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Royal Institute of Technology (KTH)

Master of Science Thesis

Adiabatic Capture

Heavy Ion Beams in RF Buckets of

Author: Supervisor:

Katarina Samuelsson Prof. Lars Blomberg

Assoc. Prof. Em. Staan Rosander Dr. Oliver Boine-Frankenheim

January 29, 2009

Royal Institute of Technology Report XR-EE-SPP-2009:001

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Abstract

When accelerating particles in a synchrotron a bunched beam is needed, which means that the particles have to be divided into groups, bunches. To form a bunched beam, the voltage over the cavity, one of the structures in a synchrotron, is raised. The voltage can be raised as a step or as a ramp. The ramps can of course have dierent shapes and duration. If one raises the the voltage as a step the enlargement of the beam area will be very large. If instead one raises the voltage with a ramp the area enlargement will be smaller. In this work the eects of an iso-adiabatic ramp and of a linear ramp on the beam will be studied for both single and dual harmonic.

From the work it can be concluded that the dierence between the dierent ramps is small and the ramp time has a small inuence for medium heavy particles if the initial voltage over the cavity is non-zero.

Zusammenfassung

Für die Beschleunigung von Teilchen in einen Synchrotron wird ein gebuncht Strahl benötigt, bedeutend dass die Teilchen in kleine Gruppen eingeteilt werden, Bunches.

Um ein gebuncht Strahl zu erzeugen, wird die Spannung über die Kavität, eines be- deutenden Teil des Synchrotrons, erhöht. Die Spannung kann als einen Sprung oder eine Rampe erhöht. Die Rampe kann selbstverständlich unterschiedliche Formen und Laufzeit haben. Ob die Spannung mit einem Sprung erhöht wird, ist die Ver- gröÿerung der Strahläche sehr groÿ. Aber wird stattdessen die Spannung mit eine Rampe erhöht wird die Vergröÿerung der Strahläche geringer. In dieser Arbeit wird die Wirkungen von einer iso-adiabatic Rampe und eine lineare Rampe auf die Strahl untersucht für beide Single und Dual harmonic. Ergebnis von die Arbeit ist das die unterschied zwischen die Rampen ist klein und die Rampzeit hat eine kleinen Einwirkung auf mittle schwer Teilchen ob die Anfangsspannung über die Kavität ungleich Null ist.

Sammanfattning

För att accelerera partiklar i en synkrotron behövs en bunchad stråle, vilket betyder att partiklarna måste delas i grupper, bunchar. För att forma en bunchad stråle höjs spänningen över kaviteten, en av beståndsdelarna i en synkrotron. Spänningen kan höjas som ett steg eller som en ramp. Ramperna kan naturligtvis ha olika form och stigtid. Om spänningen höjs som ett steg blir förstoringen av strålarean mycket stor.

Om spänningen istället höjs med en ramp kommer förstoringen av strålarean att bli i

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ii

mindre. I detta arbete kommer eekterna av en iso-adiabatisk ramp och en linjär ramp på strålen att studeras för både "single harmonic" och "dual harmonic". Från arbetet kan slutledas att skillnaden mellan olika ramper är liten och att ramptiden har en liten betydelse för medeltunga partiklar om initialspänningen över kaviteten är nollskild.

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Acknowledgment

I would like to thank Dr. Oliver Boine-Frankenheim at GSI, for the opportunity to write my Master thesis at GSI and for his help and support throughout the work. I would also like to thank Dr. Oleksandr Chorniy for his help and support throughout my work. I would also like to thank the HF Gruppe at GSI for their support during the experiment. I would also like to thank the entire FAIR Accelerator Theory group for making me feel welcome into the group during my stay at GSI.

I would also like to thank Associate Professor Em. Staan Rosander at KTH for his willingness to help with knowledge and time. I would also like to thank Professor Lars Blomberg at KTH for making it possible for me to write my Master thesis at GSI and his help and support throughout the work.

Finally, I would like to thank my family for their love and support throughout the years.

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Contents

1 Introduction 1

2 GSI and SIS-18 3

2.1 SIS-18 . . . 4

2.1.1 Beam parameters . . . 5

2.2 Cavity . . . 5

3 Theory 7 3.1 Particle parameters . . . 7

3.2 Motion in rf elds . . . 7

3.2.1 Denitions . . . 7

3.2.2 Synchrotron frequency . . . 9

3.3 Rf capture . . . 12

3.3.1 Adiabatic voltage ramp . . . 12

3.3.2 Energy amplitude, bunch area and length . . . 13

3.3.3 Dual harmonic rf bucket . . . 14

Adiabatic voltage ramp . . . 14

Bunch area . . . 16

4 Simulation 19 4.1 Single harmonic rf buckets . . . 19

4.1.1 Results . . . 20

4.2 Dual harmonic rf buckets . . . 21

4.2.1 Results . . . 21

4.3 Simulations with experiment parameters . . . 22

4.3.1 Single harmonic . . . 22

4.3.2 Dual harmonic . . . 23

5 Experiment 25 5.1 Experimental setup . . . 25

5.2 Experimental evaluation . . . 26

5.3 Results . . . 27

5.3.1 Single harmonic . . . 27

5.3.2 Dual harmonic . . . 29

6 Discussion 31

v

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vi CONTENTS

7 Summary and Conclusions 33

7.1 Summary . . . 33 7.2 Conclusion . . . 33 7.3 Future work . . . 33

A 37

B 39

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Chapter 1 Introduction

GSI, Gesellschaft für SchwerIonenforschung, is a research center for nucelar physics.

For the experiments, accelerators are used to produce ions of dierent energy. At GSI all elements up to uranium can be accelerated. In the future, the facility will be extended with the international facility FAIR, Facility for Antiprotons and Ion Research. One of the accelerators at GSI is the heavy ion synchrotron SIS-18, SchwerIonenSynchrotron. When the FAIR accelerator SIS-100 is built, SIS-18 will function as a pre-accelerator to SIS-100. Since the design goal is that all ions up to uranium shall be accelerated also in SIS-100, uranium is used a reference ion during the design process.

