desy summer student lecture 31 st july 2002
DESCRIPTION
An Introduction to the Physics and Technology of e+e- Linear Colliders Lecture 7: Beam-Beam Effects Nick Walker (DESY). DESY Summer Student Lecture 31 st July 2002. USPAS, Santa Barbara, 16 th -27 th June, 2003. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
An Introduction to thePhysics and Technologyof e+e- Linear Colliders
Lecture 7: Beam-Beam Effects
Nick Walker (DESY)
DESY Summer Student Lecture31st July 2002USPAS, Santa Barbara, 16th-27th June, 2003
Introduction
• Beam-beam interaction in a linear collider is basically the same Coulomb interaction as in a storage ring collider. But:– Interaction occurs only once for each bunch (single
pass); hence very large bunch deformations permissible.– Extremely high charge densities at IP lead to very
intense fields; hence quantum behaviour becomes important
• Consequently, can divide LC beam-beam phenomena into two categories:– classical– quantum
Introduction (continued)
• Beam-Beam Effects– Electric field from a “flat” charge bunch– Equation of motion of an electron in flat bunch– The Disruption Parameter (Dy)– Crossing Angle and Kink Instability– Beamstrahlung – Pair production (background)
Ebeam x y x y z Ne
GeV mm mm nm nm m 1010
NLC 250 8 0.1 245 2.7 110 0.75
TESLA 250 14 0.4 550 5 300 2.0
CLIC 1500 8 0.15 43 1 30 0.42
Storage Ring Collider Comparison
Linear beam-beam tune shift ,,
,2 ( )x ye b
x yx y x y
r N
Putting in some typical numbers (see previous table) gives:
0.541.44
x
y
Storage ring colliders try to keep , 0.05x y
Electric Field from a Relativistic Flat Beam
• Highly relativistic beam E+vB 2E• Flat beam x y (cf Beamstrahlung)• Assume
• infinitely wide beam with constant density per unit length in x ( (x))
• Gaussian charge distribution in y:• for now, leave (z) unspecified
1( )2 x
x
2
1 1( ) exp22 yy
yy
Electric Field from a Relativistic Flat Beam
0 ' 0
0
( ) ( )( , ) ( ') '
( , ) Erf ( )2 2 2
y
yy
yx y
qN x z x zE y z x z y dy
qN yE y z z
Use Gauss’ theorem:0s
q
E ds
Assuming Gaussian distribution for z, the peak field is given by
0
ˆ4y
x z
qNE
Electric Field from a Relativistic Flat Beam
10 5 5 10
2000
1000
1000
2000
10
500 nm5 nm
300μm
10
x
y
z
N
/ yy
MV/cmyE
0z
Note: 2Ey plotted
Assuming a Gaussian distribution for (z)
Electric Field from a Relativistic Flat Beam
10 20 30 40 50
500
1000
1500
10
500 nm5 nm
300μm
10
x
y
z
N
/ yy
MV/cmyE
0z
Assuming a Gaussian distribution for (z)
effect of x width
Equation Of Motion
F = ma:
Changing variable to z:
0
2 ( , )( ) yqE y t
y tm
20
2
20
2 ( , )( )
( )Erf (2 )2
2
y
zy
x o
qE y zy z
m c
y zq N z
m c
2
20 04eqrm c
2 2 ( )( ) Erf (2 )2
ez
x y
Nr y zy z z
2( ) ( )y t c y z
whyz(2z)?
