Seminar Ia, cetrti letnik, stari program

LONGITUDINAL DYNAMICS OF PARTICLES IN ACCELERATORS

Author: Ursa Rojec

Mentor: Simon Sirca

Ljubljana, November 2012

AbstractThe seminar focuses on longitudinal motion of charged particles in particle accelerators. The technique ofacceleration 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 deviationsfrom its motion.

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Contents

1 Introduction 2

2 Acceleration methods 22.1 Some comments on acceleration by static fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Acceleration by radio-frequency (RF) fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Equations of motion in phase space 43.1 Path length and momentum compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Small oscillation amplitudes 64.1 Phase Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Phase Space Motion 85.1 Phase Space Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.1.1 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.1.2 Momentum Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.1.3 Emittance, Momentum Spread and Bunch Length . . . . . . . . . . . . . . . . . . . . . . 105.1.4 Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.2 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 Effect of RF Voltage on Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.4 Phase Space Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.5 Longitudinal Gymnastics: Debunching and Bunch Rotation . . . . . . . . . . . . . . . . . . . . . 13

6 Conclusion 15

1 Introduction

Particle accelerator physics primarily deals with interaction of charged particles with electromagnetic fields.The force that the field exerts on the charged particle is called Lorentz force and can be written as

F = q [E + v ×B]

Transverse fields1 are used to guide particles along a prescribed path but do not contribute to their energy.Acceleration is achieved by longitudinal fields and this seminar will focus on interaction of charged particleswith longitudinal fields.

In the simplest case, acceleration is achieved by static electric fields. Particle that travels through a potentialdifference of V0 gains qV0 energy, if q is the particle’s charge. This way of acceleration is simple but limitedto ∼ 106 V due to voltage breakdown. It is still widely used for acceleration of low energy particles at thebeginning of acceleration to higher energies. Somewhat higher voltages can be achieved by pulsed applicationof such fields, but for acceleration to higher energies, different methods must be exploited.

Most common and efficient way for particle acceleration are high frequency electromagnetic fields in acceler-ating structures and this is the topic covered in the seminar. In a very general way, equations of motion will bederived and stability limits will be inspected. For a more rigorous treatment of interaction of charged particleswith longitudinal fields one must turn to higher order equations and take into account losses due to interactionsof 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 fields

Since we are limited by maximum V0 because of voltage breakdown and accelerator’s cost goes up with everymeter, the first thing that comes to mind would be to curve the particle trajectory to a circle, so that itpasses the same accelerating section repetitively, as is shown in Fig. 1. The field required to bend the particle

1Mostly magnetic, because for perpendicular orientation of the fields, this is true: FE ∝ qE and FB ∝ qvB. In accelerators weare mostly dealing with particles with velocities close to the speed of light

2

Figure 1

trajectory is a static magnetic field. To ensure, that the electric field is non zero only between the plates, theymust extend to infinity. In reality this is not possible, and we have an electric field outside the capacitor. Thisfringe field will decelerate the particle when it approaches or departs the capacitor. As we anticipate accordingto Farraday’s law ∮

C

E ds = − ∂

∂t

∫S

B dS,

there will be no net acceleration, which is consistent with the conservative nature of the electrostatic field.

2.2 Acceleration by radio-frequency (RF) fields

With electromagnetic waves, accelerating voltages far exceeding those obtainable by static fields can be achieved.This method of acceleration is used in linear as well as in circular accelerators. For practical reasons, specificallyin circular accelerators, particle acceleration occurs in short straight accelerating sections placed along theparticle path.

Since a free electromagnetic wave does not have a longitudinal electrical field component, special boundaryconditions must be enforced. This is done by accelerating structures, called wave-guides or resonant cavities,providing a travelling or standing EM wave respectively. In a crude approximation, a waveguide is a pipe madeof conducting material and a cavity is a waveguide closed at both ends. Waveguides are used primarily in linearaccelerators (linacs), while resonant cavities are used in both linacs and synchrotrons. In both cases TM modesare used, because magnetic field cannot accelerate particles. The modes are found from Maxwell’s equations,with boundary conditions E‖ = 0, B⊥ = 0 at the cavity/waveguide walls. The two most commonly used modesare represented on Fig. 2.

