CAS Zakopane October 2006

Oliver Bruning / CERN AP−ABP

Imperfections

Linear

integer resonances

local orbit bumps

BPMs & dipole correctors

orbit correction

quadrupole perturbations

beta−beat

half−integer resonances

one−turn map & tune error

Linear Imperfections

equation of motion in an accelerator

Hills equation

sine and cosine like solutions

closed orbit

sources for closed orbit perturbations

dipole perturbations

closed orbit response

dispersion orbit

+ K(s) x = G(s);

d sd x

K(s) =

K(s) = K(s + L);2

2+ K(s) x = 0;

d sd x2

2 G(s) =

dds= d

dtdsdt

v

Hill’s Equation:

p[GeV]B[T/m]

ρ 2

0.3 quadrupole

dipole1/

drift0

v p0

F(s)Lorentz

Variable Definition

x s

ρ

sy

Perturbations:

Variables in moving coordinate system:

ddsx = x

x = p

p

0

x

Y (0) =

cos ( K s)

− K sin ( K s)

y (s) = a Y (s) + b Y (s)

=

0 K y = 0 y + y =

x

x

2Y (s) =

2

2

2

=

1

1Y (0) = 1

1

=

0

cos ( K [s−s ])

− K sin ( K [s−s ])0

sin ( K s)

K cos ( K s)

0

1

y (s) = M(s − s ) y (s ) 0 0

1

sin ( K [s−s ])

K cos ( K [s−s ])0

0

2

Y (s) =

0

Y

Y

Y

Y

10

1

K = const

initial conditions:

and

Sinelike and Cosinelike Solutions

system of first order linear differential equations:

general solution:

transport map:

with:

s 0φ( ) + φ β( )s

β( )ss 0φ( ) + φ

s 0φ( ) + φ

Y (s) =1

s 0φ( ) + φ

= (s )0

(s )00S(s ) =

(s )0

(s )0=

0C (s ) =

β( ) s 0s φ( ) + φsin ( )

α( )

β( )s cos ( )Y (s) =2

2

s 0φ( ) + φ[cos ( ) + s sin( )] /

α( )−[sin ( )+ s cos( )]/

1C (s) = a Y (s) + b Y (s) 1 2S(s) = c Y (s) + d Y (s)

1

C

S

S0

C 0

1

β1φ (s) = ds; β1

2

Floquet theorem:

Sinelike and Cosinelike Solutions

α (s) =− (s)

´sinelike´ and ´cosinelike´ solutions:

β β(s) = (s + L);

and

to the case with constant K(s)!one can generate a transport matrix in analogy

with:

0

C (s) =0φ( ) + φ s 0φ( ) + φ

s 0φ( ) + φ

0(α −α)

1

1 0

0 I =

−α

β

−γJ =

α2; γ = [1 + α ] / β

M = I cos(2 Q) + J sin(2 Q) π π

β( )s

s sβ( )β( )

α( )s 0φ( ) + φφ( ) + φ

sin ( )

[cos ( ) + s sin( )]/

0

[sin ( )+ cos( )]/

[cos ( ) + sin( )]/

0 y (s) = M(s, s ) y (s ) 0 0

M = C (s)

C (s)

S (s)

S (s)

sβ( )0

sβ( )0

S (s) =β( )ss 0φ( ) + φ

s 0φ( ) + φ

β β0

s 0φ( ) + φ α( )s0

0−(1+αα ) s

transport map for s = s + L:

´sinelike´ and ´cosinelike´ solutions:

transport map from s to s:

Sinelike and Cosinelike Solutions

with:

B = g x + g x

B DB

FBDB

dipole component

x = x + x0y

x

y

x

0

FB

DB F

B = g y

B = g x

B = g y

Closed Orbit

particles oscillate around an ideal orbit:

additional dipole fields perturb the orbit:

energy error

α = ρl = 1− q B l

error in dipole field

offset in quadrupole field

Δ p + pq B l Δ p

p p

d sd x2

2+ K(s) x = G(s); G(s) =

g [T/m]p [GeV]

p[GeV]

g[T/m]Δ

B [T]

p [GeV]

x

k (s) = 0.3

k (s) = 0.3

k (s) = 0.3

v p0

00

x

y x

yF = −q v B

F = q v BN S

NS

y

x R

x

y

B = g y

B = g x

F(s)Lorentz

normalized fields:

quadrupole misalignment:

0

0

0

dipole:

quadrupole: 10

0

Quadrupole Magnet

the perturbation adds up

Q: βnumber of −oscillations per turn

Q = N + 0.5

the perturbation cancelsafter each turn

Kick

watch out for integer tunes!

Dipole Error and Orbit Stability

Q = N

Kick

amplitude growth and particle loss

Quadrupole Error:

amplitude increase

1. Turn: x > 0

Q = N + 0.5

F

kick

amplitude increase

beam offset in quadrupole

F

x < 02. Turn:

watch out for half integer tunes!

