2 - CERN Accelerator School

Introduction 

Emittance mittance growth in single- 

passage systems 

Sources of emittance 

growth 

M. Giovannozzi 

CERN- CERN AB Department 

Summary: 

Scattering through thin foils 

Emittance growth in multi- 

passage systems 

Injection process 

Scattering processes 

Others 

Emittance manipulation 

Longitudinal 

Transverse 

Acknowledgements: 

D. Brandt and D. Möhl 

hl 

Massimo Giovannozzi CAS - 2-14 14 October 2005 1

Introduction - I 

The starting point is the well-known well known Hill’s Hill s 

equation 

x ″ (s) + K(s) 

x(s) 

= 

Such an equation has an invariant (the so- 

called Courant-Snyder Courant Snyder invariant) 

2 

A = γ x + 2α x x′ 

+ β x′ 

Parenthetically: in in a bending-free bending bending-free free region 

the following dispersion invariant exists 

2 

A = γ 

D + 2α D D′ 

+ β D′ 

Massimo Giovannozzi CAS - 2-14 14 October 2005 2 

2 

2 

0

Introduction - II 

In the case of a beam, i.e. an ensemble of 

particles: 

Emittance: Emittance: 

value of the Courant-Snyder 

Courant Snyder 

invariant corresponding to a given fraction of 

particles. 

Example: rms emittance for Gaussian beams. 

Why emittance can grow? 

Hill equation is linear -> > in the presence of 

nonlinear conserved. 

effects emittance is no more 


Introduction - III 

Why emittance growth is an issue? 

Machine performance is reduced, e.g. in the 

case of a collider the luminosity (i.e. the rate 

of collisions per unit time) is reduced. 

It can lead to beam losses. 


Introduction - IV 

Filamentation is one of the key concepts for 

computing emittance growth 

Due to the presence of nonlinear imperfections, 

the rotation frequency in phase space is amplitude- 

dependent. 

After a certain time the initial beam distribution is 

smeared out to fill a phase space ellipse. 


Typical situation: 

Incident 

beam 

Scattering through thin foil - I 

Vacuum window between the transfer line and a target (in 

case of fixed target physics) 

Vacuum window to separate standard vacuum in transfer line 

from high vacuum in circular machine 

Emerging 

beam 

The beam receives an 

angular kick 


x 

p 

i 

xi 

Multiple Coulomb scattering 

due to beam-matter 

beam matter 

interaction is described by 

means of the rms 

scattering angle: 

14 MeV / c L 

θ = q 1 + 

rms 

p 

pβ 

L 

Downstream of the foil the 

transformed coordinates 

are given by 

→ 

→ 

x 

i 

p 

xi 

= 

+ 

A 

p 

io 

Δ 

sin( ψ 

p 

= 

A 

i 

io 

) 

Scattering through thin foil - II 

rad 

cos( ψ 

( ε ) 

i 

corr 

) + β θ 

Normalised coordinate 

= α x + β x′ 


p x 

α = 0 at the 

location of the foil

By assuming that: 

Scattering through thin foil - 

III 

Scattering angle and betatronic phase are uncorrelated 

Averaging of the phase, i.e. assuming filamentation in the 

transfer line or the subsequent machine 

2 

i 

2 

i 

2 

xi 

Therefore the final result is 

Taking into account the statistical definition of emittance 

π 

ε rms ( θ 

2 

= Δ 


rms 

2 

) 

2 

i0 

A = x + p = A + β θ 

2 

x 

2 

x0 

2σ = 

2σ 

+ β θ rms 

2 

β 

2 

2 

2 

i


IV 

Few remarks 

A special case with α = 0 at the location of the 

thin foil is discussed -> > it can be generalised. 

The correct way of treating this problem is (see 

next slides): 

Compute all three second-order second order moments of the beam 

distribution downstream of the foil 

Evaluate the new optical parameters and emittance using 

the statistical definition 

The emittance function! 

growth depends on the beta- 

THE SMALLER THE VALUE OF THE BETA- BETA 

FUNCTION AT THE LOCATION OF THE FOIL 

THE SMALLER THE EMITTANCE GROWTH 



V 

Correct computation (always for α=0): =0): 

x 

= 

x′ 

= 

− 

A 

o 

A 

o 

β 

o 

1/ 

β 

o 

By squaring and averaging over the beam distribution 

Using the relation 

cos( ψ ) 

o 

sin( ψ ) + θ = − A 

o 1 

2 

2 < A0 

> 

< x > = 

2 

′ 2 < A > 0 

< x > = 

2β 

2 

0 

β 

0 

1/ 

β 

1 

cos( ψ ) 

