Beam Transfer Lines

• Distinctions between transfer lines and circular machines

• Linking machines together

• Blow-up from steering errors

• Correction of injection oscillations

• Blow-up from optics mismatch

• Optics measurement

• Blow-up from thin screens

Verena Kain

CERN

(based on lecture by B. Goddard and M. Meddahi)

Injection, extraction and transfer

CERN Complex

LHC: Large Hadron Collider

SPS: Super Proton Synchrotron

AD: Antiproton Decelerator

ISOLDE: Isotope Separator Online Device

PSB: Proton Synchrotron Booster

PS: Proton Synchrotron

LINAC: LINear Accelerator

LEIR: Low Energy Ring

CNGS: CERN Neutrino to Gran Sasso

Transfer lines transport the

beam between accelerators,

and onto targets, dumps,

instruments etc.

• An accelerator has limited dynamic range

• Chain of stages needed to reach high energy

• Periodic re-filling of storage rings, like LHC

• External experiments, like CNGS

Normalised phase space

• Transform real transverse coordinates x, x’ by

'1

1

xx

x

SS

S

S

'X

X

'

011

' x

x

x

x

SSS

N'X

X

Normalised phase space

1

x

x’

1max'x

maxx

Area = p

maxX

max'X

Area = p

22 ''2 xxxx 22 'XX

Real phase space Normalised phase space

X

'X

General transport

1

1

1

1

21'

2

2

'''' x

x

SC

SC

x

x

x

xM

x

x

y

y

s s

1

2

n

i

n

1

21 MM

Beam transport: moving from s1 to s2 through n elements, each with transfer matrix Mi

sincossin1cos1

sinsincos

22

12121

21

2111

2

21M

Twiss

parameterisation

Circular Machine

Circumference = L

• The solution is periodic

• Periodicity condition for one turn (closed ring) imposes 1=2, 1=2, D1=D2

• This condition uniquely determines (s), (s), (s), D(s) around the whole ring

QQQ

QQQ

L ppp

ppp

2sin2cos2sin11

2sin2sin2cos2021 MMOne turn

Circular Machine

• Periodicity of the structure leads to regular motion

– Map single particle coordinates on each turn at any location

– Describes an ellipse in phase space, defined by one set of and

values Matched Ellipse (for this location)

x

x’

2

max

1'

ax

ax max

Area = pa

2

22

1

''2

xxxxa

Circular Machine

• For a location with matched ellipse (, , an injected beam of

emittance , characterised by a different ellipse (*, *) generates

(via filamentation) a large ellipse with the original , , but larger

x

x’

After filamentation

,

, ,

, ,

x

x’

After filamentation

,

, ,

, ,

Matched ellipse

determines beam shape

Turn 1 Turn 2

Turn 3 Turn n>>1

Transfer line

sincossin1cos1

sinsincos

22

12121

21

2111

2

21M

'

1

1

21'

2

2

x

x

x

xM

'

1

1

x

x

• No periodic condition exists

• The Twiss parameters are simply propagated from beginning to end of line

• At any point in line, (s) (s) are functions of 1 1

One pass:

'

2

2

x

x

Transfer line

L0M

• On a single pass…

– Map single particle coordinates at entrance and exit.

– Infinite number of equally valid possible starting ellipses for single particle

……transported to infinite number of final ellipses…

x

x’

x

x’

1, 1

1, 1

Transfer Line

2, 2

2, 2

'

2

2

x

x

'

1

1

x

x

Entry Exit

Transfer Line

• Initial , defined for transfer line by beam shape at entrance

• Propagation of this beam ellipse depends on line elements

• A transfer line optics is different for different input beams

x

x’

,

x

x’,

Gaussian beamNon-Gaussian beam

(e.g. slow extracted)

x

x’

,

x

x’,

Gaussian beamNon-Gaussian beam

(e.g. slow extracted)

1500 2000 2500 30000

50

100

150

200

250

300

350

S [m]

Beta

X [

m]

Horizontal optics

Transfer Line

• The optics functions in the line depend on the initial values

• Same considerations are true for Dispersion function:

– Dispersion in ring defined by periodic solution ring elements

– Dispersion in line defined by initial D and D’ and line elements

- Design x functions in a transfer line

x functions with different initial conditions

1500 2000 2500 30000

50

100

150

200

250

300

350

S [m]

