Lattice Design in Particle
Accelerators
Bernhard Holzer, DESY
Historical note:
... Particle acceleration where lattice design is not needed
1
N ntZ e
*
( 8πε ) r K sin ( θ / 2 )
2
N (θ ) =
4
i
2
2
2
4
0
N(θ)
Rutherford Scattering, 1906
Using radioactive particle sources:
α-particles of some MeV energy
θ
Historical note:
II.) The „infancy of particle accelerators“
1923 Betatron,
1928 Cockroft-Walton Generator,
1929 Cyclotron
1932 Van de Graaf Generator, etc
Diagram of the first cyclotron
by Lawrence and Livingstone
12 MV-Tandem van de Graaf
Accelerator
1
1952: Courant, Livingston, Snyder: Theory of strong focusing in
particle beams à Ted Wilson in this school
Lattice design: design and optimisation of the principle
elements of an accelerator ... the lattice cells
Lattice Design: Prerequisites
... I will start very easy ... and slowly the topic will become more and more
difficult
... interesting
Lorentz Force:
r
r r r
F = q* ( E + v × B )
neglecting electrical fields:
r
r r
F = q* ( v × B )
High energy accelerators à circular machines
somewhere in the lattice we need a number of dipole magnets,
that are bending the design orbit to a closed ring
2
In a constant external magnetic field the particle trajectory will be a part of a
circle and ...
... the centrifugal force will be equal to the Lorentz force
e*v * B =
mv
ρ
2
→e* B =
→
mv
=p/ρ
ρ
B * ρ = p /e
p = momentum of the particle,
ρ = curvature radius
B*ρ is called the “beam rigidity”
Example: Heavy Ion Storage Ring: TSR
8 dipole magnets of equal bending
strength
ρ
Circular Orbit:
α=
ds dl
≈
ρ ρ
α=
α
ds
B * dl
B* ρ
The angle swept out in one revolution
must be 2π, so
Bdl
α= ò
= 2π
B* ρ
... for a full circle
Example HERA:
920 GeV Proton storage ring
number of dipole magnets N = 416
l = 8.8m
q = +1 e
field map of a storage ring dipole magnet
→ ò Bdl = 2π *
p
q
ò Bdl ≈ N * l * B = 2π p / q
B≈
2π * 920 * 10 9 eV
≈ 5 .15 Tesla
m
416 * 3 * 10 8 * 8.8 m * e
s
3
Focusing forces and particle trajectories:
Magnetic field in a quadrupole magnet
Bx = − g * z ,
Bz = − g * x
leads to a linear retrieving force on the particle.
Relating the fields to their optical effect: normalise to the particles
momentum:
dipole magnet :
1
B
=
ρ p/e
quadrupole lens : k =
g
p/e
focal length : f :=
1
k*l
Under the influence of the focusing and defocusing forces the differential
equation of the particles trajectory can be developed:
x' ' + K * x = 0
K = −k + 1 / ρ
K =k
2
hor. plane
vert. plane
Example:
HERA Ring: Circumference:
Circumference:
C0 =6335 m
Bending radius:
ρ = 580 m
Quadrupol Gradient: G= 110T/m
à
k = 33.64*10-3 /m2
à 1/ρ2 = 2.97 *10-6 /m2
the two storage rings
of the HERA collider
!
For estimates in large accelerators the weak focusing term 1/ρ2 can
in general be neglected ...
