collective effects in particle accelerators

collective effectsin particle accelerators

Frank Zimmermann

Bodrum Summer School

September 2007

what is an accelerator?

• charged particles moving in electromagnetic fields• these fields can be

- static or time dependent

- externally applied or beam generated

- linear (restoring force proportional to displacement) or nonlinear

• motion can be classical or quantum mechanical• combinations of all these occur and can be important

examples

• they have good memory• they won’t forgive you• easily perturbed and mistakes add up

beam particles are like elephants…

… and they are not alone!

particles do not move independently; many of the limits of accelerator performance arise from interactions between beam particles = collective effects

various types of collective effects• beam instabilities (coherent, motion of many particles

is correlated), due to – self-interaction [space charge]– interaction with vacuum chamber [impedance]– interaction with other beam [beam-beam effects]– interaction with “foreign” particles [ions or electrons]

• poor beam lifetime or emittance growth (incoherent), due to – scattering of individual particles off each other [intrabeam

scattering, Touschek effect]– motion of individual particles in the nonlinear electromagnetic

field generated by one of the four interactions above [space charge, beam-beam, incoherent electron cloud,…]

2 particles at rest or traveling

many charged particles traveling in an unbunched

beam with circular cross-section

K.-H. Schindl

βr

space chargespace chargecartoon - in reality transverseelectric (and magnetic) fields increase for relativstic particles

Quadrupole(F-type)

FocusingLinear

Uniform

DefocusingLinear

Gaussian

DefocusingNon-Linear

Proton

Beam

K.-H. Schindl

magnet space charge

space-charge force is defocusing in both x & y direction, unlike a quadrupole

BvEeF srr

22

02

2

0

1

21

2 a

r

c

eI

a

r

c

eIFr

consider as example:uniform round continuous charged beam of radius a & current I

space-charge force:

schematic of betatron oscillation around storage ringbetatron tune = number of oscillations per turn

interlude: betatron motion

C s

dsCQ

)(2

1

2

)(

space-charge effect on betatron motion

x

y ring effectivefocusing force

space chargedefocusing force

the global force on each particle shouldbe focusing! in storage rings this is always true.

tunes are dependent on transverse (and longitudinal) position through the global Coulomb-force effect of the beam

find a “jungle” ofcoherent and incoherent effects

G. Franchetti

2,;

032

,

0, 22 yxNyxyx

NrNrQ

space-charge tune shift - proportional to intensity N- inversely proportional to emittance - proportional to beam brightness N/N

- it decreases like 1/2 or 1/3

(important for low only)- it does not depend on machine radius

dsssKQ yx

C

yxscyx ,

0

,;, 4

1

change of betatron tune due to defocusing force:

from spacecharge force

100% (norm.) beam emittance for our example

cm

FK yxsc

yxsc0

,;,;

interlude: emittance

x

x’“area in phase space”occupied by the beam= x

rms emittance 222 '' xxxxrms

for Gaussian distributionrms ~ 39%, 4rms ~ 86%, 6rms ~95% of the beam

incoherent tune shift due to conducting walls

a line charge representing the particle beam between parallel conducting plates of distance 2h; electric field parallel to the conducting plates have to be zero; this is achieved by introducing negative image line charges (right)

K.H. SchindlJ.R. Laslett

incoherent tune shift due to conducting walls - 2

K.H. SchindlJ.R. Laslett

22

00 41

2

1

2

1

21

hn

y

ynhynhE nniny

y

hy

nhEE

n

inyiy 124

1

4

2

201

2201

0

y

E

x

EE iyixi

xh

Eix 124

2

20

K.H. Schindl

2

2

223

0

482

1

haec

ICrQ x

x

2

2

223

0

482

1

haec

ICrQ

y

y

direct s.c. imagefeatures: - electric image field is vertically defocusing but horizontally focusing

(typical for most vacuum beam pipes) - field is larger for smaller chamber height h- image effects decrease as 1/, much weaker than 1/3 for direct space charge; they are of some concern for high-energy p machines and e rings

incoherent tune shift due to conducting walls - 3

d.c. beam current is accompanied by dc magnetic field, which is not shielded by beam pipe, but influenced by ferromagnetic boundaries, like magnets, representedby mirror currents → incoherent tune shift due to magnetic images

incoherent and coherent tune shift K.H. Schindl

incoherent betatron motion of a particleinside a static beam with its center of massat rest

amplitude and phase are distributed at random over all particles

coherent motion of the whole beam after having received a transverse kick

the source of the direct space charge is now moving, individual particles still continue incoherent motion around the common coherent trajectory

coherent tune shift due to conducting wallexample: round, perfectly conducting beam pipe

coherent oscillation of the beam of the beam inside a circular perfectlyconducting beam pipe and its oscilllating image charge

