Dynamics of laser-cooled ion beams

Dynamics of laser-cooled ion beams Niels Madsen

Institute of Physics and Astronomy University of Aarhus, Denmark September, 1998

"While I'm still confused and uncertain, it's on a much higher plane, you see, and at least I know I'm bewildered about the really fundamental and important facts of the universe."

(Terry Pratchett, Equal Rites)

Acknowledgements The present thesis deals with the dynamics of stored laser-cooled ion beams. The thesis is based on work done by myself and coworkers during my 4 years as a Ph.D. student at the Institute of Physics and Astronomy, University of Aarhus, Denmark, under the supervision of Professor Jerey S. Hangst. Experimental work as presented in this thesis is seldom the work of one man alone. This work is no dierent and I therefore feel that I should mention some of the people I have had the pleasure to work with during the last 4 years. I would rst of all like to thank my supervisor for his support and many fruitful discussions about physics and life in general and for introducing me to the fascinating world of beam physics. I have also had the pleasure to work with A. Labrador and J.S. Nielsen, whose enthusiasm never fail. More recent L.H. Hornekr, M. Drewsen, N. Kjrgaard and P. Bowe have joined the group and I have enjoyed working with them. Furthermore I have had the pleasure to work with J.P. Schier during a stay at Argonne National Laboratory and when he participated in our work in Aarhus, and S. Maury and D. Mohl during a 6 month stay at CERN. Also I should mention Karsten and Jens from ISA who can always nd time for a helping hand and especially H.J. Larsen who always found time for doing projects which I wanted nished yesterday. Finally a thanks to my parents for the never ending support and to my brother for always showing up at strange hours of the day.

Contents 1 Introduction

1

2 The physics of storage rings

5

1.1 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basics of particle motion . . . . . . . . . . 2.1.1 Particles with momentum deviation 2.1.2 Resonances . . . . . . . . . . . . . 2.2 Tune shifts from self elds . . . . . . . . . 2.2.1 Uniform Density Beam . . . . . . . 2.2.2 Gaussian beam . . . . . . . . . . . 2.3 Crystallization . . . . . . . . . . . . . . . . 2.3.1 The ground state . . . . . . . . . . 2.3.2 Stability . . . . . . . . . . . . . . . 2.3.3 Reaching the crystalline state . . .

3 Beam cooling

3.1 Liouville's Theorem . . . . . . 3.2 Standard Cooling mechanisms 3.2.1 Electron Cooling . . . 3.2.2 Stochastic Cooling . . 3.2.3 Radiative Cooling . . . 3.3 Laser-cooling . . . . . . . . . 3.3.1 Doppler cooling . . . . 3.3.2 Coasting beams . . . . 3.3.3 Bunched beams . . . .

4 Experimental Setup

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4.1 ASTRID . . . . . . . . . . . . . . 4.2 Laser-cooling setup for ASTRID . 4.2.1 Frequency Drift . . . . . . 4.2.2 Cavity Locking . . . . . . 4.2.3 Present Performance . . . 4.3 Longitudinal Diagnostics . . . . .

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vii

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5 9 10 11 11 12 14 14 16 16

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31 31 32 34 35 38 38

4.3.1 Bunch Shapes . . . . . . . . . 4.3.2 Schottky noise . . . . . . . . . 4.3.3 Laser induced uorescence . . 4.4 Absolute Current . . . . . . . . . . . 4.5 Transverse Diagnostics . . . . . . . . 4.5.1 Requirements . . . . . . . . . 4.5.2 Residual Gas Pro le Monitor 4.5.3 Beam Imaging . . . . . . . . .

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5 Longitudinal Dynamics

5.1 Coasting Beams . . . . . . . . . . . . . . 5.1.1 Cooling a coasting beam . . . . . 5.1.2 Suppression of Landau Damping . 5.1.3 Diusion . . . . . . . . . . . . . . 5.1.4 Touschek Scattering . . . . . . . 5.2 Bunched Beams . . . . . . . . . . . . . . 5.2.1 Cooling a bunched beam . . . . . 5.2.2 Bunch Pro les . . . . . . . . . . . 5.2.3 Pro le Measurements . . . . . . . 5.3 Discussion . . . . . . . . . . . . . . . . .

6 Transverse Dynamics

6.1 Beam Pro les . . . . . . . . . . . . . 6.1.1 Space charge limited . . . . . 6.1.2 Finite Temperature . . . . . . 6.2 Position Dependence . . . . . . . . . 6.2.1 Drift and oscillations . . . . . 6.2.2 Sympathetic cooling . . . . . 6.3 Time Scale . . . . . . . . . . . . . . . 6.3.1 Measurements . . . . . . . . . 6.3.2 Discussion . . . . . . . . . . . 6.4 Density Limitations . . . . . . . . . . 6.4.1 Longitudinal Velocity Spread 6.4.2 Laser power . . . . . . . . . . 6.4.3 Beam Current . . . . . . . . . 6.4.4 Discussion . . . . . . . . . . . 6.5 Ribbon Beams . . . . . . . . . . . . . 6.5.1 Tapered Cooling . . . . . . .

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39 40 43 45 46 46 47 48

65 65 65 67 69 75 79 79 81 84 87

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89 90 91 92 92 94 98 98 100 100 100 103 105 112 112 114

7 Conclusions and outlook

117

8 Dansk Resume

121

7.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 viii

A Calculations

125

B Technical Speci cations

129

A.1 Velocity and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.2 Bunch Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 B.1 CCD Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 B.2 Optical Scanners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 B.3 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

C Intra-Beam Scattering

C.1 Introduction . . . . . . . . . . . C.2 Detailed approach . . . . . . . . C.3 The simple approach . . . . . . C.3.1 Diusion in the plasma . C.3.2 Friction in the plasma . C.4 Calculations for ASTRID . . . . C.4.1 Emittance Evolution . .

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References Index

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133 133 133 134 135 137 138 139

141 148

ix

x

Chapter I Introduction Accelerated beams of particles are among the most important tools in physics. Almost all disciplines in physics use moving beams of particles for one purpose or another, whether for implantation, diraction, fragmentation or other applications. For most applications a high intensity as well as a high brightness1 is important for the success of the experiment. Thus physicists strive to achieve higher currents/intensities as well as higher density/brightness in their experiments. At the heart of many of these eorts lies beam cooling. By beam cooling we mean a reduction, without loss of particles, in the total (6D) phase space volume the beam occupies2. During transport, storage and acceleration, beams are transversely con ned. With transverse con nement the beam particles will undergo oscillations in transverse phase space, thus a reduction in the phase space volume, which is the same as an increase in the phase space density (we have no losses), will result in a decrease of both the transverse momentum spread and the transverse beam size. Beam cooling therefore leads to increased particle spatial density. In a particle beam collisions with the rest gas, intra-beam scattering and interactions with the surroundings can cause heating of the beam, and therefore decrease of the particle spatial density. Beam cooling counters these eects and is therefore necessary for achieving and sustaining high densities. The interaction with surroundings and self elds can furthermore lead to instabilities. Instabilities are often more pronounced in high intensity beams, where the elds are stronger. Instabilities are a fast kind of heating and may lead to beam losses in the severe case. Beam cooling is therefore also a means by which instabilities can be damped, an important facet of its application in high intensity beams3. High intensity beams have for instance been proposed as energy suppliers for nuclear waste transmutation and inertial con nement fusion. In a dense beam the motion of individual particles may be perturbed or even 1 It is common in beam physics to talk about intensity as the absolute number of particles pr. unit

time, and brightness as the intensity per unit area. 2 Note that the de nition of beam cooling implies that a reduction in the longitudinal velocity spread leading to an increase in for example the transverse spread in such a way that the total (6D) phase space volume of the beam is constant, is not cooling 3 Sometimes, however, beam cooling also leads to changes which drive an instability instead of damping it.

2

Introduction

dominated by the Coulomb repulsion from the other particles in the beam (these forces are also referred to as space-charge). Cooling results in increased spatial density, and to understand how the cooling works it is therefore interesting and important to understand what in uence on the dynamics space-charge may have. Furthermore, space-charge forces induce changes in the beam spatial density distribution which may introduce limitations to the attainable densities and intensities through instabilities. Thus the study of the in uence of space-charge is important both for understanding the cooling process itself and for identifying the limitations to the achievable intensities and densities in particle beams. The present thesis is a study of the physics of beam cooling as well as the physics of dense particle beams. There are presently only three ways to cool charged particle beams; Electron cooling [11], stochastic cooling [99] and laser-cooling [34]. In the experiments presented in this thesis we have studied the in uence of laser-cooling on a charged particle beam. Laser-cooling is an intriguing new tool both for cooling trapped particles and beams of particles. It was for instance via the application of laser-cooling that researchers achieved low enough temperatures to observe Bose-Einstein condensation for the rst time in 1995 [1]. And in 1997 physicists W. Philips, S. Chu and C. Cohen-Tannoudji received the Nobel Prize in physics "for development of methods to cool and trap atoms with laser light"4. As laser-cooling is limited to ions with a suitable electronic structure it is somewhat restricted for general use for cooling of particle beams. However, laser-cooling provides much faster cooling times and much lower temperatures than other particle beam techniques, and it therefore oers the possibility of achieving unprecedented beam quality [94, 28]. In a very cold beam it may be that the thermal energy of the particles becomes so low that they can no longer overcome the Coulomb barrier induced by the surrounding particles. If this is the case we may have a situation where all the particles are locked relative to each other, and thus occupy xed sites about which they oscillate. This state has been called a crystalline beam [91]. The structural properties of cylindrically con ned Coulomb crystals have been studied extensively [86, 93, 89, 35, 92], and in 1992 the rst experimental evidence for the predicted shell structures was found in a quadrupole ring trap [6]. However, in a storage ring, where the crystal would move at high velocity, the bending and periodic focusing might prevent the beam from crystallizing , as was realized early [86]. Recently though, the impact of the periodic motion and bending has been investigated in more detail in Molecular Dynamics (MD) simulations, and necessary criteria for crystallization have been found [104, 105]. Unfortunately, laser-cooling in a storage ring has the drawback that it is only directly applicable in the longitudinal dimension (the direction of beam motion) where good overlap between the laser-beam and the ion beam can be accomplished. It is therefore vital, not only to reach crystallization but also to the applicability of laser-cooling for cooling in general, that heat can be coupled from the transverse degrees of freedom to the longitudinal laser-cooled degree of freedom (this process is sometimes referred to as sympathetic cooling5). The bulk of the work covered in this thesis has been focused at understanding 4 See for example the Nobel Foundations Web site http://www.nobel.se/ 5 Sympathetic cooling may also refer to the cooling process in which one cold species of particles is used

1.1 Structure of the thesis

3

the mechanisms for this coupling and the strength of it, as well as determining whether it is strong enough that the crystalline beam state can eventually be reached by the application of laser-cooling. We have also investigated what limits the achieved temperatures and densities. Limits to the cooling process are important for the applicability of laser-cooling, as well as for the ability to attain a crystalline state.

1.1 Structure of the thesis The thesis has eight chapters. Chapters 2 and 3 introduce the physics of stored beams, and cooling of stored beams. Chapter 2 is a discussion of the physics of particle motion in a storage ring, with emphasis on subjects which are important for understanding the experiments reported in later chapters. This also includes recent developments in the knowledge of beam crystallization. In Chapter 3 beam cooling is discussed. First the general concept is presented, and then three standard cooling mechanisms are discussed. The last part of the chapter is devoted to laser-cooling, both the general theory of Dopplercooling and the speci cs of the application of laser-cooling in a storage ring are discussed. Chapter 4 contains a description of the setup for the experiments. This includes some details on the storage ring ASTRID used for the experiments, as well as a description of the laser system used for laser-cooling, including recent improvements to the laser system. Then the diagnostics used for studying ion beams is presented. This discussion includes a detailed presentation of a new system developed at ASTRID for transverse diagnostics of a circulating ion beam [57]. The new system utilizes the uorescence light from the laser-excited ion beam to monitor the transverse particle density distribution, and has demonstrated high sensitivity and resolution compared to other systems implemented for storage ring use. Chapter 5 deals with the longitudinal dynamics of laser-cooling studied before the implementation of the new system for transverse diagnostics. This covers both work which I have participated directly in [29, 48] and work initiated before I joined the group actively [30, 47]. Chapter 6 treats the recent results in transverse dynamics during laser-cooling [56, 55]. In Chapter 7 the presented results are summarized and it is discussed where and how the laser-cooling experiments should proceed in the future. Chapter 8 is a summary in Danish written for people only vaguely familiar with physics.

to cool another species.

4

Introduction

Chapter II The physics of storage rings A storage ring is a device used to store beams of charged particles, as opposed to traps which store particles at rest. This chapter gives an introduction to the basic concepts of particle and beam motion in a storage ring. A few more advanced concepts which are relevant for the laser-cooling experiments in this thesis are also described. This includes the most recent results on crystalline beams.

2.1 Basics of particle motion In a storage ring particles can be con ned by either magnetic or electric forces. Magnetic forces provide the possibility of reaching the highest energies, and have therefore been the standard choice for storage rings for many years, as storage rings were mostly applied in the high energy community. The Aarhus storage ring ASTRID used for the experiments discussed in this thesis is based on the same principle, and the discussion in this chapter therefore focuses on particle motion in a magnetic storage ring. In low energy rings (or traps) electric con nement is possible and it oers some advantages for atomic and molecular physics experiments. Recently an electrostatic storage ring was built in Aarhus to be used for experiments in atomic and molecular physics [68]. To store a beam of particles it is necessary to guide the particles on a closed orbit. We will assume, which is usually the case, that this orbit is in the horizontal plane, and de ne a longitudinal dimension s which is in the direction of beam motion, and then a horizontal dimension x and vertical dimension y which are locally perpendicular to the longitudinal. The horizontal and longitudinal dimensions change with respect to the labframe as a function of the position in the ring, as they depend on the local direction of the beam. An illustration of the coordinate system in which the beam is described is given in Figure 2.1. A vertical dipole eld is used to generate the necessary centripetal force to de ne a closed orbit for the particles. However, small deviations in energy and momentum would prevent the beam from being con ned for very long, and would result in unreasonably large beams. It is therefore necessary to provide some transverse con nement. The transverse con nement introduced

6

The physics of storage rings y s

s0

s x

Figure 2.1: Coordinate system in which a beam in a storage ring is described. The coordinate s measures the position along the beam orbit from an arbitrary position s0. The direction of s is changing around the ring (and therefore also the direction x, which is perpendicular to the s direction and lies in the plane of the orbit.). consists of alternately focusing and defocusing magnetic quadrupoles [18, 19] and for this reason this type of storage ring is called alternating gradient (AG). The alternating forces arises because a quadrupole, which generates the necessary linear con nement force, only focuses in one plane at a time (from r B = 0), as shown in Figure 2.2. But by placing two

Figure 2.2: Schematic drawing of a magnetic quadrupole, illustrating how the eld causes focusing in one plane and defocusing in the other (dark arrows indicate the forces and gray arrows indicate the elds). quadrupoles which are focusing in perpendicular dimensions at a distance l which ful lls

l < 2jf j;

(2.1)

where f is the focal length, the net eect is that the system is focusing [18]. When the beam passes a quadrupole the change it undergoes is similar to the change in a beam of light passing through a lens. We can illustrate beam passage through a FODO cell (Focusing, drift space, Defocusing, drift space) as in Figure 2.3. The general behavior of a beam in a storage ring, consisting of many FODO cells1, can be described by Hill's equation (the vertical direction is used for illustration) [88]

y00 + K (s)y = 0 ; K (s + L) = K (s)

(2.2)

1 The arrangement of magnetic elements in a storage ring is often referred to as the magnetic lattice of

the machine.