The accelerating part of a synchrotron is a cavity. A voltage is applied across the cavity which aects the particles in the beam passing the cavity. For acceleration to occur the particles passing the cavity have to pass when the voltage is positive.

For this to happen the particle beam is divided into bunches, which is accomplished by raising the voltage over the cavity. The voltage can be raised as a step or as a ramp. The ramps can of course have dierent shapes and duration.

Our goal is to examine how the ramping time in SIS-18 aects the bunch quality.

Therefore simulations are done for uranium since it is the reference ion, but these simulation results can not be veried for practical reasons. To be able to verify the simulations with experiments simulations are done also with argon. In this work we look at how the ramping time aects the eciency of the bunching for both single harmonic voltage forms and dual harmonic voltage forms. We also look at how the unavoidable initial voltage of the ramp inuences the bunch quality.

The work has the following outline. In chapter 2 a short presentation of GSI and the SIS-18 is made. In chapter 3 some important theoretical aspects are treated.

Chapter 4 contains the simulation results. In chapter 5 the experimental work done is presented together with the comparison with simulation results. The last two chapters include discussion, summary and conclusions.

1

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2 CHAPTER 1. INTRODUCTION

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Chapter 2

GSI and SIS-18

GSI, Gesellschaft für SchwerIonenforschung, is an accelerator facility for heavy ions.

Here all elements up to uranium can be accelerated. The beam ions can have any charge state and can be accelerated almost to the speed of light. (One also has the possibility to create and store beams of radioactive nuclei at the facility.) Figure 2.1 shows a schematic picture of the GSI accelerator facility as of today [1].

Figure 2.1: GSI accelerator facility

UNILAC - is a 120 m long linear accelerator. It is used as a pre-accelerator for SIS-18 and accelerates the ion beam to an energy of 11.4 MeV/u which corresponds to ≈ 15% of the speed of light.

SIS-18 - is a heavy ion synchrotron with a circumference of 216 m, which accelerates the beam to the speed of light for lighter ions and for uranium to about 90% of the speed of light.

ESR - is an experimental storage ring where accelerated ions can be stored before they are used in experiments. During the storage the ions will complete millions of revolutions at a constant energy [1].

In the future the facility of GSI will be extended with the international accelerator facility FAIR , Facility for Antiprotons and Ion Research. The SIS-18 will then work

3

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4 CHAPTER 2. GSI AND SIS-18 as a pre-accelerator for the FAIR accelerator SIS-100. Within the FAIR project also a proton linear accelerator will be built. By building FAIR new elds of research will be possible; among the elds are high energy physics and quantum chromo-dynamics studies with anti-proton beams, nuclear structure and nuclear astrophysics with nuclei far o the stability domain. Extended capabilities for high density plasma physics, atomic physics and applied research will also be available. The design of the new facility will be such that up to four research programs can run in parallel [2].

2.1 SIS-18

The SIS-18 consists of 24 bending dipole magnets. For focusing there are 24 hori- zontal focusing quadrupole magnets and 12 vertical focusing quadruplole magnets.

The horizontally focusing magnets are de-focusing in the vertical direction and vice versa for vertical focusing magnets. The behavior in the horizontal direction decides if the magnet is called a focusing or de-focusing magnet. The SIS-18 is divided in 12 periods, with each period consisting of two bending magnets and one quadrupole focusing structure. The focusing structure used is a triplet structure, which consists of two focusing magnets and one de-focusing magnet [2].

Figure 2.2: Schematic picture of SIS-18

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2.2. CAVITY 5

2.1.1 Beam parameters

SIS-18 has a circumference, L, of 216 m and a gamma transition, γt, of 5.4, which is a constant that is specic for each accelerator and is dependent on the circumference and transverse focusing lattice of the accelerator. The beam parameters of SIS-18 are given in tables 2.1 and 2.2.

SIS-18 presently SIS-18 after upgrade

Reference ion U+73 U+73

Repetition rate 0.3 Hz 1 Hz

Maximum intensity < 7 · 109 2 · 1010

Maximum energy 1GeV/u 1GeV/u

Table 2.1: Beam parameters of SIS-18 as of today and requirements for FAIR.

Particle energy Number of ions per cycle U: 50-1400 MeV < 7 · 109

Ne: 50-2000 MeV 5 · 1010

p: 4.5 GeV 2 · 1010

Table 2.2: Energies achieved in SIS-18.

2.2 Cavity

In SIS-18 there are two identical rf cavities which are designed for an accelerating voltage amplitude of 16 kV and a working frequency range of 0.8 − 5.6 MHz. In

gure 2.3 the design principle can be seen. Each cavity consists of two coaxial quarterwave-resonators, which operate on a common accelerating gap. In the space between the inner and outer conductor of the resonator a ferrite material is used as

lling. To be able to vary the resonance frequency of the cavity, the permeability of the ferrite material is changed. This change is done by superimposing a dc bias

eld, which is realized by 6 high current windings in a gure-of-eight conguration.

This conguration compensates for the rf voltage induced in the windings by the

eld [3].

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6 CHAPTER 2. GSI AND SIS-18

Figure 2.3: Design principle of a cavity

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Chapter 3 Theory

3.1 Particle parameters

The total energy ,Es, of a particle, the energy at rest ,E0, its kinetic energy Ekin = Es− E0, the Lorentz factor ,γ, and the relativistic parameter ,β = v/c, are related by:

Es = γm0c2 (3.1)

E0 = m0c2 (3.2)

Ekin = Es− E0 = γm0c2− m0c2 = m0c2(γ − 1) (3.3) γ = 1 + Ekin

m0c2 (3.4)

β = s

γ2− 1

γ2 (3.5)

where m0is the mass at rest and c is the speed of light. Then at the injection energy of SIS-18 γ = 1.01 and β = 0.14.