4
Single particle trajectories:
Equation of motion
y( s ) = y0 * cos(
y'' + K ( s ) * y = 0
y' ( s ) = − y0 *
Or written more convenient in matrix form:
y'0
* sin(
K
K * s)+
K * sin(
K * s)
K * s ) + y'0* cos(
K * s)
æ yö
æ yö
ç ÷ = M*ç ÷
y
'
è øs
è y' ø0
Matrices of lattice elements
æ
ç cos( K * l )
M QF = ç
ç − K sin( K * l )
è
Hor. focusing Quadrupole Magnet
Hor. defocusing Quadrupole Magnet
Drift space
1
sin(
K
cos(
ö
K * l )÷
÷
K * l ) ÷ø
1
ö
sinh( K * l ) ÷
K
÷
cosh( K * l ) ÷ø
æ
ç cosh( K * l )
M QD = ç
ç K sinh( K * l )
è
æ1 sö
÷
M Drift = ç
è 0 1ø
Periodic Lattices:
In the case of periodic lattices the transfer matrix can be expressed
as a function of a set of periodic parameters α, β, γ
æ cos µ + α S sin µ
M ( s ) = çç
è − γ S sin µ
s+ L
β S sin µ
ö
÷
cos( µ ) − α S sin µ ÷ø
µ= ò
s
dt
β(t )
μ = phase advance
per period:
For stability of the motion in periodic lattice
structures it is required that
trace ( M ) < 2
In terms of these new periodic parameters the solution of the equation
of motion is
y( s ) = ε * β ( s ) * cos( Φ ( s ) − δ )
y' ( s ) =
− ε
* {sin( Φ ( s ) − δ ) + α cos( Φ ( s ) − δ )}
β
5
The new parameters α, β, γ can be transformed through the lattice via the
matrix elements defined above.
æ C2
æβ ö
ç
ç ÷
ç α ÷ = ç − CC'
2
ç
çγ ÷
è ø S è C'
− 2 SC
SC' + S' C
− 2 S' C'
S2 ö æ β ö
÷ ç ÷
− SS' ÷ * ç α ÷
S' 2 ÷ø çè γ ÷ø0
Question: What does that mean ????
... and here starts the lattice design !!!
Question: What does that mean ????
æC
M =ç
è C'
Most simple example: drift space
particle coordinates
S ö æ1 l ö
÷
÷=ç
S' ø è 0 1ø
æ x ö æ1 l ö æ x ö
÷* ç ÷
ç ÷ =ç
è x' ø l è 0 1 ø è x' ø 0
→ x( l ) = x0 + l * x0'
→ x' ( l ) = x0'
transformation of twiss parameters:
æ β ö æ 1 − 2l
ç ÷ ç
1
çα ÷ = ç 0
ç γ ÷ ç0
0
è øl è
l2 ö æ β ö
÷ ç ÷
− l ÷* çα ÷
1 ÷ø çè γ ÷ø0
β ( s ) = β 0 − 2l * α 0 + l 2 * γ 0
Stability ...?
trace( M ) = 1 + 1 = 2
àA periodic solution doesn‘t exist in a magnetic
lattice built exclusively of drift spaces.
6
Lattice Design:
Arc: regular (periodic) magnet structure:
bending magnets à define the energy of the ring
main focusing & tune control, chromaticity correction,
multipoles for higher order corrections
Straight sections: drift spaces for injection, dispersion suppressors,
low beta insertions, RF cavities, etc....
... and the high energy experiments if they cannot be avoided
The FoDoFoDo-Lattice
A magnet structure consisting of focusing and defocusing quadrupole lenses in
alternating order with nothing in between.
(Nota bene: nothing = elements that can be neglected on first sight: drift, bending magnet,
RF structures ... and especially experiments...)
QF
QD
QF
L
Starting point for the calculation: in the middle of a focusing quadrupole
Phase advance per cell µ = 45°,
à calculate the twiss parameters for a periodic solution
7
Periodic solution of a FoDo Cell
β
x
β
y
Output of the optics program:
NR
TYP
0
1
2
3
4
QX=
IP
QFH
QD
QFH
IP
LENGTH STRENGTH BETAX ALFAX PHIX
0
0.25
3.251
6.002
6.002
0.00E+00
-5.41E-01
5.41E-01
-5.41E-01
0.00E+00
0.125 QZ=
11.611
11.228
5.4883
11.611
11.611
0
1.514
-0.781
0
0
BETAZALFAZ PHIZ
0
0.0035
0.0699
0.125
0.125
5.295
0
0
5.488 -0.78 0.007
11.23 1.514 0.066
5.295
0 0.125
5.295
0 0.125
0.125
0.125 * 2π = 45°
... Can we understand, what the optics code is doing?