Linear Approximation and the Disruption Parameter
Taking only the linear part of the electric field:
Take ‘weak’ approximation:y(z) does not change during interaction y(z) = y0
Thin-lens focal length:
Define Vertical Disruption Parameter
2 ( )
4 (2 )( ) ( )e
x y
k z
Nr zy z y z
0
0 0
4 (2 )
2 1
ez
x y
e
x y
Nry y z dz
Nr y yf
21 e
x y
Nrf
2 e zzy
x y
NrDf
2 e z
yx y y
NrD
exact:
Number of Oscillations
Equation of motion re-visited:
Approximate (z) by rectangular distribution with same RMS as equivalent Gaussian distribution (z)
2( ) (2 ) ( )y
zz
Dy z z y z
3 2 1 0 1 2 30
0.1
0.2
0.3
0.4
3 z3 z
12 3 z
2
3( ) ( );23
yy
z
Dy z y z z
14
22
323
30.21
2
y z
z
yy
D kk
DD
half-length!
Example of Numerical Solution
500 0 500 1000 1500
40
20
0
20
40102 10
500 nm5 nm
300 μm250 GeV
x
y
z
N
E
27.7
1.1yD
( )/ μmz ct
/ nmygreen: rectangular approximation
black: gaussian
Pinch Enhancement
• Self-focusing (pinch) leads to higher luminosity for a head-on collision.
3, ,1/ 4
, , ,3,
0.81 ln 1 2ln
1x y x y
Dx y x y x yx y z
DH D D
D
‘hour glass’ effect
Empirical fit to beam-beam simulation results
Only a function of disruption parameter Dx,y
The Luminosity Issue: Hour-Glass
= “depth of focus”
reasonable lower limit for is bunch length z
2 1 0 1 2Z
3
2
1
0
1
2
3
Y
2 1 0 1 2Z
3
2
1
0
1
2
3
Y
Luminosity as a function of y
200 400 600 800 1000
11034
21034
31034
41034
51034
300z m
100z m
500 m
700 m
900 m
( )y m
2 1( )L cm s
2
4b
x y
n N fL
1BS z
Beating the hour glass effect
Travelling focus (Balakin)
• Arrange for finite chromaticity at IP (how?)
• Create z-correlated energy spread along the bunch (how?)
*z y
Beating the hour glass effect
Travelling focus
• Arrange for finite chromaticity at IP (how?)
• Create z-correlated energy spread along the bunch (how?)
*z y
Beating the hour glass effect
Travelling focus
• Arrange for finite chromaticity at IP (how?)
• Create z-correlated energy spread along the bunch (how?)
ct
Beating the hour glass effect
foci ‘travel’ from z = 0 to z =
t = 0
3 z
f = c t t > 0
3 z
3 2z c t
3 z
RMS
2RMS
2
2
2
2
z
y
y
z
y
z
chromaticity:
y yf
travelling focus:
NB: z correlated!
1
654
32
The arrow shows position of focus for the read beam during travelling focus collision
Kink Instability
Simple model: ‘sheet’ beams with:
Linear equation of motion becomes
Need to consider relative motion of both beams in t and z:
Classic coupled EoM.
,,
12 3x z
x z
2
2 3( ) ( );6 2
yz
z
Dy z y z z
22
1 0 1 2
22
2 0 1 2
( , ) ( )
( , ) ( )
c y t z y yt z
c y t z y yt z
2 20 2
26
y
z
Dc
Kink Instability
and substituting into EoM leads to the dispersion relation:
1(2) 1(2) expy a i kz t
2 2 2 2 2 2 2 40 0 04c k c k
Assuming solutions of the form
Motion becomes unstable when 2 0, which occurs when
02kc
Kink Instability
Exponential growth rate:
Maximum growth rate when
Remember!
