Seminar I
a
,cetrti letnik,stari program
LONGITUDINAL DYNAMICS OF PARTICLES IN ACCELERATORS
Author:Ursa Rojec
Mentor:Simon
Sirca
Ljubljana,November 2012
Abstract
The seminar focuses on longitudinal motion of charged particles in particle accelerators.The technique of
acceleration by electromagnetic waves is explored and the stability of motion under such acceleration is
inspected.The seminar introduces the concept of ideal particle and develops equations that treat deviations
from its motion.
1
Contents
1 Introduction 2
2 Acceleration methods 2
2.1 Some comments on acceleration by static elds............................2
2.2 Acceleration by radiofrequency (RF) elds..............................3
3 Equations of motion in phase space 4
3.1 Path length and momentum compaction................................4
3.2 Dierence equations...........................................5
3.3 Dierential equations...........................................6
4 Small oscillation amplitudes 6
4.1 Phase Stability..............................................7
5 Phase Space Motion 8
5.1 Phase Space Parameters.........................................9
5.1.1 Fixed Points...........................................9
5.1.2 Momentum Acceptance.....................................10
5.1.3 Emittance,Momentum Spread and Bunch Length......................10
5.1.4 Acceptance............................................11
5.2 Acceleration................................................11
5.3 Eect of RF Voltage on Phase Space..................................12
5.4 Phase Space Matching..........................................12
5.5 Longitudinal Gymnastics:Debunching and Bunch Rotation.....................13
6 Conclusion 15
1 Introduction
Particle accelerator physics primarily deals with interaction of charged particles with electromagnetic elds.
The force that the eld exerts on the charged particle is called Lorentz force and can be written as
F = q [E+v B]
Transverse elds
1
are used to guide particles along a prescribed path but do not contribute to their energy.
Acceleration is achieved by longitudinal elds and this seminar will focus on interaction of charged particles
with longitudinal elds.
In the simplest case,acceleration is achieved by static electric elds.Particle that travels through a potential
dierence of V
0
gains qV
0
energy,if q is the particle's charge.This way of acceleration is simple but limited
to 10
6
V due to voltage breakdown.It is still widely used for acceleration of low energy particles at the
beginning of acceleration to higher energies.Somewhat higher voltages can be achieved by pulsed application
of such elds,but for acceleration to higher energies,dierent methods must be exploited.
Most common and ecient way for particle acceleration are high frequency electromagnetic elds in acceler
ating structures and this is the topic covered in the seminar.In a very general way,equations of motion will be
derived and stability limits will be inspected.For a more rigorous treatment of interaction of charged particles
with longitudinal elds one must turn to higher order equations and take into account losses due to interactions
of particles within the beam,because of incomplete vacuum and so on,that will not be covered here.
2 Acceleration methods
2.1 Some comments on acceleration by static elds
Since we are limited by maximum V
0
because of voltage breakdown and accelerator's cost goes up with every
meter,the rst thing that comes to mind would be to curve the particle trajectory to a circle,so that it
passes the same accelerating section repetitively,as is shown in Fig.1.The eld required to bend the particle
1
Mostly magnetic,because for perpendicular orientation of the elds,this is true:F
E
/qE and F
B
/qvB.In accelerators we
are mostly dealing with particles with velocities close to the speed of light
2
Figure 1
trajectory is a static magnetic eld.To ensure,that the electric eld is non zero only between the plates,they
must extend to innity.In reality this is not possible,and we have an electric eld outside the capacitor.This
fringe eld will decelerate the particle when it approaches or departs the capacitor.As we anticipate according
to Farraday's law
I
C
E ds =
@
@t
Z
S
B dS;
there will be no net acceleration,which is consistent with the conservative nature of the electrostatic eld.
2.2 Acceleration by radiofrequency (RF) elds
With electromagnetic waves,accelerating voltages far exceeding those obtainable by static elds can be achieved.
This method of acceleration is used in linear as well as in circular accelerators.For practical reasons,specically
in circular accelerators,particle acceleration occurs in short straight accelerating sections placed along the
particle path.