Orbit Stability

Quadrupole Error and

orbit kick proportional to

momentum distribtion

power supplies

calibration

civil engineeringseasonsmoon

slow drift

civilisation

injection energy (RF off)

RF frequency

Energy error of particles

ground motion

Sources for Orbit Errors

alignment +/- 0.1 mm

Quadrupole offset:

Error in dipole strength

Example Quadrupole Alignment inLEP

x−y coupling

aperture beam losses

filamentation beam size

aperture

energy error

field imperfections

beam separation

dispersion beam size at IP

rms < 0.5 mm x, y < 4 mmΔ

beam monitors and

injection errors:

orbit correctors

Δ

closed orbit errors:

Problems Generated by Orbit Errrors

Aim:

π

t

V

γ

a)

b)

assume: L > design orbit

the orbit determines the particle energy!

Synchrotron:

RFf = h f

rev

f = 12

qm

Brev

energy increase

Equilibrium:

E depends on orbit and magnetic field!

Moon

tide

<

Earth

orbit and beam energy modulation:

E 10 MeVΔ

0.02%

aim: Δ E 0.003%

requires correction!

tidal motion of the earth:

modf = 24 h; 12 h

Δ E

[M

eV]

November 11th, 1992

Δ E

[M

eV]

August 29th, 1993

Daytime

October 11th, 1993

-5

0

5

23:00 3:00 7:00 11:00 15:00 19:00 23:00 3:00

-5

0

5

11:0

0

13:0

0

15:0

0

17:0

0

19:0

0

21:0

0

23:0

0

18:0

0

20:0

0

22:0

0

24:0

0

2:00

4:00

6:00

8:00

Δ

energy modulation due to tidal motion of earth

E 10 MeV

orbit and energy perturbations

the position of the LEP tunnel and thus the

quadrupole positions

Δ E 20 MeV

energy modulation due to lake level changes

changes in the water level of lake Geneva change

TGV line between Geneva and Bellegarde

energy modulation due current perturbations in

the main dipole magnets

RAIL

TGV

Gen

eve

Mey

rin

Zim

eysa

LEP Polarization Team17.11.1995

LEP beam pipe

LEP NMR

E 5 MeV for LEP operation at 45 GeVΔ

with the voltage on the TGC train tracks

correlation of NMR dipole field measurements

ground motion due to human activity

quadrupole motion in HERA−p (DESY Hamburg)

RMS

peak to peak

d x0

0 1 0 K

ψ( )s

s = c(s) S(s) + d(s) C(s) ψ( )

y(s) = a S(s) + b C(s) +

G

0 y = G; G = y +

G(s) =+ K(s) x = G(s);2

2

d sk (s)

Closed Orbit Response

Δ

inhomogeneous equation:

we need to find only one solution!

variation of the constant:

φ( )φ( )s

φ( ) G(t) dt −1

s

0

s s u(s);

t u(s) =

φ( )s

ψ( ) = φ( )

s

s

G(t) dt −1

s

0

u (s) = G(s)

φ( )=s

y(s) = a S(s) + b C(s) +

S(s)

S (s) C (s)

C(s)

t

Closed Orbit Response

with

substitute into differential equation:

variation of the constant in matrix form:

φ( )

s πt β( )2 sin( Q)

s β( )π

s

y(s) =x(s)x (s)

s φ( )

t x(s) = G(t) cos[ − Q] dt

y(s) = a S(s) + b C(s) +

φ( )−φ( )

t −1

G(t) dt

s0

s0

s0+circ

x (s) = x (s + L)

Closed Orbit Response

periodic boundary conditions determinecoefficients anda b

periodic boundary conditions:

with

; x(s) = x(s + L);

t

G(t) cos[ − Q] dt s t

Δ1ρ( )t

π

1ρ( )

x(s) =

p p

φ( )−φ( )t s π cos[ − Q] dt t β( )

p[GeV]

B[T]

G(t) = 1

ρ( )t

Δ p p

β( )

ρ( )2 sin( Q)

s β( )π

D(s) =

k (s) = 0.3

t

k (s) =

φ( )−φ( )

−

Δ p p x(s) = D(s)

t 2 sin( Q)

s β( )π

Closed Orbit Response

0

Dispersion Orbit

with

Example:

normalized dipole strength:

0

particle momentum error

0

0

from the quadrupole alignment errors

magnets is dominated by the contributions

the orbit error in a storage ring with conventional

β -functions at the location of the dipole error

Orbit Correction

orbit perturbation is proportional to the local

alignment errors at QD causes mainly

horizontal orbit errors

alignment errors at QF cause mainly

vertical orbit errors

place orbit corrector and BPM next

to the main quadrupoles

vertical BPM and corrector next to QD

QF MB

horizontal BPM and corrector next to QF

QD

Orbit Correction

BPM

HX

BPM

HV

aim at a local correction of the dipole error due

to the quadrupole alignment errors

orbit in the opposite plane?

relative alignment of BPM and quadrupole?