xb 2

xbxb

xEix 2000

1

2

1

2

1

2

defocusing force

xe

xFix 20

1

2

22

,0

;, 2

NrQ

yx

cohyx

coherent tune shift due to conducting wall - 2

coherent force

coherent tune shift

for symmetry reasons the force is the same in x and y direction

Features:features: - force is linear in → there is a coherent tune shift- 1/ dependence stems from the fact that the field is proportional to the number of charged beam particles (independent of mass), but their deflection is inversely proportional to their relativistic mass m0- the coherent tune shift is never positive- effect of a thin vacuum chamber with finite coductivity are more subtle

xx

x

more realistic example: elliptical vacuum chamber,still unbunched beam with uniform density

incoherent and coherent tune shifts given by “Laslett coefficients” and

most coefficients are larger vertically; coherent coefficients all positive or 0

most rings storebunched beams;

the s.c. tune shiftfor bunched beams changes withlongitudinal coordinate z

→ tune spread

K.H. Schindl

space-charge tune spreadspace-charge tune spreadparticlein a storage ring, beam makes many turns

(e.g. PS booster ~106 turns)particles with small deviations from the design orbit oscillate

around this orbit in phase spaceinteger tunes, ½ integer tunes etc. must be avoided since

they lead to resonances and beam loss(particles will “sum up” all machine / magnet imperfection

resonances turn by turn)the space charge reduces the tune, and also leads to a

tune spread Q in the beam (for a real non-uniform and bunched beam particles at large transverse and longitudinal amplitudes will see less tune shift)

once Q becomes too big there will always be some particleson resonance and these will be lost

this is the major problem at low energy hadron accelerators M. Benedikt

Example for space-chargelimited synchrotron:betatron tune diagram andareas covered by direct tunespread at injection, intermediate energy,and extraction, for the CERN Proton Synchrotron Booster.During acceleration,acceleration gets weakerand the “necktie” areashrinks, enabling the externalmachine tunes to move the“necktie” to a region clearof betatron resonances(up to 4th order)

K.H. Schindl

nonlinear dynamics and space charge

the problem is complex

detuning resonance condition

particle amplitudegrowth

beam sizegrowth

G. Franchetti

particleloss

nonlinear dynamics in a bunch

periodic crossingof a resonance

z

x

bare tune

resonance

G. Franchetti

trapping into resonances during synchrotron motion

periodic crossingof a resonance

z

x

bare tune

resonance

G. Franchetti

trapping into a resonanceduring synchrotron motion

1 synchrotron oscillation in 6000 turns

G. Franchetti

single passage through a resonance

3rd orderresonance

+0.15-0.15

Qx = 0.1

Qy

Qx

Bare tune

role of transverse detuning when the stop-band is crossed

Bare tune

Several particles remain on one side of the resonance increasing their amplitude

3 Qx = 13

particle trapping into a resonanceduring accumulation (at injection energy)

Space charge increased in N turns

3rd orderresonance

Bare tune

Qx = 0.15

Qy

Qx

0.03

“Scattering”

N = 103 turns

Full beam emittance Test particle

N = 5 x103 turns Trapping

G. Franchetti

adiabatic / non-adiabatic regimescondition for a particle to remain trapped

Tune on theFixed point

Size of the island

Speed of the fixed point

If during 1 revolution around the fixed point the island moves less than its size than the particle can remain trapped

T << 1 characterizes the adiabatic regime

A.W. Chao and Month NIM 121, 129 (1974).A. Schoch, CERN Report, CERN 57-23, (1958)A.I. Neishtadt, Sov. J. Plasma Phys. 12, 568 (1986)

G. Franchetti

how to overcome the ‘space-charge limit’?

FNAL booster

CERN PS, BNL AGS

1. raise the injection energy!

K.H. Schindl

how to overcome the ‘space-charge limit’?2. flatten the bunch distribution!

transN

yx FCr

Q

ˆ4 2

0,

maximum line density

transverse form factor

rms emittance

Ftrans=1 for GaussianFtrans=1/2 for transversely uniform distribution

“phase-space painting”, double harmonic rf,…

how to measure the incoherent tune shift/spread?

K.H. Schindl

various types of collective effects; space charge

2) wake fields, impedances, beam instabilities, Landau damping

3) beam-beam effects

4) ions and electron cloud effects

collective effects in particle accelerators

wake fields• the real vacuum chamber (beam pipe) is not a perfectly

conducting pipe of constant aperture• a beam passing an obstacle radiates electromagnetic

fields and excites the normal modes of the object• consequences:

– beam loses energy– energy can be transferred from head to tail of a bunch– the head of the bunch can deflect the tail – energy and deflections can be transferred between bunches if

the high Q (quality factor) normal modes

• the wake fields characterize (“are”?) the beam induced energy losses and deflections

}Instabilities!

calculation by T. Weiland

wake-field properties

Particle of charge Q (=1) followed at a distance ct0 by q.Let Q travel on axis (RT=0) and let Ez be the longitudinal electric field at q.Longitudinal wake potential is

dzzEtV z0

longitudinal wake fieldin the ultrarelativistic limit (→1), V has simple properties:

V

talso, V is independent of rT

to get V(t0) for a beam, convolute V with the bunch shape

dttttVtV 00

a

now, let Q travel off axis

cos0'