Since a free electromagnetic wave does not have a longitudinal electrical eld component,special boundary
conditions must be enforced.This is done by accelerating structures,called waveguides or resonant cavities,
providing a travelling or standing EMwave respectively.In a crude approximation,a waveguide is a pipe made
of conducting material and a cavity is a waveguide closed at both ends.Waveguides are used primarily in linear
accelerators (linacs),while resonant cavities are used in both linacs and synchrotrons.In both cases TMmodes
are used,because magnetic eld cannot accelerate particles.The modes are found from Maxwell's equations,
with boundary conditions E
k
= 0;B
?
= 0 at the cavity/waveguide walls.The two most commonly used modes
are represented on Fig.2.
Figure 2:A cylindrical waveguide operating in TM
01
mode (left) and a pillbox cavity operating in TM
010
mode (right)
Waveguides The most commonly used strategy is to excite a waveguide at a frequency above the cuto
frequency of the lowest mode,TM
01
,but below the cuto frequency for other modes.This way only one mode
is propagating trough the waveguide.In order to be able to accelerate charged particles over a reasonable
distance,the wave must have the same phase velocity as the velocity of the particles.This way the particle
travels along the structure with the wave and is accelerated or decelerated at a constant rate.Since phase
velocity of such a wave is larger than the speed of light,waveguides are loaded with discs to make this happen
Cavities The most commonly used accelerating mode is TM
010
.For this mode,the electric eld is directed
longitudinally and has constant magnitude along z.It has no azimuthal dependence and has a maximum on
the axis of the cavity,decreasing in the radial direction until it is zero at the cavity walls.
Accelerating structures in a circular accelerator may be either distributed around the ring or grouped
together so that the ring only has one accelerating section.In both cases,the frequency of the voltage in the
accelerating structures must be an integer multiple of particle revolution frequency,where the integer is called
the harmonic number and denoted by h.The harmonic number is the maximum number of bunches (groups
of particles moving together in the accelerator,more on this later in the text) that can be in the accelerator at
the same time.
3
3 Equations of motion in phase space
To achieve acceleration,one must ensure constructive interaction of the particles with the wave.Because EM
elds oscillate,special synchronicity conditions must be met in order to obtain the desired acceleration.
Weather we have a standing or a travelling wave,its current value is determined by the phase.This means
that the degree of acceleration is determined by the phase.If systematic acceleration is to be achieved,this
phase must be at a specic value at the moment the particle arrives to the accelerating section.This value is
called synchronous phase and denoted by
s
.We assume that the ideal,synchronous particle arrives at each
station at the same phase and thus receives the same energy boost at each station.In a circular accelerator,
revolution frequency of the particles and the RF frequency must be related by
!
rf
= h!
rev
;(1)
where h is the harmonic number.It represents the number of times,per particle revolution period,that the RF
voltage is at the correct level to accelerate particles.
In the following discussion,we will introduce the concept of a synchronous particle with the ideal energy and
phase,and develop equations of motion that treat deviations from its trajectory.
3.1 Path length and momentum compaction
In the case of a circular
2
accelerator we come to the problem of momentumdependent path length.The
dependence arises in bending dipoles required to keep the particles on a circular path.When a charged particle
enters a homogeneous magnetic eld,the eld exerts the Lorentz force on it and bends the particle trajectory.
The radius of the bend depends on the particle's charge,as well as its velocity.Since the dipoles are tuned to
a so called ideal particle,any particle with momentum dierent than the momentum of the ideal particle will
not follow the designed path.
We will denote the deviation of the particle from its ideal path by
x = D(s)
p
p
0
;
where p
0
represents the momentumof the ideal particle and D(s) represents the dispersion function.It describes
the eect that bending magnets have on the particles'trajectory.We need not concern ourselves with more
detail,and can take it as a machine parameter.The total path length can now be written as
L =
Z
L
0
0
1 +
x(s)
ds;
where is the radius of the bend.We can immediately see that for an ideal particle with p = 0 the path
length is just L
0
.which is the ideal design circumference of the accelerator.The deviation from the ideal path
can thus be obtained by integrating the second term.
