Download Introduction to Collective Effects in Particle Accelerators

Introduction to Collective Effects
in Particle Accelerators
Part 1: Space Charge and Scattering Effects
Andy Wolski
University of Liverpool
Outline
• Space charge




Principles of space-charge forces
The envelope equations
Space-charge tune shifts
Longitudinal space-charge effects
• Beam-beam effects
 Beam-beam tune shifts
• Scattering effects
 Touschek effect and Touschek lifetime
 Intrabeam scattering
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Learning Objectives
By the end of this lecture, you should be able to:
• describe some of the main effects of space-charge on transverse and
longitudinal beam dynamics, including betatron tune shifts, emittance
growth, and longitudinal instability;
• explain what is meant by the “perveance” of a beam, and determine, with
reference to the envelope equation, whether the transport of a beam with
given parameters is dominated by emittance or space-charge effects;
• explain the significance of the beam-beam interaction in colliders, and
estimate the beam-beam tune shift for the collision of beams with a given
set of parameters;
• describe some of the effects of scattering between particles within a
bunch, including emittance growth (intrabeam scattering), and particle
loss (Touschek effect), and explain how these effects depend on
parameters such as beam energy, beam size, and bunch charge.
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Space-charge forces
• Charged particles in an accelerator generate electromagnetic
fields that can be calculated from Maxwell’s equations:
 
E 
0

 1 E

 B  2
 0 J
c t
• The fields affect the motion of other particles in the
accelerator, through the Lorentz force:


  
F  q E vB
Collective Effects in Accelerators: Part 1

4
Fields around ultra-relativistic charges
• Solving Maxwell’s equations for an ultra-relativistic charge
moving in a straight line shows that the fields become
“flattened” in a plane perpendicular to the direction of
motion of the charge.
• For two (like) charges moving parallel to
each other:
• the electric force pushes the charges apart;
• the magnetic force pulls the charges together.
• At ultra-relativistic energies ( >> 1), the
electric and magnetic forces almost
cancel.
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Space-charge forces in a uniform beam
• The simplest realistic case in an accelerator is a continuous
beam with uniform density within a circular cross-section:
radius a
velocity v
nq particles (with charge q) per unit length
• Each charge experiences a net radial force:
q 2 nq
r
Fr 
2 0  2 a 2
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Equation of motion with space charge
Consider motion in the horizontal transverse (x) direction.
The equation of motion for a particle in the beam is:
d 2x K
 2x
2
ds
a
where s is the distance along the beamline, and K is the
perveance of the beam:
2I
K 3 3
b  Ic
b = v/c, I is the beam current, and Ic is the critical current:
4 0 mc3
Ic 
q
For electrons, Ic is the Alfvén current, IA  17.045 kA.
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Equation of motion with space charge
• Notice that the space-charge force varies linearly with
transverse position in the beam.
 This applies only to the special case of a uniform transverse
distribution.
• For a uniform transverse distribution, space-charge forces act
like forces from quadrupole magnets, except that the spacecharge forces are simultaneously defocusing in x and y.
• It follows that important properties, in particular conservation
of emittance, apply to beam transport in the presence of
(linear) space-charge forces.
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The envelope equation
• Since we know the equation of motion for individual particles,
we can work out the equation of motion for the beam
distribution.
• The equation of motion for the rms beam size is known as the
envelope equation:
d 2 x
 x2
K
 k1 x  3 
0
2
ds
 x 2 x   y 
where k1 is the quadrupole focusing strength, and:
x 
x
2
and
x 
x
2
x  xx
2
2
• Similar equations apply for the vertical (y) motion.
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The envelope equations
• So far we have considered only the special case of a beam
with uniform charge density within a circular cross-section.
• In this case, the horizontal and vertical rms beam sizes evolve
according to the envelope equations:
d 2 x
 x2
K