Figure 2: A cylindrical waveguide operating in TM01 mode (left) and a pillbox cavity operating in TM010 mode (right)

Waveguides The most commonly used strategy is to excite a waveguide at a frequency above the cut-offfrequency of the lowest mode, TM01, but below the cut-off frequency for other modes. This way only one modeis propagating trough the waveguide. In order to be able to accelerate charged particles over a reasonabledistance, the wave must have the same phase velocity as the velocity of the particles. This way the particletravels along the structure with the wave and is accelerated or decelerated at a constant rate. Since phasevelocity 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 TM010. For this mode, the electric field is directedlongitudinally and has constant magnitude along z. It has no azimuthal dependence and has a maximum onthe 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 groupedtogether so that the ring only has one accelerating section. In both cases, the frequency of the voltage in theaccelerating structures must be an integer multiple of particle revolution frequency, where the integer is calledthe harmonic number and denoted by h. The harmonic number is the maximum number of bunches (groupsof particles moving together in the accelerator, more on this later in the text) that can be in the accelerator atthe same time.

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3 Equations of motion in phase space

To achieve acceleration, one must ensure constructive interaction of the particles with the wave. Because EMfields 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 meansthat the degree of acceleration is determined by the phase. If systematic acceleration is to be achieved, thisphase must be at a specific value at the moment the particle arrives to the accelerating section. This value iscalled synchronous phase and denoted by ψs. We assume that the ideal, synchronous particle arrives at eachstation 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

ωrf = hωrev, (1)

where h is the harmonic number. It represents the number of times, per particle revolution period, that the RFvoltage 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 andphase, and develop equations of motion that treat deviations from its trajectory.

3.1 Path length and momentum compaction

In the case of a circular2 accelerator we come to the problem of momentum-dependent path length. Thedependence arises in bending dipoles required to keep the particles on a circular path. When a charged particleenters a homogeneous magnetic field, the field 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 toa so called ideal particle, any particle with momentum different than the momentum of the ideal particle willnot follow the designed path.

We will denote the deviation of the particle from its ideal path by

x = D(s)∆p

p0,

where p0 represents the momentum of the ideal particle and D(s) represents the dispersion function. It describesthe effect that bending magnets have on the particles’ trajectory. We need not concern ourselves with moredetail, and can take it as a machine parameter. The total path length can now be written as

L =

∫ L0

0

(1 +

x(s)

ρ

)ds,

where ρ is the radius of the bend. We can immediately see that for an ideal particle with ∆p = 0 the pathlength is just L0. which is the ideal design circumference of the accelerator. The deviation from the ideal pathcan thus be obtained by integrating the second term.

∆L =∆p

p0

∫ L0

0

D(s)

ρ(s)ds,

To measure the variation of the path length with momentum, we define the momentum compaction factor

αc =∆L/L0

∆p/p0=

1

L0

∫D(s)

ρ(s)ds = 〈D(s)

ρ〉, (2)

The momentum compaction factor is non-zero only in curved sections, where ρ is finite. In the case of a LINACthe curvature (κ = 1/ρ) is 0, ρ =∞ and the momentum compaction factor vanishes.

However, when dealing with phase advance from one accelerating section to the other, we are not so muchinterested in the deviation of the path length, but would rather know the time it takes for a particle to travelbetween two successive accelerating sections separated by a distance L. This time is given by the equationτ = L/v. By differentiating the logarithmic version of the equation we get:

∆τ

τ=

∆L

L− ∆v

v.

2Path 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.

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The first term on the right is just momentum dependent path length that we derived earlier. To connect thesecond term to momentum deviation, wee need only to differentiate the momentum (p = mγv), and we get,after some manipulation:

∆v

v=

1

γ2

∆p

p,

where γ is the Lorentz factor. We can see that both αc and γ appear as a factor before momentum deviation.We can thus put them together to form the momentum compaction

ηc =1

γ2− αc, (3)

and get the final expression for time deviation

∆τ

τ= −ηc

∆p

p. (4)

Momentum compaction vanishes when

γt =1√αc

(5)

This is the transition Lorentz factor. From special relativity we know that

γ =Etotal

Erest,

and so γt is usually referred to as transition energy. Transition energy is very important when it comes to phasefocusing.