The action of a magnet on the beam
is given by the transfer matrix:
M QF
æ
ç cos( K * l )
=ç
ç − K sin( K * l )
è
Input: strength and length of the FoDo elements
K =+/- 0.54102 m-2
lq = 0.5 m
ld = 2.5 m
1
sin(
K
cos(
ö
K * l )÷
÷
K * l ) ÷ø
æ1 l ö
÷
M Drift = ç
è 0 1ø
The matrix for the complete cell is obtained by multiplication of the element matrices
M FoDo = M QFH * M LD * M QD * M LD * M QFH
Putting the numbers in and multiplying out ...
æ 0.707 8.206 ö
÷
M FoDo = ç
è − 0.061 0.707 ø
8
The transfer matrix for 1 period gives us all the information that we need !
1.) is the motion stable?
trace( M FoDo ) = 1.415
→
< 2
2.) Phase advance per cell
æ cos µ + α sin µ
M( s ) = ç
è − γ sinµ
β sin µ
ö
÷
cos( µ ) − α sinµ ø
3.) hor β-function
cos( µ ) =
→
1
* trace( M ) = 0.707
2
1
µ = a cos( * trace( M )) = 45°
2
4.) hor α-function
M ( 1 ,2 )
β=
= 11.611 m
sin( µ )
α=
M ( 1 ,1 ) − cos( µ )
=0
sin( µ )
Some perls of wisdom about lattice cells:
L
1.) think first ...
* a first estimate of the FoDo parameters can and should be done before we
run our optics codes.
* we can learn a lot without doing to many too sophisticated calculations
* the optic codes will never tell us whether a lattice cell is technically feasible
* at the very beginning we have to define parameters that lead at least to a stable
periodic solution
2.) some rules of „thumb“... to start with: the tune Q
s+ L
dt
β(t )
phase advance per cell (i.e. )period:
µ= ò
Tune := phase advance of the machine in
units of 2π
Q := N *
→Q≈
s
µ
1
ds
=
*ò
2π 2π
β(s)
1 2πR R
*
=
β
2π
β
The tune is roughly given by the mean bending radius of the circular
accelerator divided by the mean β-function
9
3.) can we do it a little bit easier ?
We can: the thin lens approximation
Matrix of a focusing quadrupole magnet:
æ
cos( K * l )
ç
M QF = ç
ç − K sin( K * l )
è
1
ö
sin( K * l ) ÷
K
÷
cos( K * l ) ÷ø
If the focal length f is much larger than the length of the quadrupole magnet,
f = 1
klQ
>> lQ
klQ = const , lQ → 0
the transfer matrix can be aproximated using
æ 1
M =ç 1
ç f
è
0ö
÷
1÷
ø
FoDo in thin lens approximation
lD
L
Calculate the matrix for a half cell, starting in the middle of a foc. quadrupole:
M halfCell = M QD / 2 M lD M QF / 2
0ö
0ö æ 1 l ö æ 1
æ 1
D
÷* ç
÷
÷* ç − 1
M halfCell = ç 1
1
~ 1÷
ç ~
÷ è0 1 ø ç
f
è f
ø
è
ø
M halfCell
æ1 − lD
l D ö÷
~
ç
f
÷
=ç
çç − l D ~ 2 1 + l D ~ ÷÷
f
fø
è
lD indicates the length of the drift
which is now just half the cell length
lD = L / 2
and starting at the middle of a
quadrupole the focal length of the
half quad is
~
f =2f
For the second half cell set f à -f
10
FoDo in thin lens approximation
Matrix for the complete FoDo cell:
Multiplying out we get ...