k



0
1 x
2
3
ds
 x 2 x   y 
d 2 y
ds 2
 y2
K
 k1 y  3 
0


 y 2  x  y
where the emittances x and y are constant.
• It turns out that the envelope equations apply to any
continuous beam with elliptical symmetry – although in
general, the emittances are not conserved.
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Emittance and space charge
• Notice that the terms in the envelope equation have a clear
physical origin:
d 2 x
 x2
K
 k1 x  3 
0
2
ds
 x 2 x   y 
quadrupole
focusing
emittance
space charge
• If the emittance term is much larger than the space-charge term,
then the beam transport is said to be emittance dominated.
• If the space-charge term is much larger than the emittance term,
then the beam transport is said to be space-charge dominated.
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Beam transport with space charge
Example: transport of a beam with emittance 10 mm mrad
through a drift space of length 1 m.
Initial conditions chosen to minimise the maximum beam size.
K = 5×10-4
K = 2×10-4
K=0
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Space-charge tune shifts
• The defocusing effects of space-charge forces lead to changes
in the betatron oscillation frequencies.
• For a continuous beam with uniform charge density, all
particles experience the same frequency shift.
• For a bunch in a storage ring, the situation is much more
complicated:
 the space-charge forces are nonlinear
 the local charge density seen by a particle varies in a way that depends
on the amplitude of the synchrotron oscillations of the particle.
particles near the centre of
a bunch experience a large
tune depression
particles at the head or tail
of a bunch experience
almost no tune depression
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Incoherent space-charge tune shifts
• Although each particle within a bunch experiences a betatron
frequency shift from space-charge forces, the net spacecharge force on a bunch is zero.
• This kind of tune shift is sometimes called an incoherent tune
shift: it cannot be measured by observing the coherent
motion of a bunch of particles.
• For particles in a bunch with non-uniform density, there is in
addition a spread in the tune shifts for different particles. The
vertical tune spread can be estimated from the formula:
by
K
 y  
ds

4  y  x   y 
where K is the peak perveance in the bunch, by is the vertical
beta function, and the integral is over the full circumference.
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Laslett tune shifts
• In addition to direct forces between particles in a bunch,
particles experience forces due to image charges in the walls
of the vacuum chamber.
• The forces from image charges depend on the geometry of
the vacuum chamber, and the position of the bunch within
the chamber.
• Tune shifts resulting from image charges are known as Laslett
tune shifts.
• In general, there will be coherent Laslett tune shifts as well as
incoherent Laslett tune shifts.
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Laslett tune shifts
Beam on the mid-plane
between infinite plane parallel
plates: incoherent tune shift.
Collective Effects in Accelerators: Part 1
Beam displaced from the midplane between infinite plane
parallel plates: incoherent and
coherent tune shifts.
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Longitudinal space-charge effects
• In a continuous beam with uniform charge density (within a
given cross-section) the longitudinal fields are zero, by
symmetry.
• Longitudinal components of the fields occur when there is
some variation in the density of particles along the beamline.
High particle density

E
radius a
velocity v = bc
Low particle density
nq(s) particles (with charge q) per unit length
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Longitudinal space-charge fields
• The longitudinal components of the fields are related to the
transverse components through Maxwell’s equations: this
provides one route to calculating the longitudinal fields.


B
 E  
t

 
 

C E  dl   t  B  dA
• In a chamber (of radius b) with perfectly conducting walls, the
longitudinal component of the electric field vanishes at the
walls.
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Longitudinal space-charge effects
The longitudinal electric field is conveniently expressed in terms
of the Fourier components of the particle density.
nq(s)
s
radius a
velocity v = bc

nq ( s, t )   n~q ( )ei ( ks t ) d

Es r  0, s, t   i
qg

k
~ ( ) ei ( ks t ) d

n
 q
4 0 b 2c 
Collective Effects in Accelerators: Part 1

 bc
b
g  1  2 ln  
a
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Longitudinal space-charge effects
• The main effect of the longitudinal space-charge fields is to
change the energy of the particles.
• The change in energy of a particle over the distance from s1 to
s2 is found by integrating the force on the particle over that
distance:
s
q 2 
sz
 ds
 ( z ) 
Es  s, t 

E0 s1 
bc 

E  E0 
  

E0 

velocity v = bc
s1
z
reference particle
at position s at time t
Collective Effects in Accelerators: Part 1
s2
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Longitudinal space-charge impedance
Combining the expressions for the longitudinal electric field and
the energy change, we find that in a storage ring, the energy
change over one revolution can be written:
 z   
q
E0

~
i z c




I

Z

e
d
|| sc


~
where I   is the Fourier spectrum of the beam current
observed at a fixed location in the ring, and Z|| sc   is the
longitudinal space-charge impedance, given by:
gZ 0 
Z|| sc    i
2b 2 0
g is the geometry factor, and 0 is the ring revolution frequency.
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Longitudinal space-charge in a storage ring
In a synchrotron storage ring, the longitudinal space-charge
impedance leads to a change in the synchrotron frequency and
the equilibrium longitudinal distribution.
Storage ring
below transition.
equilibrium distribution
without space-charge
equilibrium distribution
with space-charge
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Beam-beam effects
• Particles in colliding beams experience electromagnetic forces
from the particles in the opposing beam.
• For particles moving in opposite directions, the electric and
magnetic forces add up rather than tending to cancel out.
• Beam-beam forces can be significant even at energies large
enough for space-charge effects to be negligible.
• For colliding beams of