L =
p
p
0
Z
L
0
0
D(s)
(s)
ds;
To measure the variation of the path length with momentum,we dene the momentum compaction factor
c
=
L=L
0
p=p
0
=
1
L
0
Z
D(s)
(s)
ds = h
D(s)
i;(2)
The momentum compaction factor is nonzero only in curved sections,where is nite.In the case of a LINAC
the curvature ( = 1=) is 0, = 1and the momentum compaction factor vanishes.
However,when dealing with phase advance from one accelerating section to the other,we are not so much
interested in the deviation of the path length,but would rather know the time it takes for a particle to travel
between two successive accelerating sections separated by a distance L.This time is given by the equation
= L=v.By dierentiating the logarithmic version of the equation we get:
=
L
L
v
v
:
2
Path length along a straight line also depends on the angle that the particle trajectory encloses with the line.This,however,
is a second order correction,so it will be neglected here.
4
The rst term on the right is just momentum dependent path length that we derived earlier.To connect the
second term to momentum deviation,wee need only to dierentiate the momentum (p = m v),and we get,
after some manipulation:
v
v
=
1
2
p
p
;
where is the Lorentz factor.We can see that both
c
and appear as a factor before momentum deviation.
We can thus put them together to form the momentum compaction
c
=
1
2
c
;(3)
and get the nal expression for time deviation
=
c
p
p
:(4)
Momentum compaction vanishes when
t
=
1
p
c
(5)
This is the transition Lorentz factor.From special relativity we know that
=
E
total
E
rest
;
and so
t
is usually referred to as transition energy.Transition energy is very important when it comes to phase
focusing.
3.2 Dierence equations
In order to derive the equations in longitudinal phase space,we take a look at phase and energy advance between
successive passes through the RF cavities.The energy deviation and phase at the entrance to the (n + 1)
th
cavity can be expressed as:
n+1
=
n
+!
rf
( +)
n+1
;
=
n
+!
rf
n+1
1 +
n+1
!
;
E
n+1
= E
n
+e [V (
n
) V (
s
)];
(6)
where
s
is the synchronous phase and V ( ) is the RF waveform.Since the synchronous particle always stays
in phase and!
rf
is the phase advance of the synchronous particle,we can rewrite the phase advance in the
equation (6) as
n+1
=
n
+!
rf
n+1
n+1
:(7)
Here!
rf
is the phase advance from cavity to cavity.For a circular accelerator with only one accelerating
section,it must be an integer multiple of 2 (harmonic number),so that we satisfy the synchronicity condition
The nal form of the equations can be obtained by using equation (4) and the relationship E=E =
2
p=p
in (7)
n+1
=
n
!
rf
c
p
n+1
p
s
E
n+1
= E
n
+e [V (
n
) V (
s
)];
(8)
where p
s
is the momentum of the synchronous particle.Both equations are coupled.This can be seen if we
replace the energy deviation with momentum deviation in (8),by noting that cp = E.
p
n+1
= p
n
+
e
c
[V (
n
) V (
s
)]:(9)
5
3.3 Dierential equations
Typically the phase and energy change by small amounts at each pass of the accelerating section,which allows
us to treat them as continuous variables.We can then approximate the dierence equations by dierential ones,
using n as the independent variable.We can rewrite equations (8) as:
d
dn
=
c
!
rf
p
s
p;
dp
dn
=
e
c
[V () V (
s
)]:
(10)
In most practical cases,parameters like particle velocity or its energy vary slowly during the acceleration,
compared to the rate of the change in phase.We can thus consider them constant an obtain a single second
order dierential equation from equations in (10):
d
2
dn
2
+
c
!
rf
e
cp
s
[V () V (
s
)] = 0:(11)
We can not go much further without making an assumption about V ().Since RF elds are created in
accelerating cavities,we will assume a sine function.