2.1 Basics of particle motion

7 s

F

O

D

O

Figure 2.3: Beam passage through a FODO cell where K(s) is a periodic function describing the transverse forces, and y00 is the second derivative with respect to the longitudinal position s. The period L may be as large as the circumference C of the storage ring. A particle trajectory is a real solution to Hill's equation, and can be written as q y(s) = a (s) cos((s) , ) (2.3) where  is an arbitrary initial phase, and (s), called the beta function, describes the changing beam envelope due to the varying con nement forces around the ring. The betatron phase advance (s) can be computed from (s) Zs 1 (2.4) (s) = (t) dt s0 where s0 is some position along the orbit. The tune or Q value is de ned as I 1  ( C ) 1 Q = 2 = 2 (s) ds (2.5) and is the number of transverse oscillations around the ideal orbit (called betatron oscillations) per revolution in the ring. In calculations on beams we often use the smooth approximation which assumes a constant transverse con nement. Furthermore the con nement is often assumed to be axisymmetric, in which case we use the term Q? for the transverse tune. In beams the most commonly used transverse parameters are the divergence and the size, therefore the transverse phase space is often depicted as in gure 2.4. The divergence is given by py y0 = dy = (2.6) ds mc

where m is the particle mass, c the speed of light, = vs=c and = (1 , v2=c2),1=2 are relativistic factors, and py is the momentum. In the transverse phase space a particle will describe an ellipse as shown in gure 2.4. This ellipse can in Cartesian coordinates be written as

(s)y2 + 2(s)yy0 + (s)y02 = a2

(2.7)

8

The physics of storage rings y’(s ) 1

turn 3 turn 4 turn 5

turn 2 turn 1

y(s ) 1

Figure 2.4: Phase space ellipse for one particle. For each revolution in the ring the point (y, y') makes Q turns in the phase plane. y0 is the divergence, see explanation in the text. where primes mark derivation with respect to s, (s) = , 21 (s)0  , 21 d ds(s) and 2

(s) = 1 + (s)(s) (2.8) and where a2 is a constant called the Courant-Snyder invariant, as it is a constant around the ring as a consequence of Liouville's theorem (Section 3.1). In a real machine it turns out not to be strictly constant, as we will come back to later, however it is usually constant enough to be a good gure of merit and therefore deserves some attention. The parameter a2 is also called the single particle emittance. If we have a beam of particles the area in phase space they cover (divided by ) is called the emittance  of the beam. I.e. the emittance of a beam of particles is de ned as area of ellipse = 
 (2.9) The density distribution in phase space is often Gaussian, and covers therefore in principle an in nite area. The emittance is therefore de ned as the area in phase space of some fraction of the beam. In this report the emittance will be de ned as the phase space area of 68% (1 sigma in a 1D Gaussian) of the beam. Furthermore, in order to avoid confusion with de nitions which state the emittance as the area of the ellipse without dividing by , emittances in this report will be stated with  written explicitly in front of the emittance (f.ex. 2  mm mrad means  = 2
10,6 rad m, where  is de ned above). In case of acceleration the emittance decreases due to the fact that only the longitudinal momentum increases during acceleration, and thus the beam divergence shrinks. This process is called adiabatic damping. Sometimes one introduces an invariant emittance which is constant during acceleration. The invariant emittance is given by  =  (2.10) where and are the longitudinal relativistic factors. In the experiments discussed in this thesis the energy is always constant. We can now write the particle trajectory for the ellipse that encloses the beam using the emittance q y(s) = y (s) cos((s) , ); (2.11)

2.1 Basics of particle motion thus the beam envelope is and the divergence

9 q ymax(s) = py (s)

(2.12)

v u q 2 u p 0 ymax(s) = y t 1 + (s)(s) = py (s)

(2.13)

q from which is seen that at a local extreme in envelope, the divergence is given by y = . Later we will see that our transverse beam size diagnostics occurs at such a position. It should at this point be noted that sometimes a transverse temperature is de ned via the mean kinetic energy T = mhv2i=kB , where kB is the Boltzmann constant, m the ion mass, and hv2i the rms velocity spread [61]. However, as the mean transverse kinetic energy varies around the ring due to the varying focusing this is a bad gure of merit for a stored beam.

2.1.1 Particles with momentum deviation If particles have momentum deviation from the ideal particle, they will describe an orbit with a dierent average radius. This is easily understood in an accelerator consisting only of a bending eld, as the bending depends on the particle momentum. In an AG storage ring the 'radius' deviation varies around the ring due to the focusing elements, and one therefore de nes a closed orbit dispersion D(s) by the equation ) = Dh (s)
dp x(s; dp p p

(2.14)

which (to rst order) describes the horizontal displacement of the mean orbit of a particle with momentum deviation dp from the ideal particle with momentum p. A particle with momentum deviation will therefore have a dierent orbit length than the ideal particle. This deviation is described by the momentum compaction factor  dC =  dp ;  = 1 I Dh (s) ds (2.15) C p C (s) where (s) is the local radius of curvature. From this we can nd the change in revolution frequency with momentum (assuming constant magnetic elds)

df =  dp ;  = 1 ,  f p

2

(2.16)

where is the relativistic factor. The momentum compaction is also written  = 1= tr2 , where tr is called the transition energy and is the energy above which an increase in

10

The physics of storage rings

momentum leads to a decrease in revolution frequency . This can give rise to instabilities, as for instance the negative mass instability2.  is called the slip factor [40]. A particle with momentum deviation will, apart from having a dierent orbit length, also be focused dierently in the quadrupoles, this leads to a shift in the tune
Q, which can be written as I
Q = 
pp where  = , 41 (s)K (s) ds (2.17) where  is called the chromaticity (as it is similar to chromatic aberration in a normal lens). This tune shift causes the beam to have a tune spread, something which might be a problem in low density beams.

2.1.2 Resonances

Consider a particle with an integer tune, i.e., with an integer number of betatron oscillations per revolution. This means that the particle orbit will close on itself, and thus a small perturbation some place in the orbit will build up coherently, and the particle might eventually be lost from the machine. This eect is called a tune resonance, in this case of rst order. Resonances can be of any order. A magnetic 2n-pole perturbation will drive a resonance of n'th order. The speed with which a perturbation in the beam will grow decreases with the order of the resonance. Resonances above the 2nd order are called nonlinear resonances, and their growth rate depends on the betatron amplitude [108]. When running a storage ring, it is important that one chooses an operating tune as far from these resonances as possible. A general expression for the resonance condition, also including coupling between the two transverse dimensions is [108]

nQh + mQv = p

(2.18)

where n, m and p are integers and where jnj + jmj is the order of the resonance. A plot showing lines corresponding to ful llment of this resonance condition is usually referred to as a working diagram, an example of which is given in gure 2.5. The resonances are also referred to as stop bands3 . A working point far from as many resonances as possible is usually chosen in order to avoid beam loss. Chromaticity may induce large tune spreads in the beam. If the beam density is low individual particles move independently of each other, and the chromaticity may cause the tune of some particles to be close to a resonance, and thus cause them to be lost. This will not be a problem in our experiments, as our momentum spreads are quite small due to laser-cooling. 2 Consider a machine below transition, and a longitudinal density wave. At the front slope of the wave

crest space-charge forces will increase the particle energy, and opposite at the back slope, this will in turn increase their revolution frequency, and they will move away, thus atten out the wave. If we are above transition, the revolution frequency will instead decrease, thus amplifying the wave. This is called the negative mass instability because particles behave as if they have negative mass [37]. 3 A speci c band can be 'removed' by introducing compensating focusing and defocusing elements which interfere destructively with the resonance [108].

2.2 Tune shifts from self elds

11

3

Vertical Tune Qy

2.8

2.6

2.4

2.2

2 2

2.2

2.4

2.6

2.8

3

Horizontal Tune Qx

Figure 2.5: Working diagram in the area of tunes where ASTRID is usually operated during laser-cooling experiments. The dot shows the standard tunes for most measurements in this thesis. The solid lines are second order resonances, and the dashed third order.

2.2 Tune shifts from self elds Apart from the external elds the motion of a beam in a ring is in uenced by the self elds from the beam. The self elds will vary depending on the beam as well as on the boundary conditions given by the vacuum chamber, cavities and so on. The self elds will in uence the motion of individual particles (incoherent eect) and the motion of the beam as a whole (coherent eect) . The incoherent eect is the most important in our context, thus we will limit ourselves to a discussion of that. We consider it here, as is usually the case, in the limit where the eects can be considered as a perturbation to the particle motion, i.e. that the forces from self elds are small compared to the relative kinetic energy of the particles in the beam rest frame. In the next section the opposite limit is investigated.

2.2.1 Uniform Density Beam A particle in a beam is directly in uenced by the Coulomb force of the other particles and the magnetic eld generated by the beam. The force on a beam particle, assuming a cylindrical beam with radius a (axisymmetic with uniform particle density) of singly charged particles, is radial and given by [38]

F?(r) = e
Er + evB r (1 , 2) ; r < a = 2e 0 a2 = 2e 2 ar2 ; r < a 0

(2.19)

12

The physics of storage rings

where  is the charge density per unit length (longitudinal charge density) measured in the lab-frame, and is the relativistic factor. The Coulomb repulsion is thus defocusing and the magnetic force is focusing. The magnetic force increases with , thus the defocusing forces are strongest in non-relativistic beams. This defocusing due to space-charge counteracts the focusing from the con nement elds, thus the total con nement is weakened and the betatron frequency or equivalently the tune decreases. This tune shift is called the incoherent tune shift and is, for a beam with uniform particle density (a cylindrical beam), given by [38]

e2 n
Q = 4 m! 2 3 Q 0

(2.20)

0

where n is the density and !0 is the revolution frequency. It is interesting to note that the tune shift only depends on the number of particles through the particle density, and not on the absolute current.

2.2.2 Gaussian beam In most of our experiments the transverse beam particle distribution has been Gaussian, it is therefore natural to try to extend the above results to the Gaussian beam case. In this case we would naturally expect the tune shift to depend on the betatron amplitude, thus leading to a tune spread. Assuming a longitudinally uniform and axisymmetric beam of singly charged particles, the transverse Gaussian charge density distribution can be written 2 ! r (s; r) = e
n exp , 22 ; n = 2N2 C (2.21) ?

?

where N is the total number of particles, ? the transverse rms beam size, C the ring circumference and e the electron charge. The longitudinal direction is speci ed by s and the transverse by the distance from the center of mass r. The number n is called the peak density. In a bunched beam, where the longitudinal linear density varies, the peak density is given by p 3 (2.22) nb = 822NL ?

where L is the FWHM length of the bunch, and we have assumed a parabolic longitudinal bunch shape. The electrical eld (in the non-relativistic limit, which we use) from a Gaussian beam of constant sigma is purely radial. Its magnitude is found to be 2n " 2 !# e r ? sc E? (r) =  r 1 , exp , 22 (2.23) 0

via Gauss' law.

?

2.2 Tune shifts from self elds

13

In the smooth approximation, i.e., assuming uniform focusing, with axisymmetric con nement, the transverse con nement force can be written in terms of the betatron tune Q? as F?conf = ,k
r ; k = m
(Q?
!rev )2 (2.24) thus the total force on a single charged particle with a transverse displacement r is

F?tot(r) = e
E?sc (r) , k
r

(2.25)

With a potential which includes the space-charge eects, the time it takes for a particle at distance r0 from the beam center to do one (betatron) oscillation can be calculated to be Z 0s m 0 R T =4 dr (2.26) r 0 tot 00 00 2 ,F (r )dr r0

r0

?

where r0 and r00 are integration variables. In gure 2.6 equation (2.26) has been used to calculate the tune of a particle with a betatron amplitude equal to the rms size of the beam as a function of the beam size (and thus the peak density - as the tune shift of a particle with this amplitude only depends on the density). 107

3

Betrotron Tune

106 2 105

1.5

1 104

Peak Particle Density [cm- 3]

2.5

0.5 Density 0 10-4

Tune 10-3

103 10-2

RMS Beam Size @ 100nA [mm]

Figure 2.6: Tune shift and peak density calculation for a particle with an amplitude equal to the rms radius of the (Gaussian) beam, as a function of the beam radius for a 100nA beam (2.78
107 particles). A constant focusing with Q?=2.5 has been used. These calculations are valid as long as the tune shift can be considered a perturbation to the forces on the particles. However in gure 2.6 the tune is calculated for tune shifts up to Q?. The reason is that, even though the exact value of the tune as a function of density most likely cannot be trusted, we still nd that the peak density corresponding to a tune shift of Q? is very close to the density in a zero emittance beam as found in Section 6.1. It thus seems that the approximation may be better than expected.

14

The physics of storage rings

In reference [54] simulations show that the important parameter, when approaching a resonance through space-charge tune shift, is the rms tune shift of the beam. This may seem counter-intuitive, as one would expect the tune spread to play a role in the in uence of the resonance on the beam, i.e. that it is important whether an individual particle has been shifted close to the resonance. However when space-charge is important enough to tune shift the beam signi cantly, the heating of particles with tunes close to a resonance is damped by the interaction with the bulk of the beam. It therefore seems reasonable only to look at the tune shift of the rms particle as a rst approximation, as was done in gure 2.6 (the rst order rms tune shift inpa Gaussian beam is equal to the rst order tune shift of a uniform beam with radius a = 2 [38] 4). In bunched beams the situation is slightly more complicated, because the longitudinal density varies along the beam. However, the bunches we have studied are very long compared to their transverse size, and for tune shift calculations they will be treated as coasting beams with an increased longitudinal density.

2.3 Crystallization A measure for the in uence of space-charge in a one component plasma (like a stored beam of a single type of ions) is the relation between the potential energy from the inter particle Coulomb repulsion and the thermal energy. A parameter known as the plasma parameter , describes this  4 ,1=3 2=40 a ( Ze ) where a = 3 n (2.27) ,= k T B where Ze is the ion charge, T the temperature, kB the Boltzmann constant, a the mean inter particle distance and n is the number density. Thus when , is large the thermal energy is small compared to the potential energy from the inter particle Coulomb repulsion.

2.3.1 The ground state In principle a zero emittance beam, which corresponds to in nite ,, would have a uniform charge distribution due to the linear con nement forces5. This is however under the assumption that the beam is a continuous medium, which it is not, as it consists of a nite number of discrete charged particles. This fact has a large in uence on the lowest (coldest) state of the beam, the ground state of the stored beam, as was rst proposed by Schier and Kienle in 1985 [91]. They speculated that due to the grainy nature of a 4 The peak density is equivalent to the density in a uniform beam of radius

p

= 2. Thus the rst order tune shift in a Gaussian beam can be calculated using the expression for a uniform beam with the peak density of the Gaussian beam. 5 The uniform charge distribution is a solution to the Poisson equation with harmonic con nement along the chosen nite dimensions, consider that r2 = ,=0 where  is a constant, is solved by  = kx x2 +ky y2 . For a detailed discussion on equilibrium in three dimensions see [22]. Recently a review article on equilibria and thermodynamics of trapped plasmas has also been released [76]. a

2.3 Crystallization

15

charged particle beam, it might crystallize into an ordered state at ultra-low temperatures (, > 170), like what was expected for cold ions trapped at rest [81]. In 1986 Molecular Dynamics (MD) studies of cylindrically con ned charged particles revealed unforeseen shell structures [86], and in 1992 the rst experimental con rmation was accomplished in a ring ion trap experiment [6]. The precise structure and the number of shells depend on the particle density and the con nement forces. Using a normalized dimensionless linear density de ned by

!1=3 2 a N 3 q  = q = C 8 m!2Q2 0 0 ?

(2.28)

where  is the line charge density, a the inter particle distance in the uniform beam (the Wigner-Seitz radius), and N the number of particles, it is found in MD simulations that the structure of the beam changes as listed in Table 2.1 [35]. Linear Density 0 <  < 0.709 0.709 <  < 0.964 0.964 <  < 3.10 3.10 <  < 5.7 5.7 <  < 9.5 9.5 <  < 13

N 0 - 6.8
105 6.8
105 - 9.3
105 9.3
105 - 3.0
106 3.0
106 - 5.5
106 5.5
106 - 9.2
106 9.2
106 - 1.3
107

I [nA] 0 - 2.4 2.4 - 3.3 3.3 - 11 11 - 20 20 - 33 33 - 47

Structure 0 - String Zigzag Shell6 Shell + string 2 Shells 2 Shells + String

Table 2.1: Structures for increasing linear density [35]. The equivalent particle numbers are for ASTRID with an average transverse focusing of Q?=2.5, and the current are for a 99.1 keV 24Mg+ beam. The indicated structures are to be interpreted as follows: In an ultra-cold low density beam the particles may stabilize into a state where they are all on the same orbit, and thus constitutes a string of particles. As the number of particles is increased the Coulomb repulsion causes rst the formation of a 2D beam where the particles lie in one plane but with alternating deviations from the ideal orbit (i.e. a zig-zag). Later when more particles are added it turns out that the particles will stabilize on a shell, i.e. that the beam is a hollow cylinder with walls one particle thick. Increasing the density further will cause the structure to change by introducing rst one string and a shell, then two shells and no string, then two shells and a string and so on. The possibility of reaching a crystalline state in a storage ring is intriguing, rst of all it would open up for measurements of unprecedented energy resolution, and it would increase the luminosity in colliders as recently discussed by Wei and Sessler [106].