Lake Geneva

LEP Orbit

Horizontal Orbit:

beam offset in quadrupoles

moon

energy error

Vertical Orbit:

beam separationorbit deflection depends onparticle energyvertical dispersion [D(s)]

small vertical beam size relieson good orbit

corrections in physics1994: 13000 vertical orbit

beam offset in quadrupoles:

yσ = ε β + δ 2Dy

2

y

π

2

M = I cos(2 Q) + J sin(2 Q) π

2; γ = [1 + α ] / βJ = α

1

1

1 0

0 I =

−α

β

n+1z = M z n

x x

z =

sin( 2 Q) ππM − I cos( 2 Q)

−γ

cos( 2 Q) = trace M π

can be generated by matrix multiplication:

provide the optic functions at s

remember:

0

Quadrupole Gradient Error

the coefficients of:

one turn map:

and can be expressed in terms of the C and S solutions

1

0

1

1 m =

0

M = m m M 00

−1

Δ

−k

0 ππ π

l−[k + k ]

l

lcos(2 Q) = cos(2 Q ) − k sin(2 Q )

11

1

β Δ1 2

1 0

1 m =

0

one turn matrix with quadrupole error:

Quadrupole Gradient Error

matrix for single quadrupole with error:

transfer matrix for single quadrupole:

trace M

4

π0cos(2 Q) = cos(2 Q ) −

2 0

πsin(2 Q )

β Δ

Δ

k ds

Δ1 π4

Δ Q = − β k ds

β Δk ds

Δ

Δ

Q =

= ξ

1 π

Δ

π

distributed perturbation:

momentum error k = − k

chromaticity: k = e g p 1

1

1

1

1

1

p p

p p

p p

Quadrupole Gradient Error

2 sin( Q)

s β( )

2πt β( ) φ( )−φ( )t s = Δ 2πk(t) cos[2[ ]− Q] dt

M = I cos(2 Q) + J sin(2 Q) π π

n+1z = M z n

2; γ = [1 + α ] / β−α

β

−γJ =

α

m

m 21

1211

m 22

m

12

M =

1

1 0

0 I =

β

s Δβ( )

2π

s0+circ

msin( Q)

calculate:

s0

− Beat

quadrupole error:

with

β

− beat oscillates with twice the betatron frequency

i θ i

θ =

β β( )s − φi

φ( )s sin[ ] x(s) =

θ i

− φi

φ( )s cos[ ] s β( )iβ /x (s) =

= G (t) dt p[GeV]

i0.3 B [T] l

deflection angle:

trajectory response:

[no periodic boundary conditions]

Local Orbit Bumps I

dipole

ii

closure of the perturbation within one turn

possibility to correct orbit errors locally

local orbit excursion

π

closure with one additional corrector magnet

closure with two additional corrector magnets

three corrector bump

Local Orbit Bumps II

closed orbit bump:

- bump

compensate the trajectory perturbation with

additional corrector kicks further down stream

1

β

β 1 θ = θ

22

3 θ θ 1 2 θ

orequires 90 lattice

closure depends on lattice phase advance

sensitive to lattice errors

sensitive to BPM errors

requires large number of correctors

requires horizontal BPMs at QF and QD

s

limits / problems:

(quasi local correction of error)

QF QD QF QD

- bump:π

QF

Local Orbit Bumps III

x

β 1β

θ

β

β 2 θ = −

Δφ3-1

Δφ 3-2

Δφ 1

θ θ θ 1 2 3

θ 1

3-1 θ = 3

1

2

Δφ3-1

Δφ 3-2

3

QF QD QF

sin( )sin( )

QF QD

can not control x

- cos( )sin( )tan( )

requires only horizontal BPMs at QF

works for any lattice phase advance

limits / problems:

Local Orbit Bumps IV

x

s

3 corrector bump: (quasi local correction of error)

sensitive to BPM errors

large number of correctors

Summary Linear Imperfections

they amplifies the response to quadrupole field errors

avoid storage ring tunes near half−integer resonances:

dispersion = eigenvector of extended ´1−turn´ map

closed orbit = fixed point of ´1−turn´ map

tune is given by the trace of the ´1−turn´ map

twiss functions are given by the matrix elements

avoid machine tunes near integer resonances:

they amplifies the response to dipole field errors

a closed orbit perturbation propagates with the

betatron phase around the storage ring

discontinuities in the derivative of the closed

orbit response at the location of the perturbation

the betatron phase advance around the storage ring

betafunction perturbations propagate with twice

integral expressions are mainly used for estimates

numerical programs mainly rely on maps