Thus ‘amplification’ factor of an initial offset is:
122 2 2 4 2 2 2
0 0 04 c k c k
0 03 ;2 2
k ic
3 3 3 /2 2
z zzt t c
c c
14
0.60 1exp exp2 2 2
yDy
t D e
For our previous example with Dy ~28, factor ~ 3
Kink Instability
0.1
20
300μm
y
y
z
y
D
Pinch Enhancement
geomL L H H= enhancement factor
2
2exp4
y
y
results ofsimulations:
High Dy example: TESLA 500
24yD
Disruption Angle
Remembering definition of Dyz
yDf
The angles after collision are characterised by
02 2
( )y yx x e e
z z x y x
DD Nr Nr
Numbers from our previous example give
OK for horizontal plane where Dx<1
For vertical plane (strong focusing Dy > 1), particles oscillate:
0 467 μrad
2
3( ) ( );23
yy
z
Dy z y z z
1 1 14 4 2
0 67 μrad3 3
yy
z y
D
D
previous linear approximation:
Disruption Angle: simulation results
- 400 - 200 0 200 400
0
50
100
150
200
250
- 400 - 200 0 200 4000
200
400
600
800
1000
horizontal angle (rad) vertical angle (rad)
Important in designing IR (spent-beam extraction)
Beam-Beam Animation Wonderland
Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code
(written by D. Schulte, CERN).
Examples of GUINEAPIG Simulations
NLC parametersDy~12
Nx2Dy~24
NLC parametersDy~12
Luminosity enhancement HD ~ 1.4
Not much of an instability
Examples of GUINEAPIG Simulations
Nx2Dy~24
Beam-beam instability is clearly pronounced
Luminosity enhancement is compromised by higher sensitivity to initial offsets
Examples of GUINEAPIG Simulations
Beam-beam deflection
Sub nm offsets at IP cause large well detectable offsets (micron scale) of the beam a few meters downstream
Beam-beam deflectionallow to control collisions
Examples of GUINEAPIG Simulations
Examples of GUINEAPIG Simulations
Beam-Beam Kick
Beam-beam offset gives rise to an equal and opposite mean kick to the bunches – important signal for feedback!
For small disruption ( D1) and offset (y/y<<1)
012bb
y
y
For large disruption or offset, we introduce the form factor F:
01 /2bb yF y
( )F
Long Range Kink Instability & Crossing Angle
To avoid parasitic bunch interactions in the IR, a horizontal crossing angle is introduced:
angle c
2bctl
2b cctX
parasitic beam-beam kick: ,2 e
x yNrr rr
Long Range Kink Instability & Crossing Angley
y
IP
small vertical offset ( = y/y) gives rise to beam-beam kick
resulting vertical offset at parasitic crossing gives next incoming bunches additional vertical kick: IP offset increases
instability
01 ( )2ip F
e+e
02 2
2 ( )ipe e
c
lNr Nr FX l
l
Long Range Kink Instability & Crossing Angle
offset at IP of k-th bunch:
distance from IP to encounter with l-th bunch (l < k)k k k
,0 ,
0,0 2
/
( )
k k lk lk y
ek l
c y
l
Nr F
0,0 2
2 ( )
k k k
ek l
c y
Nr F
( )2
blk
k l t cl
contribution from encounter with l-th bunch:
NB. independent of llk
total offset:21
0,0 2
1
2 /( );k
e x zk k i x y
i c y c
NrC F C D D
Long Range Kink Instability & Crossing Angle
Assuming all bunches have same initial offset:
For small disruption and small offsets,,0 0k
( )i iF
10(1 )k
k C
since2
/ 1x zx y
c
C D D
0
1 ( 1)k k C
thus ( 1) 1bm C where mb = number of interacting bunches
example:Dx,y = 0.1, 10x = 220 nmz = 100 mc = 20 mrad
C = 0.012 mb < 80 for factor of 2 increase
Crab Crossing
2 2 2,
20mr 100μm 2μm
x projected x c z
c z
factor 10 reduction in L!