16

The physics of storage rings

2.3.2 Stability

There are several considerations that need to be done before we can be sure that the crystalline state observed in traps is possible in a storage ring. First of all, as was pointed out early [86], the ground state might not be stable due to the periodic focusing, or due to the shear in the bending dipoles. Shear arises from the fact that the path length is larger for the outer particles than for the inner, thus the crystal will be exposed to shear stresses. This has been investigated in much detail using Molecular Dynamics (MD), and the most recent results give the following criteria for a stable crystalline ground state [104, 105]  The storage ring must be alternating gradient (AG) and operated below transition energy. p  The periodicity of the machine must be more than 2 2 times the betatron tune. The rst condition arises from the criterion of stable kinematic motion under the Coulomb interaction when particles are subject to bending (related to the negative mass instability). The second arises from the criterion that the phonon modes of the crystal are not resonant with the machine lattice periodicity (the highest frequency of the crystal is the monopole p mode, which can be shown to have a frequency of 2!? [31]). If these conditions are ful lled the crystalline state can in principle last for a very long time. The periodic focusing and bending do not cause heating of the crystal but make it 'breathe' when moving around the ring [104]. However in a real machine various eects will cause heating which must be compensated by continuous cooling.

2.3.3 Reaching the crystalline state

Unfortunately ASTRID does not ful ll the criteria for a stable crystalline state, as the periodicity is only four, however this might not yet be a cause of worry, as it might not even be possible to reach the required densities. As was mentioned in Section 2.2 the space-charge causes a tune shift. The crystalline beam is the extreme in this sense, as the incoherent motion is that the particles oscillate around their respective lattice sites, thus the incoherent tune has been shifted to virtually zero. This means that we have to cross several integer and half-integer resonances, which might not be possible at all, as noted in [31]. Another point, realized early but only treated in detail recently, is the shear. As mentioned in the previous section the crystal can survive shear, but stability implies that in the ground state, the angular velocity of the particles should be constant, i.e. the revolution frequency. However, the laser force tends to give the beam a constant linear velocity, thus laser-cooling in its present form would heat the crystalline ground state. A solution to this is so called tapered cooling , which takes the local dispersion into account, and thus cools to constant angular velocity [90, 103]. 6 In the region 0.964 <  < 3.10, the surface distribution of particles on the shell has been studied, and

varies between dierent structures [35].

2.3 Crystallization

17

When considering how to reach the crystalline state we should also consider the heating due to the coupling of energy from the center of mass motion into the beam rest frame due to the Coulomb interactions between the particles. In the low density limit where only binary encounters are important this is what we call intra-beam scattering. When spacecharge forces are no longer negligible and we thus approach the density of the crystalline beam the simple models (which we will discuss in more detail in Chapter 5) for the heating due to intra-beam scattering no longer apply. This situation has been studied in MD simulations, and it has been found that the heating increases when the density is increased until the crystallization starts, at which point the particles start to freeze into the crystalline structure and the heating is observed to drop signi cantly [105]. Thus in order to reach the crystalline state a cooling force large enough to overcome the maximum heating is needed [105].

18

The physics of storage rings

Chapter III Beam cooling Cooling mechanisms are of vital importance for many storage ring applications. Without cooling the ion beam will heat up due to collisions with the rest gas, and diusion due to intra-beam scattering, which couples the lab-frame kinetic energy into the rest frame of the beam. Eventually these heating mechanisms will lead to beam loss due to the ever increasing beam size in the nite vacuum chamber. In this chapter the method of laser-cooling used for the experiments presented in this thesis is described in detail. Furthermore a short introduction to other cooling methods in storage rings is given. The discussion includes application areas as well as advantages and drawbacks of the various methods.

3.1 Liouville's Theorem In order to understand why cooling is necessary and how it is accomplished it is important to understand the physics behind the systems of interest. The discussion here is a short version of a good review given in [62]. The dynamics of ions which are electromagnetically con ned can be described by a Hamiltonian, which in general may be time-dependent. The motion of N particles can be described as a phase space trajectory of a single point in 6N dimensions (3 generalized positions and 3 conjugate momenta per particle). If we have an ensemble of Hamiltonian systems having a distribution in 6N phase space, Liouville's theorem states that the phase space density function f is constant df = 0 (3.1) dt However this information is not easily applicable in practice, as we usually consider a set of particles, each represented by a point in 6 dimensional phase space. It turns out that for non-interacting particles an analogue of Liouville's theorem holds. This states that the density of particles in 6-dimensional phase space f (p; q; t) is a constant in time. As interactions are in fact present, especially in a crystalline beam where the interactions are dominant and thus leads to crystallization, it is more appropriate to consider this

20

Beam cooling

alternate version as stating that the density is constant on some time scale (in a crystal this time scale is extremely short). As discussed in the previous chapter we often see that two dimensional projections of 6D phase space are independently stationary (the emittances are constant if we only consider the con nement forces). As we have discussed, and will become evident from the experimental results, particle interactions are important, thus it is relevant to consider how Liouville's theorem is altered by this fact. Liouville's theorem reduces to the Boltzmann transport equation when interactions are included ! @f + v
r f + F
r f = @f (3.2) r @t m v @t collisions

where F includes all external forces and the bulk force from the beam's own electromagnetic elds. The evaluation of the collision term on the right is in general rather complicated, and usually only multiple small-angle binary collisions are considered. In storage rings the changing transverse velocity spread and the changing particle density around the ring as well as the dispersion have to be taken into account, and an extensive theory was rst developed by Piwinski [80], and a computer code which implements this has been developed at CERN [24, 25]. This is discussed in detail in Appendix C. As will be discussed later, collective eects as well as small and large-angle scattering events are important for laser-cooling of ion beams.

3.2 Standard Cooling mechanisms In order to put laser-cooling into perspective two standard cooling mechanisms for heavy ions are discussed as well as the auto-cooling feature of extremely relativistic particles. The discussion of electron cooling and stochastic cooling is based on a good review of beam cooling given in [32]. When talking about beam cooling in this context we are interested in cooling times and ultimate temperatures (emittances). By cooling time we mean the e-folding time, which is given by !,1 1 d  = ,  dt (3.3)

where  is the rms spread of the velocity distribution (or betatron amplitude if it is the emittance which is in question).

3.2.1 Electron Cooling

Electron cooling was rst proposed in 1967 by Budker [11], and is to date the most generally applicable method for cooling of charged particle beams. The principle is quite simple, and an illustration of an electron cooler is shown in Figure 3.1. The principle is that a beam of cold electrons is accelerated to match closely the velocity of the desired particle beam, and then merged with the particle beam over some distance.

3.2 Standard Cooling mechanisms

21 toroids

electron collector

ion beam

solenoids electron gun

Figure 3.1: Schematic diagram of an electron cooler. Through the Coulomb interaction the relatively hot particle beam will transfer energy to the cold electron beam, and thus be cooled in the moving frame of the particles. Thus in essence the ion beam is coupled to an external heat sink. Due to the initial acceleration of the ion beam the electron velocity distribution is

attened, i.e. the longitudinal temperature is much smaller than the transverse [59]. In reference [59] a typical electron cooler is set to have a longitudinal temperature of 2
10,4 eV ( 5K) and a transverse temperature of 0.5eV ( 6000K). Electron cooling is therefore much more eective in the longitudinal dimension than in the transverse, and the high transverse temperature also decreases the eciency in the longitudinal dimension somewhat due to intra-beam scattering. Electron cooling times depend on a lot of factors, in general the cooling time scales as [59] 2 lab  nmZem2 p (3.4) e

where mp is the mass of the cooled particles, Z their charge, and ne the electron density. Typical cooling times for an ecient system are in the range from tens to hundreds of milliseconds. Apart from cooling of particle beams, electron coolers are also used for internal targets in atomic and molecular physics, as very low energy collisions can be accomplished with good energy resolution.

3.2.2 Stochastic Cooling Stochastic cooling is an idea due to van der Meer [99], and is the heart of many high energy particle accumulation facilities. Stochastic cooling, schematically illustrated in Figure 3.2, is based on a feedback system to correct errors in the particle trajectories. The idea is that a pickup is positioned in a place which has a large value of the beta function, thus large emittance causes large position deviation. The pickup measures the position deviation, and at a position an uneven number of 90 degree phase advances away this position deviation will be an angle deviation, which is corrected by a kicker. In a continuous beam the signal on the pickup would just be constant. However, the beam consists of particles, and thus if the bandwidth of the pickup

22

Beam cooling

kicker Amplifier Pickup

Figure 3.2: Schematic drawing of a stochastic cooling system system is large enough, and we therefore probe a small enough fraction of the particles the method works. In practice it is somewhat more complicated, as intra-beam scattering and momentum spread in the beam causes phase mixing, which in the bad case can cause us to correct the wrong particles. It turns out that the optimum cooling time of a stochastic cooling system can be described rather well by [100] (3.5)  = 2NW where W is the bandwidth and N the number of particles. Bandwidths are typically of order GHz, so for 109 particles we will have a cooling time of order 1s. Stochastic cooling is thus comparatively slow, however in high energy machines it is often the only choice, as electron coolers are dicult to make at these energies and their eciency decreases with energy. A detailed review of stochastic cooling is given in [65].

3.2.3 Radiative Cooling

For completeness we should also mention how cooling is accomplished in electron synchrotrons. Of course one could use stochastic cooling, but fortunately it turns out that circulating high energy electron beams are self-cooling. The principle is quite simple and relies on the fact that the synchrotron radiation emitted by the electrons scales with their energy. The energy release due to synchrotron radiation per unit time is given by [41] 2 P = 23 e2c 4 4

(3.6)

where  is the radius of curvature. At the energies reached in accelerators to date this energy loss is negligible for protons and heavier particles due to the 4 scaling. The energy loss in each revolution is compensated by an RF accelerating voltage, and the synchronous particle is the particle whose energy loss in one revolution is exactly

3.3 Laser-cooling

23

compensated by the accelerating cavity. Particles with dierent phase or momentum will oscillate in phase and energy around the synchronous particle (synchrotron phase space). As the energy loss due to synchrotron radiation increases with energy it will provide damping of the oscillations, i.e. cooling. Damping of the betatron oscillations arises because the momentum loss due to synchrotron radiation occurs in the direction of the motion of the individual particles, but the momentum gain in the RF cavity is purely longitudinal - thus the divergence is decreased [101].

3.3 Laser-cooling Laser-cooling is a means to cool atoms and ions using the momentum changes occurring when these systems absorb or emit photons. The rst steps for development of this technique were taken by Ashkin in 1970 [2], who demonstrated that a resonant absorption combined with spontaneous emission of photons could lead to a net force in the direction of the incoming (resonant) photons. In 1975 Hansh and Schawlow [34] showed how this radiation pressure could be used to cool a low-density gas, and at the same time Wineland and Dehmelt proposed cooling of ions bound in an electromagnetic trap [109]. Shortly after, in 1978, Wineland et al. [110] presented the rst experimental observation of this new cooling technique. Since then laser-cooling has developed rapidly, and facilitated in 1995 the attainment of a Bose-Einstein condensate [1].

3.3.1 Doppler cooling

Laser-cooling relies on the fact that photons carry momentum. When an ion absorbs a photon its momentum therefore changes by the amount of the absorbed photon h k. When the ion later decays the photon is reemitted and the momentum of the atom is therefore changed by the same amount again. But the radiation pattern of the spontaneous decay is spatially symmetric, and by repeating the above process over and over the spontaneous emissions will not contribute any net momentum, and the ion will gain a net momentum increase in the direction of the incident photons. Thus by shining a laser resonant with a closed1 transition onto an ion it can be accelerated. The process is illustrated in Figure 3.3. Due to the Doppler shift this force will be velocity dependent, and have a form described by the expression [42] =2)2 (3.7) F (v) = 21 h kS , ( , vk)2 (, + (,=2)2 (1 + S ) where  = ! , !0 is the detuning of the laser from resonance, k = 2= is the wavenumber of the absorbed photons, , is the spontaneous decay rate from the upper to the lower 1 A closed transition is a transition where the probability that the excited state decays to another state

than the lower state is vanishing. This is the simplest system for laser cooling. Any scheme were it is possible to make the ions circulate through a nite (and small) number of levels (possibly using more laser frequencies) can be used.

24

Beam cooling

level, and S is the saturation parameter given by S = I=Is = 22=,2, where I is the laser intensity, Is = hc,=3 is the saturation intensity, and  = dE0 =h is the Rabi frequency, where E0 is the amplitude of the electric eld and d is the eective dipole matrix element [63]. a)

b) γ

V c)

V + dV d)

V + dV + dV’

V + NdV

Figure 3.3: Illustration of the laser force. a) A photon is incident on the ion. b) The photon is absorbed, and its momentum is transferred to the now excited ion. c) The excited ion decays, and the photon is reemitted. d) As the spontaneous emission process is spatially symmetric it gives no net momentum change, thus after N absorptions and emissions the momentum change of the ion is N times the momentum of each incoming photon. If we make a Taylor expansion of this force around v=0, we obtain F (v) = F0 , v + O(v2) (3.8) that is a constant force term, a friction term and some higher order terms. The constant and friction terms are as follows =2)2 (3.9) F0 = 21 h kS , 2 + (,(,=2) 2 (1 + S ) 2 (3.10) = , 12 h k2S , (2 + (,(,=2)=2)2(1 + S ))2 2

from which we conclude that for both S1 and S1 the friction term goes to zero. An optimum friction force as a function of detuning p can bepfound by dierentiating the friction term, and this optimum is given by  = ,, 1 + S=(2 3). The constant term F0 means that the ions are pushed by the lasers; we are however (usually, see Section 3.3.2) interested in a stable situation. This can be accomplished by introducing a force with equal magnitude and opposite sign, i.e. a counter-propagating laser. The resulting force from two counter-propagating lasers with the same intensity and detuning from resonance is given by " # 2 2 1 (, = 2) (, = 2) F (v) = 2 h kS , ( , vk)2 + (,=2)2 (1 + S ) , ( + vk)2 + (,=2)2 (1 + S ) (3.11)

3.3 Laser-cooling

25

where we can see that F(v=0) is zero. However we can also see that by shifting the detuning of the co-propagating laser blue (higher frequency) and the counter propagating red, we can freely select at which velocity we wish the resulting force to be zero. In this way we can decrease the velocity spread of a moving particle beam. The cooling can be interpreted physically by considering the particle rest frame. When a particle is moving towards one laser, the frequency of this laser is Doppler-shifted blue, thus by detuning both lasers red ( < 0) moving particles will always be closer to resonance with the counter-propagating laser, than with the co-propagating laser, thus the term Doppler cooling. So far we have ignored the spontaneous emission of photons when the system decays. These spontaneous emissions lead to diusion, and the ultimate temperature reachable with Doppler cooling is given by an equilibrium between the friction and the diusion, this limit, known as the Doppler limit, is given by (3.12) TD = 2hk, B where kB is the Boltzmann constant. This limiting temperature is in general very low compared to the lowest temperatures achievable with electron cooling, and for magnesium it is 1.0 mK. This limit is however not fundamental, a fact which was rst discovered experimentally [52] and several techniques have been developed which can overcome this limit [3, 20]. However in a storage ring there are other sources of diusion and heating than the spontaneous emission of photons, for example intra-beam scattering and scattering on the residual gas. These processes are in general much more important than the laser induced diusion , as we will discuss in later sections, but the achievable temperatures are still about an order of magnitude lower than with electron cooling, and the cooling times, as we will see next, are very short.

