A :

This equation and its derivativemay b e written into the compact form

C (s + L) S (s + L) C (s) S (s) a a

11 21

= ;

0 0 0 0

C (s + L) S (s + L) C (s) S (s) a a

12 22

M (s + L=s )= M(s=s )A : (A.15)

0 0

Intro ducing the transfer matrix M (s)over one p erio d L, using (A.8) and the last expres-

sion we get

t 1

M (s)=M(s=s )A M (s=s ) (A.16)

0 0

Since the determinant of a transfer matrix is equal to unity, it follows immediately that

jA j = jAj =1: (A.17)

Similarly, the characteristic equation of M (s)is

jM(s)I j = Tr[M(s)] +1=0:

then

t 1 t

jM (s) I j = jM (s=s )(A I )M (s=s ) j = jA I j

0 0

= Tr(A)+1=0;

since the trace of a matrix is equal to the trace of its transp ose. Identifying these last two

expressions yields

Tr[M(s)] = Tr(A): (A.18)

The trace M (s) is indep endent of the reference p oint s, due to A b eing a constant matrix.

Hence the ro ots and of the characteristic equation (A.14) are

1 2

= Tr[(M(s)] Tr[M(s)] 4 : (A.19)

1;2

2 87

These ro ots are related by

=1: (A.20)

1 2

Let u (s) and u (s) b e the solution u(s) with the prop erty (A.12), in which = and

1 2 1

= , resp ectively. De ne w (s) for i =1;2by

2 i

w(s)=e u (s) ;

i i

then, using (A.12), we get

ln ln

i i

(s+L) s

L L

w (s + L)=e u (s + L)=e u (s)=w(s);

i i i i i

so that w (s) is p erio dic with p erio d L:

w (s + L)=w(s): (A.21)

i i

Hence wemay write

u (s)=w(s)e ; (A.22)

i i

where

=ln : (A.23)

i i

The numb ers are called the characteristic exp onents of Hill's equation. They need not

b e real numb ers. If and are distinct, the solutions u (s) and u (s) are a fundamental

2 2 1 2

set of solutions with the prop erty

u (s + L)=e u(s): (A.24)

i i

This result is known as Flo quet's theorem. When = = ,wehave=1by means

1 2

of (A.20). There exists a fundamental set of solutions of the form

u (s) = w (s)e

1 1

u (s) = w (s)+ w (s) e ; (A.25)

2 2 1

with = 0 for = 1 and = i for = 1. In this case the solution u (s) is p erio dic,

with p erio d L when = 1 and p erio d 2L when = 1. The main diculty in the Flo quet

analysis is that the fundamental set of solutions C (s) and S (s), from whichwe derived

and , is generally unknown.

1 2

Instead of de ning the Flo quet solution u (s) and u (s)by (A.22), wemay alterna-

1 2

tively express these solutions in the generalized Flo quet form

(s)

u (s)=w(s)e ; (A.26)

i i

such that w (s) are p erio dic,

w (s + L)=w(s); (A.27)

i i 88

and

(s + L) (s) ln ; (A.28)

i i i

and b eing the characteristic exp onents (assuming and as distinct). Thus the

1 2 1 2

derivative (s) is p erio dic with p erio d L. Indeed, owing to u (s)having the prop erty

(A.12), we de ne

(s)

w (s)=e u(s)

i i

and then

(s+L) [(s+L)(s)]

i i i

w (s + L)=e u(s+L)=e u (s)=w(s);

i i i i i

since =ln , and provided (A.28) is satis ed. However, such an equation (A.28) exists

i i

since and are indep endentofsby (A.18) and (A.19). The Flo quet solutions u (s)

1 2 1

and u (s) are obviously di erent from the cosine- and sine-like solutions C (s) and S (s)

from which they are derived.

A.3 Stability of solutions

In plain words, a given solution of Hill's equation is stable (in the Liapunov sense)

if any other solution coming near it remains in its neighb orho o d. Otherwise it is said to

b e unstable. It can b e shown that a solution is stable if, and only if, it is b ounded. Some

deductions ab out the stability of the solution u(s) of Hill's equation can b e made from

the ro ots and of the characteristics equation. By (A.19) and (A.20) we see that:

1 2

1) The ro ots may b e complex conjugate on the unit circle

= e ; (A.29)

1;2

representing stable solutions

i s i s

L L

u(s)= c w (s)e +c w (s)e ; (A.30)

1 1 2 2

with

jTr[M(s)]j = j2 cos j < 2 :; (A.31) 89

2) The ro ots may b e real recipro cals

= e ;; (A.32)

1;2

representing growing and decaying solutions

s s

L L

u(s)= c w (s)e +c w (s)e ;; (A.33)

1 1 2 2

with

jTr[M(s)]j = j2 cosh j > 2; (A.34)

where the unstable solutions are divided into two p ossibili ties:

Tr[M(s)] > 2 (for > 1) and Tr[M(s)] < 2 (for < 1).

1 1

3) The ro ots may b e equal,

= = 1 ;; (A.35)

1 2

representing transition b etween the stable and unstable cases with

jTr[M(s)]j =2:; (A.36)

According to (A.25) there is one stable solution of p erio d L for = = 1 (the

1 2

other is unstable), and one stable solution of p erio d 2L for = = 1 (the other

1 2 also b eing unstable).

λ 1

λλ 12 0 1 01

λ 2

 Tr(M(s)) < 2  Tr(M(s)) > 2

Figure 36: Stability conditions for a one-p erio d transfer matrix: lo cation of eigenvalues in the

complex plane (left gure: stable motion, right gure: unstable motion).

As an example, consider again a symmetric thin-lens FODO cell of length 2L whose

transfer matrix has b een derived to b e

2 2

L L L

1 2L +

f f f

M (2L=0) =

FODO

L L

f f

The stability criterion (A.31) gives the condition

jTr[M (2L=0)]j = 2 < 2

FODO

f 90

or equivalently

L 1

1 < 1 < 1

2 f

yielding

0 < < 1

The stabilityisthus obtained for distances L between the quadrup oles up to twice their

fo cal length f . 91

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