3.2 Difference equations

In order to derive the equations in longitudinal phase space, we take a look at phase and energy advance betweensuccessive passes through the RF cavities. The energy deviation and phase at the entrance to the (n + 1)th

cavity can be expressed as:ψn+1 = ψn + ωrf (τ + ∆τ)n+1 ,

= ψn + ωrfτn+1

(1 +

[∆τ

τ

]n+1

),

∆En+1 = ∆En + e [V (ψn)− V (ψs)] ,

(6)

where ψs is the synchronous phase and V (ψ) is the RF waveform. Since the synchronous particle always staysin phase and ωrfτ is the phase advance of the synchronous particle, we can rewrite the phase advance in theequation (6) as

φn+1 = φn + ωrfτn+1

[∆τ

τ

]n+1

. (7)

Here ωrfτ is the phase advance from cavity to cavity. For a circular accelerator with only one acceleratingsection, it must be an integer multiple of 2π (harmonic number), so that we satisfy the synchronicity condition

The final form of the equations can be obtained by using equation (4) and the relationship ∆E/E = β2∆p/pin (7)

φn+1 = φn − ωrfτηc∆pn+1

ps

∆En+1 = ∆En + e [V (φn)− V (φs)] ,

(8)

where ps is the momentum of the synchronous particle. Both equations are coupled. This can be seen if wereplace the energy deviation with momentum deviation in (8), by noting that β∆cp = ∆E.

∆pn+1 = ∆pn +e

βc[V (φn)− V (φs)] . (9)

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3.3 Differential equations

Typically the phase and energy change by small amounts at each pass of the accelerating section, which allowsus to treat them as continuous variables. We can then approximate the difference equations by differential ones,using n as the independent variable. We can rewrite equations (8) as:

dφ

dn= −ηcωrfτ

ps∆p,

d∆p

dn=

e

βc[V (φ)− V (φs)] .

(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 secondorder differential equation from equations in (10):

d2φ

dn2+ηcωrfτe

βcps[V (φ)− V (φs)] = 0. (11)

We can not go much further without making an assumption about V (φ). Since RF fields are created inaccelerating cavities, we will assume a sine function.

V = V0 sin(φ).

Let us now take a look at particle movement in phase space. We will take the simplest case and say that theonly way that the particle’s energy can change, is trough interaction with the RF field. If we rewrite the phaseas φ = ϕ+ φs and expand the trigonometric term, the equation of motion in phase space becomes:

ϕ+ηcωrfeV0

τβcps(sinφs cosϕ+ cosφs sinϕ− sinφs) = 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= τ

d

dt. (13)

4 Small oscillation amplitudes

To get some insight into the solutions and the stability of motion, we first take a look at small oscillationsabout the synchronous phase. Since ϕ is small we can approximate the sine and cosine term by their Taylorexpansions. Keeping only the linear terms in ϕ we obtain an equation for a harmonic oscillator:

ϕ+ Ω2ϕ = 0, (14)

where we have defined the synchrotron oscillation frequency as

Ω2 =ηcωrfeV0

τβcpscosφs. (15)

For real values of Ω we have a simple solution

ϕ = ϕ0 cos(Ωt+ χi), (16)

where χi is some general phase that we will set to zero. Since φ = φs + ϕ and φs is constant, it is true thatφ = ϕ we can construct an equation for momentum error, from (10,13)

δ =∆p

ps= − ϕ

ηcωRF=

Ωϕ0

ηcωRFsin(Ωt+ χi) (17)

If Ω is real, both phase and particle momentum oscillate about the ideal value with synchrotron frequency - wehave stable oscillations. If we join solutions for phase and momentum error, we get an invariant of motion(

ϕ

ϕ0

)2

±(δ

δ0

)2

= 1 (18)

It describes particle trajectories in phase space. They can be ellipses or hyperbolas, depending on Ω. Ellipsesrepresent stable motion in case of a real Ω and hyperbolas represent unstable motion, when Ω is imaginary.

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Figure 3: Synchrotron oscillations in (δ, ϕ) phase space for small deviations from the synchronous phase. Trajectories for real (left) andimaginary (right) values of synchrotron frequency

In Fig. 3 on the left graph for stable oscillations we can clearly see separatrices enclosing the area of stablemotion. In accelerator physics, areas where stable motion exists are called buckets. All the particles sharinga bucket are called a bunch. Maximum number of bunches that can be in a circular accelerator at the sametime 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 establishedthat in order to have stable oscillations, synchrotron frequency needs to be real. Taking a closer look at theexpression for Ω2 (15) we can see that besides cosφs and ηc all other quantities are non-negative. Thereforethese two will determine whether the motion is stable or unstable.

Momentum compaction, ηc, goes to zero (3) when particles cross the energy of

γt =1√αc. (19)

When this happens, the travel time from one accelerating gap to another does not depend on the particlemomentum. There is no phase stability at this energy. This is a machine dependent parameter. LINACs donot have a transition energy because they are straight.