Beta

X [

m]

Horizontal optics

Transfer Line

• Another difference….unlike a circular ring, a change of an element

in a line affects only the downstream Twiss values (including

dispersion)

10% change in

this QF strength

- Unperturbed x functions in a transfer line

x functions with modification of one quadrupole strength

Linking Machines

• Beams have to be transported from extraction of one machine to

injection of next machine

– Trajectories must be matched, ideally in all 6 geometric degrees of freedom

(x,y,z,q,f,y)

• Other important constraints can include

– Minimum bend radius, maximum quadrupole gradient, magnet aperture,

cost, geology

Linking Machines

1

1

1

22

22

2

2

2

'''2'

''''

2

SSCC

SSSCCSCC

SCSC

Extraction

Transfer

Injection

1x, 1x , 1y, 1y x(s), x(s) , y(s), y(s)

s

The Twiss parameters can be propagated

when the transfer matrix M is known

'

1

1

'

1

1

21'

2

2

'' x

x

SC

SC

x

x

x

xM

2x, 2x , 2y, 2y

Linking Machines

• Linking the optics is a complicated process

– Parameters at start of line have to be propagated to matched parameters

at the end of the line

– Need to “match” 8 variables (x x Dx D’x and y y Dy D’y)

– Maximum and D values are imposed by magnet apertures

– Other constraints can exist

• phase conditions for collimators,

• insertions for special equipment like stripping foils

– Need to use a number of independently powered (“matching”)

quadrupoles

– Matching with computer codes and relying on mixture of theory,

experience, intuition, trial and error, …

Linking Machines

• For long transfer lines we can simplify the problem by designing the

line in separate sections

– Regular central section – e.g. FODO or doublet, with quads at regular

spacing, (+ bending dipoles), with magnets powered in series

– Initial and final matching sections – independently powered quadrupoles,

with sometimes irregular spacing.

0

50

100

150

200

250

300

0 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000

S [m]

[

m]

BETXBETY

SPS LHC

Regular lattice (FODO)

(elements all powered in series

with same strengths) Final

matching section

SPS to LHC Transfer Line (3 km)

Extraction

point Injection

point

Initial matching

section

• Magnet misalignments, field and powering errors cause the

trajectory to deviate from the design

• Use small independently powered dipole magnets (correctors) to

steer the beam

• Measure the response using monitors (pick-ups) downstream of the

corrector (p/2, 3p/2, …)

• Horizontal and vertical elements are separated

• H-correctors and pick-ups located at F-quadrupoles (large x )

• V-correctors and pick-ups located at D-quadrupoles (large y)

Trajectory correction

p/2

Corrector dipole Pickup

Trajectory

QF

QD QD

QF

Trajectory correction

• Global correction can be used which attempts to minimise the RMS

offsets at the BPMs, using all or some of the available corrector

magnets.

• Steering in matching sections, extraction and injection region

requires particular care

– D and functions can be large bigger beam size

– Often very limited in aperture

– Injection offsets can be detrimental for performance

Trajectory correction

Uncorrected trajectory.

y growing as a result

of random errors in the

line.

The RMS at the BPMs

is 3.4 mm, and ymax is

12.0mm

Corrected trajectory.

The RMS at the BPMs

is 0.3mm and ymax is

1mm

Steering (dipole) errors

• Precise delivery of the beam is important.

– To avoid injection oscillations and emittance growth in rings

– For stability on secondary particle production targets

• Convenient to express injection error in s (includes x and x’ errors)

X

'X

a

a [s] = ((X2+X’2)/) x2 2xx’+ x’2/

Bumper

magnets

Septum

kicker Mis-steered

injected beam

Steering (dipole) errors

• Static effects (e.g. from errors in alignment, field, calibration, …) are

dealt with by trajectory correction (steering).

• But there are also dynamic effects, from:

– Power supply ripples

– Temperature variations

– Non-trapezoidal kicker waveforms

• These dynamic effects produce a variable injection offset which can

vary from batch to batch, or even within a batch.