200

0 tVa

RrtVtV TT

on-axis wake potential modes that have Ez=0 on axis are excited

V’(t0) must have same qualitative behavior as V(t0)

components azimuthal and radial have ,00' BEr

EV z

→ deflections are possible

dzBvER

tWT

1' 0

transverse wake field

0

'' 0

t

dttVtW

from Maxwell”s equations:

longitudinal-transverse wake relation

0

'' 0

t

dttVtW Panofsky-Wenzeltheorem

transverse wakeis defocusing

emittance growth in linacs & linear colliders

• 1st example of impact of wake fields• advantage of 2-particle model for getting insight

single particle injected on axis travels down linac

if injected off-axis, quadrupoles surrounding linac→ oscillation about axis

cartoon- scales are not correct!

2nd particle follows 1st, wake from 1st deflects 2nd

deflected outward, amplitude grows

amplitude of second particle (bunch tail) grows, effective emittance growth

multi-particle simulationby Karl Bane,

large growth in effectiveemittance

possible solutions:1.BNS damping (Balakin, Novokhatsky, Smirnov)2. reduce wake fields

BNS damping – analogy classical driven oscillator

response

drive

natural frequency

drive=head, nat=tail

responsedriven with beatsn oscillatio tailinitial 2)

reduced is response 1)

:effects two if tailhead

multi-particle simulationby Karl Bane,

SLC with BNS damping

(SLC) 2)

CLIC) of version (previous squadrupole rf 1)

:by achieved becan

tailhead

tailhead

EE

instabilities in circular accelerators

• stick with 2-particle model

• head produces wake that acts on the tail

of charge q/2 each

• head and tail interchange due to synchrotron oscillations

HEADTAIL

WAKE FIELD

HEAD

TAIL

TAIL

HEAD

1/2 Ts later

example : “fast head-tail” instability

• simplified transverse wake field

W

W

2/

0

122

2

12

1

Wqyyy

yy

for 0 < t < Ts/2: SHM

driven harmonic oscillator

head

tail

2/

0

212

1

22

2

qWyyy

yy

For Ts/2 < t < Ts: head

tail

look for solutions:

dt

dY

dt

Yd ,

dt

dY ,

2

2

2,12,1 YetYty ti

when 0<t<Ts/2: 1) Y1 is constant2)

8

2/2

12

12

s

titi

TiWqYY

eWqYeYi

when Ts/2<t<Ts: 1) Y2 is constant2)

8

2/2

21

21

s

titi

TiWqYY

eWqYeYi

• a common technique for assessing stability of this and similar systems is to write as a matrix

02

1

02

1

2

1

1)8/(

01

10

)8/(1

Y

YM

Y

Y

iWqT

iWqT

Y

Y

s

s

Ts

02

1

2

1

Y

YM

Y

Y N

NTs

for N synchrotron periods:

the motion is stable if all elements of MN remain bounded as N→infinity

look at eigenvalues of M. They are exp(+/-i ) where cos = ½ Trace(M) =1/2( 2-(WqTs/(8 ))2)

• motion is stable if |cos |<1

→ q<16 /(WTs)

note contrast with linac; synchrotron motion in ring gives stability below “threshold”

“fast head-tail instability”(also called “transverse mode coupling instability”)

1. a better calculation2. past and future importance

1. A better calculation – use many particles and time-dependent wakes

results for PEP

the threshold can be calculated with zero free parameters –wakes are determined by geometryright answer within a factor of two for e+/e- machines

past & future importance• past: PEP & PETRA & LEP performances were limited

by fast head-tail instability• present: the electron cloud gives rise to a similar

instability limiting KEKB and PEP-II (only the “wake field” is due to electron cloud and not due to the chamber wall)

• future: the single bunch proton current in the LHC is expected to be limited by the fast head-tail instability in the SPS (which serves as LHC injector)

• future: the damping rings of ILC and CLIC may be limited by this instability, in particular the e-cloud driven version

beam size

center of mass

beam sizew/oSR part

multiparticlesimulationfor PEP by R. Siemann

multi-bunch instabilities• in the linac example it does not matter whether

the two particles are in the same bunch or not, but for the ring the head-tail exchange is central → particles in same bunch

• if beam induced fields last long enough different bunches can communicate

• an example: Robinson instability

RF

Npart0

00

Tc

turnsprev

tprfrf tV

cm

eN

cm

eV

...

02

0

22

20

arc:

rf:

beam induced voltageacting on beam particle

turnsprevtp

crf

rfcc

turnsprevtprf

rf

c

tVTcm

eN

Tcm

eV

dt

d

dt

d

tVTcm

eN

Tcm

eV

dt

d

T

dt

d

T

...

0

002

0

22

002

002

2

...

0

02

0

22

02

00

0

0

0

turnsprev

tps tVTcm

Ne

dt

d

...