Synchronous phase must be selected according to γ 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 < φ < π, hence

0 < φs <π

2for γ < γt,

π

2< φs < π for γ > γt.

For electrons the transition energy is in the range of MeV and for protons in in the range of GeV. Crossingthe transition presents us with many technical problems. Since γt for electrons is relatively small, electrons areinjected 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, soprotons are usually injected into the synchrotron below γt.

An oscillating accelerating voltage, together with a finite 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 effect that going over transition energy (5) has on phase focusing. The momentumcompaction ηc (3) changes sign when γ = γt:

ηc > 0 for γ < γt and ηc < 0 for γ > γt

Below the transition energy, the arrival time is determined by the particle’s velocity. After transition, theparticle has a velocity close to c and its arrival time depends more on the path length than on its speed. Thekey difference is this:

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Figure 4: Phase focusing principle. If the particle is lagging behind the synchronous particle it will see a higher accelerating voltage. Thiswill 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 accelerationsection faster than the ideal particle if we are below the transition. If we are above the transition,because ηc 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 canclearly see that in order to obtain focusing, slow particles must arrive later and faster particles sooner. If theparticles were to cross the γt 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, ϕ) phase space. There are two distinct cases,one where synchronous phase is set to 0, and the other when it is not. If φs = 0, the synchronous particle willexperience the voltage of V = V0 sinφs = 0 when it passes the accelerating section. We call this the stationarycase. For all values of φs that are not integer multiples of π, 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 nπ.

Particle accelerators consist of many elements. For example, a light source would consist of an injector linacthat would accelerate particles to Ei. After that, the particles are transferred to a booster synchrotron thataccelerates them from Ei to their final energy, Ef . When Ef is reached, the particles are again transferred toa storage ring. The function of the storage ring is to keep the particles orbiting at constant speed. Becauseof synchrotron radiation, electron storage rings also contain accelerating sections to make up for energy lostdue to synchrotron radiation. Since protons are much heavier (factor of 103 eV) and the energy lost per turnfor synchrotron radiation scales as γ4/ρ effect of synchrotron radiation on their energy is negligible. Still RFsections are needed to provide phase focusing.

By using difference equations (8) we can make simulations to take a look at particle motion in stationaryand accelerating case. The following simulations were done for the Fermilab Tevatron ring, which is a protonaccelerator. On the left graph in Fig. 5 we can see an example of a stationary bucket. Particles with φ = φs

Figure 5: Phase space plot for a stationary (left) and an accelerating bucket (right).

are not accelerated. The phase stable region is 2π in extent and particles that find themselves out of the bucketwill undulate in energy and diverge in phase. They may stay in the ring indefinitely.

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Comparing the phase space plot for an accelerating bucket to the one for the stationary bucket on Fig.5, we can see a significant change in the shape of the separatrix as well as its origin. The center of the bucket isno longer at (0, 0) but is shifted by φs. Particles that find themselves out of the bucket will diverge in phase aswell as in energy. In contrast to the stationary bucket case, these particles will eventually gain too much energyand leave the circular accelerator.

The graphs in Fig. 5 were plotted for a proton energy below the transition energy of the accelerator, γ < γt.For γ > γt 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 =ϕ2

2− Ω2

cosφs[cos(φs + ϕ)− cosφs + ϕ sinφs] . (20)

For the stationary case, Hamiltonian (20) simplifies and is identical to that of a mechanic pendulum:

H =ϕ2

2− Ω2 cosϕ. (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 canbe seen from equations (16,17). Within the stable regions, we can find two fixed points, one stable and oneunstable. They can be calculated from

∂H

∂ϕ= 0,

∂H

∂ϕ= 0. (22)

The two fixed points correspond to minima (sfp - stable fixed point) and saddles (ufp - unstable fixed point) inthe potential represented on Fig. 7. From conditions (22) we obtain coordinates for fixed points in (ϕ, ϕ). sfpis located at (φs, 0) and ufp at (π − φs, 0).

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Figure 8: Characteristic bucket and separatrix parameters

5.1.2 Momentum Acceptance

ϕ is proportional to ∆p/p0. Maximum momentum acceptance can thus be found by differentiation of thehamiltonian, (20), with respect to ϕ. At the extreme points, where the momentum reaches a minimum or amaximum, we have ∂ϕ/∂ϕ = 0 and the contition is ϕ = 0. From Fig. 8 we see that the maximum phaseelongation occurs at the ufp where ϕ is zero. Maximum momentum acceptance can then be found by equatingthe values of the hamiltonian for the ufp and the derived condition. We get:

1

2ϕ2 = 2Ω2

[1−

(π2− φs

)tanφs

]. (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 Hamiltonianequations. Following the Liouville theorem that states that the area in phase space is conserved for a systemthat can be described by Hamiltonian equations (this follows directly from Hamilton equations, since ∇v = 0),we define 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 abunch. The derivations are valid for a distribution where the emittance is much smaller than the bucket area, sothat effects due to non-linearities of the RF focusing and large tails can be ignored, and where the distributionof the bunch is assumed to be Gaussian.