• An injection damper system is used to minimize effect on

emittance

Inje

cti

on

osc.

bunch number

LHC injected

batch

Beam 2

Blow-up from steering error

• Consider a collection of particles with max. amplitudes A

• The beam can be injected with a error in angle and position.

• For an injection error ay (in units of sigma = ) the mis-injected

beam is offset in normalised phase space by L = ay

X

'XMisinjected

beam

Matched

particles

L

A

Blow-up from steering error

• The new particle coordinates in normalised phase space are

q

q

sin

cos

new

new

LXX

LXX

'

0

'

0

2/2

'222

A

XXA

• For a general particle distribution,

where A denotes amplitude in

normalised phase space

Misinjected

beam

Matched

particles

L

A

q X

'X

Blow-up from steering error

• So if we plug in the new coordinates….

2/1

2/2/

0

0

2

22

a

LA

newnew

• Giving for the emittance increase

0 0

Blow-up from steering error

A numerical example….

Consider an offset a of 0.5 sigma for

injected beam

For nominal LHC beam:

norm = 3.5 m

allowed growth through LHC cycle ~ 10 %

Misinjected beam

Matched

Beam

0

0 2/1

1.125

2anew

0.5

X

'X

Injection oscillation correction

• x, x’ and y, y’ at injection point need to be corrected.

• Minimum diagnostics: 2 pickups per plane, 90° phase advance

apart

• Pickups need to be triggered to measure on the first turn

• Correctors in the transfer lines are used to minimize offset at these

pickups.

• Best strategy:

– Acquire many BPMs in circular machine (e.g. one octant/sextant of

machine)

– Combine acquisition of transfer line and of BPMs in circular machine

• Transfer line: difference trajectory to reference

• Circular machine: remove closed orbit from first turn trajectory pure

injection oscillation

– Correct combined trajectory with correctors in transfer line with typical

correction algorithms. Use correctors of the line only.

Example: LHC injection of beam 1

Display from the LHC control room to correct injection oscillations

Transfer line TI 2 ~3 km

Injection point in LHC IR2

LHC arc 23 ~3 km

closed orbit subtracted

-6 t

o +

6 m

m

Example: LHC injection of beam 1

• Oscillation down the line has developed in horizontal plane

• Injection oscillation amplitude > 1.5 mm

• Good working range of LHC transverse damper +/- 2 mm

• Aperture margin for injection oscillation is 2 mm

correct trajectory in line before continue LHC filling

• Optical errors occur in transfer line and ring, such that the beam can be

injected with a mismatch.

Blow-up from betatron mismatch

• Filamentation will produce an

emittance increase.

• In normalised phase space, consider

the matched beam as a circle, and the

mismatched beam as an ellipse.

Mismatched

beam

Matched

beam

X

'X

Blow-up from betatron mismatch

x2 = a2b2 sin(j +jo ), x '2 = a2 b2 cos(j +jo)-a2 sin(j +jo)[ ]

2

2

111 '

011

x

x

2

2

'X

X

2

121

1

2

1

2

2

2

121

1

2

2

122

22

22

22 'XX'XXA

General betatron motion

applying the normalising transformation for the matched beam

an ellipse is obtained in normalised phase space

2

2

121

1

2

2

1

1

2

2

121

1

2

newnewnew , ,

characterised by new, new and new, where

Blow-up from betatron mismatch

AA

ba

Mismatched

beam

Matched

Beam

a b

112

112

HHA

bHHA

a ,

1

2

2

2

121

1

2

2

1

2

1

2

1

newnewH

generally

From the general ellipse properties

where

giving

112

1111

2

1 HHHH

,

A

X

'X

)cos(1

),sin( 1new1new ff

ff A'XAX

Blow-up from betatron mismatch

)(cos1

)(sin22

02

220

2211newnew ff

ff AA'XXA 22

new

1

2

2

2

121

1

2

2

1002

2

02

11

2

1

Hnew

We can evaluate the square of the distance of a particle from the origin as

The new emittance is the average over all phases

If we’re feeling diligent, we can substitute back for to give

2

20

2

2

22

2

2

22

1

2

1

)(cos1

)(sin2

1

)(cos1

)(sin2

1

2

1

ff

ff

ff

ff

11

11new

20

20

20

2

A

AAAnew

where subscript 1 refers to matched ellipse, 2 to mismatched ellipse.