3.3.2 Coasting beams In a storage ring the initial velocity spread may be much wider than the Doppler width of the cooling transition. In general it is therefore necessary to nd some way of collecting the particles into a narrow distribution which can be kept by the lasers. The small friction term discussed before is therefore not very important in a storage ring compared to the direct force term. Laser-cooling a coasting beam can be accomplished in a couple of dierent ways.

 Using two counter-propagating lasers to decelerate particles moving too fast, and

accelerate slow particles.  Using an external acceleration force, such as an induction accelerator to accelerate the beam into resonance with a counter-propagating laser, thus generating a stable point in the accelerating reference frame.

26

Beam cooling  Accelerating the beam by frequency chirping the laser2. This will, in the accelerating

reference frame, be equivalent to the previous method. The dierent methods are illustrated in Figure 3.4. The two last methods are special in the sense that they do not imply a stable situation, in the sense that at some point the maximum velocity possible by the chosen acceleration method is reached and thus the cooling has to be stopped. In our experiments we have used the rst method which is stable in the sense that there is no intrinsic limitation to the time the cooling can be kept active. a)

F

b+c)

F

v

v stable point

Figure 3.4: Three methods of coasting beam cooling. a) Two counter propagating laserbeams shown in the lab frame of reference. b+c) Accelerated beam or chirped laser, forces shown in accelerated frame of reference. There are actually two ways to use the rst cooling method. One is to detune both lasers far red from resonance before the beam is injected, and then during some time chirp the laser frequencies blue until the desired nal detuning from resonance is reached. This method is necessary if we initially have a large velocity spread compared to the desired detuning, as the narrow range of the laser force limits the collection eciency of ions with velocities below or above the velocities resonant with the two lasers. If on the other hand the longitudinal velocity spread of the injected ion beam is smaller than the desired longitudinal velocity spread to be achieved by laser-cooling, the cooling lasers can be set to the desired (small) detuning from before injection. In this case the lasers are used to prevent the blow up of the longitudinal velocity distribution due to heating. The cooling time with laser-cooling depends on the details of the transition, and the initial velocity spread. With the standard setup we use for ASTRID the cooling time is a few milliseconds, and the theoretical best total cooling time of an initially warm beam is about 4ms. Laser-cooling is thus about an order of magnitude faster than electron cooling. The range of the laser-force is, as mentioned, rather short. This means that the cooling system will not recollect particles in the tails of the velocity distribution. As we will discuss in Chapter 5 energetic collisions of transversely hot particles may cause particles to gain a longitudinal momentum change which brings them out of the range of the laser force. This 2 In this thesis 'chirp' is used to describe change of the laser frequency, in order not to be confused with

'scan' which will refer to the spatial scan of the probe laser by the galvo system described in Chapter 4.

3.3 Laser-cooling

27

problem can be solved by extending the range of the laser force, however it will still be desirable to have as sharp an edge as possible in the force to reach the lowest temperatures. There are a couple of ways to extend the range of the laser force. One method, demonstrated at the TSR in Heidelberg, utilizes optical adiabatic excitation [102]. Another method was proposed in 1992 [43] and demonstrated in practice recently [4]. This method consists of generating a frequency comb by letting the laser pass multiple times through an acousto-optic modulator. This can be done in such a way that a sharp edge is still kept to one side, such that low temperatures can be reached. A third method, inspired by the Zeeman slowers used to decelerate atomic beams [78], would be to introduce a longitudinal magnetic eld in the ion beam which Zeeman shifts the magnetic sublevels of the electronic con gurations. In magnesium we nd that we can shift the transition frequency by one transition line width (45MHz) with a eld of 30 Gauss - which makes this method feasible. The advantage of using this method is that it will be possible to use in connection with the tapered cooling scheme which will be discussed in section 6.5.1, and an implementation of this system in ASTRID is therefore being considered. A disadvantage of the frequency comb scheme is that it does not utilize the available laser power very economically, as a lot of modes may have more power than needed to collect tails, and power is lost due to the losses during the round trips in the cavity. This problem does not arise with the two other schemes where the force can be tailored.

3.3.3 Bunched beams As mentioned above, intra-beam scattering may cause tails in the longitudinal velocity distribution of the circulating beam when laser-cooling a coasting beam to ultra low temperature. This is due to the narrow capture range of the laser force, making it unable to recollect particles which end up in the tails. If however, the beam is longitudinally bunched by a sinusoidally varying longitudinal electrical eld with a frequency which is a harmonic of the revolution frequency, the particles will undergo longitudinal phase oscillations around the synchronous particle, and tail particles may be recollected by the lasers [30]. Figure 3.5 shows the forces on a circulating charged particle in such a sinusoidally varying eld. We observe that due to the nature of the forces the particles will cluster into bunches around the stable points, and thus a coasting beam will be split into h bunches, where h is the harmonic number given by !RF = h
!0 where !0 is the revolution frequency and !RF is the frequency of the longitudinal eld variation. In earlier work we used an RF cavity to generate the longitudinal electric eld [30]. However, as the supply for this cavity turned out to be noisy, and the elds necessary for our purposes are quite small, we decided to switch to a dierent bunching method. In the work presented here the bunching has been performed by exciting a cylindrical drift tube with a sinusoidally varying voltage. The average force on a particle passing through the tube arises due to the change in the potential between entry and exit and it is given by

F (s) = ,F0
sin( 2Ch s) ; F0 = q 2VCrf  sin( h C L)

(3.13)

28

Beam cooling Too slow

t Too fast

Potential [arb. units]

F

-1.5

-1.0

-0.5 0.0 0.5 Relative position in bunch [m]

1.0

1.5

Figure 3.5: Left: The stable point of a stored particle exposed to a sinusoidally varying longitudinal force. Right: The same eld averaged over many revolutions corresponds to a pseudo potential of sinusoidal form. The dotted line shows the harmonic potential extracted from a rst order Taylor expansion of the sinusoidal eld. where q is the charge of the circulating particles, Vrf is the amplitude of the applied sinusoidal voltage, C is the ring circumference, L is the tube length, s is the relative displacement of the particle from the center of mass of the bunch, and  = 1= 2 ,  is the slip factor. In the rst order approximation where sin(x)  x the potential is harmonic. The dierence between the harmonic and the sinusoidal potential is illustrated in Figure 3.5. The harmonic portion of the potential can be written s 1 0 (3.14) Uk(s) = 2 m!s2(s2 , s20) ; !s = 2hF mC where !s is called the synchrotron frequency, and the longitudinal oscillations called synchrotron oscillations. As the potential is not quite harmonic, the synchrotron frequency will depend on the amplitude. There are two ways in which to laser-cool a bunched beam. In the rst method the laser is frequency chirped, as when cooling coasting beams. In the second the laser is kept at a small red detuning (given by the desired nal temperature) from the beginning of the cooling. In Figure 3.6 it is illustrated what happens in longitudinal (or synchrotron) phase space during cooling with a xed laser. The particles oscillate in energy (momentum) and time (position) relative to the synchronous particle, which has the correct revolution frequency and phase relative to the applied RF eld. Therefore the particles with velocity amplitudes larger than the amplitude resonant with the laser will be brought into resonance twice per synchrotron revolution, and thus have their amplitude decreased, i.e. be cooled. This means that it is not necessary to chirp the laser frequency to cool a beam with a large velocity spread compared to the range of the laser force. The time scale for the xed laser cooling process is strongly related to the synchrotron frequency. We can calculate the time is takes to cool a particle with velocity amplitude
v by integrating the laser force on the particle over a synchrotron period, assuming a

3.3 Laser-cooling

29 ∆v

Laser

∆t

Laser

Figure 3.6: Illustration of longitudinal phase space. Particles oscillate in longitudinal phase space with the synchrotron frequency. The illustration shows how a co-propagating laser detuned to be resonant with particles with velocity lower than the mean velocity damps the synchrotron oscillations. harmonic potential and thus ignoring the discrete nature of the physical process. The synchrotron frequency, may however be much slower than the time it takes the laser to accelerate all resonant particles out of resonance (empty a velocity class). In that case the cooling time is dominated by the synchrotron frequency, and therefore rather long. In ASTRID we usually have a synchrotron frequency of  150 Hz, which is slow compared to the acceleration induced by the laser force, and the total cooling time of an initially warm beam can be found to be about 0.6 seconds - much slower than the few milliseconds necessary for cooling with chirped lasers. In ASTRID it would therefore be advantageous to use the chirped laser technique we often use in coasting beams. However the xed laser technique has the advantage that it is much simpler to use, and does not rely on a tunable laser system. The experiments in this work have therefore been done with xed laser cooling of bunched beams.

30

Beam cooling

Chapter IV Experimental Setup The experiments discussed later in this thesis have been conducted at the storage ring ASTRID in Aarhus. In this chapter the parameters of the storage ring are presented, and the experimental setup necessary for the study of laser-cooling is presented. In order to perform experiments with laser-cooling and study the behavior of the ion beam during laser-cooling, a laser system has been developed which can be used both for cooling of bunched and coasting beams. The laser system and recent improvements to it are described. It is also necessary to be able to observe the impact the laser-cooling has on the ion beam, and for this purpose a range of dierent diagnostics techniques are utilized. These techniques, of which some are unique to beams which can be laser-cooled, are described. The description of diagnostics includes a detailed description of a novel system for transverse diagnostics developed for the experiments. The new system utilizes the uorescence light from the laser-excited ion beam to measure the transverse beam density pro le. Understanding the diagnostics as well as the cooling system is important for understanding how the measurements are done, and how to extract information from the measurements. Furthermore, the new system for transverse diagnostics is rather dierent in many respects from ordinary storage ring diagnostics, dierences which make a detailed discussion necessary.

4.1 ASTRID The relevant ring parameters for the experiments performed for this thesis on the storage ring ASTRID are given in Table 4.1. A schematic drawing of the machine is shown in Figure 4.24 at the end of this chapter. ASTRID [67] is a four-fold symmetric machine, with four 90 degree bends, and four identical sections of two FODO cells each. There are 16 combination sextupole/correction dipole magnets for chromaticity and orbit control. The ion beam is de ected into the storage ring using a magnetic septum magnet. Located diametrically opposite to the septum is an electrostatic kicker, which kicks the injected beam onto the closed orbit. The

32

Experimental Setup

lattice parameters of one of the four identical sections corresponding to the data in Table 4.1 are shown in Figure 4.1. Parameter Symbol Value Circumference C 40.0 m Transition Energy

T 4.183 Momentum Compaction Factor  0.0572 Slip factor  0.943 24 Ion Species Mg+ Ion Kinetic Energy E 99.1 keV Revolution Frequency frev 22325 Hz Number of particles N 2.8
105 - 2.8
108 Equivalent ion current I 1nA - 1A Typical storage lifetime   30s Typical average pressure P 10,11 Torr Horizontal Betatron Tune Qh 2.27 Vertical Betatron Tune Qv 2.83 Horizontal at center of straight section h;0 12.1 m Vertical at center of straight section v;0 2.61 m Horizontal Dispersion at section center Dh;0 2.74 m

Table 4.1: Standard values for ASTRID when used for laser-cooling. Figure 4.24 also indicates the position of various beam diagnostics. This includes the 8 horizontal and 8 vertical position pickups used to monitor the beam position. Using the pickups in combination with shunts on the quadrupoles the beam can be centered in the quadrupoles to better than 1 mm [71]. The other parts of the diagnostics are discussed later in this chapter. The magnesium ions used for the experiments are generated in a so-called 'Nielsen' ions source, which is a standard plasma ion source [72]. The ions are accelerated electrostatically to an energy of 99.1 keV. This is the only energy used in the experiments presented in this thesis. The ions are mass separated in a 45 degree analyzing magnet, and pass through an injection beam line. The injection beam line has two sets of electrostatic quadrupoles to match the beam into the storage ring. Magnetic correction dipoles provide horizontal and vertical ne control over the beam position and angle at injection. An electrostatic chopper is used to select the desired length of beam to inject.

4.2 Laser-cooling setup for ASTRID Table 4.2 lists some characteristic values for the magnesium ions used for the experiments presented in this thesis. So far only two other ions have been successfully cooled in storage rings: 9Be+ and 6;7Li+ [26], however Hg+ has also been proposed as a possible candidate

4.2 Laser-cooling setup for ASTRID

33

15

Lattice Function Value [m]

Beta - X

Beta - Y

Dispersion - X

10

5

0

0

2.5

5

7.5

10

Relative Position in Ring [m]

Figure 4.1: Beta and dispersion functions for one of the four identical sections. The center of a dipole is located at 5m. [39]1. As discussed earlier the longitudinal velocity spread of the particles is easily much larger than the range of the friction force with Doppler cooling. It is therefore often desirable to have a laser system with tunable lasers. The system used for ASTRID consists of two identical laser systems with an Ar-ion laser pumping a tunable dye laser, which is, in turn, frequency doubled in an external cavity [69]. A schematic drawing of the setup is shown in Figure 4.4, and the detailed speci cations are listed in Section B.3. The UV laser beams from the cavities are directed into the storage ring through windows in the straight sections. The typical laser beam con guration at the storage ring is shown in the drawing of ASTRID (Figure 4.24). For the experiments to be possible it is mandatory that the laser system is stable, both in the sense that the laser power should be stable, thus the locking system should be good, and also that the frequency of the dye laser is stable compared to the transition line width of 42.7 MHz. Apart from these points, which will be discussed shortly, the spatial pro le of the laser beam is also important. It is of course desirable, to obtain as homogenous a cooling force as possible, that the intensity pro le of the laser beam is as homogeneous as possible. In the ideal case we would have a Gaussian beam, however, due to walk o2 in the very long (55 mm) KDP crystals used for the second harmonic generation, the laser intensity pro le is distorted by points of destructive interference as well as hot spots, and varying divergence for dierent parts of the beam. An example of a beam intensity pro le is shown in Figure 4.2. This is of course undesirable, especially when transverse cooling is studied. A new cavity using much shorter crystals of BBO is therefore under construction [44]. 1 In ASTRID 166Er+ was tested, but no cooling was ever detected [31]. 2 Walk o means that the second harmonic light is generated with an angle to the primary beam.

34

Experimental Setup Parameter Symbol Value 24 Ion Mg+ , 26 Ion mass m 3.9828
10 kg Ion Energy E 99.1 keV Ion Velocity v0 8.93
105 m/s Relative velocity 3
10,3 Transition 32S1=2 $ 32P3=2 Frequency in Vacuum vac 1.072
1015 Wavelength in Vacuum vac 279.6345 nm Wavelength in Air air 279.553 nm Doppler-shift at 8.93
105 m/s
 3.19
103GHz
 0.835 nm Transition linewidth , 2
42.7 MHz vl 11.9 m/s Saturation Intensity Isat 8.07 mW/mm2 Doppler cooling limit TD 1.0 mK

Table 4.2: Parameters for the ions used for laser-cooling experiments in ASTRID.

Figure 4.2: Intensity pro le of one of the lasers used for cooling. The side length of the laser spot is usually  3mm.

4.2.1 Frequency Drift In early experiments it was realized that the frequency stability of the dye lasers was not sucient. The problem arose because the frequency of the light emitted by the dye lasers changed slowly over time (see Figure 4.3). This induced problems when experiments were done where a longitudinally cold distribution ( 1 K) was maintained, as the drift could cause changes in the relative detuning of the two lasers, thus changing the conditions for the measurement (the longitudinal temperature). A simple improvement consisting of a thermal insulation of the reference cavities for the dye lasers was implemented. Figure 4.3 illustrates the improvement from putting extra thermal insulation around the reference cavity of the dye lasers. The curves show the beat frequency of a mix of the dye laser and a reference laser (a stabilized He-Ne laser ( = 543.5150 nm)) as a function of time. The measurements were conducted under similar laboratory conditions (room

4.2 Laser-cooling setup for ASTRID

35

temperature, noise level etc.). These curves were measured by shining the light from the two lasers simultaneously onto a fast photo diode and analyzing the photo diode output to extract the beat frequency. 180

230

Beat Frequency [MHz]

No insulation of ref. cavity

Ref. cavity insulated

170

220

160

210

150

200

140

190

130

0

60 120 Time [min.]

180

180

0

60 120 Time [min.]