We start by evaluating (20) for small angles and obtain

ϕ2 + Ω2ϕ2 = 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

ϕ = ωrf∆t,

and from (10) we obtain

ϕ =1

τ

dϕ

dn=

∆p

p0ηcωrf =

1

β2

∆E

Eηcωrf.

We are left with the equation for particle trajectory in (∆E,∆t) phase space

(∆E)2

+β2EseV0ω

2rf cosφs

2πhηc(∆t)

2= const. (24)

which is the equation of an ellipse. To evaluate the constant, we need to find the trajectory, that encloses95% of the particles. Since we have assumed a Gaussian distribution, the radius that corresponds to this isapproximately

√6σ. Our constant is than just 6σ2

E , where σE is the rms of energy deviation. We can rewritethe 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 = πAB. The area of this ellipse is thencalled 95% longitudinal emittance, denoted by εl, and has units of eVs.

εl =π

ωRF

√2πhηcE

3s

β2eV0 cosφs

(σEEs

)2

. (25)

The emittance can be compactly written as a product of the momentum spread and the bunch length (σt) as

εl = 6πσEσt = 6πβcσpσt. (26)

Given the emittance and the RF parameters, we can now express the momentum spread and bunch length andsee that they scale as

σp ∝ 4√V0, (27)

σt ∝ 4

√1

V0.

5.1.4 Acceptance

We must distinguish between acceptance and emittance. The acceptance is associated with the bucket (availablestable phase space area) whereas the emittance is associated with the bunch (actual phase space occupied bythe beam). The acceptance is the maximum allowed value of emittance and is determined by the design of thetransport or accelerating lines.

Emitance can be obtained from the hamiltonian (20). We know that the total energy of the system is aconserved quantity. To evaluate the constant, we use the ufp, because we know that ϕ is zero at these locations.To get the acceptance we need to integrate

A =

∫S

∆E

ωRFdϕ, (28)

where the integral must be taken over the separatrix. The integral can be solved analytically only for stationarybuckets with φs = 0, π. In this case we get the stationary acceptance, denoted by εsta:

εsta = 8

√2eV0E0β

2

πh|ηc|ω2rf

(29)

For other values of φs the integral can be solved numerically. From Fig. 9 we can see how acceptance varieswith φs, with the largest value for the stationary bucket.

Figure 9: The emittance for a moving bucket εmov with respect to the emittance for a stationary bucket εsta

5.2 Acceleration

We will now take a closer look at the particle motion during acceleration. In a synchrotron, particles are injectedat some initial energy Ei. They are then slowly, through a number of turns, accelerated to their final energy,Ef . Because the acceleration should be adiabatic, synchronous phase is slowly increased. The simulation tracksa single particle and varies the synchronous phase from 0 to its final (arbitrarily chosen) value of π/6. Bychanging the time (number of turns) it takes the particle to reach the final acceleration, we can observe theinfluence of non-adiabatic effects. In Fig. 10 we can see how fast changes of phase affect the longitudinalemittance. The top left graph represents adiabatic acceleration and we can clearly see how trajectory followsφs. We can tell that the acceleration is adiabatic, since the area of the particle’s phase space ellipse remainsessentially constant. If we change the number of turns it takes for the particle to reach the acceleration at φs

to a smaller value, the process becomes non-adiabatic and emittance is not preserved. The smaller the number,the bigger the final emittance. In general, if the motion is to be adiabatic, the system parameters must changemore slowly than the period of motion.

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Figure 10: Trajectories for a particle underging acceleration from φs = 0 to φs =π

6. Number of turns is 3000, number of turns to reach

final acceleration φs is ??,500,100,25

5.3 Effect of RF Voltage on Phase Space

As can be seen from Eqs. (27), (28), (23), the RF voltage has an effect on the bunch length and the momentumacceptance of the accelerator. Even for a stationary bucket we can observe that with V0 too high, the wholephase 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. TheRF systems of both machines are phase locked and bunches are transferred directly from the bucket of onemachine to the other. If we want the longitudinal emittance to stay the same, the bunch must be centred inthe bucket of the final machine and both machines must be longitudinally matched, meaning that they havethe sameLongitudinal Twiss Parameter, βL.

