Introduction to Transverse Beam Optics II · Problem: chromaticity is generated by the lattice...
Transcript of Introduction to Transverse Beam Optics II · Problem: chromaticity is generated by the lattice...
Introduction to Transverse Beam Optics II
Bernhard Holzer, CERN
I. Reminder: the ideal world
0s
x xM
x x
0
1cos( ) sin(
sin( ) cos( )
foc
K s K sKM
K K s K s
z
x
ρ
s
θ
z
Beam parameters of a typical
high energy ring: Ip = 100 mA
particles per bunch: N ≈ 10 11
The Beta Function
... question: do we really have to calculate some 1011 single particle trajectories ?
Example: HERA Bunch pattern
( ) ( ) cos( ( ) )x s s s
0
( )( )
sds
ss D
general solution of Hills equation:
* ε is a constant of the motion … it is independent of „s“
* parametric representation of an ellipse in the x x‘ space
* shape and orientation of ellipse are given by α, β, γ
equation of motion:
2
1( ) ( )
2
1 ( )( )
( )
s s
ss
s
""mmbeam size:
0)()()( sxsksx
))(cos()()( sssx
x´
x
?
?
?
?
?
?
x´
x
?
x´
x
?
/
)()()()()(2)()( 22sxssxsxssxs
Beam Emittance and Phase Space Ellipse
13.) Liouville during Accelerationx´
xBeam Emittance corresponds to the area covered in the
x, x´ Phase Space Ellipse
Liouville: Area in phase space is constant.
But so sorry ... ε ≠ const !
Classical Mechanics:
phase space = diagram of the two canonical variables
position & momentum
x px
EnergypotEnergykinVTLq
Lp
j
j ..;
)()()()()(2)()( 22sxssxsxssxs
II ... the not so ideal world
According to Hamiltonian mechanics:
phase space diagram relates the variables q and p
Liouvilles Theorem: p dq const
for convenience (i.e. because we are lazy bones) we use in accelerator theory:
xdx dx dtx
ds dt dswhere βx= vx / c
1x dx
the beam emittance
shrinks during
acceleration ε ~ 1 / γ
q = position = x
p = momentum = γmv = mcγβx
2
2
1
1
c
v c
xx
;
dxmcpdq x
dxxmcpdq
ε
Nota bene:
1.) A proton machine … or an electron linac … needs the highest aperture at injection energy !!!
as soon as we start to accelerate the beam size shrinks as γ -1/2 in both planes.
2.) At lowest energy the machine will have the major aperture problems,
here we have to minimise
3.) we need different beam
optics adopted to the energy:
A Mini Beta concept will only
be adequate at flat top.
ˆ
LHC injection
optics at 450 GeV
LHC mini beta
optics at 7000 GeV
Example: HERA proton ring
injection energy: 40 GeV γ = 43
flat top energy: 920 GeV γ = 980
emittance ε (40GeV) = 1.2 * 10 -7
ε (920GeV) = 5.1 * 10 -9
7 σ beam envelope at E = 40 GeV
… and at E = 920 GeV
Linear Accelerator
1928, Wideroe
+ + + +- - -
* RF Acceleration: multiple application of the same acceleration voltage;brillant idea to gain higher energies ... but changing acceleration voltage
Energy Gain per „Gap“:
tUqW RFsin0
500 MHz cavities in an electron storage ring
drift tube structure at a proton linac
14.) The „ Δp / p ≠ 0“ Problem
Problem: panta rhei !!!(Heraklit: 540-480 v. Chr.)
Z X Y( )
Bunch length of Electrons ≈ 1cmExample: HERA RF:
U0
t c
MHz500cm60
cm60
994.0)84sin(
1)90sin(
o
o
3100.6U
U
typical momentum spread of an electron bunch: 3100.1
p
p
15.) Dispersion: trajectories for Δp / p ≠ 0
vBex
mvx
dt
dmF y
2
2
2
)( yρ
s
x
remember: x ≈ mm , ρ ≈ m … develop for small x
veBxmv
dt
xdm y)1(
2
2
2
consider only linear fields, and change independent variable: t → s
mv
gxe
mv
Bexx 0)1(
1
p=p0+Δp
Force acting on the particle
… but now take a small momentum error into account !!!