The maximum energy deviation, Em, can be calculated from the ideal particle in a coasting beam with the knowledge of the momentum spread, ∆p/p. Since Es= γm0c2 and p = γm0v, it can be shown that the momentum spread is given by

∆p p = c2

v2

∆E Es = 1

β2

∆E

Es (3.6)

from which we can derive Em as

Em = ∆E = ∆p

p β2Es. (3.7)

3.2 Motion in rf elds

3.2.1 Denitions

Some important concepts within accelerator physics that are relevant to the present work are explained below. The coordinate system considered is always in phase

7

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8 CHAPTER 3. THEORY space (∆E, φ). ∆E is the energy deviation of a particle in relation to an ideal particle. φ is the angular dierence in position between a particle and the ideal particle in relation to the accelerating station, which is illustrated in gure 3.1. φm

is referred to as the bunch length and is obtained as

φm = h

R(s − s0) (3.8)

where s is the position along the orbit of a particle, s0 is position of the ideal particle, h is the harmonic number of the cavity and R is the radius of the synchrotron. In

s s0 Accelerating

station

Φ

Figure 3.1: Graphical explanation of φ

longitudinal phase space there is a boundary between stable and unstable motion of the particles in the beam. This boundary is called separatrix. The area inside the separatirix, where the particles preform a stable motion, is referred to as the bucket.

The particles that share a particular bucket are called a bunch. These terms are graphically explained in gure 3.2.

DE

Φ Bucket

Bunch

Φm Π

Em

Figure 3.2: Denition of bucket and bunch

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3.2. MOTION IN RF FIELDS 9

3.2.2 Synchrotron frequency

When accelerating particles to high energies, the particles have to pass through sev- eral high frequency cavities, where the accelerating elds are produced. A "station"

is a single cavity or a system of cavities. In a linear accelerator the particles travel along a straight line, thus there are several stations placed in a line along which the particles travel. In a synchrotron the particles travel in a circle, thus one only needs one station which the particles travel through several times. A schematic picture can be seen in gure 3.3. Suppose that every station is excited by a source of radio frequency power at angular frequency ωrf. Further, it is assumed, that the ideal particle always arrives at the same phase and receives the same amount of energy at every station. The non-ideal particles will deviate from the path of the ideal particle, and it is necessary to describe these deviations.

Accelerating station

Figure 3.3: Layout of a synchrotron and linear accelerator

If the distance between two stations is L and the velocity of the particle is v then the time, τ, needed for a particle to pass between the stations is

τ = L

v. (3.9)

A particle moving faster than the ideal particle will need less time to pass between two stations, under the assumption that the path length is constant. A particle with a longer path length than the ideal particle will need more time to pass between the two stations. So the fractional change in τ dependent on deviations in L or v is given by

∆τ

τ = ∆L L −∆v

v . (3.10)

The second term in (3.10) can be expressed using a fractional momentum deviation

∆v v = 1

γ2 µ∆p

p

. (3.11)

Also the rst term of (3.10) can be dependent on the fractional momentum deviation.

In a synchrotron one can expect that the circumference is larger for a particle of a

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10 CHAPTER 3. THEORY momentum higher than that the ideal particle. So the rst term can be expressed

as ∆L

L = 1 γt2

µ∆p p

(3.12) where γt is a synchrotron dependent constant. For further simplication one can introduce the slip factor η as

η = 1 γt2 1

γ2. (3.13)

The fractional change (3.10) can then be written as

∆τ

τ = η∆p

p . (3.14)

Suppose a particle of arbitrary phase and energy arrives at the entrance of the nth cavity with energy En and phase φn. At the entrance to the (n + 1)th cavity the phase would be

φn+1 = φn+ ωrfτn+1 µ∆τ

τ

n+1

= φn+ ηωrfτn+1

µ∆p p

n+1

(3.15) which is illustrated in gure 3.4.

Station n Station n+1

En Φn

En+1 Φn+1

Figure 3.4: Particle entrance of station n and n + 1

For simplicity it can be assumed that ωrfτn+1 is independent of n and hence the subscript can be dropped. For a circular accelerator the product ωrfτ is an integral multiple of 2π. This multiple is called the harmonic number, h, and is given by

h = ωrfτ

(3.16)

If the energy of the ideal synchronous particle, with charge q, at the entrance of the nth cavity is Es then

(Es)n+1 = (Es)n+ qV sin φs (3.17) where V is the eective voltage seen by the particle in the cavity and φs, the syn- chronous phase, is the phase of the ideal particle on arrival at the nth cavity. For an arbitrary particle the corresponding equation is

En+1 = En+ qV sin φn (3.18)

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3.2. MOTION IN RF FIELDS 11 The dierence in energy between the arbitrary particle and the ideal particle, ∆E = E − Es, must satisfy (3.19). One can rewrite (3.15) as (3.20) using (3.16) and (3.6).

(3.19) and (3.20) are referred to as the two dierence equations of motion for a particle with respect to the ideal particle.

∆En+1 = ∆En+ qV (sin φn− sin φs) (3.19)

φn+1= φn+2πhηc2

v2Es ∆En+1 (3.20)

To be able to study the dierence equations analytically they may be approxi- mated by dierential equations, since the change of phase and energy is small be- tween successive turns. The turn number n can thus be related to the independent variable time t = nτ and the dierence equations can be expressed as

d∆E dt = qV

τ (sin φ − sin φs) (3.21)

dt = 2πhηc2 τ v2Es

∆E (3.22)

These two rst order dierential equations can be rewritten as a single second order dierential equation:

d2φ

dt2 = 2πhηqV c2

τ2v2Es (sin φ − sin φs) . (3.23) By multiplying with dφ/dt and then integrating over t, a rst integral is given by

1 2

µ dt

2

+2πhηqV c2

τ2v2Es (cos φ + φ sin φs) = constant. (3.24) An expression for the contour describing particle motion in phase space is obtained by combining (3.22) and (3.24). The resulting expression is

∆E2+ v2EsqV

πηhc2 (cos φ + φ sin φs) = constant (3.25) illustrated for a stationary bunch in gure 3.5.