To obtain the βL we rewrite the equation for particle trajectories in (∆E,∆p) phase space (24) as(∆E

A

)2

+

(∆t

B

)2

= 1 −→ ∆E2B

A+ ∆t2

A

B= AB,

Since we previously defined the emittance as πAB we obtain the final expression for phase-space trajectories:

βL (∆E)2

+1

βL(∆t)

2=εlπ

(30)

Figs. 12, 13, 14 track the motion of a bunch for a 100 turns after transfer. They describe the motion afterthe transfer for a matched case, a case when the transfer occurs with a phase error of π/3, and a transfer witha βL error of factor three.

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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 π/6

Figure 14: First hundred turns after transferring the beam from linac to the synchrotron for a transfer with βL mismatch by a factor ofthree

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 RFvoltage. From (27) we can see that the momentum spread scales as 4

√V0 and the bunch length as 1/ 4

√V0. Since

the longitudinal emittance is preserved when motion is adiabatic, we can see that the shape of the bunch canbe manipulated by changing the RF voltage. The change in voltage always results in one of the δ or ϕ spreadsgetting larger and the other one getting smaller - there is no way to shrink them both, which is consistent withemittance preservation.

Debunching We adiabatically reduce the voltage over many synchrotron periodes, untill finally it is turnedoff. 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 isdebunched, no RF voltage is applied to it. This is why it is not possible to debunch an electron beam at anysignificant energy. Because of synchrotron radiation, electron beams always need to be accelerated in order to

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compensate for radiation losses.

Bunch Length Manipulation The figures from the section on mismatched transfer give a hint that a bunchshape could be manipulated. We can see that the bunch rotates if mismatches occur - these mismatches are aconsequence of phase/momentum offset or different RF parameters (meaning different buckets). We are moreinterested in the latter, since the RF parameters are something that we can change while the particles arecircling in the accelerator. Let us assume that we have a bunch with a small momentum spread and long bunchlength. To change the bunch to a short one, we would increase the RF voltage in the time that is short comparedto the synchrotron oscillation period. Since the bucket has changed, the bunch is essentially mismatched andstarts 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 first plot, the bunch has its original shape. Itthan 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 narrowestpoint and, unless the RF voltage is increased again, it will continue to rotate and start getting longer. Thebunch rotates because its boundary does not coincide with a phase space trajectory, so the second time the RFvoltage is ramped up, it must increase to such a value that the particles at the edge of the bunch will followthe same phase space trajectory. From (17) we get the relation

δ0 = | Ω

ωrfηc|ϕ0. (31)

To get the overall bunch reduction factor, we proceed as follows. Before the rotation, the relation betweenmomentum and phase deviation is

δ1 = | Ω1

ωrfηc|ϕ1,

where Ω1 denotes the synchrotron frequency with RF voltage V1. After rotating, the phase deviation ϕ1 istransformed into momentum deviation

δ2 = | Ω2

ωrfηc|ϕ1,

and the original momentum deviation is transformed into phase deviation

ϕ2 = |ωrfηc

Ω2|δ1.

We now need to stop rotation. This is achieved if the new momentum error and the new phase error are on thesame phase space trajectory. The required RF voltage can be obtained from

δ2 = | Ω3

ωrfηc|ϕ2.

To get the ratio of the bunch lengths we take the quotient

ϕ2Ω3

ϕ1Ω1=δ2δ1

=Ω2ϕ1

Ω2ϕ2

Since l ∝ ϕ0 and Ω ∝√V we get the overall bunch length reduction factor for this process:

l1l0

= 4

√V1

V3, (32)

where V1 is the initial and V3 final RF voltage.

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The bunch length manipulation described in this section is applicable only to non-radiating particles. Forparticles that radiate bunch manipulation is easier due to damping effects. Relation (31) still holds, but themomentum spread is independently determined by synchrotron radiation and the bunch length scales propor-tionally to

√V .

6 Conclusion

The first idea to use RF cavities instead of static fields was to achieve higher energies. Throughout the seminarthe stability of motion of particles interacting with such fields was inspected and it was shown that besideshigher accelerating voltages, oscillating fields also provide an additional stabilizing force, that results in phasefocusing. The result of oscillating fields are bunched beams, within which the particles oscillate about the idealvalues 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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