x
BxBB
y
y 0
y
Dispersion:
develop for small momentum error2
000
0
11
p
p
ppppp
10xk
2
00
02
00
0
2
1
p
pxeg
p
xegeB
p
p
p
Bexx
1
0
2p
pkx
xx
1)
1(
0
2p
pkxx
Momentum spread of the beam adds a term on the r.h.s. of the equation of motion.
inhomogeneous differential equation.
xkp
pxk
p
eB
p
pxx *
1**
)(*
00
0
0
2
1
2
1 1( )
px x k
p
general solution:
( ) ( ) ( )h ix s x s x s
( ) ( ) ( ) 0h hx s K s x s
1( ) ( ) ( )i i
px s K s x s
p
Normalise with respect to Δp/p:
( )( ) i
pp
x sD s
Dispersion function D(s)
* is that special orbit, an ideal particle would have for Δp/p = 1
* the orbit of any particle is the sum of the well known xβ and the dispersion
* as D(s) is just another orbit it will be subject to the focusing properties of the lattice
.ρ
xβ
Closed orbit for Δp/p > 0
( ) ( )i
px s D s
p
Matrix formalism:
( ) ( ) ( )p
x s x s D sp
0 0( ) ( ) ( ) ( )p
x s C s x S s x D sp
0s
x C S x Dp
x C S x Dp
DispersionExample: homogenous dipole field
00 0 1p p
p ps
x C S D x
x C S D x
Example HERA
3
1...2
( ) 1...2
1 10
x mm
D s m
pp
Amplitude of Orbit oscillation
contribution due to Dispersion ≈ beam size
Calculate D, D´
1 1
0 0
1 1( ) ( ) ( ) ( ) ( )
s s
s s
D s S s C s ds C s S s ds
D
Example: Drift
1
0 1Drift
lM
1 0
0 1 0
0 0 1
Drift
l
M
1 1
0 0
1 1( ) ( ) ( ) ( ) ( )
s s
s s
D s S s C s ds C s S s ds
0 0
Example: Dipole
cos sin
1sin cos
Dipole
l l
Ml l
( ) (1 cos )
( ) sin
lD s
lD s
p
p*)s(Dx
D
Dispersion is visible
HERA Standard Orbit
dedicated energy change of the stored beam
closed orbit is moved to a dispersions trajectory
HERA Dispersion Orbit
Attention: at the Interaction Points
we require D=D´= 0
16.) Momentum Compaction Factor:
The dispersion function relates the momentum error of a particle to the horizontal
orbit coordinate.
1( )*
px K s x
p
( ) ( ) ( )p
x s x s D sp
xβ
inhomogeneous differential equation
general solution
But it does much more:
it changes the length of the off - energy - orbit !!
ρ
ρ
dsx
dl
design orbit
particle trajectory
particle with a displacement x to the design orbit
path length dl ...
1( )
xdl ds
s
dl x
ds
1( )
EE
xl dl ds
s
circumference of an off-energy closed orbit
remember:
( ) ( )E
px s D s
p
( )
( )E
p D sl ds
p s
* The lengthening of the orbit for off-momentum
particles is given by the dispersion function
and the bending radius.
o
o
o
cp
l p
L p
1 ( )
( )cp
D sds
L s
For first estimates assume: 1
const
1 1 1 12cp dipolesl D D
L L
2cp
DD
L R
Assume: v c
cp
lT p
T L p
Definition:
αcp combines via the dispersion function
the momentum spread with the longitudinal
motion of the particle.
o
dipoledipoles
dipoles
DldssD *)()(
Question: what will happen, if you do not make too
many mistakes and your particle performs
one complete turn ?
17.) Tune and Quadrupoles
0 0
0
0 0 0
0
cos sin sin
( ) cos (1 )sincos sin
ss s s s
s s s ss s s
s
M
s
Transfer Matrix from point „0“ in the lattice to point „s“:
cos sin sin
sin cos sin
turn turn turnturn
turn turn turn
C SM
C S
Definition: phase advance
of the particle oscillation
per revolution in units of 2π
is called tune
2 2
turnQ
x(s)
s
0
Matrix for one complete turn
the Twiss parameters are periodic in L:( ) ( )
( ) ( )
( ) ( )
s L s
s L s
s L s
Quadrupole Error in the Lattice
optic perturbation described by thin lens quadrupole
0
rule for getting the tune
Quadrupole error Tune Shift
ideal storage ringquad error
turnturnturn
turnturnturn
kdistKds
MMMsincossin
sinsincos*
1
01* 0
turnturnturnturnturnturn
turnturnturn
distKdsKds
Msincossin*sin)sin(cos
sinsincos
00 sincos2cos2)( KdsMTrace
remember the old fashioned trigonometric stuff and assume that the error is small !!!