l
æ1 + lD
l D ÷ö æç 1 − D ~
l D ö÷
~
ç
f
f
÷* ç
÷
M =ç
çç − l D ~ 2 1 − l D ~ ÷÷ çç − l D ~ 2 1 + l D ~ ÷÷
f
f
f
f
è
ø è
ø
æ
2l 2
l
ç 1 − ~D2
2l D ( 1 + ~D
f
f
M =ç
ç l D2
l D2
lD
1 − 2 ~2
ç 2( ~ 3 − ~ 2 )
f
f
è f
ö
)÷
÷
÷
÷
ø
Now we know, that the phase advance is related to the transfer matrix by
1
4l 2
2l 2
cos µ = trace ( M ) = 1 * ( 2 − ~D2 ) = 1 − ~D2
2
2
f
f
After some beer and with a little bit of trigonometric gymnastics
cos( x ) = cos 2 ( x ) − sin 2 ( x / 2 ) = 1 − 2 sin 2 ( x )
2
2
We can calculate the phase advance as a function of the FoDo parameter
2l 2
cos( µ ) = 1 − 2 sin 2 ( µ / 2 ) = 1 − ~D2
f
~ L
sin( µ / 2 ) = l D / f = Cell
~
2f
L
sin( µ / 2 ) = Cell
4f
Example: 45-degree Cell
LCell =
lQF + lD + lQD +lD =
0.5m+2.5m+0.5m+2.5m = 6m
1/f = k*lQ = 0.5m*0.541 m-2 = 0.27 m-1
sin( µ / 2 ) ≈
LCell
= 0.405
4f
→ µ ≈ 47 . 8 °
→ β ≈ 11 . 4 m
Remember:
Exact calculation yields:
µ = 45 °
β = 11 .6 m
11
Stability in a FoDo structure
æ
l
2l 2
ç 1 − ~D2
2l D ( 1 + ~D
f
f
M =ç
ç l D2
lD
l D2
1 − 2 ~2
ç 2( ~ 3 − ~ 2 )
f
f
è f
transfer matrix for the complete cell
Stability requires:
ö
)÷
÷
÷
÷
ø
4l 2
trace( M ) = 2 − ~D2 < 2
f
trace( M ) < 2
→ f >
L
4
cell
For stability the focal length
has to be larger than a
quarter of the cell length !!
SPS Lattice
Transformation matrix in terms of the Twiss parameters
en detail
Transformation of the coordinate vector (x,x´) in a magnet
æx ö
æ x( s ) ö
ç
÷ = M çç '0 ÷÷
è x´( s ) ø
è x0 ø
M QF
æ
ç cos( K * l )
=ç
ç − K sin( K * l )
è
1
ö
sin( K * l ) ÷
K
÷
cos( K * l ) ÷ø
General solution of the equation of motion
x( s ) = ε * β ( s ) * cos(φ ( s ) + ϕ )
x' ( s ) = − ε
{
}
β ( s ) * α ( s ) cos(φ ( s ) + ϕ ) + sin( φ ( s ) + ϕ )
Transformation of the coordinate vector (x,x´) expressed
as a function of the twiss parameters
M 0→ S
æ
βS
ç
(cos φ + α S sin φ )
β0
= çç
( α − α S ) cos φ − ( 1 + α 0α S ) sin φ
ç 0
β S β0
è
ö
÷
÷
÷
β0
(cos φ − α S sin φ ) ÷
βS
ø
β S β 0 sin φ
12
Transfer matrix for half a FoDo cell:
M halfCell
æ 1 − lD
l D ö÷
~
ç
f
÷
=ç
çç − l D ~ 2 1 + l D ~ ÷÷
f
fø
è
lD
L
æ
β
ç
(cos φ + α sin φ )
β0
ç
M =ç
( α − α ) cos φ − ( 1 + α 0α ) sin φ
ç 0
ββ 0
è
Compare to the twiss
parameter form of M
where Φ denotes the
phase advance through
the half cell
ö
÷
÷
÷
β0
(cos φ − α sin φ ) ÷
β
ø
ββ 0 sin φ
In the middle of a foc (defoc) quadrupole of the FoDo we allways have α=0,
and the half cell will lead us from βmax to βmin
æC
M =ç
è C'
∨
æ
ç
β
cos φ
Sö ç
βˆ
÷=ç
S' ø ç − 1
sin φ
ç
∨
ç βˆ β
è
Solving for βmax and βmin and remembering that ….