To obtain a simple expression for the frequency of phase oscillations it is of useful to linearize the equation of motion, (3.23). To do this one needs to assume that

∆φ = φ − φs is small. With this assumption the expression in parentheses on the right hand side of (3.21) can be approximated as

sin(φs+ ∆φ) − sin φs ≈ cos φs∆φ (3.26) and then equation of motion can be rewritten as

d2∆φ

dt2 = 2πhηqV c2cos φs

τ2v2Es ∆φ. (3.27)

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12 CHAPTER 3. THEORY

-150 -100 -50 0 50 100 150

Φ

-1.0 -0.5 0.5 1.0

DE

Figure 3.5: Location of a stationary bunch in the bucket (γ < γt) which, by introducing the angular synchrotron frequency, ωs,

ωs= s

−2πηhc2qV cos φs

τ2v2Es (3.28)

can be written as

d2∆φ

dt2 = −ωs2∆φ. (3.29)

The synchrotron period Ts is dened as Ts =

ωs (3.30)

The synchrotron tune is dened as the number of synchrotron oscillations per turn and is given by

νs= ωs

ω0 (3.31)

where ω0 is the revolution frequency given by ω0 = 2π/τ [4].

3.3 Rf capture

3.3.1 Adiabatic voltage ramp

When particles are formed into a bunch it is said that they are capture. For a capture to be adiabatic it has to occur slowly. This means that the ramping time T, the amount of time used to change the voltage over the cavity, has to be longer than the synchrotron period Ts. The adiabaticity parameter µ can then be expressed as

µ = Ts

T ¿ 1 (3.32)

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3.3. RF CAPTURE 13 which has to be much smaller than 1 for an adiabatic capture to occur. A so called adiabatic voltage ramp can be constructed from the condition as

1 V0

dV0 dt = 1

Ts

. (3.33)

For the angular synchrotron frequency, (3.28), the adiabatic voltage ramp can be expressed as

1 V0

dV0

dt = αp

V0 (3.34)

where α is a constant. Integrating (3.34) using the initial conditions

V0(t = 0) = Vi (3.35)

V0(t = T ) = Vf (3.36)

where Vi and Vf respectively are the initial and nal rf voltage amplitudes in the cavity, gives

V0 = Vi µ

1 −Tt

Vf Vi

Vf

2 (3.37)

This equation is the so-called iso-adiabatic voltage ramp for a single rf bucket. This ramp is only valid for short bunches when single harmonic are used [5]. In gure 3.6 the iso-adiabatic voltage ramp for the single harmonic bucket and a linear ramp can be seen. The initial and nal voltage used in the plot is Vi = 100 V and Vf = 15 kV respectively and the ramping time T = 40 ms.

0.00 0.01 0.02 0.03 0.04

0 2000 4000 6000 8000 10 000 12 000 14 000

t @sD

U@VD

Voltage ramp, single harmonic

linear adiabatic

Figure 3.6: Adiabatic and linear voltage ramp for the single harmonic bucket.

3.3.2 Energy amplitude, bunch area and length

To be able to calculate the coasting beam area one needs to know the bunch length, φm, and the maximum energy deviation, Em, at the beginning. The maximum

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14 CHAPTER 3. THEORY energy deviation can be calculated from the momentum spread, ∆p/p, using (3.7).

The harmonic number of a synchrotron is equal to the number of buckets available, but does not mean that all buckets are occupied. To be able to compare dierent machines one bucket is considered to occupy the entire ring. Thus the total length of a beam is 2π. A coordinate system is introduce with the zero point at the midpoint of the beam and the bunch length will be π. The momentum spread can be chosen as desired. The coasting beam area Acoast is then obtained from

Acoast= 2π2Em (3.38)

After bunching the bunch will have an elliptic form, with the half axes Em and φm

according to gure 3.2. The contour is given by equation

∆E2 Em2 + φ2

φ2m = 1 (3.39)

and the area of the ellipse, thus the bunch area Abunch for the single harmonic case, can be shown to be as

Abunch = πEmφm. (3.40)

3.3.3 Dual harmonic rf bucket

So far only a single harmonic bucket has been used. Now we also look at the case of dual harmonic buckets. Here not one but two cavities are used and because of that everything is more complicated when it comes to calculations, because of nonlinear motion, although the principle is the same. With a single harmonic bucket the voltage over the cavity gap is varying sinusoidally. With a dual harmonic bucket the voltage is a combination of two sine voltages with dierent amplitude and phase.

The most advantageous combination is described by

V (φ) = sin (φ) − 0.5 sin (2φ) (3.41) and this combination is only valid for a stationary bucket, which means that no acceleration occurs.

If one adds a higher harmonic rf voltage to the main voltage, the resulting voltage is attened according to [7], p. 299. The dual rf system give a larger bunching factor which means that the peak current of the beam is reduced.

Adiabatic voltage ramp

According to the work done in [8] it can be shown that the synchrotron frequency is given by

ωs = ωs0 π

2

sinφ2 K

·q

1 2

¡1 + sin2 φ2¢¸ (3.42)

where K (x) is the complete elliptic integral of the rst kind, and can be approxi- mated to

ωs = ωs0 π 23/2

φ K(1/√

2) (3.43)

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3.3. RF CAPTURE 15 if small amplitudes, φ ¿ φcrit, are assumed. φcrit is located at approximately 117. In gure 3.7 one can see the synchrotron frequency for the single and dual case along with the critical frequency. The voltage, Yrf, is dened as

0 50 100 150

0.0 0.2 0.4 0.6 0.8 1.0

Φ @degD sΩs0

Φcrit

Double Single

Figure 3.7: Synchrotron frequency for single and dual harmonic, the vertical line is the critical amplitude for the dual harmonic case.