0

002

0

22

2

2

2/1

002

0

Tcm

eV rfcrfs

simpleharmonicoscillator withfrequency

“force” due to wake field

for e+ or e- beams;[for protons replace c→c-1/2]

concentrate on force – make some approximation to see essential physics 1.only 1 previous turn2.single mode in cavity with natural frequency N,

quality factor Q,not necessarily N=rf

decay due to QV(tp.t.)

t

rings atN

initial negative valuefrom energy conservation T0+

0

0

00

0

0

...

.

T

T

dt

dV

dt

dTTV

dt

dVTV

TVturnprevV

neglect constant,it does not affectstability

022

00

2

2

2

0

s

T dt

d

dt

dV

cm

Ne

dt

d

If [ ] > 0: damped motionIf [ ] < 0: unstable motion

A. N=rf, NT0=n 2dV/dt|T=0

B. N>rfdV/dt|T>0, unstable

C. N<rfdV/dt|T<0, damped

stability is extremely sensitive to mode frequency

multiparticlesimulationby R. Siemann

microwave instability• longitudinal analog of fast head-tail instability

- single bunch - longitudinal- no centroid motion

• self limiting due to nonlinearity of rf wave • important in hadron colliders

+ in SLC damping ring+ B factories+ ILC/CLIC damping rings

observationof microwave instabilityat SPEAR

instability categories

instabilities can be classified as1. single bunch (e.g. fast head tail) or multi-bunch [multiple turn]

instability (e.g. Robinson instability) 2. transverse (fast head tail) or longitudinal (Robinson) instability3. with (Robinson) or without (fast head-tail) centroid motion

all of these 1. occur2. have one or several names – impact on literature3. have different degrees of importance depending on accelerator4. have different cures (basic design strategy is reducing the

size of the wake fields by the design of vacuum vessel and theamount of ions or electrons by surface treatments (e.g. coatingwith low-SEY material) or clearing electrodes

“sawtooth” instability in the SLC damping ring (B. Podobedov, R. Siemann)complex behavior, relaxation oscillations, non-monotonic dependence on N

tWedtiZ

tVedtZ

ti

ti

'1

'||0

remark: Fourier transform of the wake function is the impedance

usually the real part of the impedance is related toinstability growth (or damping) rates; the imaginary part shifts the mode frequencies; energy lossis due to the real part of Z0

||

some references• accelerator-physics lectures by R. Siemann,

1998• CERN summer school lectures by E. Metral

and S. Gilardoni, 2007• several CERN lectures by K.-H. Schindl and

M. Benedikt• summer student talk by I. Santiago 2007• GSI acceleratpr palaver 2005, G. Franchetti• HB2006 talk by G. Franchetti

Why do accelerators work?

• Large number of collective instability mechanisms.

• But the beam seems to be basically stable.

Natural stabilizing mechanism:

Basic idea: swing…

• Stiff frame. Frequency spread.• COHERENT C.M. motion decays quickly compared to INCOHERENT motion of children.• STABLE MOTION

• Flexible frame • INSTABILITY GROWTH RATE higher than FREQUENCY SPREAD.• INSTABILITY

A. Hofmanncoupled oscillators → new eigenmodes with shifted frequencies

Basic idea (still)

Landau dampingeffect occurs in systems consisting of a high number of oscillators having different oscillation frequencies and performing a collective motion

L.D. Landau, 1946, J. Phys. USSR 10 (1946)plasma physicsN.G. van Kampen, 1955, Physica 21 (1955)mathematics C.E. Nielsen, A.N. Sessler, K.R. Symon, HEACC Geneva (1959) accelerator physicsR.D. Kohaupt, “What is Landau Damping? Plausibilities, Fundamental Thoughts, Theory”DESY M-86-02accelerator physics

if a system of many oscillators (protons) with different oscillation frequencies is excited (kicked) their centroid motion decays in time as a result of the frequency spread,which extinguishes the coherent motion

on the other hand, a beam can be driven unstable by self-excited electromagnetic fields (impedance) which act back on the beam; instability rise time g

the beam is Landau damped if the decay time due to thefrequency spread is ‘shorter’ than the instability rise time

g

1

sN

t

edxeqx tiN

k

tik

743

11

01

0

1031060210

10

we can replace discrete sum by integral,for times which are not extremely long

max

min

max

min

,

txdtD

Nd

WDxx 2

frequency distribution

centroid

linear “wake force”

2

WNcoh coherent frequency shift in the absence of

frequency spread

0

11

220

20

NW

eD

NWDDDti

special case: single frequency

ansatz harmonic oscillation

solution exists for arbitrary small W

general case

00

~

~

2

1~

2

1

izttii

C

izttii

etdtDetdtDiD

ezDdeiDdtD

z

C

xdzD ~~

Fourier-Laplace-transform

Bromwich integral(all singularities above the path C)

22

2

1

))0(')0((~

zdW

dz

xizx

zD

)0(')0(~~22 xizxDWxz

C

cut –(max)<Rez<(max)

singularities at zk

analytic otherwise

zk

cut

tizk

ti

tizkcutC

k

k

eceDDd

ec

~~

initial conditions

C

iztezDdtD~

2

1

(1) 1

22

zdW

singularities zk in the upper z-plane describe coherentstability and those in the lower z-plane coherent instability; the zk correspond to solutions of

the integral has a finite value for any reasonablefrequency distribution for all z values

there is no solution to (1) for small W!