0.5 0.5

Blow-up from betatron mismatch

0

22

0

67.1

12

1

new

o

A numerical example….consider b = 3a for the mismatched ellipse

Mismatched

beam

Matched

Beam

a b=3a

3/ ab

Then

X

'X

OPTICS AND EMITTANCE

MEASUREMENT IN TRANSFER

LINES

Dispersion measurement

• Introduce ~ few permille momentum offset at extraction into transfer

line

• Measure position at different monitors for different momentum offset

– Linear fit of position versus dp/p at each BPM/screens.

Dispersion at the BPMs/screens

Data: 25.10.2009

x(s) = xb (s)+D(s) ×dp

p

Position at a certain BPM

versus momentum offset

Optics measurement with screens

• A profile monitor is needed to measure the beam size

– e.g. beam screen (luminescent) provides 2D density profile of the beam

• Profile fit gives transverse beam sizes s.

• In a ring, is ‘known’ so can be calculated from a single screen

Optics Measurement with 3 Screens

• Assume 3 screens in a dispersion free region

• Measurements of s1,s2,s3, plus the two transfer matrices M12 and

M13 allows determination of , and

s1 s2 s3

s1 s3 s2

e =s1

2

b1

=s 2

2

b2

=s 3

2

b3

21M

Optics Measurement with 3 Screens

• Remember:

1

1

1

1

21'

2

2

'''' x

x

SC

SC

x

x

x

xM

b2

a2

g2

é

ë

êêêê

ù

û

úúúú

=

C2

2 -2C2S2 S2

2

-C2C2 ' C1S1 '+ S1C1 ' -S2S2 '

C2 '2 -2C2 'S2 ' S2 '2

é

ë

êêêê

ù

û

úúúú

×

b1

a1

g1

é

ë

êêêê

ù

û

úúúú

b2 =C2

2 × b1 -2C2S2 ×a1 +S2

2 ×g1

b3 =C3

2 × b1 -2C3S3 ×a1 +S3

2 ×g1 ✕

s 2

2 =C2

2 × b1e -2C2S2 ×a1e +S2

2 ×g1e

s 3

2 =C3

2 × b1e -2C3S3 ×a1e +S3

2 ×g1e

Square of beam sizes as

function of optical functions at

first screen

Optics Measurement with 3 Screens

• Define matrix N where S = NP

• Measure beam sizes and want to calculate 1, 1,

• Solution to our problem S’ = N-1S

– with 11-12 = 1 get 3 equations for 1, 1 and – the optical functions at

the first screen

S =

s1

s 2

s 3

æ

è

çççç

ö

ø

÷÷÷÷meas

P =

b1e

a1e

g1e

æ

è

çççç

ö

ø

÷÷÷÷

b1 = A / AC -B2

a1 = B / AC -B2

e = AC -B2

with

A = ¢S1

B = ¢S2

C = ¢S3

…and if there is dispersion at the screens

• Dispersion and momentum spread need to be measured

independently at the different screens

• Trajectory transforms with Ti transport matrix for d ≠ 0

xi is the contribution to the dispersion between the first and the ith screen

• Define 6 dimensional S and P and respective N and then same

procedure as before

xi

¢xi

d

æ

è

çççç

ö

ø

÷÷÷÷

=

Ci Si xi

¢Ci ¢Si ¢xi

0 0 1

æ

è

çççç

ö

ø

÷÷÷÷

×

x1

¢x1

d

æ

è

çççç

ö

ø

÷÷÷÷

Di =CiDi +Si ¢Di +xi

S =

s 2

1

s 2

2

s 2

3

D1d2

¢D1d2

d 2

æ

è

ççççççççç

ö

ø

÷÷÷÷÷÷÷÷÷

P =

b1e +D1

2d 2

a1e -D1 ¢D1d2

g1e + ¢D1

2d 2

D1d2

¢D1d2

d 2

æ

è

ççççççççç

ö

ø

÷÷÷÷÷÷÷÷÷

6 screens with dispersion

• Can measure , , , D, D’ and d with 6 screens without any other

measurements.