180

Figure 4.3: Laser frequency drift as a function of time for two cases. The dierence between the insulated and the un-insulated cavity is clear - a decrease in the frequency drift over time of approximately four is observed with the insulation installed. For the insulated case we observe that the dierence between the maximum and the minimum beat frequency is approximately 12 MHz, thus reasonably small compared to the transition line width of the 24Mg+ cooling transition (42.7 MHz). The beat frequency was measured by using a fast counter. The fast oscillations observed in both measurements is frequency jitter in the dye laser (the few large spikes are caused by bubbles in the dye supply).

4.2.2 Cavity Locking As explained earlier the dye lasers are used for pumping light into two frequency doubling cavities. Due to temperature variations, acoustic noise and dye laser drift which might bring the cavities out of resonance, the round trip length of the cavities can be changed to keep the cavities resonant at all times. This is done by having one of the cavity mirrors mounted on a piezo electric crystal. Keeping the cavity on resonance is called locking. When chirping the laser frequency for cooling the needed change in cavity length to stay on resonance may be too much for the fast piezo used for locking. Therefore the fast piezo is mounted on a slow piezo whose movement is correlated with the chirp signal for the dye laser.

36

Experimental Setup

The original AM locking3 technique for the UV-cavities was not stable enough for our experiments, therefore a polarization locking system, based on the technique of polarization spectroscopy of a re ecting reference cavity [33], has been implemented. The polarization locking technique, in contrast to the FM locking technique [7], was chosen mainly because the polarization technique oers a very wide error signal as compared to the FM technique and it was therefore expected to be better to re-lock after large frequency jumps [33]. Furthermore it was expected to be easier to work with due to its simplicity (i.e. the error signal is generated by simple optical components). Theory of polarization locking

Following the theory given in [33] an experimental setup as illustrated in Figure 4.4 is considered. The dye laser generates vertically polarized light. As the cavity transmits Ia

λ/2 λ/4 Ib UV-Light 280nm

Ar-ion Laser

Dye Laser

λ/2 SHG - Cavity

Figure 4.4: Setup for polarization locking of a SHG cavity horizontally polarized light, the light is passed through a =2-retarder4 to rotate the polarization. The light is rotated so that its polarization forms a (small) angle  with the transmission plane of the cavity (horizontal). Consider the slightly skew-polarized light as two polarization components, one parallel with the cavity transmission axis, and one perpendicular to it. The light from the dye laser is directed onto the cavity input mirror and it is investigated how the re ected light depends on the transmission of the cavity. The perpendicular component of the incident light will not be transmitted by the cavity (make a round trip) thus only the direct re ection contributes to the re ection, whereas the parallel component, apart from the fraction re ected o the input mirror, experiences a frequency dependent phase shift due to its passage through the cavity. Thus the resulting re ected light will be elliptically polarized, and the eccentricity will depend on the frequency of the light. When the cavity is on resonance, the phase shift of the parallel component is a multiple of 2, and the transmitted 3 AM locking: The position of the piezo mounted mirror is modulated. By analyzing the power build

up in the cavity it is possible to continuously update the mean position of the mirror to keep the cavity on resonance (maximum power build up). 4 A retarder is a birefringent crystal with the feature that light polarized perpendicular to the so-called fast axis is retarded a speci c fraction of a wavelength during passage compared to light polarized parallel to the fast axis. Light which is polarized at 45 to the fast axis of a =2 retarder will have its polarization rotated 90 .

4.2 Laser-cooling setup for ASTRID

37

light will be linearly polarized. Thus the polarization of the light re ected o the input mirror contains information, which after the following analysis turns out to be useful for locking the cavity to resonance. The light incident on the cavity is decomposed into two components, parallel and perpendicular to the transmission axis of the cavity, this can be written in the plane wave approximation Ek(i) = E (i) cos ; E?(i) = E (i) sin  (4.1) where E (i) is the amplitude of the incoming beam. The re ected parallel component can, neglecting losses induced by the frequency doupling process in the cavity, be calculated by the standard approach in [8]. (q i ) re T 1 ( r) ( i) Ek = Ek R1 , pR 1 , Re,i 1 (q ) T R cos  , R + i sin  1 ( i ) = Ek R1 , p (1 , R)2 + 4R sin2 =2 (4.2) R1 where  is the phase shift per roundtrip, R1 and T1 are the re ectivity and transmitivity of the input mirror (in our case R1  2%), and R the loss per roundtrip in the cavity. For simplicity (of calculation) the =4 retarder's fast axis is parallel to the transmission axis of the cavity. When the elliptically polarized light passes through this retarder it's transformed into two orthogonal linear polarized components with an angle of 45 to the horizontal. The =2 retarder is used to rotate the components to be parallel and perpendicular to horizontal, for ease of splitting in the polarization beam splitter5. The intensities of these two components are measured and the dierence signal is6  Ia , Ib = I (i) cos  sin  (1 , RT)21R+1 4sin (4.3) R sin2 =2 where I (i) is the intensity of the incident beam. An experimental measurement of this signal can be seen in Figure 4.5, and is ideal for locking of a cavity. Experimental results

The described system has successfully been implemented for locking the UV cavities (see Figure 4.5). The stability of the system has been probed over long time spans during many experiments, and the system has shown the expected ability to re-lock after large frequency jumps. Furthermore the ability to lock has improved to a degree, that the cavities can be locked even if the dye lasers are not locked. The diculties of locking at high pump powers, as mentioned in [70], have only improved slightly. The reasons for these diculties are not completely clear (when scanning the cavity length it shows up as a decrease in nesse for high pump powers), but it is assumed to be due to heating of the KDP crystal. The 5 This addition to the setup makes it equivalent to the one described in [33] 6 Note: Formula (6) in [33] is misprinted, the correct form is stated above

38

Experimental Setup B)

Transmitted Power

Intensity

Polarization signal amplitude [arb. units]

A)

Intensity

Polarization Signal

0.0

15.0

30.0 45.0 60.0 Misalignment angle [degrees]

75.0

90.0 Cavity length [arb. units]

Figure 4.5: A: Amplitude of polarization signal Ia , Ib as function of misalignment angle  (grey line: t to sin  cos ). B: Example of locking signal and cavity transmission signal. frequency doubling process may be in uencing this decreasing ability to lock when the pump power is increased. The intensity of second harmonic light increases as the square of the intensity of the pump, which means that at some point it will start having a signi cant in uence on the cavity losses (as mentioned earlier this was ignored in the calculations), which might give a contribution to the instability of the lock at high powers (50mW of UV at 1000mW of pump  5% losses - compared to the default intra-cavity loss of  2%).

4.2.3 Present Performance With these improvements, and the addition of a ow box over each dye laser generating a laminar ow of dust free air, which keeps the dye lasers from becoming dirty with dust, the laser system has become extremely stable. During the latest runs we have been able to run for many hours without any attention to the lasers necessary. This should be compared to earlier when the mirrors in the dye lasers needed to be cleaned at least once a day, and the UV-cavities needed attention more than once per hour. This stability has been a major step forward, and the results presented in the coming chapters could not have been done without these improvements.

4.3 Longitudinal Diagnostics Longitudinal diagnostics are naturally important for the study of laser-cooling as lasercooling is a means to cool an ion beam in the longitudinal dimension. Therefore the methods below have been around for some time, and been described elsewhere, thus only a brief introduction into each method is given.

4.3 Longitudinal Diagnostics

39

4.3.1 Bunch Shapes

As the dynamics and equilibrium conditions in ion beams depend heavily on the density of particles, it is crucial to the study of bunched beams to be able to measure the bunch length and shape. In our case this is done simply by recording the voltage induced on a capacitor through which the beam passes (it is actually the sum signal from a transverse pickup [45]). 1

1.2

1

Induced Voltage [V]

Measured Bunch Length [m]

Real Bunch Width Pickup measured Bunch Width

0.1

0.8

0.6

0.4

0.2

0.01 0.01

0 0.1

Actual Bunch Length [m]

1

1

2

3

4

5

6

7

Relative Position [m]

Figure 4.6: Left: Calculation of the eective FWHM extracted from the induced voltage on a 10cm square pickup from the passage of a parabolic bunch. Right: Example measurement of a laser-cooled bunched beam including parabolic ts to the bunch shapes. Each bunch consisted of  3.8
106 particles. Now, it is clear that if the bunches which pass the pickup are much shorter than the pickup, we only measure the pickup and not the bunch. In earlier studies the bunches were quite short [30], and an approximate formula for taking the pickup shape into consideration was developed [70]. In appendix A.2 the equations describing this are reproduced. In the experiments presented in this thesis the bunches were quite long, and, as we have illustrated in Figure 4.6, the in uence on the FWHM, which is the main gure of merit for the bunches, is negligible for bunches longer than  25 cm. We have therefore chosen to ignore the nite pickup length in the FWHM measurements presented in this thesis. In Figure 4.6 we have shown an example of a bunch measurement. The induced voltage on the pickup can be calculated from the following equation (4.4) Vind = QCind p where Qind is the total charge induced on the pickup by the passing charges, and Cp is the capacitance of the pickup. What we actually measure in the experiment is however the output voltage after some ampli cation circuitry, which can be written as Vout = gVind , where g is the gain of the ampli cation. With this system we can measure the bunch shapes. Furthermore the area of the bunches in this measurement is proportional to the number of particles in the bunch. Thus

40

Experimental Setup

we can, as discussed in Section 4.4 measure the beam current as well as the bunch lengths simultaneously if we calibrate the area of the bunches by some other current monitor (see Section 4.4 and appendix A.2).

4.3.2 Schottky noise

Consider one particle circulating in a ring. If we let this particle pass a perfect pickup with an in nite bandwidth the particle will induce delta functions in the time spectrum separated by its revolution time. In the frequency domain this is equal to delta functions at the revolution frequency and all higher harmonics (assuming in nite lifetime). Figure 4.7 shows how the signal would look for two particles. Note that the separation in the frequency domain and the time domain increases with harmonic number. V

V

n=1

n=2

n=3

n=4

n(f1-f2)

T1

T2

time

f1

f2

frequency

Figure 4.7: Two particles with revolution frequencies f1 and f2 seen in time domain and frequency domain. A beam of particles with uniformly distributed charges will induce an AC component on the Schottky pickup. A typical magnesium beam in ASTRID have about 107 particles and with the standard revolution frequency of 22 KHz the AC component induced by a beam of uniformly distributed particles would be of order 1011 Hz, and it would not be resolved in ordinary systems which have bandwidths of order GHz. However particles are normally distributed in a random manner, and the statistical uctuations in the number of particles detected will lead to a noise in the DC component on the pickup with frequency components in the bands discerned above. The density of the noise in these so-called Schottky bands can be found to be [64] Sn (!) = Nn f (!=n) (4.5) where n is the harmonic number, N the number of particles in the beam and f the normalized particle distribution function in revolution frequency. Thus measuring Schottky noise not only gives information about the frequency, and thus the velocity distribution, but also about the beam current (in an uncorrelated beam).

4.3 Longitudinal Diagnostics

41

The Schottky spectra discussed in this thesis were measured by measuring the noise spectrum at the 23rd harmonic of the revolution frequency (514 kHz). In order to improve resolution and save measurement time, the Schottky signal was mixed down to a low frequency using a mixing frequency of 490 kHz, then it was digitized and Fourier transformed (FFT) to give the frequency distribution. Several FFT spectra were then averaged to increase the signal to noise ratio. The mixed Schottky signal was digitized on a fast oscilloscope for typically 0.2 s, giving a frequency resolution of 5 Hz. Dense and cold beams

In a dense and cold beam the particle motion will be strongly in uenced by other particles in the beam. In this case the Schottky noise spectrum will be distorted by two peaks located symmetrically around the harmonic. These peaks are associated with co- and counterpropagating charge density waves in the beam, rst described in [77], and observed rst in [21]. It was shown in [77] that the noise spectral density at the nth harmonic of the revolution frequency is ) (4.6) Sn (!) = Nn jf(!=n 2 (!) j n

where n(!) is the dielectric permittivity of the beam (plasma), and was shown for a Gaussian frequency distribution, to be given by  Zy 2 2  p

2 2 n , y , y , x n (!) = 1 + n2!2 1 , 2ye e dx , i ye ; y = p! 2 (4.7) 0 n 2!

where ! is the rms spread in revolution frequency, and n is the coherent tune shift7 s

n = 2nv (4.8) C where C is the circumference of the ring, and vs is the velocity of the charge density waves giving rise to the peak structure. The velocity is given by s 22 e g0 (4.9) vs = 4NZ  mC 2 0

where g0 is a geometric factor (see Section 5.2.2), Ze the charge of each particle, the relativistic factor, and  the slip factor. In conclusion one can still extract the velocity spread of the beam even though it is strongly coupled. The total integrated Schottky power follows from equation (4.6) 2 ( ! ) hSn i = N (!)2 + =n n

(4.10)

7 In this expression for the coherent tune shift, it has been assumed that space charge is the only

contribution to the storage ring impedance, and that the machine is operated below transition [47]

42

Experimental Setup

thus when the temperature is high the power is proportional to the number of particles, and when it is low to the spread in revolution frequency. Before the introduction of the laser velocimetry discussed in the next section, this was the only way in which to measure the momentum distribution of a cold ion beam directly. Using laser velocimetry we could for the rst time directly test this model and con rm it [29]. A measurement of the Schottky noise spectra from a cold beam together with a velocity distribution measurement is shown in Figure 4.8. 40 12

30

Counts [102]

Relative Schottky Power

35

25

-vs

8

+vs

4 0 -20

-10

0

10

20

∆v/v [10-4]

20 15 10 5 0 -60

-40

-20

0

20

40

∆f/f [10 ] -4

Figure 4.8: An experimental Schottky spectrum taken 0.4s after injection of an uncooled (but cold) beam, together with a t to the spectrum. The inset shows the measured velocity distribution [29]. Figure 4.8 shows the characteristic double structure of a cold beam. The Schottky spectrum was measured 0.4 seconds after injection of the ions into the ring. The injector is a simple electrostatic accelerator with a very stable acceleration voltage, and the injected beam is therefore so cold and dense that even without cooling, the eects of correlations can be seen. The gure also shows a t to equation (4.6). The beam current was 2A, corresponding to 5.5
108 stored particles, giving a sound velocity of  730 m/s. The t to equation (4.6) gave a velocity spread of fit;Schottky = 488  30 m/s, in good agreement with the velocity spread LIF = 467  13 m/s, measured using laser-induced uorescence (described next). The velocity measurement is shown in the inset in Figure 4.8, where also the position of the sound velocity is indicated.

4.3 Longitudinal Diagnostics

43

4.3.3 Laser induced uorescence During laser-cooling the laser continuously excites the ions, which in turn decay by spontaneous emission. Using this uorescent light (LIF) we can measure the longitudinal velocity distribution of an ion beam. Photomultiplier Tube MCS

PAT

v

v+dv U

Figure 4.9: Illustration of the experimental con guration for measurement of longitudinal velocity distributions using laser induced uorescence. The uorescence at each voltage step is counted by a photomultiplier (PMT), which, via some ampli cation and pulse shaping, is connected to a multichannel scaler (MCS). We let the ions pass through a drift tube which can be excited by a DC voltage. When the tube is excited the ions inside the tube are accelerated to a dierent velocity, and thus the Doppler shifted transition frequency is changed. The DC voltage brings a dierent velocity class of ions into resonance with the laser, and by sweeping the voltage on the drift tube (called Post Acceleration Tube (PAT)) and simultaneously monitoring the

uorescence we can measure the velocity distribution of the ions with a resolution limited by the transition linewidth8. The setup used for LIF based longitudinal velocity distribution measurements is illustrated in Figure 4.9. The ion beam uorescence is not monitored directly by the photomultiplier (PMT) as indicated in the drawing, but rather, the photomultiplier monitors the light which passes through a spectrometer. The uorescence from the ion beam is guided into the spectrometer by a simple lens system. This addition to the setup is used in order to lter away scattered laser light in the vacuum chamber. Due to the large ion beam velocity the dierence between the wavelength of the scattered laser light and the wavelength of the light from the uorescing ion beam is 0.8nm, which the spectrometer, which has a resolution of about 0.1 nm, can separate. A velocity distribution measurement usually consists of a sum of several single measurements (one PAT sweep) measured at the same time relative to injection in order to increase the photon statistics. 8 The measurement will be the result of a folding of the Lorentzian transition line shape and the actual

velocity distribution. This is called a Voight pro le if the velocity distribution is Gaussian [82].