Yrf = 1 V0

Z

V (φ) dφ. (3.44)

If φ ¿ φcrit one can approximate the voltage to Yrf = φ4m

8 (3.45)

and the bunch area according to

AB = 2 Z φm

0

Em(φ)dφ = 2ωs0 Z φm

0

q

Yrf(φ) − Yrfm)dφ ≈ 2Emφm. (3.46) From [8] one can show that the synchrotron frequency at zero amplitude follows

ωs02 = Em2

2Yrfm) (3.47)

which can be rewritten using (3.44) and (3.46) to ωs02 = A2B

φ6m. (3.48)

If small amplitudes are assumed then (3.43) can be used, and expressed as

ωs(V0, AB) = πω2/3s0 A1/3B 23/2K(1/√

2) (3.49)

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16 CHAPTER 3. THEORY Then the adiabatic voltage ramp for dual harmonic can be constructed as

1 V0

dV0

dt = ωs(V0, AB) = αV02/3 (3.50) where α is a constant. Rewriting and integrating on both sides will lead to

V0 = µ

3

2(αt + C)

3/2

(3.51) where C is a constant. Using initial conditions (3.35) and (3.36) one can derive the adiabatic voltage ramp for dual harmonic to

V0 = Vi

µ

1 − Tt V

2/3 f −V i2/3

Vf2/3

3/2. (3.52)

In gure 3.8 the adiabatic voltage ramp for the dual harmonic case and a linear ramp can be seen. The initial and nal voltage used in the plot are Vi = 100 V and Vf = 15 kV, respectively, and the ramping time T = 40 ms. In gure 3.9 a

0.00 0.01 0.02 0.03 0.04

0 2000 4000 6000 8000 10 000 12 000 14 000

t @sD

U@VD

Voltage ramp, double harmonic

linear adiabatic

Figure 3.8: Voltage Ramp, dual harmonic

comparison between the adiabatic ramp in the single harmonic case and the dual harmonic case has been performed. The initial and nal voltage used in the plot are Vi = 100 V and Vf = 15 kV, respectively, and the ramping time is T = 40 ms.

Bunch area

The coasting beam area Acoast, see equation (3.38), is the same as for the single harmonic case, since a coasting beam is used at the beginning in both cases. After bunching the dual harmonic bunch form is not an ellipse as in the case of the single harmonic. Instead the bunch form can be assumed to be of the form of a attened ellipse. The contour of such a bunch is given by

∆E2 Em2 + φ4

φ4m = 1 (3.53)

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3.3. RF CAPTURE 17

0.000 0.01 0.02 0.03 0.04

2000 4000 6000 8000 10 000 12 000 14 000

t @sD

U@VD

Voltage ramp

Double Single

Figure 3.9: Adiabatic voltage ramp for single and dual harmonic and by solving for ∆E one will get

∆E = s

Em2 −Em2φ4

φ4m . (3.54)

Integrating over ∆E and multiplying by four gives the bunch area Abunch for the dual harmonic case as

Abunch = 4 Z φm

0

∆Edφ = 8

πEmφm 3

Γ¡5

4

¢ Γ¡3

4

¢ (3.55)

where Γ (x) is the Euler gamma function.

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18 CHAPTER 3. THEORY

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Chapter 4 Simulation

To simulate the expected results from the experiment a program called ESME [9]

was used. It is developed at Fermilab and is used to model the longitudinal motion of a beam in a synchrotron. The program iterates a map corresponding to the single particle equation of motion. In this way the evolution of a distribution in energy-phase coordinates can be followed. Two single particle dierence equations

ϑi,n=

·τs,n−1

τs,n ϑi,n−1+ 2π µτi,n

τs,n − 1

¶¸

(4.1) Ei,n= Ei,n−1+ eV (ϕs,n+ hϑi,n) − eV (ϕs,n) (4.2)

where τs,n and τi,n are the circulation period for the beam and i−th particle respec- tively, give the change in phase and energy for particle i during the n−th turn of the synchronous particle. At each iteration the particle distribution is mapped to the previous distribution by the dierence equations. The initial distribution of parti- cles can be chosen in many dierent ways in ESME. In this work we have chosen a random uniform distribution in φ and a Gaussian distribution in ∆E.

To be able to compare dierent bunching processes the dilution factor is in- troduced, which is dened as Abunch/Acoast. Using the dilution factor the optimal bunching process can be chosen.

4.1 Single harmonic rf buckets

For single harmonic two dierent ramps were used to change the voltage. A linear ramp and the iso-adiabatic ramp, see equation (3.34). The ramps can be seen in

gure 3.6. It is expected that the iso-adiabatic ramp will give a better result than the linear ramp. The simulations were done for both uranium and argon. If the parameters are chosen in a way that the synchrotron frequency is the same for both cases it is expected to receive similar results. The parameters used in the simulations are given in table 4.1.

19

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20 CHAPTER 4. SIMULATION Uranium

Vi 100 V

Vf 15 kV

∆p/p 1 · 10−3 ωs/2π 1.47 kHz charge of ion 28+

Argon

Vi 100 V

Vf 3.9 kV

∆p/p 1 · 10−3 ωs/2π 1.47 kHz charge of ion 18+

Table 4.1: Parameters used in the simulation

4.1.1 Results

Figure 4.1 shows the results for uranium. As expected the results for argon are similar. In appendix tables A.1, A.2 the numerical values used in the gure are found.

0 5 10 15 20 25 30 35 40 45

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

Ramp time T/T s

Dilution factor

Adiacatic Linear

Figure 4.1: Results of single harmonic simulation for uranium.

It can be seen that the linear ramp leads to better results for short ramping times.

This is because the iso-adiabatic ramp is only valid for short bunches, but for the chosen parameters one deals with long bunches and then the 'iso-adiabatic ramp' is not valid.

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4.2. DUAL HARMONIC RF BUCKETS 21

4.2 Dual harmonic rf buckets

Also in the case of dual harmonic two dierent ramps were used to change the voltage. The ramps used were a linear ramp and an adiabatic ramp, see equation 3.52. The ramps can be seen in gure 3.8. Also here the simulations were done for both uranium and argon. With the same synchrotron frequency for the bunch in the two dierent cases similar results were received. The parameters used for the simulations are given in table 4.2.

Uranium Vi 100 V, 50 V Vf 15 kV, 7.5 kV

∆p/p 1 · 10−3 ωs/2π 1.47 kHz Ion U18+

Argon Vi 100 V, 50 V Vf 3.9 kV, 1.95 kV

∆p/p 1 · 10−3 ωs/2π 1.47 kHz Ion Ar18+

Table 4.2: Parameters used in the simulation

4.2.1 Results

Figure 4.2 shows the results for uranium. As expected the results for argon are similar. In appendix tables A.3,A.4 the numerical values used in the gure are found.