x
xuse transverse (crab) RF cavity to ‘tilt’ the bunch at IP
ˆ22ˆ( ) sin
ˆ4
RFRF
RFc cav ip z
RF
V zzV z V
V
RF kick
Beamstrahlung
Magnetic field of bunch B = E/c
Peak field:
From classical theory, power radiated is given by
For E = 250 GeV and z = 300m:
maxmax
0
2 1160Tesla2 x z
E qNBc c
45 -3
2 ; 8.85 10 m..GeV2cC EP C
-11 1.4 mcBE
32 GeVzradE P t P
c
E/E ~ 12%note: / 200μrad 1/ 2μradz
Beamstrahlung
Most important parameter is
2 2
20
2 23
c e
s
B e F pE B m
c
e
s
BBF
p
critical photon frequencyCompton wavelengthlocal bending radiusbeam magnetic fieldSchwinger’s critical field (= 4.4 GTesla)em field tensorelectron 4-momentum
Beamstrahlung: photon spectrum
NOT classical synchrotron radiation spectrum!Need to use Sokolov-Ternov formula:
5 233
2
( ) ( )1
2 13 1 1
BS
c
yF K d Ky
yE
yy y
classical quantum theoretical
0.001 0.0050.01 0.05 0.1 0.5 10.01
0.05
0.1
0.5
1
5
10
0.1
1
10
1/3
/y E
Beamstrahlung Numbers
56 ( )
2.4
e eavr
z x y
max avr
Nr
average and maximum
photons per electron: 2/32.54
1avrz
e avr
n
2
22/31.24
1 (1.5 )avrz
BSe avr
EE
average energy loss:
BS and n (and hence ) talk directly to luminosity spectrum and backgrounds
Beamstrahlung energy loss
56 ( )
e eavr
z x y
Nr
2
22/31.24
1 (1.5 )avrz
BSe avr
3 2
2 20
0.862 ( )
e cmBS
z x y
er E Nm c
In lecture 1, we used the following equation to derive our luminosity scaling law
Now we have this:
with
under what regime is our original expression valid?
Luminosity scaling revisited
BSbeam
,y nL P
low beamstrahlung regime 1:
32
BSbeam
,z y nL P
high beamstrahlung regime 1:
*y z
homework: derive high BS scaling law
Beamstrahlung Numbers: example
10
500GeV220nm2nm
110μm
0.75 10
x
y
x
E
N
,0.28 210GeV
0.661.3
7.5%
avr c
max
BS
E
n
TESLA
Why Beamstrahlung is bad
• Large number of high-energy photons interact with electron (positron) beam and generate e+e pairs– Low energies (0.6), pairs made by incoherent process
(photons interact directly with individual beam particles)
– High energies (0.6100), coherent pairs are generated by interaction of photons with macroscopic field of bunch.
– Very high energies (100), coherent direct trident production e e e+ e
TESLA 0.5TeV ~ 0.06NLC 1TeV ~ 0.28CLIC 3TeV ~ 9
• Beamstrahlung degrades Luminosity Spectrum
Pair Production
• Incoherent e+e pairs 0.6– Breit-Wheeler:– Bethe-Heitler:– Landau-Lifshitz:
• Coherent e+e pairs 0.6100
– threshold defined by
e e e e e e
e e e e e e
20
2
1
s
Bm c B
E
0.001 0.005 0.01 0.05 0.1 0.5 10.01
0.05
0.1
0.5
1
5
10
0.1
1
10
1/3
/y E
for >1 / ~ (1)E O 1
for intermediate colliders (Ecm<1TeV), incoherent pairs dominate
Pair Production
0.1 1
0.01
0.1
1
PT (
GeV
/c)
(rad)
Td
z
Pr rcB
pairs curl-up (spiral) in solenoid field of detector
rd
ld
d
vertexdetector
e+e pairs are a potential major source of background
Most important: angle with beam axis () and transverse momentum PT.
Summary
• Single pass collider allows us to use very strong beam-beam to increase luminosity
• beam-beam is characterised by following important parameters:– Dy = z/f defines pinch effect (HD), kink
instability, dynamics QM effects, backgrounds, – BS [=f(av)] energy loss, lumi spectrum
• strong-strong regime requires simulation(e.g. GUINEAPIG). Analytical treatments limited.