V = V
0
sin():
Let us now take a look at particle movement in phase space.We will take the simplest case and say that the
only way that the particle's energy can change,is trough interaction with the RF eld.If we rewrite the phase
as ='+
s
and expand the trigonometric term,the equation of motion in phase space becomes:
'+
c
!
rf
eV
0
cp
s
(sin
s
cos'+cos
s
sin'sin
s
) = 0;(12)
where we have also changed from number of turns n to time t as independent variable,by noting that
d
dn
=
dt
dn
d
dt
=
d
dt
:(13)
4 Small oscillation amplitudes
To get some insight into the solutions and the stability of motion,we rst take a look at small oscillations
about the synchronous phase.Since'is small we can approximate the sine and cosine term by their Taylor
expansions.Keeping only the linear terms in'we obtain an equation for a harmonic oscillator:
'+
2
'= 0;(14)
where we have dened the synchrotron oscillation frequency as
2
=
c
!
rf
eV
0
cp
s
cos
s
:(15)
For real values of
we have a simple solution
'='
0
cos(
t +
i
);(16)
where
i
is some general phase that we will set to zero.Since =
s
+'and
s
is constant,it is true that
_
= _'we can construct an equation for momentum error,from (10,13)
=
p
p
s
=
_'
c
!
RF
=
'
0
c
!
RF
sin(
t +
i
) (17)
If
is real,both phase and particle momentum oscillate about the ideal value with synchrotron frequency  we
have stable oscillations.If we join solutions for phase and momentum error,we get an invariant of motion
'
'
0
2
0
2
= 1 (18)
It describes particle trajectories in phase space.They can be ellipses or hyperbolas,depending on
.Ellipses
represent stable motion in case of a real
and hyperbolas represent unstable motion,when
is imaginary.
6
Figure 3:Synchrotron oscillations in (;') phase space for small deviations from the synchronous phase.Trajectories for real (left) and
imaginary (right) values of synchrotron frequency
In Fig.3 on the left graph for stable oscillations we can clearly see separatrices enclosing the area of stable
motion.In accelerator physics,areas where stable motion exists are called buckets.All the particles sharing
a bucket are called a bunch.Maximum number of bunches that can be in a circular accelerator at the same
time is determined by the harmonic number (1).
4.1 Phase Stability
In the previous sections,when deriving the equation for small oscillation amplitudes,we have already established
that in order to have stable oscillations,synchrotron frequency needs to be real.Taking a closer look at the
expression for
2
(15) we can see that besides cos
s
and
c
all other quantities are nonnegative.Therefore
these two will determine whether the motion is stable or unstable.
Momentum compaction,
c
,goes to zero (3) when particles cross the energy of
t
=
1
p
c
:(19)
When this happens,the travel time from one accelerating gap to another does not depend on the particle
momentum.There is no phase stability at this energy.This is a machine dependent parameter.LINACs do
not have a transition energy because they are straight.
Synchronous phase must be selected according to of the particles to obtain stable oscillations and acceler
ation.If RF frequency is represented by a sine wave,we have V > 0 if 0 < < ,hence
0 <
s
<
2
for <
t
;
2
<
s
< for >
t
:
For electrons the transition energy is in the range of MeV and for protons in in the range of GeV.Crossing
the transition presents us with many technical problems.Since
t
for electrons is relatively small,electrons are
injected to electron synchrotrons above the transition energy,thus avoiding stability problems during acceler
ation.This is not the case with protons.A LINAC with proton energy of 10 GeV would be very costly,so
protons are usually injected into the synchrotron below
t
.
An oscillating accelerating voltage,together with a nite momentum compaction produces a stabilizing fo
cusing force in the longitudinal degree of freedom.This is the principle of phase focusing represented in Fig.
4.
We now explore the eect that going over transition energy (5) has on phase focusing.The momentum
compaction
c
(3) changes sign when =
t
:
c
> 0 for <
t
and
c
< 0 for >
t
Below the transition energy,the arrival time is determined by the particle's velocity.After transition,the
particle has a velocity close to c and its arrival time depends more on the path length than on its speed.The
key dierence is this:
7
Figure 4:Phase focusing principle.If the particle is lagging behind the synchronous particle it will see a higher accelerating voltage.This
will cause it to gain more energy and because it will travel faster,deviation of its phase with respect to synchronous particle will decrease.