1 

sin( ψ 

1 


+ 

ε rms = 

π 

θ 

= 

2 

rms 

= 

A 

1 

1 

β 

1 

< A2 

> 

2 

1 

β 

1 

< A 2 > 1 = 

2β 

2 

A 

2 

)

The solution of the system is 


VI 

This can be solved exactly, or by assuming that the 

relative emittance growth is small, then 

emittance growth is now only half 

π 

Δε 

rms = θ 2β 

rms 0 

4 

1, β1 

Twiss 

⎡ 

⎤ 

⎢ π θ 2 ⎥ 

β = β ⎢1− 

rms 

1 0 

⎥ 

⎢ 4 ε ⎥ 

⎢⎣ 

rms ⎥⎦ 

NB: NB: the the emittance 

growth is now only half 

of of the the previous estimate! 

Downstream of of the the foil foil the the transfer line 

line 

should be be matched using the the new new Twiss 

parameters α1, α1, β1 


A 

β1 

β 

0 

2 

1 

2 

= 

− 

A 

2 

0 

A 

A 

A 

2 

0 

2 

0 

2 

1 

2 

= 

2β 

0 

θ 

2 

rms

70 

50 

30 

10 

-10 

β H 

Stripper location 

D V 

Example: ion stripping for LHC 

lead beam between PS and SPS 

β V 

D H 

0 50 100 150 200 250 300 

Courtesy M. M. Martini - CERN 

Pb 54+ -> > Pb 82+ 

A low-beta low beta 

insertion is 

designed (beta 

reduced by a 

factor of 5) 

Stripping foil, 

0.8 mm thick 

Al, is located in 

the low-beta low beta 

insertion 


Exercises… 

Exercises 

Compute the Twiss parameters and emittance 

growth for a THICK foil 

Compute the Twiss parameters and emittance 

growth for a THICK foil in a quadrupolar field 


Two sources of errors: 

Injection process - I 

Steering errors. 

Optics errors (Twiss ( Twiss parameters and dispersion). 

In case the incoming beam has an energy 

error, then the next effect will be a 

combination of the two. 

In all cases filamentation, filamentation, 

i.e. nonlinear 

imperfection in the ring, is the source of 

emittance growth. 


Steering errors: 

Injection process - II 

Injection conditions, i.e. position and angle, do not 

match position and angle of the closed orbit. 

Consequences: 

The beam performs betatron oscillations around the 

closed orbit. The emittance grows due to the 

filamentation 

Solution: 

Change the injection conditions, either by steering 

in the transfer line or using the septum and the 

kicker. 

In practice, slow drifts of settings may require 

regular tuning. In this case a damper (see course 

on feedback systems) is the best solution. 


Analysis in normalised phase 

space (i ( stands for injection 

m for machine): 

x 

p 

m 

m 

= 

= 

r 

r 

i 

i 

cosψ 

sinψ 

Squaring and averaging gives 

< 

< 

r 

r 

2 

m 

2 

m 

> 

> 

= 

= 

< 

< 

After filamentation 

2 

x 

r 

2 

i 

i 

2 

m 

2 

i 

+ Δr 

cosψ 

Injection process - III 

+ Δr 

cosψ 

+ 

p 

2 

m 

> + Δr 

< x > = 1 < rm 

> = 1 < ri 

> + 1 Δr 

2 

2 

2 

> 

2 

2 


σ 

2 

= σ 

2 

i 

+ 

1 2 

Δr 

2

Example of beam 

distribution 

generated by 

steering errors 

and filamentation. 

filamentation 

The beam core is 

displaced -> > large 

effect on 

emittance growth 

Injection process - IV 


Dispersion mismatch: 

analysis is similar to 

that for steering errors. 

In this case 

Δr 

2 

⎡ 

= ⎢ΔD 

⎣ 

2 

+ 

1 

2 

( β ΔD′ 

+ α ΔD) 

The final result is 

ε 

after fil. 

rms 

= ε 

rms 

Injection process - V 

2 

⎤ ⎛ σ 

⎥ 

⎜ 

⎦ 

⎜ 

⎝ p 

p 


⎞ 

⎟ 

⎠ 

2 

⎧ 

2 

⎪ 1 [ ( ) ] 

2 

2 ⎛ σ p ⎞ 

⎨1 

+ ΔD 

+ β ΔD′ 

+ α ΔD 

⎜ ⎟ 

⎜ ⎟ 

/ σ 0 

⎪⎩ 

2 

⎝ p ⎠ 

2 

⎫ 

⎪ 

⎬ 

⎪⎭

Optics errors: 

