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 radio-frequency (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 radio-frequency (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 wave-guides 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 cut-o

frequency of the lowest mode,TM

01

,but below the cut-o 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 momentum-dependent 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 non-zero 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 non-negative.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 non-linearities 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 non-adiabatic 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 non-adiabatic 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 bucket-to-bucket.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 phase-space 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 non-radiating 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 (Wiley-VCH,

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)

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