The same principle can be applied to particles that are too fast,except that they gain less energy.
A particle with momentum higher than that of the ideal particle will arrive at the acceleration
section faster than the ideal particle if we are below the transition.If we are above the transition,
because
c
is negative,it will arrive after the ideal particle.
This can be clearly seen from equation (4).In Fig.4 the basic idea of phase focusing is introduced.We can
clearly see that in order to obtain focusing,slow particles must arrive later and faster particles sooner.If the
particles were to cross the
t
and RF voltage would remain the same,the motion would no longer be stable.
Slower particles would be accelerated less,and faster particles more than the ideal particles.
5 Phase Space Motion
The equation of motion (12) describes the particle motion in (p;') phase space.There are two distinct cases,
one where synchronous phase is set to 0,and the other when it is not.If
s
= 0,the synchronous particle will
experience the voltage of V = V
0
sin
s
= 0 when it passes the accelerating section.We call this the stationary
case.For all values of
s
that are not integer multiples of ,particles will be accelerated or decelerated,
depending on the phase.
In order to accelerate the particles,the synchronous phase must be set to a value other than n.
Particle accelerators consist of many elements.For example,a light source would consist of an injector linac
that would accelerate particles to E
i
.After that,the particles are transferred to a booster synchrotron that
accelerates them from E
i
to their nal energy,E
f
.When E
f
is reached,the particles are again transferred to
a storage ring.The function of the storage ring is to keep the particles orbiting at constant speed.Because
of synchrotron radiation,electron storage rings also contain accelerating sections to make up for energy lost
due to synchrotron radiation.Since protons are much heavier (factor of 10
3
eV) and the energy lost per turn
for synchrotron radiation scales as
4
= eect of synchrotron radiation on their energy is negligible.Still RF
sections are needed to provide phase focusing.
By using dierence equations (8) we can make simulations to take a look at particle motion in stationary
and accelerating case.The following simulations were done for the Fermilab Tevatron ring,which is a proton
accelerator.On the left graph in Fig.5 we can see an example of a stationary bucket.Particles with =
s
Figure 5:Phase space plot for a stationary (left) and an accelerating bucket (right).
are not accelerated.The phase stable region is 2 in extent and particles that nd themselves out of the bucket
will undulate in energy and diverge in phase.They may stay in the ring indenitely.
8
Comparing the phase space plot for an accelerating bucket to the one for the stationary bucket on Fig.
5,we can see a signicant change in the shape of the separatrix as well as its origin.The center of the bucket is
no longer at (0;0) but is shifted by
s
.Particles that nd themselves out of the bucket will diverge in phase as
well as in energy.In contrast to the stationary bucket case,these particles will eventually gain too much energy
and leave the circular accelerator.
The graphs in Fig.5 were plotted for a proton energy below the transition energy of the accelerator, <
t
.
For >
t
the orientation of the buckets changes.Fig.6 shows the bucket shape before and after the transition:
Figure 6:Shape of non stationary buckets before and after transition
5.1 Phase Space Parameters
The equation of motion (12) can be derived from a Hamiltonian
H =
_'
2
2
2
cos
s
[cos(
s
+') cos
s
+'sin
s
]:(20)
For the stationary case,Hamiltonian (20) simplies and is identical to that of a mechanic pendulum:
H =
_'
2
2
2
cos':(21)
In Fig.7 potential for accelerating and stationary case is shown,lines representing equipotential surfaces.
Figure 7:Potential well for a stationary (left) and accelerating bucket (right).Accelerating potential is tilted compared to stationary,
which is the result of an additional linear term in (20)
5.1.1 Fixed Points
In the stable phase space regions,particles oscillate about the synchronous phase and ideal momentum as can
be seen from equations (16,17).Within the stable regions,we can nd two xed points,one stable and one
unstable.They can be calculated from
@H
@'
= 0;
@H
@ _'
= 0:(22)
The two xed points correspond to minima (sfp  stable xed point) and saddles (ufp  unstable xed point) in
the potential represented on Fig.7.From conditions (22) we obtain coordinates for xed points in (';_').sfp
is located at (
s
;0) and ufp at (
s
;0).