44

Experimental Setup The induced velocity change for a singly charged positive ion can be calculated from

E0 + eU = 12 m(v0 +
v)2 , e U
v  mv 0

(4.11)

in the nonrelativistic case where we operate. If the laser used to probe the velocity distribution is used solely for this purpose the PAT can be oset with a large DC voltage, and the laser detuned only to be resonant with the ions when inside the PAT. This technique has the advantage that the laser only interacts with the ions inside the PAT, and the in uence of the laser on the beam can therefore be kept negligible by using a short PAT. The PAT currently used in ASTRID is 40 cm long, and this technique is used for all bunched beam measurements if otherwise is not stated. The typical oset used is 400V corresponding to a Doppler-shift of 6GHz. A dierent way of measuring the velocity distribution using the laser uorescence would be to chirp the laser frequency instead of sweeping the voltage on the PAT. Using a chirped laser technique was necessary for measuring the transverse velocity distributions of uncooled ion beams in reference [70]. However, this technique may in uence the beam more than desired, as the laser will be resonant with the ions at all positions (mainly a problem if the technique is used for longitudinal velocity distribution measurements, as the laser overlaps with the ion beam a large fraction of the ring). Furthermore the voltage on the PAT can be changed much faster than the frequency of the laser, which means that we can probe fast changes in the velocity distribution when we use the PAT technique. The maximum sweep range of the PAT with the current setup is  400 V, corresponding to  6 GHz, which corresponds to  1800 m/s, or
v=v in the range  2
10,3 . In typical measurements the voltage on the PAT is swept in 0.5s, but measurements where the voltage was swept in 8ms have been conducted - thus very fast probing of the velocity distribution is possible. The maximum chirp rate of the laser frequency is about 200 GHz/s, this would correspond to a minimum sweep time for the PAT of  60ms for the full range of  400V. Table 4.3 lists some relevant conversion factors for the velocity distribution measurements. Parameter Value Voltage to laser frequency 16.1 MHz/Volt Voltage to Velocity 4.50 m/s/Volt FWHM Voltage to temperature 10.6mK
(Ufwhm=[V])2 Velocity spread (rms) to temperature 2.88mK
(v=[m/s])2 Table 4.3: Useful conversion factors for the velocity distribution measurements. 99.1 keV 24Mg+ ions with a Gaussian velocity distribution have been assumed.

4.4 Absolute Current

45

4.4 Absolute Current In order to understand the behavior of a stored ion beam we saw in an earlier chapter that the beam current as well as density were important parameters. This is of course obvious in the case of a crystalline beam where the structure of the crystal depends critically on the number of particles. But also in less space-charge in uenced beams the current is important, as self elds dependent on current and density induce tune shifts, and the heating due to IBS depends on the beam density. It is therefore vital to our investigations that we have an accurate measure of the absolute number of particles in the beam. Unfortunately, the ion currents which can be stored in ASTRID at the low energy we use are rather low compared to the current resolution of standard non-destructive beam current monitors ( 1 A). This means, that as our beam currents are typically of order 1A or less, we have to rely on a measurement of the current to be injected, which can be easily measured by a faraday cup [45], together with measurements of the beam lifetime, measured by measuring the amount of neutralized ions impinging on a channel-plate detector at the end of one of the straight sections. This method has been used for all the coasting beam measurements presented in later chapters. In the bunched beam we can use a standard beam current transformer. The reason this is possible, is that the time variation in the beam induces much larger signals than for the DC beam. In a bunched beam a resolution of order 5nA can be obtained, which is sucient for most of our measurements. The beam current transformer however caused some operational problems, and was therefore not used directly for current measurements during our experiments. Rather, it was used to calibrate the capacitance of the position pickup used to measure the bunch shapes. Thus by integrating the bunch signal we could determine the absolute number of particles in each bunch. In appendix A.2 an example of such a calibration is given. The noise level on the position pickup used to measure bunch shapes is such that we can measure bunch shapes in beams with currents above 10nA, corresponding to about 2.8
106 particles circulating in the ring (about 1.8
105 particles per bunch with 16 bunches as we normally use). In coasting beams we have extrapolated beam currents down to about 0.7nA, corresponding to about 2
105 particles. The absolute uncertainty in beam current is about 10%, and usually slightly higher in coasting beams due to the uncertainty in the life time of the beam. As the density in the beam is an important parameter we would eventually like to have less uncertainty in the current measurements. The rst objective in this respect would be to nd a direct way to measure the current in a bunched beam. In principle the current can be extracted from the uorescence intensity of the laser-excited ion beam. However, this requires extremely good control of the laser intensity, laser pro le, beam pro le and the relative laser/beam position. An alternative possibility would be to install a system using a cryogenic current comparator in ASTRID. Such a system uses a SQUID9 to measure the magnetic eld induced by the beam [27]. The hitherto best reported resolution of a system based on this tech9 SQUID is an acronym for Superconducting QUantum Interference Device.

46

Experimental Setup

nique is 1nA for a coasting beam [97]. In our experiments 1nA corresponds to 2.79
105 particles, which in ASTRID gives a linear density more than a factor 2 below the phase transition density for the shift from string regime to zigzag regime (see table 2.1). The system however incorporates a cryostat with liquid helium and is thus rather costly.

4.5 Transverse Diagnostics As laser-cooling of heavy ion beams is mainly a method for cooling the longitudinal degree of freedom, and the desire is to cool all three dimensions, it is important in the attempt to study various indirect methods for transverse cooling that the transverse beam temperature can be observed. As discussed earlier the transverse temperature is not a constant around the ring, thus we usually use the emittance to characterize the beam. The transverse emittance can be found by observing either the transverse beam size or the transverse velocity distribution, while assuming knowledge of the local beta function. The transverse velocity distribution can be measured using a principle similar to the way the longitudinal velocity distribution was measured. A laser is shined onto the ion beam at right angles to the ideal orbit. By monitoring the intensity of the uorescence while chirping the laser frequency the velocity distribution can be measured. This method has been used to study the heating after injection [70]10. However the method requires a laser which is not Doppler shifted as the lasers for cooling and longitudinal diagnostics must be. This may be a problem, as the scattered light from the laser beam then has the same frequency as the uorescence from the ion beam. When measuring the longitudinal velocity distribution the scattered light could be ltered away by the use of a spectrometer. Furthermore both the available laser systems are used when laser-cooling coasting beams, and therefore a new system would be necessary in order to measure the transverse velocity distribution. What is more important, is that space charge eects would not be directly evident in the transverse velocity pro le. As we are interested in how the beam changes due to space charge, especially in the crystalline beam state, it was decided that we needed a system to monitor the transverse beam size and not the velocity spread.

4.5.1 Requirements

Measuring the size of a heavy ion beam can be done in various ways [45]. There were two main concerns in choosing what system to use. As it would be interesting to work with relatively sparse beams (down to string regime of 106 particles) it is necessary that the developed system is able to 'see' such weak beams, thus has a high sensitivity. A high spatial resolution is needed as cold beams in ASTRID are rather small (rms size  1mm), and in due time the system should be able to observe crystalline structures, which have a length scale of order microns [6]. As we wish to study dynamics and at some point crystallization, only non-destructive methods have been considered. 10The laser was not shined perpendicular to the ion beam in these measurements but rather at an angle

of 56 , which meant that knowledge of the longitudinal velocity pro le had to be included in the analysis

4.5 Transverse Diagnostics

47

4.5.2 Residual Gas Pro le Monitor The most commonly used way of measuring the size of a circulating heavy ion beam nondestructively is to use the fact that the beam ionizes the rest gas in the vacuum chamber. There are two slightly dierent ways of doing this, each of these principles are shown in Figure 4.10. The fundamental principle is the same: The circulating beam ionizes the rest gas, and by applying a transverse electric eld, the fragments can be extracted from the beam region and made to impact on a position sensitive channel-plate detector. Detector ions

Ions

Beam

Beam

ε

HV e-

ε

HV

B

eDetector

Figure 4.10: Two variants of residual gas ionization beam pro le monitors. The left is the simplest and detects the ionized rest gas atoms only. In the right a magnetic eld is employed parallel to the electric, which makes it possible to look at the electrons coming out of the ionization too. More details appear in the text. The rst, which is employed for instance at the TSR in Heidelberg [36], accelerates the ionized rest gas atoms via a transverse electric eld onto a channel plate detector and can thus measure the transverse density distribution of the circulating ion beam. In order to measure both transverse pro les two detectors are mounted, one rotated 90 degrees with respect to the other. The second method monitors the electrons stemming from the ionization [85]. This is usually rather dicult, as electrons produced outside the detector volume disturb the measurements. This problem is overcome by applying a strong magnetic eld parallel to the accelerating electric eld (for instance by positioning the detector in a bending dipole), which causes the electrons to spiral around the magnetic eld lines. The magnetic eld has the further advantage that the electron collection eciency is increased and that the position resolution is enhanced due to the spiraling. The technique has recently been improved to incorporate time-of- ight measurements in order to detect the perpendicular transverse density pro le simultaneously [84]. These methods have been applied successfully, and the reported spatial resolutions are of order 200 m and 60m respectively. The above experiments are performed at energies of order several MeV/nucleon, whereas the experiments done in this work are at an energy of some keV/nucleon. In low energy beams the quality of the vacuum is crucial, and we therefore operate at a very low pressure, something which decreases the feasibility of an ionization system. Furthermore, the electric elds needed for the extraction

48

Experimental Setup PAT

UV-Filter

shutter

CCD

Lens Mount

Ion Beam

Beam Pipe

Bellows

Dewar

Figure 4.11: The experimental con guration of the beam imaging system. The bellows indicated in the gure is cylindrical, made of black rubber and surrounds the lens system to block out room light. of fragments are quite high, using a likely eld of 40 V/mm and a 10 cm long eld section, our 24Mg+ beam would be de ected by 0.1 mrad. Thus there are several problems, of which the resolution and the sensitivity are probably the most critical, as we would eventually like to observe crystalline structures, which have detailed structural properties with length scales of order 20 m [6], as well as beams well into the string regime. These methods were therefore discarded.

4.5.3 Beam Imaging

The basic principle behind the new diagnostics we have chosen to develop for storage ring use is well know, and has been and is being applied in ion trap experiments for imaging of for example ion crystals [6]. We utilize the uorescence light from the laser-excited ion beam to monitor the beam size. The uorescence light is imaged onto a high resolution, low noise CCD via a simple optical system, thus the transverse density pro le of the ion beam is probed. The following text describes the system in detail in order to give insight into the problems/limitations of the implemented system. Experimental Con guration

An overview of the experimental con guration for the system is shown in Figure 4.11. The con guration consists of the following main components  Post Acceleration Tube (PAT). The PAT is simply a drift tube which can be excited by a DC voltage and thereby change the local ion velocity and thus the local Doppler shifted resonant frequency. Thus by exciting the PAT we can select which velocity class of ions should be resonant with the laser. This was described in more detail in Section 4.3.3

4.5 Transverse Diagnostics

49

 Lens doublet The lens doublet consist of two spherical lenses, which are mounted

with facing concave sides for minimizing the rst order spherical aberration.  UV lter The UV lter is there to minimize the in uence from room light scattering into the vacuum chamber through the dierent viewports into the storage ring.  Shutter A mechanical shutter is mounted in front of the CCD. This shutter can be controlled externally, and is used to control when light should be incident on the CCD.  CCD camera Currently we have two identical systems, though with two slightly dierent CCD cameras. The detailed speci cations of the camera systems are given in Appendix B.1. The light from the ion beam is, as mentioned, imaged onto a CCD. A CCD11 is a metaloxide semiconductor (MOS) optical detector that is composed of up to several million independent sites where photon-induced charge is stored (as electrons). These photosites are called picture elements or pixels. When an arrangement of pixels is exposed to light a charge pattern accumulates that corresponds to the illumination pattern [79]. Before we implemented the system we did some calculations as to what we could expect in performance in order to establish whether the system would indeed be better than a rest gas ionization system. Performance expectations

The performance of the system is characterized by two main factors. The rst is the sensitivity of the system, i.e. the lowest ion beam current for which a beam pro le can be measured. The second is the spatial resolution of the system, which determines the smallest structures which can be detected. Sensitivity

In order to estimate the sensitivity of the system it is necessary to know the geometry of the system, as the geometry determines how much of the light from the spontaneous emission of the ions is imaged onto the CCD. In Table 4.4 the important distances in the mechanical setup, earlier shown in Figure 4.11, are given. As the system exhibits cylindrical symmetry, i.e. the beam size is constant along the short part of the orbit which can be observed, we rst considered using cylindrical lenses instead of spherical lenses. However, we would like to have the con guration in which as much as possible of the light from the ion beam reaches the CCD. A geometric factor  has therefore been calculated using the distances in table 4.4. The geometric factor is given by   =
lCvisible (4.12) 11CCD is an acronym for Charge Coupled Device.

50

Experimental Setup Diameter of lenses 50 mm Radius of post acceleration tube (PAT) 62.5mm Radius of vacuum chamber 75mm Width of square side hole in PAT 40.0mm Lens to beam distance range 78 - 104 mm CCD to beam used distance range 208 - 234 mm Size of CCD chip 25 mm  25 mm

Table 4.4: Dimensions of the camera setup. The lens/CCD ranges equals a magni cation range from 1 to 2. where C is the ring circumference, and lvisible is the length of the imaged part of the ion beam.  is the average solid angle. The solid angle is the fraction of light from a given point in the beam which reaches the CCD. As the solid angle changes as a function of position in the beam it is necessary to average over the part of the beam which contributes light to the CCD. The geometric factor represents the fraction of photons emitted from an evenly excited ion beam which reaches the CCD, where we have assumed that the transverse size of the ion beam is small compared to all other distances. The geometric factor as a function of the chosen magni cation is shown in Figure 4.12. Figure 4.12 shows that with a spherical lens system a larger amount of the ion beam contributes light on the CCD, thus system gives a better sensitivity, and was therefore implemented. An imaging system with spherical lenses furthermore has the advantage that it is easier to use in calibrations as the images are easier to interpret. 9.0e-06

Geometric Factor

8.0e-06

7.0e-06

Spherical Lens System Cylindrical Lens System

6.0e-06

5.0e-06

4.0e-06 1.0

1.2

1.4 1.6 Magnification

1.8

2.0

Figure 4.12: Geometric factor  (see text) as function of magni cation. Apart form the geometric con guration we also need to take into account how many of the ions are resonant with the laser in order to calculate the photon ux on the CCD.