Optical parameters (Twiss ( Twiss and 

dispersion) of the transfer line 

at the injection point are 

different from those of the ring 

Consequences: 

The beam performs quadrupolar 

oscillations (size changes on a 

turn-by turn by-turn turn basis). The 

emittance grows due to the 

filamentation 

Solution: 

Tune transfer line to match 

optics of the ring 

Injection process - VI 


x’ 

Ring ellipse 

Transfer 

line ellipse 

x

Injection process - VII 

In normalised phase space (that of the ring) the 

injected beam will fill an ellipse due to the 

mismatch of the optics… 

optics 


ε 

after fil. 

rms 

= 

ε 

rms 

⎛ 

⎜ 

⎝ 

l 

i 

Injection process - VIII 

Given the initial condition of the beam end of the transfer 

line 

x = A b / a sinψ 

, p = A a / b cosψ 

b / a 

2 

< 

2 2 

ri >= < Ai 

+ a / b 

⎞ 

⎟ 

⎠ 

i 

By squaring and averaging over the particles’ particles distribution 

> 


l 

i 

( b / a + a / b) 

The emittance after filamentation is given by 

ε 

F 

2 

after fil. 

rms 

= 

= 

ε 

rms 

1 

⎛ 

⎜ β l 

2 ⎜ β m ⎝ 

+ 

F 

β m 

β 

l 

i 

⎛ α 

+ 

⎜ 

⎝ β 

m 

m 

αl 

⎞ 

− 

⎟ 

β l ⎠ 

2 

β 

m 

⎞ 

β 

⎟ 

l ⎟ 

⎠

Scattering processes - I 

Two main categories considered: 

Scattering on residual gas -> > similar to scattering 

on a thin foil (the gas replaces the foil…) foil 

Δε 

kσ 

= 

2 

2 2 14 MeV / c β pc 

t 

k q p 

β 

2 pβ 

⎟ 

p Lrad 

⎟ 

⎛ 

⎞ 

⎜ 

⎝ 

⎠ 

π 

NB: βpct ct represents the 

scatterer length until time t 

NB: β p ct represents the 

where 

average beta 

Intra-beam Intra beam scattering, i.e. Coulomb scattering 

between charged particles in the beam. 


β 

This is is why good 

vacuum is is necessary!

Intra-beam Intra beam scattering 

Multiple (small angle) 

Coulomb scattering 

between particles. 

charged 

Single scattering 

events lead to 

Touscheck effect. 

All three degrees of 

freedom are affected. 

Scattering processes - II 


Scattering processes - III 

How to compute IBS? 

Transform momenta of colliding particles into their centre 

of mass system. 

Rutherford cross-section cross section is used to compute change of 

momenta. momenta 

Transform the new momenta back to the laboratory 

system. 

Calculate the change of the emittances due to the change 

of momenta at the given location of the collision. 

1. For each scattering event compute average over all 

possible scattering angles (impact parameters from the 

size of the nucleus to the beam radius). 

2. Take the average over momenta and transverse position of 

the particles at the given location on the ring 

circumference (assuming Gaussian distribution in all phase 

space planes. 

3. Compute the average around the circumference, including 

lattice functions, to determine the change per turn. 


Scattering processes - IV 

Features of IBS 

For constant lattice functions and below 

transition energy, the sum of the three 

emittances is constant. 

Above transition the sum of the emittances 

always grows. 

In any strong focusing lattice the sum of the 

emittances always grows. 

Even though the sum grows from the 

theoretical point of view emittance reduction 

in one plane predicted by simulations, but 

were never observed. 


Scattering processes - V 

Scaling laws of IBS 

Accurate computations can be performed only with 

numerical tools. 

However, scaling laws can be derived. 

Assuming 

1 1 

= 

Then 

1/ 

τ0 

= 

( 4/ 

2 

π 

* 

x 

* 

y 

τ 

2 

2 

2⎛q 

⎞ 

Nbr0 

⎜ 

A ⎟ 

⎝ ⎠ 

) γ ε ε ε / E 

* 

l 

x, 

y, 

l 

0 

τ 

∝ 

0 

F x, 

y, 

l 

2 ⎛q 

⎞ 

Nb 

⎜ 

A ⎟ 

⎝ ⎠ 

γ ε ε ε 

* 

x 


* 

y 

2 

* 

l 

Strong charge 

dependence on 

on 

ε x,y,l* x,y,l* 

are x,y,l* are 

emittances 

normalised 

N b of b of particles/bunch 

r 0 classical 0 classical proton radius

Diffusive phenomena: 