9
Figure 8:Characteristic bucket and separatrix parameters
5.1.2 Momentum Acceptance
_'is proportional to p=p
0
.Maximum momentum acceptance can thus be found by dierentiation of the
hamiltonian,(20),with respect to'.At the extreme points,where the momentum reaches a minimum or a
maximum,we have @ _'=@'= 0 and the contition is'= 0.From Fig.8 we see that the maximum phase
elongation occurs at the ufp where _'is zero.Maximum momentum acceptance can then be found by equating
the values of the hamiltonian for the ufp and the derived condition.We get:
1
2
_'
2
= 2
2
h
1
2
s
tan
s
i
:(23)
5.1.3 Emittance,Momentum Spread and Bunch Length
The dynamics of both the stationary and the accelerating case can be described with the help of Hamiltonian
equations.Following the Liouville theorem that states that the area in phase space is conserved for a system
that can be described by Hamiltonian equations (this follows directly from Hamilton equations,since rv = 0),
we dene the longitudinal emittance as the area of phase space enclosed by the beam.
In what follows,we will derive the relationships between the rms momentum and the 95% emittance of a
bunch.The derivations are valid for a distribution where the emittance is much smaller than the bucket area,so
that eects due to nonlinearities of the RF focusing and large tails can be ignored,and where the distribution
of the bunch is assumed to be Gaussian.
We start by evaluating (20) for small angles and obtain
_'
2
+
2
'
2
= const:
Since the bunch spread is usually measured in t instead of in terms of phase deviation,we will switch to
(E;t) phase space.This is important,because Liouville theorem holds only for conjugate pairs of variables.
The transformation is done by substituting
'=!
rf
t;
and from (10) we obtain
_'=
1
d'
dn
=
p
p
0
c
!
rf
=
1
2
E
E
c
!
rf
:
We are left with the equation for particle trajectory in (E;t) phase space
(E)
2
+
2
E
s
eV
0
!
2
rf
cos
s
2h
c
(t)
2
= const:(24)
which is the equation of an ellipse.To evaluate the constant,we need to nd the trajectory,that encloses
95% of the particles.Since we have assumed a Gaussian distribution,the radius that corresponds to this is
approximately
p
6.Our constant is than just 6
2
E
,where
E
is the rms of energy deviation.We can rewrite
the equation in the form of
E
A
2
+
t
B
2
= 1;
10
and obtain the emittance by calculating the area it encloses from S = AB.The area of this ellipse is then
called 95% longitudinal emittance,denoted by
l
,and has units of eVs.
l
=
!
RF
s
2h
c
E
3
s
2
eV
0
cos
s
E
E
s
2
:(25)
The emittance can be compactly written as a product of the momentum spread and the bunch length (
t
) as
l
= 6
E
t
= 6c
p
t
:(26)
Given the emittance and the RF parameters,we can now express the momentum spread and bunch length and
see that they scale as
p
/
4
p
V
0
;(27)
t
/
4
r
1
V
0
:
5.1.4 Acceptance
We must distinguish between acceptance and emittance.The acceptance is associated with the bucket (available
stable phase space area) whereas the emittance is associated with the bunch (actual phase space occupied by
the beam).The acceptance is the maximum allowed value of emittance and is determined by the design of the
transport or accelerating lines.
Emitance can be obtained from the hamiltonian (20).We know that the total energy of the system is a
conserved quantity.To evaluate the constant,we use the ufp,because we know that _'is zero at these locations.
To get the acceptance we need to integrate
A =
Z
S
E
!
RF
d';(28)
where the integral must be taken over the separatrix.The integral can be solved analytically only for stationary
buckets with
s
= 0;.In this case we get the stationary acceptance,denoted by
sta
:
sta
= 8
s
2eV
0
E
0
2
hj
c
j!