4.5 Transverse Diagnostics

51

The amount of light emitted from the beam is determined by the fraction of the ion distribution the laser interacts with, as well as the laser intensity. The amount of photons emitted per second by one ion is given by (consider equation (3.7))

=2)2 Nion (v) = 21 S , (vk)2 + (, (,=2)2 (1 + S )

(4.13)

and the longitudinal velocity distribution assuming that it is Gaussian can be written as 0 21 N fk(v) = p exp @, 2v2 A (4.14) 2k k where k is the longitudinal velocity spread, from which the temperature is de ned as Tk = mk2=kB (see Section A.1). The distribution is normalized such that an integration over all velocities gives N which is the number of particles in the beam. The light ux from the ion beam can now, assuming that all of the ion beam is embedded in the laser eld, be calculated by folding the two distributions above. For all of our measurements the saturation parameter of the probe laser used to excite the beam for imaging ful lls S  1. We can therefore ignore S in the denominator, i.e. ignore power broadening. This simpli es the calculation of the photon ux, as we can take the laser intensity outside the integral by writing =2)2 Nion = N
(vk)(, (4.15) 2 + (,=2)2 where N = 12 S , is the number of photons emitted per second by an ion resonant with the laser. Thus we can now write the total photon ux on the CCD as  = 
N
N
(Tk)

(4.16)

where N is the number of particles in the beam, and the factor (Tk) called the velocity space overlap factor has been introduced. The velocity space overlap is given by 0 21 Z1 2 (, = 2) 1 @, v 2 A dv p exp (Tk) =
(4.17) 2 2 2k ,1 (vk ) + (,=2) 2k where it for simplicity has been assumed that the laser detuning is zero (i.e. the laser is resonant with particles in the center of the velocity distribution of the ion beam). In table 4.5 some ux estimates for the chosen system have been calculated. In these calculations, losses in lenses have been ignored (including them would just be a scaling of the laser power), and an isotropic radiation distribution has been assumed. This is not strictly correct, as the radiation pattern from the spontaneous emission in 24Mg+ is a mixture of a dipole radiation pattern and an isotropic radiation pattern. The spatial variation in intensity due to the radiation pattern can be estimated by considering the

52

Experimental Setup Number of particles Laser power Laser beam diameter Saturation parameter Photons pr. ion on resonance Photon ux on CCD (Tk=1400K) Electron generation on CCD Photon ux on CCD (Tk=5K) Electron generation on CCD

N P d S N  e,  e,

Standard String 8 2.8
10 6.3
105 20mW 20mW 2cm 2cm 0.008 0.008 1.1
106 s,1 1.1
106 s,1 1.4
106 s,1 3.2
103 s,1 0.35 e,/pixel/s 8.0
10,4 e, /pixel/s 2.0
107 s,1 4.7
104 s,1 5.2 e,/pixel/s 0.012 e, /pixel/s

Table 4.5: Photon ux estimates for two situations ( = 8
10,6 ). The laser intensity distribution is assumed to be homogeneous. To demonstrate the dependence on the velocity space overlap factor  the calculation has been done for two longitudinal temperatures. m=

+1/2

-1/2

-3/2

σ+

σ-

σ2 π 3

1

σ+ π

1 3 m=

-1/2

+3/2

+1/2

2 3

3P

1

3S

Figure 4.13: Transition scheme for the (32S1=2) $ (32P3=2) transition [107]. The numbers next to the transitions are the CG coecients. The Greek letters indicate the type of polarization which is needed to drive the transition. The arrows mark the possible transitions when driving the system with linearly polarized light as done in the experiments in this thesis. (32S1=2) $ (32P3=2) electronic transition used for the cooling. In Figure 4.13 the transitions and Clebsch-Gordan (CG) coecients for the (32S1=2) $ (32P3=2) electronic transition are shown. From Figure 4.13 we can see that the spontaneous emission, when driving the transition with linearly polarized light (), consists of both circular polarized light (+;,) and linearly polarized light (). The radiation pattern of the linearly polarized light is dipolar and for the circular polarized light it is a combination of an isotropic part and a dipole part [107]. The probability of each decay type is equal to the square of the CG coecients. The total radiation pattern can be written [107] !2 2  ! 2  3 3 1 1 + cos 2  = 1 + 3 sin2 ; +
sin (4.18) I ( )  p
4 2 3 4 4 8 3 The calculation shows that the dipole part has an amplitude of 3=8, whereas the isotropic part has an amplitude of 1=4. Hence a factor 2.5 is expected to be the dierence

4.5 Transverse Diagnostics

53

between maximum and minimum intensity. We only image one transverse dimension per injection, we can therefore ip the polarization of the probe laser each injection in order to increase the light intensity in the direction which is imaged. The ux of light on the CCD we has just calculated, but, in order to determine the sensitivity of the imaging system, we also need information about possible background contributions and noise. For estimating the sensitivity there are three main factors which have to be considered: 1. Laser light scattered on the vacuum chamber can, due to the small Doppler shift (0.8nm) not be ltered eciently from the uorescence. However, this does not necessarily cause problems, as background measurements without ions can be made, and subtracted from the ion beam measurements. 2. Read out noise in the CCD system. There is an intrinsic noise in the conversion of charge in the CCD to a signal for the computer. This can be reduced by increasing the gain (see table B.1), or its relative in uence can be reduced by binning the pixels, as the noise is in the conversion and not a noise on the charge in each pixel. Binning is explained below. 3. Dark counts. Due to the nite temperature of the CCD a small amount of electrons is constantly generated even though the CCD is not illuminated. These will probably not cause any problems as the rate is as low as 0.26 e, /hour/pixel, and exposure times making this contribution signi cant are avoided due to stability of the laser/ring systems. When a CCD is struck by light, the resulting electronic charge is collected into a twodimensional imaging area called the parallel register. The parallel register consists of a number of rows, which during readout are transferred to a register called the serial register as illustrated in Figure 4.14. Binning is a process by which the charge on adjacent pixels can be added before read-out. Parallel binning, which consists of accumulating charge in the cells of the serial register before read out, is illustrated in Figure 4.14 [79]. As there Serial reg.

parallel register

Output Node

Binning with n=2. Black/gray dots represent photoelectronic charge.

Charge is shifted up the parallel register into the serial register.

Charge is again shifted parallel to fulfil n=2.

Charge is serially shifted into the output node.

Figure 4.14: CCD Readout with parallel binning [79].

54

Experimental Setup

is cylindrical symmetry in the system, binning can be done in the longitudinal direction without loss of information, and thus decrease the in uence of the noise caused by readout. The camera shutter can be triggered in phase with the injection cycle, making exposure of the CCD over several injection cycles possible (as for velocimetry or Schottky measurements). Due to the low dark current this can be done for long times without having to consider in uence from the dark current. Examples of realistic measurements for the situations in table 4.5 are given in table 4.6. Except for the 34 minutes for a hot dilute beam Standard String , , Electron generation on CCD (Tk = 1400 K) 0.35 e /pixel/s 8.0
10 4 e,/pixel/s Integration time with 20 pixel binning 4.7 s 34 min , , Electron generation on CCD (Tk = 5 K) 5.2 e /pixel/s 0.012 e /pixel/s Integration time with 20 pixel binning 0.32 s 142 s Table 4.6: Estimated integration times necessary for S/N  10 (gain 4) these times are all realistic, and have been accomplished in experiment. The dilute hot beam is probably not very interesting, but by increasing the binning it could, if necessary, be measured with realistic exposure times. In theory all the pixels (1024) could be binned to achieve minimum noise in uence. In practice however it is nice to have some notion of structure. Furthermore the scattered light often has some structure, thus to increase the signal to noise ratio some parts of the CCD are usually avoided. The usual binning factor is therefore 16, but could be increased somewhat in uncooled beams, where it is most likely to be needed, due to lower photon ux. Spatial Resolution

Of interest for the experiments (and an argument for choosing the CCD system) is the resolution of the system. The ultimate limit of the system is the diraction limit. The nite size of an imaged point source due to diraction of the aperture de ned by the lenses is given by [8] diff  0:61
sin  p 2 2 
L +r = 0:61
r  0:73m (4.19) where  is the wavelength in vacuum, r=25mm is the radius of the aperture (lens in our case) and L=104mm is the distance to the aperture. The calculation is thus done for the worst case (largest possible L). This resolution is sucient, but the lens system also sets other limitations to the resolution, as the lens system exhibits spherical aberration, which although compensated

4.5 Transverse Diagnostics

55

somewhat by the double lens structure, still limits the resolution, as well as changes the focal distance for dierent positions in the object plane (the beam). A ray-tracing routine was made which could calculate the position on a chosen image plane of a ray starting with a radial oset and an angle to the normal to the lenses from a chosen object plane. The lens con guration is as illustrated in Figure 4.15. Object Plane

Image Plane α

h -R

a

R

t

d

t

b

Figure 4.15: Double lens system to minimize spherical aberration. The lenses are Anti Re ection coasted for 280nm. We use U312242 lenses from Spindler Hoyer, with nsilica (280nm) = 1.494, and R = 51.82mm and t = 8.42 mm. The distance d between the lens surfaces is 2mm. First we look at how a Gaussian intensity distribution is imaged through the lens system. At a magni cation of 1 we have calculated the width as a function of distance from the exit lens for dierent apertures, in order to estimate the depth of focus. In Figure 4.16 the relative deviation from the size at the image plane as a function of the distance from the image plane is plotted. Relative deviation from focused size

0.10

0.08

Aperture 25mm Aperture 20mm Aperture 15mm Aperture 10mm

0.06

0.04

0.02

0.00 -10.0

-5.0 0.0 5.0 Relative Distance From Focus [mm]

10.0

Figure 4.16: Relative deviation from minimum (focused) size as a function of the distance from the image plane. The dierent curves are for various aperture radii at the rst lens. The imaged Gaussian distribution had a  of 2mm. If we decide, considering the precision in the other measurements, to accept a deviation

56

Experimental Setup

of 4%, we nd that for the free lens (aperture 25 mm) the focal depth is 4mm, which is reasonable considering that typical ion beam sizes are of this order. However, we decided, in order that this part should not limit the performance, and the photon ux was more than sucient for the needed sensitivity, that for the vertical imaging we would use an aperture of 12.5 mm and for the horizontal 10.0 mm, giving us an ideal depth of focus of order 10 mm. Apart from the depth of focus, the resolution is important. The resolution can be calculated by calculating the image size of a point light source in the object plane. We choose to de ne the resolution as the FWHM of this image. With this de nition the resolution de nes how far two point sources have to be from each other in order for the intensity to drop to 50% in between them. A ray tracing calculation of the intensity distribution in the image plane from a point source in the object plane has been done. As the trace is only two dimensional, the intensity has been compensated to take the 3D nature into account by multiplying with the term 
 (4.20)  = sin 2r
r where  is the azimutal angle and r is the distance in the object plane of the ray point of incidence to the point of incidence of a ray with =0. The distributions calculated did, when properly focused, have widths (fwhm) of the same order as the pixel size of 24m with M=1. Thus with M=1 we would expect the pixel size to be the primary limit, which gives a resolution of order 2 pixels. By changing the magni cation we should be able to reach a better resolution. We have demonstrated that our system should have very high resolution, which is also what would be expected from earlier measurements using similar systems in traps [6]. In case it turns out that the spherical aberration should impose too large limitations on the resolution, we can in time improve the lens system by using aspheric lenses and/or smaller apertures. The limitation imposed by the pixel size can, if necessary be overcome by magnifying more, which seems a feasible solution, as the high resolution is most likely to be necessary in low current beams, which are rather small. With the achievable resolution is should be possible to observe shell structures in a crystalline beam. Galvanometric Optical Scanners

During the rst measurements with the camera system it was quickly realized that generating a homogenous laser intensity distribution was not as easy as rst thought. A homogeneous intensity distribution is extremely important to the operation of the system, as it is very dicult and time consuming to compensate for a non-uniform intensity distribution when analyzing the acquired data. We therefore implemented a laser scanning system. The scanning system generates a \one-dimensional" homogenous intensity distribution by physically scanning a highly focused laser beam in the desired plane. A schematic drawing of the setup is shown in Figure 4.17. The scanning system needs to be able to

4.5 Transverse Diagnostics

57

generate a homogenous laser distribution which has a width of up to 3cm at the camera position, and at the same time a stability and reproducibility of better than 0.1mm is needed, as the laser should be stable when doing velocimetry, which happens at the same place in the ring as the imaging. For stability, and to avoid too large angles, which may cause light to be scattered in the vacuum chamber, the scanners should be positioned in the laboratory, which is about 10 meters away from the entry window to the diagnostics section of ASTRID. Thus we need an optical angular range of 1 mrad and a stability and reproducibility better than 7 rad. A galvo system which matched these conditions was purchased from Camtech, the speci cations are listed in appendix B.2. Vacuum Tube

Ion beam Scanner

Viewport

Laser

Figure 4.17: Schematic drawing of the optical galvanometer scanner setup used for scanning the lasers. The typical scanning frequency is  100 Hz Implementing this system has two further advantages. As the laser is focused to a FWHM of order 1 mm, the problems with depth of focus vanish (however, the aberration for o center particles still needs to be compensated). Furthermore the average intensity is increased as we only spread the laser power out in one dimension, which implies that more of the laser beam is at high ion density, as the laser is positioned in the center of the ion beam. It is also possible to decrease the scanning range when the beams become small, and thereby increase the light intensity, without changing the laser position. Earlier we discussed the expected uorescence on the CCD, and found that this relies on the velocity space overlap  of the laser with the ion beam. Now that the laser is scanned in one dimension and kept small in the other, the average intensity increases, but in order to estimate the uorescence it is now also necessary to have information about the laser beam pro le and the ion beam pro le, as well as their relative position. Taking these considerations into account is important if one wishes to extract a beam current from the

uorescence, something which may become necessary at low currents where ordinary beam current monitors can not be used. Calibration of Magni cation

In order to extract size information from the imaging system we need a procedure to calibrate the magni cation of the system. This is accomplished by having calibration plates mounted on a metal rod which can be moved into the center of the vacuum chamber (see Figure 4.18). As the index of refraction changes a lot with wavelength, and we are

58

Experimental Setup

interested in focusing the camera on the ion beam, the procedure for calibration is the following. Calibration Rod Viewport

Camera Laser and Ion beam

Figure 4.18: Drawing of the calibration rod for the imaging system. The stick has two plates mounted at angles to the ion beam direction. One plate has a surface which can be imaged in the horizontal plane and the other in the vertical plane. The plates are not at 45 degrees to the beam in order to avoid directly re ected laser light in the imaging system. On the calibration plates a set of lines has been drawn with well de ned distances. First the laser is positioned roughly in the center of the ion beam by monitoring the ion beam uorescence on the photomultiplier system used for velocimetry. Then the rod is inserted and the cameras are focused on their respective plates at the laser position. Then the procedure is repeated, now however the laser position relative to the ion beam is optimized by imaging the laser as well as the ion beam. The laser pro le and position are measured by injecting a large current into the machine which causes the (uncooled) ion beam to be much larger than the very narrow laser. Another procedure for optimizing the relative position of the laser beam to the ion beam could be only to maximize the intensity of the ion beam uorescence reaching the photomultiplier. However, this procedure has been found not to be very precise. The lack of precision stems from the fact that the photomultiplier itself is not very well aligned with relation to the ring as well as with relation to the physical position of the ion beam. Therefore the photomultiplier system might not be \looking" at the center of the ion beam, and not see maximum uorescence when the ion beam laser beam overlap is optimized. The photomultiplier signal is therefore maximuzed after the laser beam ion beam overlap has been optimized. Figure 4.19 shows images of the horizontal and vertical calibration plates respectively. Using the information from the calibration plates, magni cations of Mh = 0.744 and Mv = 1.16, using a pixel size of 24 m, can be extracted from the images. The apparent inhomogenity of the laser intensity distribution is caused by inhomogenity in the re ectivity of the calibration plates. There is however a slight tendency for the intensity pro le to

4.5 Transverse Diagnostics

59

Figure 4.19: Images of the stick. The left is a calibration of the horizontal imaging, and the right of the vertical. The laser was scanned in the horizontal and vertical dimension respectively while recording these images. The thinnest lines are about 3 pixels wide, signifying very good resolution. The apparent inhomogenity of the laser intensity is explained in the text. be slightly more intense at the ends of the scan range, for this reason we usually scan the laser further than the width of the ion beam. The rst measurements

The next step is to measure the beam dimensions under various conditions and at various times in the injection cycle. As mentioned earlier, this is done by controlling the mechanical shutter in front of the camera. Before a measurement is initiated, the CCD chips are cleared, and the control of the external shutter is given to an injection control program. This program is triggered at injection, and its main function is to send triggers to the dierent parts of the system (measuring velocity distributions, chirping the laser, and so forth) at speci c times depending on what is being measured. The trigger for the cameras and galvoscanners is synchronized for the galvo to scan in the dimension to be imaged, and it is controlled in such a way that it images alternating dimensions in each injection. The imaging occurs simply by opening the mechanical shutter and exposing the CCD. This can be done over many injections because the dark current of the CCD is extremely low. When the number of desired injections is reached, the CCD is read out with the desired binning. Figure 4.20 shows the important events during a single injection of a measurement consisting of many injections. The laser frequency is xed during the injection cycle, and the detuning is therefore not shown in the gure. At some time after injection, depending on the desired study, the probe laser scanner is activated, and the probe laser intensity pro le becomes homogeneous in one dimension (in the gure it is the horizontal dimension). The camera shutter on the appropriate camera is opened during this time, and the CCD thus exposed to the light from the ion beam. Sometimes the probe laser is not resonant with the particles in the center of the longitudinal velocity distribution (for instance when the probe is also used for cooling, and therefore need to be detuned red) - therefore the PAT

60

Experimental Setup V

PAT

PAT Voltage

Probe Laser Profile

injection

exposure

velocimetry

time

Figure 4.20: Time line for a typical measurement. See discussion in the text. is excited to alter the local velocity in order that the laser is resonant with the particles in the center of the longitudinal velocity distribution. When the CCD exposure is nished, the galvo scanners stop, and the system is ready for velocimetry. The velocimetry is done as explained earlier by sweeping the voltage on the PAT and simultaneously measuring the intensity of the ion beam uorescence. In Figure 4.21 a sample image of the vertical dimension of a laser-cooled coasting ion beam is shown12. The measurement was done with the procedure described above, i.e. with the two cooling lasers detuned red the desired amount from before injection. The longitudinal velocity spread was measured to be  63 m/s, corresponding to a longitudinal temperature of approximately 1 K. From a Gaussian t a width of v = 0.45mm  0.01mm can be extracted. The background level is observed to be quite high, but it is extremely constant, and is therefore, as predicted, easily compensated by subtracting a background measurement (not shown in the gure). The power in the probe laser was 3 mW, a binning factor of 16 was used, and the measurement is the sum of 10 exposures of the CCD (10 injections), with one second exposure per injection. The background level in Figure 4.21 was quite high, but did not pose any problems, as it was caused by scattered laser light in the vacuum chamber and was therefore constant in time. Using mechanical apertures which are mounted in the vacuum chamber we have been able to screen o much of the scattered laser light, and obtain signal to background ratios of 20 or more, this was however not done in the measurement above. The signal to noise ratio on the other hand is very good in this case, but the ion beam current was also quite high. The resolution and sensitivity of the system has not been tested thoroughly, but in Figure 4.22 a measurement of a low current beam of approximately 700pA is shown. 700pA is about a factor of four below the limit for the string regime in a crystalline beam. The image in Figure 4.22 is taken of a laser-cooled coasting beam at 4 seconds after injection. 100 injections with a one second exposure time per injection, were done to make the picture. The signal to background ratio is worse than before, but due to the constancy of the background it can be compensated (we have in this case been especially unlucky, as 12Note that both curves in the gure are on the same scale with the same zero point.