µ
~
βˆ
1 + l D / f 1 + sin 2
S'
= ∨ =
~=
µ
1 − lD / f
C
1 − sin
β
2
∨
S
~
= βˆ β = f 2 =
C'
l D2
sin 2
→
ö
∨
÷
βˆ β sin φ ÷
÷
÷
βˆ
∨ cos φ ÷
÷
β
ø
sin
βˆ =
∨
µ
2
µ lD
L
= ~=
2
f 4f
β=
µ
)L
2
sin µ
!
µ
)L
2
sin µ
!
( 1 + sin
( 1 − sin
The maximum and minimum values if
the β-function are solely determined by
the phase advance and the length of the cell.
Longer cells lead to larger β
typical shape of a proton
bunch in the HERA FoDo Cell
(Z , X , Y)
13
Optimisation of the FoDo Phase advance: Beam dimension
In both planes a gaussian particle distribution is assumed given by the beam
emittance ε and the β-function
σ = εβ
measured beam size
at the HERA IP
In general proton beams are „round“ in the sense that
εx ≈ εy
So for highest aperture we have to minimise the β-function
in both planes:
r 2 = ε xβ x + ε yβ y
typical beam envelope, vacuum chamber and pole
shape in a foc. Quadrupole lens in HERA
r 2 = ε xβ x + ε yβ y
Optimising the FoDo phase advance
search for the phase advance µ that results in
a minimum of the sum of the beta’s
∨
2L
βˆ + β =
sin µ
20
∨
βˆ + β =
d 2L
(
)= 0
sin µ
dµ
20
18
β ges ( µ )
µ
µ
) * L ( 1 − sin ) * L
2
2
+
sin µ
sin µ
( 1 + sin
L
* cos µ = 0 → µ = 90°
sin µ
16
2
14
12
10
10
0
0
36
72
108
µ
144
180
180
14
Nota bene: electron beams are typicalle flat, εy ≈ 2 … 10 % εy
à optimise only βhor
µ
d ˆ
d L( 1 + sin 2 )
(β )=
= 0 → µ ≈ 76°
dµ
dµ
sin µ
20
20
16
βmax( µ ) 12
βmin( µ )
red curve: βmax
blue curve: βmin
as a function of the phase advance µ
8
4
0.013
0
0
1
36
72
108
µ
144
180
180
Orbit distortions in a periodic lattice
field error of a dipole/distorted quadrupole
→ δ ( mrad ) =
ds ò Bds
=
ρ
p/e
●
●
●
the particle will follow a new closed trajectory, the distorted orbit:
x( s ) =
1
β(s)
* ò β(~
s ) ~ cos( φ ( ~
s ) − φ ( s ) − πQ )d~
s
2 sin πQ
ρ( s )
* the orbit amplitude will be large if the β function at the location of the kick is large
β ( s~ ) indicates the sensitivity of the beam à here orbit correctors should be
placed in the lattice
* the orbit amplitude will be large at places where in the lattice β(s)
is large à here beam position monitors should be installed
15
Chromaticity in the FoDo Lattice
Definition
∆Q = ξ *
∆p
p
The chromaticity describes an optical error of quadrupole lenses: For a given magnetic
field, i.e. gradient particles with smaller momentum will feel a stronger focusing force
and vice versa.