Others 

Resonance crossings 

Ripple in the power converters 

Collective effects 

Space charge (soft part of Coulomb interactions 

between charged particles in the beam) -> > covered 

by a specific course. course 

Beam-beam 

Beam beam -> > covered by a specific course. course 

Instabilities -> > covered by a specific course. 

course 


Emittance manipulation 

Emittance is (hopefully) conserved… 

conserved 

Sometimes, it is necessary to manipulate the 

beam so to reduce its emittance. 

emittance 

Standard techniques: electron cooling, stochastic 

cooling. 

Less standard techniques: longitudinal or transverse 

emittance splitting. 


PS ejection: 

72 bunches 

in 1 turn 

Acceleration 

to 25 GeV 

PS injection: 

2+4 bunches 

in 2 batches 

320 ns beam gap 

Quadruple splitting 

at 25 GeV 

Triple splitting 

at 1.4 GeV 

Longitudinal manipulation: 

LHC beam in PS machine - I 

Empty 

bucket 

72 bunches 

on h=84 

18 bunches 

on h=21 

6 bunches 

on h=7 

0.35 eVs (bunch) 

1.1 × 10 11 ppb 

1 eVs (bunch) 

4.4 × 10 11 ppb 

1.4 eVs (bunch) 

13.2 × 10 11 ppb 

40 % 

Blow-up 

(pessimistic) 

115 % 

Blow-up 

(voluntary) 

Courtesy R. R. Garoby - CERN 


Courtesy R. R. Garoby - CERN 

Longitudinal manipulation: 

LHC beam in PS machine - 

II 

Measurement results 

obtained at at the CERN 

Proton Synchrotron 


Transverse manipulation: 

novel multi-turn multi turn extraction – 

I 

The main ingredients of the novel extraction: 

The beam splitting is not performed using a 

mechanical device, thus avoiding losses. Indeed, the 

beam is separated in the transverse phase space 

using 

Nonlinear magnetic elements (sextupoles ad 

octupoles) to create stable islands. islands 

Slow (adiabatic ( adiabatic) ) tune-variation tune variation to cross an 

appropriate resonance. 

NB: third-order third third-order order extraction is is based on 

separatrices (see O. Brüning Br Brüning ning lecture) 


1 

X’ 

(a) 

-1 

-1 X 

1 

Four islands 

require 

tune = 0.25 

Transverse manipulation: novel 

multi-turn multi turn extraction – II 

Left: initial phase 

space topology. No 

islands. 

Right: intermediate 

phase space topology. 

Islands are created 

near the centre. 

1 

X’ 

-1 

-1 X 

1 

-1 

-1 X 

1 


(c) 

1 

X’ 

(b) 

Bottom: final phase 

space topology. 

Islands are separated 

to allow extraction.

Tune variation 

Phase space 

portrait 

Transverse manipulation: 

novel multi-turn multi turn extraction – 

III 

Simulation 

parameters: 

Hénon non-like like 

map (i.e. 2D 

polynomial – 

degree 3 - 

mapping) 

representing 

a FODO cell 

with 

sextupole and 

octupole 


A movie to show the evolution of 

beam distribution 

A series of 

horizontal beam 

profiles have been 

taken during the 

capture process. 

Measurement results 

obtained at the CERN 

Proton Synchrotron 


Some references 

D. Edwards, M. Syphers: Syphers: 

An introduction to the 

physics of high energy accelerators, J. Wiley & Sons, 

NY 1993. 

P. Bryant, K. Johnsen: Johnsen: 

The principles of circular 

accelerators and storage rings, Cambridge University 

press, 1993. 

P. Bryant: Beam transfer lines, CERN yellow rep. 94- 94 

10 (CAS, Jyvaskyla, Jyvaskyla, 

Finland , 1992). 

J. Buon: Buon: 

Beam phase space and emittance, emittance, 

CERN 

yellow rep. 91-04 91 04 (CAS, Julich, Julich, 

Germany, 1990). 

A. Piwinski: Piwinski: 

Intra-beam Intra beam scattering, CERN yellow rep. 

85-19 85 19 (CAS, Gif-sur Gif sur-Yvette, Yvette, France 1984). 

A. H. Sorensen: Sorensen: 

Introduction to intra-beam 

intra beam 

scattering, CERN yellow rep. 87-10 87 10 (CAS, Aarhus, Aarhus, 

Denmark 1986).

2 - CERN Accelerator School

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