interpretation: system cannot organize a collective motion if interaction not strong enough

iz

22

x

N

1

2

1

2

11

2

2

2

11

22

211

11

2

2

11112

1111

2

3

0)Im(,

0Im,22

i

ii

Nii

N

i

iiN

i

i

iiii

iiii

i

iN

iiiii

N

diiii

N

zd

x

x

x

x

xx

xx

res xx

C xx

example:Lorentz spectrum

X

X

X

X

x+i

x-i

i

i Re

Im

i

u x

dispersion relation

>0: instability growth rate, ->0: border of stability;above relation may not apply for <0

11

1

1 22

ui

u

ucoh

consider =0 absolute value of dispersion relation simplifies to |coh|= ; if coherent tune shift less than spread the beam is stable

exampleof Landau-dampedsystem inBodrum?!


wake fields, impedances, beam instabilities, Landau damping

3) beam-beam effects



beam-beam effectsLHC as example

• incoherent beam-beam effects– lifetime & dynamic aperture

• PACMAN effects– bunch-to-bunch variation

• coherent effects – oscillations and instabilities

(W. Herr, LHC Design Report, Chapter 5)

beam-beam forcecalculation is similar as for space charge, but

- electric and magnetic forces add s.c.: (1-2)=1/2 → beam-beam: (1+2)=2 - beams move in opposite direction; interaction

time is a factor 2 smaller than the time duration of a single bunch passage in lab frame

- interaction is localized at one or a few places around the ring → force does not only cause a a tune shift with amplitude but it also excites many resonances mQx+nQy+p=0

k p

ipek

2

12

2

22

220

2

22

220

2exp12'

2exp12'

yx

yx

yNry

yx

yx

xNrx

beam-beam kick for head-on collision of round Gaussian beams:

force is nonlinear and couples x and y motion;(head-on) beam-beam tune shift from linear expansion

424

2 02

*0

,.

rNNrQ

Nyxyx

the beam-beam tune shift depends only on the beam brightness and is independent of *;NCP collision points → total beam-beam tune shift = Ncp x there is a maximum acceptable value for this tune spread

→ “beam-beam limit”

if beam is notround, there isan analytical expressioninvolving the complex errorfunction(Bassetti-Erskine)

beam-beam deflection curve

W. Herr

head-oncollision

LRcollision

opposite sign of slope for long-range collisions in plane of offset

luminosity at the beam-beam limit

LR reaction rate luminosity

cross section

*0

*2

2 1

44

Nnf

r

NnNfNnfL brev

totb

N

revbrev

total beam-beamtune shiftlimited to ~0.01 (protons)~0.1 (e+/e-)

grows linearlywith energy

beam current

IP beta function

# bunches

maximum luminosity:many bunchessmall *+ max. bunch charge compatible with bb limit

two high luminosityIPs (IP1 ATLAS & IP5 CMS)

two lower-luminosityIPs (IP2 ALICE & IP8 LHCb)

3 (4) head-on collisions &~120 long-range collisions

30 long-range collisions per main IP

partial mitigation by alternating planes of crossing at IP1 & 5 etc.

#LR encounters

SPS 9

Tevatron Run-II 70

LHC 120

2

2

d

nQ LR

LR

N

bpHO

NrQ

4

head-on tune shift(with zero crossing angle)

long-range tune shiftif beams are separatedby d/(thanks to a crossing angle)

head-on & long-range tune shifts & tune spread ~ similar to space charge

HOLR QQ 33.0for nominal LHC:

LHC design criterion J. Gareyte, J.-P. Koutchouk)

avoid resonances < order 13 & |QH-QV|~0.01

→ nominal total tune spread (up to 6in x&y) from all IPs and over all bunches, including long-range effects, should be less than 0.01-0.012

notes: • this limiting value comes from SPS & Tevatron;• 6 is empirical to match results of Ritson & Chou

for “ultimate” LHC, |QH-QV|~0.005, and the total tune spread should be less than 0.015-0.017

nominal 7-TeV collision parameters* offset angle

IP1 0.55 m 16.7 m 0 285 rad (y)

[9.4]

IP2 10 m 70.9 m 355 m

[5]

300 rad (y)

[42.3]

IP5 0.55 m 16.7 m 0 285 rad (x)

[9.4]

IP8 10 m 70.9 m 0 400 rad (x)

[56.4]3 “head-on” collisions with crossing angle1 halo collision with 5- separation at IP260 long-range collisions with on average ~9.5 separation60 negligible long-range collisions

tune footprints due to head-onand long-range collisions in IP1 and IP5 [courtesy H. Grote]

total LHC tune footprint forregular and PACMAN bunch[courtesy H. Grote]