Also in the dual harmonic case the adiabatic ramp has no advantage over the linear ramp because the bunch is long and not short.

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22 CHAPTER 4. SIMULATION

0 5 10 15 20 25 30 35 40 45

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

Ramp time T/T s

Dilution factor

Adiacatic Linear

Figure 4.2: Results of dual harmonic simulation for uranium.

4.3 Simulations with experiment parameters

The parameters used during simulation and experiment are not the same, so the simulations were also done for the parameters that resulted from the experiment to see whether the simulated results and the experimental results correspond with each other. The parameters used during the experiment are given in table 4.3. The momentum spread used for the simulations is an approximation, more to that in the section about the results of the experiment, see section 5.3.

Single harmonic Vi 300 V Vf 6 kV

∆p/p 1.34 · 10−3 ωs/2π 1.8 kHz Ion Ar18+

Dual harmonic Vi 300 V, 300V Vf 6 kV, 3 kV

∆p/p 1.34 · 10−3 ωs/2π 1.8 kHz Ion Ar18+

Table 4.3: The parameters used during the experiment.

4.3.1 Single harmonic

The initial voltage over the cavities is dicult to measure and is approximated to be 300 V, therefore simulations were done with dierent initial voltages to ascertain

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4.3. SIMULATIONS WITH EXPERIMENT PARAMETERS 23 this approximation. The results of the simulations can be seen in gure 4.3. The numerical values are found in appendix table B.1.

0 10 20 30 40 50 60

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

Ramp time T/T s

Dilution factor

Simulation Vi=200V Simulation V

i=300V Simulation V

i=500V

Figure 4.3: Simulation for argon with nal voltage set to 6 kV and dierent initial voltages

By comparing gure 4.1 and gure 4.3 it can be seen that the ramping time inuences the dilution factor in the rst gure but not in the second. Since argon is a much lighter ion than uranium, a higher initial voltage inuences the results for argon more than for uranium. The higher initial voltage aects the argon beam to the extent that the subsequent ramp will have little inuence.

4.3.2 Dual harmonic

SIS-18 will be used in a dual harmonic mode for FAIR experiments, and it is therefore of interest to see how the initial voltage aects the dilution factor. The eect of the initial voltage on the dilution factor can be seen in gure 4.4. Also in this case one wants to know if the initial voltage is 300 V, so simulations were done for dierent initial voltages. The results of the simulations can be seen in table gure 4.5. The numerical values used in the gures can be found in appendix tables B.2 and B.3.

Comparing gure 4.2 and gure 4.5 one can see that the ramping time inuences the dilution factor in the rst gure but not in the second. The reason for this is the same as for the single harmonic case.

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24 CHAPTER 4. SIMULATION

200 250 300 350 400 450 500

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

Voltage

Dilution factor

Figure 4.4: Simulated dilution factor for argon as a function of the initial voltage

0 10 20 30 40 50 60

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

Ramp time T/Ts

Dilution factor

Simulation Vi=200V Simulation V

i=300V Simulation V

i=500V

Figure 4.5: Simulation for argon with nal voltage set to 6 kV and dierent initial voltages

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Chapter 5 Experiment

5.1 Experimental setup

To be able to obtain the bunch area, which is of interest in this work, one needs to know the momentum spread or the bunch length and the voltage V0. Measuring the Schottky spectrum we obtain the momentum spread for a DC coasting beam and from the Beam Position Monitor measurements we obtain the bunch length.

For the single harmonic case one cavity was used, to which cavity a function generator was connected. In the dual harmonic case two cavities were used, so a function generator was connected to each of the cavities. The function generated could either be controlled directly at the generator or remotely with the help of a signal generator. One could thus chose to use one of the common functions, such as a sine or a rectangular function, but also to load a table of which value was to be generated at when. The latter was a very useful feature for us, we could with the help of an Excel le generate a table of desired voltages at specic times and load these data to the generator. The generator could then produce the wanted voltage ramp.

One of the diagnostic tools available in SIS 18 is the Beam Position Monitor. The idea is to measure the charges that the electric eld of the beam induces in a metal plate (electrode). A high impedance senses the charge as a voltage over the capacity between the electrode and the surrounding vacuum chamber. The BPM consists of four plates that simply explained are located so that they form a shoe-box cut diagonally, which is shown in gure 5.1. When the beam passes the BPM it will induce a charge in the electrodes. The charge will be higher in the closer electrode, but the sum of the two charges is always the same and thus V1+ V2 is constant. This means that the sum is independent of the transverse position of the beam and only dependent on the beam current. With an oscilloscope the voltages were converted to digital form and stored. The stored data can then be post processed and one can receive the bunch form and thus also the bunch length [10].

Another diagnostic tool is the Schottky-measurement, which is developed by GSI.

On a microscopic level an electrical current has small irregular uctuations to the

ow. These uctuation are known as the physical eect Schottky noise. This noise can be detected by a probe placed at one point in the accelerator ring. The detected signal can then be evaluated with a spectrum analyzer. From these data we can

25

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26 CHAPTER 5. EXPERIMENT

Beam

(a)

V

1

V

2

Beam

(b)

Figure 5.1: Picture of BPM

extract the momentum spread of the beam using post processing [11].

A schematic picture of the setup can be seen in gure 5.2

SIS-18

Cavity Function Generator

Oscilloscope

BPM Spectrum

Analyser

Schottky

Figure 5.2: Schematic picture of the experimental setup

5.2 Experimental evaluation

The data acquired from the Schottky measurements have to be converted to ASCII format with a program from Tektronix. With this program one could also chose if all of the data in one le or just a part of it where to appear in the converted

le. For this analysis of the data two programs written by S. Paret at GSI was used. The rst program rene_formerIQT.cpp makes an average of the spectrum.