For small momentum errors ∆p/p the focusing parameter k can be written as
k( p ) =
k( p ) ≈
e
g
= g*
p0 + ∆p
p/e
∆p
e
(1 −
) * g = k 0 + ∆k
p0
p
∆k = − k 0
→
∆p
p
This describes a quadrupole error that leads to a tune shift of ...
∆Q =
− 1 ∆p
1
ò ∆kβ ( s )ds =
ò k0 β ( s )ds
4π
4π p
ξ contribution in the lattice
ξ =−
Chromaticity in the FoDo Lattice
ξ ≈ −
βˆ − β
1
N *
fQ
4π
∨
= −
1
ò β ( s ) * k ( s )ds
4π
ξ =−
1
ò β ( s ) * k ( s )ds
4π
µ
µ ü
ì
ï L ( 1 + sin 2 ) − L ( 1 − sin 2 ) ï
1
1
N *
* í
ý
fQ ï
sin µ
4π
ï
î
þ
using some trigonometric transformations ... ξ can be expressed in a very simple form:
ξ = −
µ
µ
L sin
1
1 2 L sin 2
1
1
2
= −
N
N
fQ
f Q f sin µ cos µ
sin µ
4π
4π
Q
2
2
ξ Cell = −
µ
1 1 L tan 2
4π f Q f sin µ
Q
2
Contribution of one FoDo Cell to the
chromaticity of the ring:
remember ...
x
x
sin x = 2 sin cos
2
2
putting ...
ξ Cell = −
sin
µ
L
=
2 4 fQ
µ
1
* tan
π
2
16
Resumé
1.) Dipole strength:
ò Bds = N * B0 * leff = 2π
p
q
leff effective magnet length, N number of magnets
2.) Stability condition:
Trace( M ) < 2
for periodic structures within the lattice / at least for the transfer matrix of
the complete circular machine
æ cos µ + α ( s ) sin µ
3.) Transfer matrix for periodic cell M ( s ) = ç
è − γ ( s ) sin µ
β ( s ) sin µ
ö
÷
cos( µ ) − α ( s ) sin µ ø
α,β,γ depend on the position s in the ring, µ (phase advance) is independent of s
æ 1
4.) Thin lens approximation: M QF = çç 1
ç
è fQ
0ö
÷
1 ÷÷
ø
fQ =
1
kQ l Q
focal length of the quadrupole magnet fQ = 1/(kQlQ) >> lQ
5.) Tune (rough estimate):
Q≈
R
β
R , β average values of radius and β-function
6.) Phase advance per FoDo cell
sin
(thin lens approx)
µ LCell
=
2 4 fQ
LCell length of the complete FoDo cell, fQ focal length of the
quadrupole, µ phase advance per cell
7.) Stability in a FoDo cell
fQ >
(thin lens approx)
8.) Beta functions in a FoDo cell
(thin lens approx)
LCell
4
βˆ =
µ
) LCell
2
sin µ
( 1 + sin
∨
β =
µ
) LCell
2
sin µ
( 1 − sin
LCell length of the complete FoDo cell, µ phase advance per cell
17
Resumé
periodic section („the arc“ in a large accelerator):
à parameters of the lattice cell defined by 2 quadrupole strengths
(sometimes even only by one if in series)
à for a given tune (phase advance) the twiss parameters are fixed
et vice versa.
àtune adjustments, variations of cell lengths, matching of twiss parameters
creation of symmetry points in the lattice ...
.... need more degrees of freedom, i.e. additional free quadrupole lenses,
varying drift lengths etc.