1
and referring to Q instead of ψ:
2
sincos)cos( 0
00
Kds
2
sincossin*sincos*cos 0
000
Kds
2
Kds
Q2
Ls
s
dsssKQ
0
04
)()(
a quadrupol error leads to a shift of the tune:
! the tune shift is proportional to the β-function at the quadrupole
!! field quality, power supply tolerances etc are much tighter at places where β is large
!!! mini beta quads: β ≈ 1900
arc quads: β ≈ 80
!!!! β is a measure for the sensitivity of the beam
Example: measurement of β in a storage ring:
GI06 NR
y = -6.7863x + 0.3883
y = -3E-12x + 0.28140.2800
0.2850
0.2900
0.2950
0.3000
0.3050
0.01250 0.01300 0.01350 0.01400 0.01450
k*L
Qx
,Qy
tune spectrum ... tune shift as a function of a gadient change
44
)()(0
0
quadLs
s
KldsssKQ
18.) Chromaticity:
A Quadrupole Error for Δp/p ≠ 0
Influence of external fields on the beam: prop. to magn. field & prop. zu 1/p
dipole magnet
focusing lensg
kp
e
particle having ...
to high energy
to low energy
ideal energy
ep
dlB
/ p
psDsxD )()(
definition of chromaticity:
gk
pe
Chromaticity: Q'
in case of a momentum spread:
… which acts like a quadrupole error in the machine and leads to a tune spread:
0p p p
kkgp
p
p
e
pp
egk 0
000
)1(
0
0
kp
pk
dsskp
pQ )(
4
10
0
p
pQQ '
Problem: chromaticity is generated by the lattice itself !!
ξ is a number indicating the size of the tune spot in the working diagram,
ξ is always created if the beam is focussed
it is determined by the focusing strength k of all quadrupoles
k = quadrupole strength
β = betafunction indicates the beam size … and even more the sensitivity of
the beam to external fields
Example: HERA
HERA-p: Q' = -70 … -80
Δ p/p = 0.5 *10-3
Q = 0.257 … 0.337
Some particles get very close to
resonances and are lost
odssskQ )()(*
4
1
dsssDkskQ )()()(4
121
N
Sextupole Magnets:
Correction of Q':
1.) sort the particles acording to their momentum ( ) ( )D
px s D s
p
2.) apply a magnetic field that rises quadratically with x (sextupole field)
xB gxz
2 21( )
2zB g x z
linear rising
„gradient“: x zB B
gxz x
S
SN
corrected chromaticity:
k1 normalised quadrupole strength
xkep
xgsextk *
/
~)( 21
p
pDksextk **)( 21
k2 normalised sextupole strength
D
x
D
xsext
D
sextD
F
x
F
xsext
F
sextF
x DlkDlkdssskQ 2214
1
4
1)()(*
4
1
Chromaticity in the FoDo Lattice
(1 sin )2ˆ
sin
L (1 sin )2
sin
Lβ-Function in a FoDo
x
y
dssskQ )()(*4
1
QfNQ
ˆ
*4
1
sin
)2
sin1()2
sin1(
*1
*4
1LL
fNQ
Q
remember ...
sin 2sin cos2 2
x xx
putting ...
sin2 4 Q
L
f
contribution of one FoDo Cell to the chromaticity of the ring:
using some TLC transformations ... ξ can be expressed in a very simple form:
sin
2sin2
*1
*4
1L
fNQ
Q
2cos
2sin
2sin
*1
*4
1L
fNQ
Q
2sin
2tan
*4
1L
fQ
Q
cell
2tan*
1cellQ
19.) Resume´:1
beam emittance:
2
0 0 0( ) 2s s s
2
0
0
( )s
s
beta function in a drift:
… and for α = 0
1)
1(
0
2p
pkxxparticle trajectory for Δp/p ≠ 0
inhomogenious equation:
( ) ( ) ( )p
x s x s D sp
… and its solution:
momentum compaction: p
p
L
lcp
R
DD
Lcp
2
quadrupole error:
chromaticity:
ls
s
dsssKQ
0
04
)()(
dsssKQ )()(4
1