Q from LR collisions is approximately cancelled by alternating crossing[D. Neuffer, S. Peggs, SSC-63 (1986)]

tune footprints & alternating crossing

x

zcR

2

;1

12

“Piwinski angle”

luminosity reduction factor

nominal LHC

crossing angle

c/2

effective beamsize →/R

note: the tune shift is reducedby roughly the same factor

dynamic aperture

simulation by L. Evans = dynamically stable region in phase space

outside: global chaos, rapid diffusion, losses

sepda xx

N

bN

“dynamic aperture”

diffusive aperture due to long-range encounters, new regime of hadron beam-beam

withindependent of * and energy

for nominal LHC: xsep~9.5, xda~6J. Irwin, SSC-223 (1989)Y. Papaphilippou & F.Z., PRST-AB 2, 104001 (1999)Y. Papaphilippou & F.Z., PRST-AB 5, 074001 (2002)

LHC filling pattern

lack of 4-fold symmetry → some bunches encounter abort gap in IP2 or 8 and have missing head-on collisions; in addition IP8 is displaced by 11.22 m and also 3 bunches in each train miss head-on collisions in IP2

All encounters in the straight sections are taken into account. Each bunch in the LHC is represented as a dot. The angular co-ordinate is the initial position of the bunch around the circumference. There is a one-to-one correspondence between beam-beam equivalence class and the radius in the plot. The classes are sorted according to the population of the class. Thus, classes containing a single bunch, of which there are several, lie towards the centre of the plot. Tomake adjacent classes easier to distinguish they are also colored differently (although the colours are used several times over at clearly distinguishable radii).Here there are 171 equivalence classes.

Beam-Beam Equivalence Classes for LHCr [J. Jowett, LHC’99]

only ~half of the bunchesare regular

PACMAN effects

• expect bunch-to-bunch variation of orbit, tune and chromaticity

• partial compensation by alternating crossing in IP1 and 5

bunch-to-bunch orbit variation

beam1

beam2

beam1

beam2

orbit displacements at IP1

HH crossing HV crossing

W. Herr

only half of bunch pattern shown;collisions are head-on in the other plane;in addition ground motion will separate the two beams

by 5 during 8 hours

bunch-to-bunch Q, Q’ variation

HV crossing

HV crossing

HH crossing

HH crossing

W. Herr

first 3 bunchesin each train

abort gap

emittance growth from noise• LHC beams are stored in two separate beam pipes; orbit

perturbations are independent and can steer the beams out of collision

• transverse feedback, rf, wire compensator, crab cavity…• emittance growth due to random beam-beam offset including

decoherence and feedback [Y. Alexahin]:

where g is a feedback gain factor (typically g~0.2), || the total beam-beam tune-shift parameter assumed equal 0.01, x* the horizontal IP beam size, nIP the number of IPs (taken to be two), and s0~0.645

• emittance growth < 1%/hr:→ tolerance: x < 2.6 nm for g=0.2, x < 0.6 nm for g=0.0

22

20

21

1

4

11

g

xsnf

dt

d

xIPrev

consistent withsimulations byK. Ohmi

data from various locations 1989-2001

A. SeryiNanobeam’2002Lausanne

ground motion

HERA

LEP

coherent beam-beam effects

• unlike SPS and Tevatron, LHC will operate in the strong-strong regime

• Y. Alexahin predicted that Landau damping of the mode may be lost

• Landau damping can be restored by symmetry breaking– different intensities– different tunes– broken symmetry for multiple interaction regions

or by overlap with synchrotron sidebands

two-beam system can show dipole-like instabilities (where one beam oscillatesagainst the other) unlike for direct space charge with a single beam

mode mode

continuum

equal intensity intensity ratio 0.55

frequency spectrum of dipole oscillations

mode not Landau damped mode Landau dampedM.P. Zorzano & F.Z., PRST-AB 3, 044401 (2000)W. Herr, M.P. Zorzano, F. Jones, PRST-AB 4, 054402 (2002)

beam-beam tune spread can also do something good – namely provide Landau damping against impedance-driven instabilities

max. octupoles 0.00012

nominalLHC: LRin IP1andIP5

ultimateLHC: HO+LRin IP1andIP5

W. Herrand L. VosLHC ProjectNote 316(2003)

LR RHIC experiments in 2005 and 2006

single off-center collision

one collision with 5-6 offset strongly increases RHIC beam loss rate; worse at smaller offsets

(W. Fischer et al.)

24 GeV

100 GeV

APC meeting, 19.09.03, LRBB J.P. Koutchouk, J. Wenninger, F. Zimmermann, et al.