This average is then used in the second program t_schottky.pro, which makes a Gaussian t of the averaged spectrum. The rms-width and the center frequency of the spectrum, which are given from program two, can then be used to determine

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5.3. RESULTS 27 the rms-momentum spread according to

µ∆p p

rms

= µ∆f

f

rms

1

η (5.1)

which is stated in the users manual to the Schottky measurement. The relation between the rms-momentum spread and the momentum spread of a parabolic dis- tribution is a factor

5, so ∆pp = 5

³∆p p

´

rms. The Schottky measurement is only valid for coasting beams.

To evaluate the data collected from the BPM one needs to use LCProg_v1.pro a translation of the original code written by R. Bear et al. This program ex- tracts one bunch from the many bunches contained in each trace from the oscil- loscope. The extracted bunch prole was then further analysied with Mathemat- ica, a mathematical analysis tool. Two dierent programs dual_prol_t.nb and from_nonlin_to_gauss.nb, written by O. Chorniy at GSI, were used. In the rst program, the le with the bunch data were imported and then used. For both cases a gaussian t was made to the data and from the σ, calculated by the program,

∆p/pcan be expressed as

∆p p =

5s

hηβcσ. (5.2)

For the single harmonic case the given σ was inserted in the second program and this program then gives (φm)rms. Next, φm can be calculated as φm =

5 (φm)rms. For the dual harmonic case a parabolic t was also made and from this one receives φm.

The coasting beam area before and the bunch area after bunching can then be calculated according to equations (3.38),(3.40) and (3.55) and the dilution factor is dened as the quotient Abunch/Acoast.

5.3 Results

Unfortunately the evaluation of the momentum spread of the coasting beam was not possible. This because the number of coasting beam frames collected per measure- ment was too small. But it can be approximated by doing simulation for dierent momentum spreads, and then compare the beam currents to the experimental beam current.

5.3.1 Single harmonic

Figure 5.3 shows a comparison between the bunch form received in the simulations and the experiment. Figure 5.4 shows the dierent bunch forms from the simula- tions. Figure 5.5 shows the dilution factor observed in experiments compared to the simulations.

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28 CHAPTER 5. EXPERIMENT

-150 -100 -50 0 50 100 150 0.0

0.2 0.4 0.6 0.8 1.0

Angle, @DegreeD

Amplitude

Simulation Experiment

Figure 5.3: The bunch form from the experiment compared with the simulated bunch form

-150 -100 -50 0 50 100 150 0

1 2 3 4 5 6

Angle, @DegreeD

Amplitude

Simulation, Vi=500 V Simulation, Vi=300 V Simulation, Vi=200 V

Figure 5.4: The bunch form from the dierent simulations

0 10 20 30 40 50 60

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Ramp time T/Ts

Dilution factor

Experiment Simulation Vi=200V Simulation Vi=300V Simulation Vi=500V

Figure 5.5: The dilution factor from the experiment and from the simulations for the single harmonic case.

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5.3. RESULTS 29

5.3.2 Dual harmonic

Figure 5.6 shows a comparison between the bunch form received from the simula- tions and experiment. In gure 5.7 one can see the dierent bunch forms from the simulations. Figure 5.8, nally, gives the dilution factor obtained in experiments and simulations.

-150 -100 -50 0 50 100 150 0.0

0.2 0.4 0.6 0.8 1.0

Angle, @DegreeD

Amplitude

Simulation Experiment

Figure 5.6: The bunch form from the experiment compared with the simulated bunch form and the theoretical bunch form

-150 -100 -50 0 50 100 150 0

1 2 3 4 5

Angle, @DegreeD

Amplitude

Simulation, Vi=200 V Simulation, Vi=200 V Simulation, Vi=200 V

Figure 5.7: The bunch form from the dierent simulations

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30 CHAPTER 5. EXPERIMENT

0 10 20 30 40 50 60

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

Ramp time T/Ts

Dilution factor

Experiment Simulation Vi=200V Simulation Vi=300V Simulation Vi=500V

Figure 5.8: The dilution factor from the experiment and from the simulations for the dual harmonic case.

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Chapter 6 Discussion

The simulations show that there is no advantage of the iso-adiabatic ramp compared to the linear ramp. This is because the iso-adiabatic ramp is only valid for short bunches. In our case we start with a coasting beam and end with a long bunch and thus the iso-adiabatic ramp is ineective. Because of this the iso-adiabatic ramp and the linear ramp give similar results. It would be interesting to see what happens with the dilution factor if the linear and iso-adiabatic ramp is combined. One would then rst use the linear ramp to form a short bunch and then the iso-adiabatic ramp to complete the bunching. Then one should be able to see the eects of the iso-adiabatic ramp and the dilution factor should thus be improved.

It is unfortunate that it was not possible to evaluate the momentum spread of the coasting beam during the experiment, so an approximation has to be used to calculate the dilution factor. With the higher initial voltage used during the experiment the ramping time seems to have no inuence on the dilution factor of medium heavy ions. This is not the case, the initial voltage aects the beam so much that the eects of the voltage ramp are not seen.

Using a lower initial voltage, as in the rst simulations, one can see that the ramping time inuences the dilution factor, and to have a bunching process which can be called adiabatic, the blow-up of the area should not be more than 10 − 20 %.

According to the simulations, T/Ts should be at least 22 which is equivalent to a ramping time of 15 ms for SIS-18. This ramping time is valid for both the single and dual harmonic case.

From the comparison of the experimental results and the simulated results, for the single harmonic case, we can see that the dilution factor is smaller in the sim- ulated case. This is because during simulation the ideal case was assumed, but in reality there are high intensity eects that inuence the particles in an accelerator.

These eects cause the dilution factor to be worsen. So, to see if the simulations and the experiments agree, these eects have to be taken into account also during the simulations. Also in the dual harmonic case there are dierences between ex- periments and simulations. But here the dilution factor is better in the experiments than in the simulation, the most probable reason for this is a deviation in momen- tum spread between the simulation and experiment. The intensity of the ion source is not constant and thus deviations in the momentum spread of the coasting beam occur.