E
opposite charge, one of
e

the principal effects is a
F
transverse focusing force
e

at the interaction point,
B e+ moving
leading to variation in
into page;
betatron tunes and
e- moving
out of page.
beta functions.
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Beam-beam tune shifts
• Beam-beam forces can be strongly nonlinear and result in
complicated effects.
• However, they are often characterised by the beam-beam
parameter, which quantifies the linear kick experienced by a
particle as a function of its transverse position:
b y dp y
y  
4 dy
y 0
• The transverse kick leads to a (coherent) tune shift of the same size
as the beam-beam parameter, e.g. for a Gaussian beam:
 y   y 
N b re b y
2 y  x   y 
Nb is the number of particles per bunch,
re is the classical radius (of the electron),
by* is the vertical beta function at the interaction point.
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Scattering effects
• When discussing space-charge effects, we considered the
impact of fields calculated with the approximation of a
smooth charge distribution.
• In reality, bunches consist of large numbers of “point-like”
particles.
• The fields close to a single particle can be very much larger
than the fields calculated in the approximation of a smooth
charge distribution.
• Occasionally, two particles within a bunch can approach each
other closely enough for the local peaks in the field strengths
to have observable effects.
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Touschek effect
• A “collision” between two electrons (or
positrons) is described as Møller
scattering.
• In the rest frame of a bunch, the
differential cross-section for the
scattering process gives the distribution
for the deflection angle.
• When transformed into the lab frame,
even a small deflection angle can lead to
a large change in energy deviation.
• If the energy deviation of a particle is
outside the energy acceptance of the
machine, the particle will be lost from
the beam: this is the Touschek effect.
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Touschek effect
• The rate at which particles are lost from a beam can be
calculated from the density of particles in the beam, and the
energy acceptance of the storage ring.
• In the simplest case (neglecting dispersion and coupling, and
assuming non-relativistic motion in the bunch rest frame), the
loss rate from the Touschek effect is given by:
2
b
2
0
dN b
N cr

dt
8 2 x y z
3
 b 

 D 
  max 
where max is the energy acceptance of the ring, and:
2
 max
b
 2 2x
b  x
Collective Effects in Accelerators: Part 1
D   
e u  u
1  u 


1

ln    du
 u 2  
2   

3/ 2
27
Touschek effect
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Touschek lifetime
• The loss rate depends on the square of the number of
particles in the bunch.
• Therefore, the bunch population does not (strictly speaking)
decay exponentially; nevertheless, we define the Touschek
lifetime:
3
2
 b 
N b cr0
1
1 dN b

 D 


2
T
N b dt
8  x y z   max 
• Other effects (in particular, gas scattering) can lead to the loss
of particles from the beam in a storage ring.
• But in low-emittance electron rings (e.g. in third-generation
synchrotron light sources) the Touschek effect is usually the
dominant lifetime limitation.
• The Touschek lifetime has a strong dependence on the energy
acceptance: usually, the goal is to achieve max > about 4%.
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Intrabeam scattering
• Even if a Møller scattering event does not result
in the loss of the particles involved, there can be
a transfer of momentum from the transverse
direction to the longitudinal.
• If the scattering takes place at a location with
non-zero dispersion, then the momentum transfer
can lead to an increase in the transverse emittance.
• Intrabeam scattering (IBS) leads to emittance
increase by a mechanism very similar to quantum
excitation by synchrotron radiation.
• The main difference between IBS and quantum excitation is that IBS
happens wherever the dispersion is non-zero: not just in dipoles.
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Intrabeam scattering
• There are two widely-used formalisms for the analysis of
intrabeam scattering:
 Piwinski (Proc. 9th Int. Conf. High Energy Accel., 1974)
 Bjorken and Mtingwa (Particle Accelerators 13, 115-143, 1983)
• Both formalisms lead to expressions for the growth rates of
the transverse and longitudinal emittances.
 Simplified formulae can be derived using various approximations.
• In an electron (or positron) storage ring, the IBS growth rates
are usually slow compared to the emittance damping rates
from synchrotron radiation.
• Increases in equilibrium emittance from IBS are usually not
observable, except with:
 high bunch population (> few × 109 particles)
 low or moderate energy (< few GeV)
 very low emittance (vertical emittance < few pm)
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Observations of IBS at the KEK-ATF
K.L.F. Bane et al, “Intrabeam scattering analysis of measurements at KEK’s Accelerator Test
Facility damping ring,” PRST-AB 5, 084403 (2002).
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Summary
• Space-charge forces have transverse defocusing effects.
 The size of the space-charge forces can be characterised by the
perveance, which is a function of the beam current and energy.
 The evolution of the transverse beam size, including focusing,
emittance and space-charge, is described by the envelope equations.
 For a uniform transverse distribution, the space-charge forces are
linear, and the emittance is conserved.
 The Laslett tune shifts describe the effects of image charges in the
walls of the vacuum chamber.
• Longitudinal variations in particle density lead to changes in
particle energy, which can be described by an impedance.
• Touschek scattering leads to loss of particle from the beam.
• Intrabeam scattering leads to an increase in beam emittance.
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