2
rf
(29)
For other values of
s
the integral can be solved numerically.From Fig.9 we can see how acceptance varies
with
s
,with the largest value for the stationary bucket.
Figure 9:The emittance for a moving bucket
mov
with respect to the emittance for a stationary bucket
sta
5.2 Acceleration
We will now take a closer look at the particle motion during acceleration.In a synchrotron,particles are injected
at some initial energy E
i
.They are then slowly,through a number of turns,accelerated to their nal energy,
E
f
.Because the acceleration should be adiabatic,synchronous phase is slowly increased.The simulation tracks
a single particle and varies the synchronous phase from 0 to its nal (arbitrarily chosen) value of =6.By
changing the time (number of turns) it takes the particle to reach the nal acceleration,we can observe the
in uence of nonadiabatic eects.In Fig.10 we can see how fast changes of phase aect the longitudinal
emittance.The top left graph represents adiabatic acceleration and we can clearly see how trajectory follows
s
.We can tell that the acceleration is adiabatic,since the area of the particle's phase space ellipse remains
essentially constant.If we change the number of turns it takes for the particle to reach the acceleration at
s
to a smaller value,the process becomes nonadiabatic and emittance is not preserved.The smaller the number,
the bigger the nal emittance.In general,if the motion is to be adiabatic,the system parameters must change
more slowly than the period of motion.
11
Figure 10:Trajectories for a particle underging acceleration from
s
= 0 to
s
=
6
.Number of turns is 3000,number of turns to reach
nal acceleration
s
is??,500,100,25
5.3 Eect of RF Voltage on Phase Space
As can be seen from Eqs.(27),(28),(23),the RF voltage has an eect on the bunch length and the momentum
acceptance of the accelerator.Even for a stationary bucket we can observe that with V
0
too high,the whole
phase space gradually becomes unstable.
Figure 11:Phase space plot for a stationary bucket with respect to increasing accelerating voltage,from 800 kV to 550 MV
5.4 Phase Space Matching
The beam transfer from one synchrotron or a linac to another synchrotron is done buckettobucket.The
RF systems of both machines are phase locked and bunches are transferred directly from the bucket of one
machine to the other.If we want the longitudinal emittance to stay the same,the bunch must be centred in
the bucket of the nal machine and both machines must be longitudinally matched,meaning that they have
the sameLongitudinal Twiss Parameter,
L
.
To obtain the
L
we rewrite the equation for particle trajectories in (E;p) phase space (24) as
E
A
2
+
t
B
2
= 1 !E
2
B
A
+t
2
A
B
= AB;
Since we previously dened the emittance as AB we obtain the nal expression for phasespace trajectories:
L
(E)
2
+
1
L
(t)
2
=
l
(30)
Figs.12,13,14 track the motion of a bunch for a 100 turns after transfer.They describe the motion after
the transfer for a matched case,a case when the transfer occurs with a phase error of =3,and a transfer with
a
L
error of factor three.
12
Figure 12:First hundred turns after transferring the beam from linac to the synchrotron for a matched transfer
Figure 13:First hundred turns after transferring the beam from linac to the synchrotron for a transfer with phase error of =6
Figure 14:First hundred turns after transferring the beam from linac to the synchrotron for a transfer with
L
mismatch by a factor of
three
5.5 Longitudinal Gymnastics:Debunching and Bunch Rotation
As was already proven in previous sections,the shape of a bucket can be manipulated by changing the RF
voltage.From (27) we can see that the momentum spread scales as
4
p
V
0
and the bunch length as 1=
4
p
V
0
.Since
the longitudinal emittance is preserved when motion is adiabatic,we can see that the shape of the bunch can
be manipulated by changing the RF voltage.The change in voltage always results in one of the or'spreads
getting larger and the other one getting smaller  there is no way to shrink them both,which is consistent with
emittance preservation.