• To correct all non-linear effects correction must be local.• Layout: 41 m upstream of D2, both sides of IP1/IP5

(Jean-Pierre Koutchouk)

Long-Range Beam-Beam Compensation for the LHC

Phase difference between BBLRC & average LR collision is 2.6o

1st Wire “BBLR” in the SPS, 2001

Tech. Coord. J. Camas &

G. Burtin/BDI

Help from many groups

two 60-cm long wireswith 267 A currentequivalent to 60 LHC LR collisions (e.g., IP1 & 5)

Iwire=Nb e c #LR/lwire

wire lengthwire current

force of current-fed wire mounted parallel to the beamis 1/r, just likethe long-range forcefrom the opposing beam

scaling from LHC to SPS and vice versa

)(

2'

dyec

Ilry wwp

da

wwp

y n

I

ec

lry~)(

2

'

'

for constant normalized emittance the effect in units of sigma is independent of energy and beta function!

relative perturbation:

perturbation by wire:

beam studies are being done in the CERN SPS at much lower beam energy (26-55 GeV) than for the future LHC (7 TeV)

for 2004 two novel 3-wire BBLRs were built; separated from 1-wire BBLR by about 2.6o

(average LR-BBLR phase advance in LHC)

3rd

10th

7th

4th

nearly perfect compensationwhat happens here?

Qx=0.31

1 wire

2 wires

no wire

vertical tune

beam lifetime

lifetime is recovered over a large tune range, except for Qy<0.285

two-wire compensation test at the SPS: tune scan

one wire modelsthe effect of the LHC long-range collisions,the second wireis used for compensation

beam-beam in linac-ring colliders

• The two beams can be optimized differently and independently. While the beam-beam tune shift for the (e+) beam in the ring is limited, the linac beam may encounter much larger beam-beam forces, thereby allowing for much larger luminosity than a ring-ring collider (TAC project)

→ Favors a high bunch charge in the ring

*,0

*2

1

44

rcoll

rtotr

N

lcollrlcoll Nf

r

NNfNNfL

beam-beam in linear colliders

• Linear colliders also have a beam-beam limit but at much higher bunch intensity than a ring collider. In linear colliders, beamstrahlung (synchrotron radiation in the field of the other beam) can lead to an intolerable degradation of the luminosity spectrum.

• Also a kink instability occurs, if particles perform several oscillations in the field of the opposing beam during a collision; this kink instability enhances small offsets and leads to a rapid decrease of luminosity.

• Beam-beam tune shift (ring) → disruption parameter (linear collider); basically the same parameter, but *→4z

some LHC beam-beam references• J. Poole and F. Zimmermann, eds., Proceedings of Workshop on beam-beam effects in Large Hadron Colliders, CERN/SL 99-039 (AP)

(1999).

• J. Gareyte, Beam-Beam Design Criteria for LHC, Proc. LHC’99

• O. Bruning et al, LHC Design Report, Vol. 1, Chapter 5 (beam-beam section by W. Herr), CERN-2004-003

• Y. Alexahin, On the Landau damping and decoherence of transverse dipole oscillations in colliding beams, Part. Accel. 59, 43 (1998).

• W. Chou and D. Ritson, Dynamic aperture studies during collisions in the LHC, CERN LHC Project Report 123 (1997).

• L. Leunissen, Influence of vertical dispersion and crossing angle on the performance of the LHC, CERN LHC Project Report 298 (1999).

• Y. Papaphilippou, F. Zimmermann, Weak-strong beam-beam simulations for the Large Hadron Collider, PRST-AB 2:104001, 1999

• Y. Papaphilippou & F. Zimmermann, Estimates of diffusion due to long-range beam-beam collisions, PRST-AB 5:074001, 2002.

• M.P. Zorzano, F. Zimmermann, Coherent beam-beam oscillations at the LHC, PRST-AB 3:044401, 2000.

• W. Herr, M.P. Zorzano and F. Jones A Hybrid Fast Multipole Method applied to beam-beam collisions in the strong strong regime, PRST-AB 4, 054402 (2001)

• H. Grote, L. Leunissen, F. Schmidt, LHC Dynamic Aperture at Collision, LHC Project Note 197 (1999).

• J. Jowett, Collision Schedules and Bunch Filling Schemes in the LHC, CERN LHC Project Note 179 (1999).

• M.P.Zorzano, T.Sen, Emittance growth for the LHC beams due to head-on beam-beam interaction and ground motion , LHC Project Note 222 (2000).

• W. Herr, L. Vos, Tune distributions and effective tune spread from beam-beam interactions and the consequences for Landau damping in the LHC, LHC Project Note 316, 2003

• W. Herr, M.-P. Zorzano, Coherent Dipole Modes for Multiple Interaction Regions, LHC Project Report 462 (2001)

• Y. Alexahin, A study of the Coherent Beam-Beam Effect in the framework of the Vlasov Perturbation Theory, NIM A 380, 253 (2002)

• W. Herr, R. Paparella, Landau Damping of Coherent Modes by Overlap with Synchrotron Sidebands, CERN LHC Project Note 304, 2002

• W. Herr, Features and Implications of Different LHC Crosing Schemes, LHC Project Report 628 (2003)

• Y. Alexahin, On the Emittance Growth due to Noise in Hadron Colliders and Methods of its Suppression, NIM A 391, 73 (1996).