18
APPENDIX
Single particle trajectories:
y'' + K * y = 0
The differential equation for the particle movement can be solved by the Ansatz ...
y = a1 * cos( ω * s ) + a 2 * sin( ω * s )
y' = − a1ω * sin( ω * s ) + a 2ω * cos( ω * s )
y' ' = − a1ω 2 * cos( ω * s ) − a 2ω 2 * sin( ω * s )
= −ω 2 * y
→ K = ω2, ω =
K
So we get for the equation of motion in a storage ring
y( s ) = a1 * cos( K * s ) + a2 * sin( K * s )
Equation of motion
y ( s ) = a * cos(
1
K * s ) + a * sin( K * s )
2
The parameters a1 and a2 refer to the individual particle and are
determined by boundary conditions.
y(0 ) = y
→a = y
0
1
0
y'
y '( 0 ) = y ' → a =
K
0
0
resulting in
2
y'
* sin( K * s )
K
y '( s ) = −y * K * sin( K * s ) + y ' * cos( K * s )
y ( s ) = y * cos( K * s ) +
0
0
0
0
Or written more convenient in matrix form:
æy ö
æy ö
ç ÷ =M * ç ÷
è y ' øs
è y 'ø
,
0
æC S ö
÷
M =ç
èC ' S ' ø
19
Matrices of lattice elements
æ
ç cos( K * l )
M =ç
ç − K sin( K * l )
è
1
ö
sin( K * l ) ÷
K
÷
cos( K * l ) ÷ø
æ
ç cosh( K * l )
M =ç
ç K sinh( K * l )
è
1
ö
sinh( K * l ) ÷
K
÷
cosh( K * l ) ÷ø
QF
QD
Hor. focusing
Quadrupole Magnet
Hor. defocusing
Quadrupole Magnet
M
Drift
æ 1 sö
=ç
÷
è 0 1ø
K = −k +
Drift space
1
ρ
in horizontal plane
2
in vertical plane
K=k
Transformation of the principal trajectories in terms of
the Twiss parameters
General solution of the equation of motion
x ( s ) = ε * β ( s ) * cos( φ ( s ) + ϕ )
(1)
x '( s ) = − ε β ( s ) * {α ( s ) cos( φ ( s ) + ϕ ) + sin(φ ( s ) + ϕ )}
Using theorems of trigonometric functions
sin(a+b) = sin(a) cos(b)+cos(a) sin(b) ...
(2)
x ( s ) = ε * β ( s ) * {cos φ ( s ) cos( ϕ ) − sinφ ( s ) sin(ϕ )}
x '( s ) = − ε β ( s ) * {α ( s ) cos φ ( s ) cos( ϕ ) − α ( s ) sinφ ( s ) sin( ϕ ) +
+ sinφ ( s ) cos( ϕ ) + cos φ ( s ) sin( ϕ )}
Set initial conditions: x(0)=x0 ,x´(0)=x´0,
β(0)= β0, α(0)= α0, Φ(0)=0
(3)
cos φ =
x
εβ
sinφ =
0
0
αx
−1
(x′ β +
)
ε
β
0
0
0
0
0
20
Inserting in (1)
β (s )
{cos φ ( s ) + α sinφ ( s )}x + β ( s )β sinφ ( s )x ′
β
x (s ) =
0
0
0
0
0
1
{( α − α ( s )) cos φ ( s ) − ( 1 + α α ( s ) sinφ ( s )}x
β ( s )β
x ′( s ) =
0
0
0
0
+
β
{cos φ ( s ) − α ( s ) sinφ ( s )}x ′
β(s )
0
0
So again we have got a matrix that transforms the orbit vector (x0, x´0) into (x(s), x´(s))
æx
æ x (s ) ö
ç
÷ = M çç
´(
)
x
s
ø
è
èx
0
'
0
ö
÷÷
ø
æ
βS
ç
(cos φ + α S sin φ )
β0
M = çç
( α − α S ) cos φ − ( 1 + α 0α S ) sin φ
ç 0
β S β0
è
ö
÷
÷
÷
β0
(cos φ − α S sin φ ) ÷
βS
ø
β S β 0 sin φ
21