Debunching We adiabatically reduce the voltage over many synchrotron periodes,untill nally it is turned
o.The beam is then distributed along the whole circumference of the accelerator.If the process is adiabatic,
the momentum spread of the beam is reduced,because the phase (time) spread gets large.When a beam is
debunched,no RF voltage is applied to it.This is why it is not possible to debunch an electron beam at any
signicant energy.Because of synchrotron radiation,electron beams always need to be accelerated in order to
13
compensate for radiation losses.
Bunch Length Manipulation The gures from the section on mismatched transfer give a hint that a bunch
shape could be manipulated.We can see that the bunch rotates if mismatches occur  these mismatches are a
consequence of phase/momentum oset or dierent RF parameters (meaning dierent buckets).We are more
interested in the latter,since the RF parameters are something that we can change while the particles are
circling in the accelerator.Let us assume that we have a bunch with a small momentum spread and long bunch
length.To change the bunch to a short one,we would increase the RF voltage in the time that is short compared
to the synchrotron oscillation period.Since the bucket has changed,the bunch is essentially mismatched and
starts to rotate.The process is shown in Fig.15.The bunch starts getting shorter by transferring some of
Figure 15:Plots for six turns (0th,2nd and 5th) of bunch manipulation process.On the rst plot,the bunch has its original shape.It
than starts to rotate until it reaches its narrowest point represented on the right graph.
its phase spread to the momentum spread.After a quarter of the synchrotron period,it reaches its narrowest
point and,unless the RF voltage is increased again,it will continue to rotate and start getting longer.The
bunch rotates because its boundary does not coincide with a phase space trajectory,so the second time the RF
voltage is ramped up,it must increase to such a value that the particles at the edge of the bunch will follow
the same phase space trajectory.From (17) we get the relation
0
= j
!
rf
c
j'
0
:(31)
To get the overall bunch reduction factor,we proceed as follows.Before the rotation,the relation between
momentum and phase deviation is
1
= j
1
!
rf
c
j'
1
;
where
1
denotes the synchrotron frequency with RF voltage V
1
.After rotating,the phase deviation'
1
is
transformed into momentum deviation
2
= j
2
!
rf
c
j'
1
;
and the original momentum deviation is transformed into phase deviation
'
2
= j
!
rf
c
2
j
1
:
We now need to stop rotation.This is achieved if the new momentum error and the new phase error are on the
same phase space trajectory.The required RF voltage can be obtained from
2
= j
3
!
rf
c
j'
2
:
To get the ratio of the bunch lengths we take the quotient
'
2
3
'
1
1
=
2
1
=
2
'
1
2
'
2
Since l/'
0
and
/
p
V we get the overall bunch length reduction factor for this process:
l
1
l
0
=
4
r
V
1
V
3
;(32)
where V
1
is the initial and V
3
nal RF voltage.
14
The bunch length manipulation described in this section is applicable only to nonradiating particles.For
particles that radiate bunch manipulation is easier due to damping eects.Relation (31) still holds,but the
momentum spread is independently determined by synchrotron radiation and the bunch length scales propor
tionally to
p
V.
6 Conclusion
The rst idea to use RF cavities instead of static elds was to achieve higher energies.Throughout the seminar
the stability of motion of particles interacting with such elds was inspected and it was shown that besides
higher accelerating voltages,oscillating elds also provide an additional stabilizing force,that results in phase
focusing.The result of oscillating elds are bunched beams,within which the particles oscillate about the ideal
values of momentum and phase with the frequency known as the synchrotron frequency.
References
[1] Helmut Wiedermann,Particle Accelerator Physics  Basic Principles and Linear Beam Dynamics (Springer
Verlag,New York,1993).
[2] D.A.Edwards,M.J.Syphers,An Introduction to the Physics of High Energy Accelerators (WileyVCH,
Weinheim,2004).
[3] William Bartletta,Linda Spentzouris,USPAS  U.S.Particle Accelerator School Slides,
http://uspas.fnal.gov/materials/12MSU/MSU
Fund.shtml (26.11.2012)
[4] M.J.Syphers,Some Notes on Longitudinal Emittance,
http://home.fnal.gov/syphers/Accelerators/tevPapers/LongEmitt.pdf (26.11.2012)
15
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