beam-beam effects



INP Novosibirsk, 1965 Argonne ZGS,1965 BNL AGS, 1965

Bevatron, 1971

ISR, ~1972 PSR, 1988

AGS Booster, 1998/99 KEKB, 2000 CERN SPS, 2000

observations of electron cloud at various accelerators

electron cloud and ionswhere do the e- (or ions) come from? ionization of residual gas

- collisional ionization ~ gas density,typical ionization cross section ~ 1 Mbarn,typically ~10-6 e-/ions per meter per beam particle- tunneling ionization in collective beam fieldif beam is sufficiently small (ILC, CLIC, FFTB)

photoemission from synchrotron radiation- typically 10-4 - 1 e- (ions) per meter per part.

avalanche build up via acceleration in the beam field - electron-cloud build up- pressure bump instability

R. Cimino, I. Collins, 2003

probability of elastic electron reflection approaches 1 forzero incident energy and is independent of *max

secondarye-yield

max

electron cloud in the LHC

schematic of e- cloud build up in the arc beam pipe,due to photoemission and secondary emission

[F. Ruggiero]

e- density at saturation

• equilibrium e- density in multipacting regime:

time average net field at wall ~ 0, or

electron line density ~ beam line density

e ~ Nb/Lsep

• in other cases from balance of e- production and loss rates

multipacting condition• kick from bunch to e- at the wall:

r’=2 re N/b

• resonance picture: e- cross the vacuum chamber (radius b) in a time equal to the bunch spacing, or r’ Lsep = 2b

• → re Lsep N = b2 (Grobner) ??

• this would suggest Nthresh~1/Lsep

• experimentally however Nthresh~ Lsep !

• threshold is determined by beam line density only

blue: e-cloud effect observedred: planned accelerators

effects of electron cloud• heat load (→ quench of s.c. magnets)

• coherent tune shift

• single and multi-bunch instabilities

• large incoherent tune shift due to e- “pinch” and local e- density spikes during a bunch passage (effects similar to space charge, but resonances are crossed only twice per synchrotron period – e- density increases roughly monotonically along the bunch) → poor lifetime, emittance growth

coherent tune shift from e-cloud

Cr

Q yxe

yxe ,,; 2

coherent tune shift due to e-cloud for a ~round beam

for a flat beam with planar symmetry, the vertical tune shift would be two times larger and the horizontal one about zero

above relation can be used to estimate the average e- densityfrom the measured tune shift

comment: the incoherent tune shift is many orders of magnitudelarger due to the e- pinch (which moves with the bunch)

K. Ohmi et al, APAC’01

insert this in the formula for fast head-tail threshold from 2-particlemodel (lecture 2):

result in good agreement with experimental data

fast head-tail instability driven by e-cloud

bey NCW /8

Cr

Q

y

sthre

0,

2

estimate of e-cloud wake field acting between bunch head and tail

incoherent effect of e-cloud

incoherent tune shift increases along the bunchresonances are crossed twice per synchrotron period

G. Franchetti

“neckties” in tune diagram

G. Franchetti

multi-particle simulation of e-cloud driven head-tail instability by K. Ohmi; characteristics of “fast head-tail”

multitude of countermeasures:• multi-bunch & intrabunch feedback

(INP PSR, Bevatron, SPS, KEKB)• clearing electrodes (ISR, BEPC, SNS)• antechamber (PEP-II)• TiN coating (PEP-II, PSR, SNS)• high Q’ (SPS)• octupoles (BEPC)• solenoids (KEKB, PEP-II, SNS)• grooved surfaces (NLC)

LHC strategy against electron cloud

1) warm sections (20% of circumference) coated by TiZrVgetter developed at CERN; low secondary emission; if cloud occurs, ionization by electrons (high cross section ~400 Mbarn) aids in pumping & pressure will even improve

2) outer wall of beam screen (at 4-20 K, inside 1.9-K cold bore) will have a sawtooth surface (30 m over 500 m) to reduce photon reflectivity to ~2% so that photoelectrons are only emitted from outer wall & confined by dipole field

3) pumping slots in beam screen are shielded to preventelectron impact on cold magnet bore

4) rely on surface conditioning (‘scrubbing’); commissioning strategy; as a last resort doubling or triplingbunch spacing suppresses e-cloud heat load

e- cloud effect may also be reduced by:• larger bunch spacing• high bunch intensity• superbunches

predicted heat load in LHC vs. bunch spacing

on a vertical log scalechange in max appears as ~constant vertical shift

WEPLT045, THPLT017

eV 9.10 E1-9

20 m103.1 cmr

E

eee

saturation of e- build up for high bunch intensities

Nb=4.6x1011

2.3x1011

~average energy of secondaryelectrons

109 m-1

0

e- line density

time10 s

schematic of e- motion during passage of long protonbunch; most e- do not gain any energy when traversingthe chamber in the quasi-static beam potential

[after V. Danilov]negligible heat load

superbunch

ion instabilities[conventional: trapped ions]single-pass: fast beam-ion

instability (FBII)



beam-beam effects

electron-cloud and ion effects


Documents

collective effects in particle accelerators