• Invert N, multiply with S to get S’

d 2 = ¢S6

D1 = ¢S4¢S6

¢D1 = ¢S5¢S6

b1 = A / AC -B2

a1 = B / AC -B2

e = AC - B2

A = ¢S1 - ¢S 2

4S6¢

B = ¢S2 + ¢S4S5¢S6

¢

C = ¢S3 - ¢S 2

5S6¢

More than 6 screens…

• Fit procedure…

• Function to be minimized: i…measurement error

• Equation (*) can be solved analytically see

– G. Arduini et al., “New methods to derive the optical and beam

parameters in transport channels”, Nucl. Instrum. Methods Phys. Res.,

2001.

¶c 2

¶Pi

= 0 (*)

In Practice….

Difficulty in practice is fitting

the beam size accurately!!!

Matching screen

• 1 screen in the circular

machine

• Measure turn-by-turn profile

after injection

• Algorithm same as for several

screens in transfer line

• Only allowed with low intensity

beam

• Issue: radiation hard fast

cameras

Profiles at matching monitor

after injection with steering

error.

Blow-up from thin scatterer

• Scattering elements are sometimes required in the beam

– Thin beam screens (Al2O3,Ti) used to generate profiles.

– Metal windows also used to separate vacuum of transfer lines from

vacuum in circular machines.

– Foils are used to strip electrons to change charge state

• The emittance of the beam increases when it passes through, due

to multiple Coulomb scattering.

qs

radrad

inc

c

sL

L

L

LZ

cMeVpmrad 10

2 log11.01]/[

1.14][

qrms angle increase:

c = v/c, p = momentum, Zinc = particle charge /e, L = target length, Lrad = radiation length

Blow-up from thin scatterer

sq

'

0

'

0

XX

XX

new

new

Ellipse after

scattering

Matched

ellipse

Each particles gets a random angle change

qs but there is no effect on the positions at

the scatterer

After filamentation the particles have

different amplitudes and the beam has

a larger emittance

2/2Anew

X

'X

Ellipse after

filamentation

Matched

ellipse

2

02

snew q

uncorrelated

Blow-up from thin scatterer

20

20

22

22

2

2

22

2

2

s

ss

ss

ss

s

q

qq

qq

qq

q

'0

'0

'200

2

'0

'200

'0

20

'222

X

XXXA

XXX

XX

XXA

new

newnewnew

0

Need to keep small to minimise blow-up (small means large spread in

angles in beam distribution, so additional angle has small effect on distn.)

X

'X

Blow-up from charge stripping foil

• For LHC heavy ions, Pb53+ is stripped to Pb82+ at 4.25GeV/u using a

0.8mm thick Al foil, in the PS to SPS line

• is minimised with low- insertion (xy ~5 m) in the transfer line

• Emittance increase expected is about 8%

0 50 100 150 200 250 3000

20

40

60

80

100

120

Stripping foil

S [m]

Beta

[m

]

TT10 optics

beta X

beta Y

Kick-response measurement

• The observable during kick-response measurement are the

elements of the response matrix R

– ui is the position at the ith monitor

dj is the kick of the jth corrector

• Cannot read off optics parameters directly

• A fit varies certain parameters of a machine model to reproduce the

measured data LOCO principle

• The fit minimizes the quadratic norm of a difference vector V

Reference: K. Fuchsberger, CERN-THESIS-2011-075

Rij =ui

d j

Vk =Rijmeas -Rij

model

s i

k = i ×(Nc -1)+ j

Rijmodel =

bib j sin(mi -m j )

0

for i>j

otherwise

si… BPM rms noise

Nc… number of correctors

Example: LHC transfer line TI 8

• Phase error in the vertical plane

• Traced back to error in QD strength in transfer line arc:

Summary

• Transfer lines present interesting challenges and differences from

circular machines

– No periodic condition mean optics is defined by transfer line element

strengths and by initial beam ellipse

– Matching at the extremes is subject to many constraints

– Emittance blow-up is an important consideration, and arises from

several sources

– The optics of transfer line has to be well understood

– Several ways of assessing optics parameters in the transfer line have

been shown

Keywords for related topics

• Transfer lines

– Achromat bends

– Algorithms for optics matching

– The effect of alignment and gradient errors on the trajectory and optics

– Trajectory correction algorithms

– SVD trajectory analysis

– Phase-plane exchange insertion solutions