31

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32 CHAPTER 6. DISCUSSION

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Chapter 7

Summary and Conclusions

7.1 Summary

In this work, performed at GSI on the synchrotron SIS-18, the inuence of dierent voltage ramps and ramping times on the dilution factor was examined. Both a linear and an iso-adiabatic ramp were used in the simulation work. In the experiments only an iso-adiabatic ramp was used because of limited beam time.

7.2 Conclusion

It can be concluded that the iso-adiabatic ramp does not have any advantages over the linear ramp, since we deal with long bunches. Further, it can be concluded that a high initial voltage inuences medium heavy ions so much that the eects of the ramping time are not seen.

7.3 Future work

The rst thing to do would be to repeat the experiment with a longer time period of coasting beam, so approximations are not needed when calculating the dilution factor. It would also be interesting to do the simulations with high intensity eects included.

33

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34 CHAPTER 7. SUMMARY AND CONCLUSIONS

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Bibliography

[1] http://www.gsi.de/beschleuniger/Beschleuniger.html, 2008-10-22

[2] A. Parfenova, Linear and nonlinear Response Matrix and its application to the SIS 18 synchrotron, Diss 2008-10 July, Gesellschaft für Schwerionenforschung mbH

[3] T. Shukla, Bucnhes with space charge for arbitrary rf wave forms, GSI-Acc- Report-2004-06-001, June 2004

[4] D.A Edwards, M.J Syphers, An introduction to the physics of high energy ac- celerators, John Wiley & Sons, 1993

[5] O. Boine-Frankenheim, Beam intensity eects in the new SIS dual harmonic rf system, GSI internal report, 27 March 2008

[6] O. Boine-Frankenheim, Script to lecture 'Introduction into Physics of Acceler- ators', Technische Univeristät Darmstadt, Darmstadt, Hesse, Germany, 2006 [7] S. Y. Lee, Accelerator Physics, World Scientic, 1999

[8] O. Boine-Frankenheim, T. Shukla, Space charge eects in bunches for dier- ent rf wave forms, Physical review special topics - Accelerators and beams 8, 034201, 10 March 2005

[9] J. A. MacLachlan, J-F. Ostiguy, User's Guide to ESME esmF95 (esme2008.2), May 2008, http://www-ap.fnal.gov/ESME/, 2008-07-03

[10] O. Chorniy, Measurement and interpretation of the bunched beam transfer func- tion in SIS-18 with space charge, Diss 2008-12 July, Gesellschaft für Schwerio- nenforschung mbH

[11] U. Schaaf, Schottky-diagnose und BTF-messungen an gekühlten strahlen im schwerionenspeicherring ESR, GSI-91-22 Report July 1991 ISSN 0171-4546, Gesellschaft für Schwerionenforschung mbH

35

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36 BIBLIOGRAPHY

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Appendix A

Iso-adiabatic ramp, single harmonic

T /Ts φm [] ∆p/p 10−3 Acoast [eV s] Abunch [eV s] Abunch/Acoast

7.3 100 3.52 50.24 77.27 1.54

14.7 104 2.87 50.24 65.56 1.30

22.0 96 2.61 50.24 55.05 1.10

29.3 92 2.63 50.24 52.75 1.05

36.7 96 2.56 50.24 53.88 1.07

44.0 92 2.60 50.24 52.38 1.04

Table A.1: Simulated results for uranium when an adiabatic ramp was used with the ramping time T/Ts

Linear ramp, single harmonic

T /Ts φm [] ∆p/p 10−3 Acoast [eV s] Abunch [eV s] Abunch/Acoast

7.3 100 2.84 50.24 62.22 1.24

14.7 96 2.63 50.24 55.44 1.10

22.0 92 2.69 50.24 54.25 1.08

29.3 92 2.60 50.24 52.38 1.04

36.7 96 2.60 50.24 54.66 1.09

44.0 92 2.60 50.24 52.38 1.04

Table A.2: Simulated results for uranium when a linear ramp was used with the ramping time T/Ts

37

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38 APPENDIX A.

Adiabatic ramp, dual harmonic

T /Ts φm [] ∆p/p 10−3 Acoast [eV s] Abunch [eV s] Abunch/Acoast

7.3 136 3.67 50.24 121.87 2.43

14.7 92 2.97 50.24 66.62 1.33

22.0 104 2.22 50.24 56.48 1.12

29.3 100 2.13 50.24 52.04 1.04

36.7 96 2.22 50.24 52.13 1.04

44.0 104 2.09 50.24 53.18 1.06

Table A.3: Simulated results for uranium when an adiabatic ramp was used with the ramping time T/Ts

Linear ramp, dual harmonic

T /Ts φm [] ∆p/p 10−3 Acoast [eV s] Abunch [eV s] Abunch/Acoast

7.3 112 2.63264 50.24 71.98 1.43

14.7 100 2.31747 50.24 56.57 1.13

22.0 104 2.13207 50.24 54.13 1.08

29.3 100 2.18769 50.24 53.40 1.06

36.7 100 2.22477 50.24 54.31 1.08

44.0 100 2.09499 50.24 51.14 1.02

Table A.4: Simulated results for uranium when a linear ramp was used with the ramping time T/Ts

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Appendix B

initial voltage = 200V

T /Ts φm [] ∆p/p 10−3 Abunch/Acoast

2.89 116 3.18 1.2

5.77 116 3.18 1.2

14.43 116 3.18 1.2

28.87 116 3.18 1.2

43.30 116 3.18 1.2

57.73 116 3.18 1.2

initial voltage = 300V

T /Ts φm [] ∆p/p 10−3 Abunch/Acoast

2.89 120 3.23 1.26

5.77 120 3.23 1.26

14.43 120 3.26 1.27

28.87 120 3.23 1.26

43.30 120 3.23 1.26

57.73 120 3.23 1.26

initial voltage = 500V

T /Ts φm [] ∆p/p 10−3 Abunch/Acoast

2.89 124 3.35 1.35

5.77 124 3.35 1.35

14.43 124 3.35 1.35

28.87 124 3.35 1.35

43.30 124 3.35 1.35

57.73 124 3.35 1.35

Table B.1: Simulated results for dierent initial voltages.

39

References

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