4.5 Transverse Diagnostics

61

Flux [Arbitrary Units]

Background + Beam Beam Only

0

5

10

15

20

Vertical Position [mm]

Figure 4.21: Vertical beam pro le. The top images shows the CCD image, with the background subtracted. The bottom graph shows the averaged beam pro le and the background of scattered laser light. The measurement was done with a 230nA beam. it seems that there is a re ection causing a hot spot in the background exactly where the beam is). The signal to noise level on the other hand is about a factor 20, thus the read out noise seems to be an insigni cant problem, and we should therefore be able to measure beams of much lower current. The measured beam size is h = 0.1 mm which corresponds to a horizontal emittance of h = 8.3
10,10  m rad. The velocity spread was only about 12 m/s which corresponds to a longitudinal temperature of 0.4K. The imaging system ful lls the expectations, as we can measure beams down to the string regime, and with a magni cation of 0.8 we can measure beams down to at least 0.1 mm sigma. At these low currents, the beam are very small, and we can, if necessary, increase the magni cation and thereby be able to observe much smaller structures. In the image in Figure 4.22 the beam is slightly skew (corresponding to about 4 mrad). This is probably because the camera and the ion beam were not aligned completely, a fact which was not noticed before we came to these very small beams (the camera mount is about 15cm long, thus a skewness of 4 mrad corresponds to one side of the camera mount being approximately 0.6 mm further away from the ion beam than the other side). The quality of the alignment should be better before smaller beams are investigated. It may however also be that some of the skewness stems from spherical aberration, as it actually seems to bend slightly. As long as the signal to noise is as good as here the problem can be circumvented by using less of the CCD area.

62

Experimental Setup

Flux [Arbitrary Units]

Background + Beam Beam Only

8

12

16

20

24

Horizontal Position [mm]

Figure 4.22: Horizontal beam pro le for a very low current beam. The top images shows the CCD image, with the background subtracted. The bottom graph shows the averaged beam pro le and the background of scattered laser light. The measurement was done with a 700pA beam. The apparent skewness of the beam in the image is discussed in the text. Dispersion

As discussed previously the beam-laser overlap in velocity space, i.e. the fraction of the beam resonant with the laser depends on the relation between the transition line width and the Doppler width of the ion beam. This means that we only image particles in a very narrow range of velocities, i.e. we are independent of the dispersion at the point of measurement, which would otherwise cause the beam to look wider than caused by emittance alone, as the average position of a particle over many orbits depends on its velocity. The dispersion for the standard lattice used for these experiments is supposed to be 2.74 m  0.3 m at the camera position. In Figure 4.23 we show measurements of the beam pro le of an uncooled bunched beam 7.0 s after injection. Identical measurements were done for three dierent probe laser frequencies. The in uence of dispersion is clearly visible in the gure. The implemented transverse beam size diagnostics thus provides us with a velocity selective probing of the transverse phase space, which enables us to measure the dispersion, a fact which might become important in light of the need for tapered cooling to reach the crystalline state (see section 2.3.3).

4.5 Transverse Diagnostics

63

Center + 1 GHz (∆p/p = 0.3*10-3) - 1 GHz (∆p/p = -0.3*10-3)

0

5

10

15

20

Position [mm]

Figure 4.23: Three horizontal distributions measured at dierent probe laser detunings. From the measurements a dispersion of D = 2.4 m  0.1 m can be extracted, which is in agreement with the theoretical of 2.7 m  0.3 m. The beam was bunched and uncooled (which explains the lower signal to noise ration than on the previous beam images), and the distributions have been normalized to the same area. The beam current was  400 nA, and the probe laser power was  30 mW. Summary

The implemented system has demonstrated a high sensitivity as well as a high resolution compared to other known non-destructive systems for transverse beam size measurements on heavy ion beams. The resolution is of order the pixel size of the CCD, 24 m, but can be increased by using a higher magni cation. The sensitivity of the system relies on the available light, which increases for cold beams. We have presently imaged cold beams down to linear densities of 695 m,1, but uncooled beams have only been imaged down to 7.0
105 m,1. However we usually only bin 16 pixels, an increase of which would increase the sensitivity. The resolution of the system has not been tested systematically but ion beams with transverse sigma down to 90 m have been measured without signs of saturation (indicating lacking resolution) at a magni cation of  1, thus the expectations seems to have been ful lled. An additional feature of the system is the independence of dispersion at the imaging position, as the laser only excites ions within a short range of velocities. This means that the emittance can be extracted directly from the beam pro le, and that we can measure the dispersion by detuning the probe laser frequency to be resonant with dierent velocity classes.

64

Experimental Setup

Injector 280nm

λ = 280nm laser light

I

Septum

Dipole 0

PMT

1

2

3

4m

Camera

PAT

Quadrupole IV

II Correction Dipole/ Sextupole Pos. sen. Pickup

Kicker

Longitudinal Schottky Pickup

III Drift tube for bunching

Figure 4.24: Schematic drawing of the ASTRID storage ring.

Chapter V Longitudinal Dynamics Laser-cooling in a storage ring only operates directly on the longitudinal velocity spread. It is therefore interesting to study the longitudinal dynamics of the ions when they are exposed to laser-cooling. In dense ion beams the longitudinal dynamics alone would not be sucient to learn about the dynamics, as the particles are strongly correlated, and therefore transverse information would be needed to describe the dynamics. However, in ASTRID we usually operate in a low density regime, where the longitudinal dynamics and the transverse dynamics are decoupled. It is therefore possible to learn about the dynamics during laser-cooling by only studying the longitudinal dynamics of the ion beam. In this chapter results from studying the longitudinal dynamics of laser-cooled ion beams are presented. The rst couple of sections describe results from measurements on coasting beams. In the last sections results from measurements on bunched beam, including measurements of the longitudinal density pro les, are presented. Most of the measurements in this chapter were done before the imaging system for measuring transverse beam pro les was implemented.

5.1 Coasting Beams 5.1.1 Cooling a coasting beam A coasting beam is a continuous beam. As explained in Section 3.3.2 a coasting beam can have an initially large velocity spread compared to the range of the laser force, which was given by the line width of the cooling transition. This makes it necessary to chirp a coand counter propagating laser from a large negative detuning initially to the desired nal detuning, and in this way collect the particles between the two velocities where the lasers are resonant. Figure 5.1 illustrates how the longitudinal velocity distribution is decreased by chirping the lasers towards being resonant with the center-of-mass particle. In magnesium the transition line width of the cooling transition is 45 MHz, this corresponds to a Doppler width of  12 m/s for the magnesium beams used in ASTRID. The typical longitudinal velocity spread one second after injection of an uncooled beam is

Longitudinal Dynamics Laser force and velocity distribution for different detunings

66

-1000

-500

0

500

1000

Velocity [m/s]

Figure 5.1: Laser-cooling with two lasers. The lasers (red - 'delta' functions) start with a large detuning (bottom of gure) and are chirped blue (higher frequency) towards being resonant with the center of the longitudinal velocity distribution (gray). In this way the longitudinal velocity spread is decreased. about 2 = 1100 m/s. This corresponds to a Doppler shift of 4GHz of the 280 nm laser used for the experiments, large compared to the transition line width. The standard procedure for cooling a coasting beam, and the procedure used throughout this chapter is therefore rst to wait a short time after injection of the ion beam, in order to let the initial injection oscillations damp out. Before injection the lasers are detuned -4 GHz +
from resonance with the ideal particle with the center-of-mass velocity. Any absolute detuning of the lasers are always with respect to resonance with this particle.
is negative. The detuning -4 GHz has been chosen because it is enough that no uorescence (i.e. interaction) is detected at injection. After the initial pause, with a standard length of one second, the laser frequencies are chirped blue (positive direction) towards resonance with the ion beam. The standard chirp length is 4 GHz, and
is therefore the nal detuning. When the chirping is stopped the measurements are conducted, if for example a measurement of the development of a cold ion beam is desired this is done by keeping the laser detuning xed for some time while performing (for instance) measurements of the longitudinal velocity distribution using the PAT.

5.1 Coasting Beams

67

An interesting point here is how fast we can expect to be able to chirp the lasers and still collect all particles. With a typical power of 30mW and a laser beam area of 1 cm2 we nd that a particle can be accelerated across the width of the cooling transition in  35 s with a laser overlapping in one fth of the ring. However, the revolution time is  45 s, thus ignoring eects from intra-beam scattering we expect to be able to scan the laser about 1000 GHz/s. Thus with a typical uncooled beam with a longitudinal velocity spread of 2 = 1100 m/s the total cooling time would be of order 4 ms. Experimentally the maximum chirp rate used has been 128 GHz/s, where no signi cant change in the cooling eciency compared to lower chirping rates was determined. However, just before injection the ions are accelerated and therefore the longitudinal velocity spread is adiabatically damped, and thus at injection the ion beam is longitudinally very cold. This means that if the laser frequencies can be set to be resonant with velocities symmetrically around the center of mass velocity of the injected beam we can cool the beam by simply using the lasers to prevent the blowup of the longitudinal velocity distribution due to heating from collisions with the rest gas as well as intra-beam scattering. This procedure for cooling has been used in the experiments described in the next chapter.

5.1.2 Suppression of Landau Damping

As discussed in Section 4.3.2, the uctuations in the induced voltage on a pickup due to uctuations in the number of detected particles can be used to extract information about the beam revolution frequency distribution. It was also mentioned brie y that if the particle beam becomes ordered the signal will disappear because the random uctuations in the number of particles detected in the pickup stops. This eect was rst discovered by researchers in Novosibirsk [21], and the experiment has been simulated and the observations ascribed to the beam obtaining longitudinal ordering, while the particles still undergo transverse oscillations, thus a string of 'disks' [103]. More recently the eect has been observed again at GSI [95]. Introducing Schottky diagnostics was supposed to serve two purposes. As explained in Section 4.3.3 the resolution of the velocity measurements using laser-induced uorescence is given by the line-width of the cooling transition. This gives a resolution of  0.1K, whereas the Doppler limit is  1 mK. The uorescence technique does therefore not cover all of the interesting temperature range. Using Schottky noise we hoped to reach lower temperatures. The second reason for trying Schottky diagnostics was, that it could be used to determine whether the beam had reached an ordered state, as this would cause the Schottky noise to drop signi cantly. However, the Schottky diagnostics did not behave as expected [29]. We measured Schottky spectra during laser-cooling using the standard cooling procedure described in Section 5.1.1. In these experiments the lasers were chirped 4 GHz at a rate of 0.5 GHz/s. Then the laser detuning was kept stable for some time during which the velocity distribution and the Schottky spectrum were measured. Thus the Schottky spectrum and velocity distribution were measured while the detuning of both lasers were
. In Figure 5.2 the spectra from a couple of such measurements with dierent detunings
are shown. The

68

Longitudinal Dynamics

measurements were done with an injected current of 5 A, and the lifetime of the beam was  10s. The laser powers were 20-40mW. a)

b)

-70 MHz

-140

-265

-455

-730

-1025 -10 0 10 ∆f/f [10-4]

-2 0 2 ∆v/v [10-4]

Figure 5.2: Measured Schottky spectra a) and measured velocity distributions b) for a number of dierent laser detunings. The vertical scale on the Schottky spectra is logarithmic and each mark represents one decade. The noise oor is represented by the dashed line [29]. The Schottky spectra in Figure 5.2 do not behave as would be expected from the calculations in Section 4.3.2. This surprising result, which has been published in detail in [47], is discussed brie y below. Two things are worth special attention in Figure 5.2. First of all the peaks in the Schottky spectra are orders of magnitude larger than the peaks in Figure 4.8 relating to charge density waves in a cold beam. Secondly, the velocity distributions for the measurements where we have these large peaks are far from being Gaussian, as is assumed in the theory discussed in Section 4.3.2. These two things are indeed connected due to a mechanism

5.1 Coasting Beams

69

called Landau damping [10]. Landau damping is a collisionless damping process in plasmas, which arises because particles moving with approximately the same velocity as a wave in the plasma can exchange energy with the wave. This energy exchange usually leads to damping of the wave. The damping depends on the shape of the velocity distribution, or more precisely on the slope [47]. If the distribution is Gaussian damping will be present. The distributions arising from laser-cooling can be far from Gaussian (Figure 5.2), and in fact the laser force is so strong that it can 'empty' a velocity class, as it drives all resonant ions out of resonance. At the velocities where ions would be resonant with the laser there are no ions, and therefore1 there will be no damping at these velocities, thus charge density waves at these velocities are allowed to grow undamped - which shows up as large peaks in the Schottky noise spectra. This is consistent with the observed behavior that the peaks shift with laser detuning, and that the eect disappears at very small detunings, where the diusion due to intra-beam scattering is fast, and thus prevents the laser from 'digging holes' in the velocity distribution. The suppression of Landau damping is interesting as Landau damping damps many dierent instabilities [14]. So far we have not seen any indication that the observed suppression causes any problems, thus the tendency of the suppression to disappear when the laser detuning is small, because the velocity distribution changes to be more Gaussian, may be rendering this possible problem insigni cant.

5.1.3 Diusion Intra-beam scattering causes velocity diusion in the beam. Intra-beam scattering depends on the particle density as well as the relative velocities of the particles in the beam. An increase in the particles density leads to an increase in the probability for collisions and thus in the intra-beam scattering rate. The same is true when the temperature is decreased. When the temperature is decreased the relative velocities of the particles is decreased, and the Rutherford cross section increases. Therefore, as beam cooling leads to both increased particle density and decreased velocity spreads, diusion due to intra-beam scattering is important in order to understand the behavior of cooled beams. In this section a series of measurements is described in which we have investigated the longitudinal diusion in an initially laser-cooled coasting beam in ASTRID. The measurements were made possible by the possibility of very fast velocity distribution measurements using the LIF technique with the PAT. The system for measuring transverse beam sizes was not implemented at the time of these measurements. 1 In fact Landau damping depends very much on the detailed structure of the velocity distribution, and

the suppression corresponds to singularities in the plasma dielectric function, the observed features also depend on the existence of tails in the velocity distribution, in this case caused by large-angle intra-beam scattering events causing particle to be scattered outside the region between the two lasers (Section 5.1.4). [47]

70

Longitudinal Dynamics

Measuring Diusion

The injected coasting beams were cooled using the same procedure as for the Schottky measurements in the previous section. However, in these measurements the laser frequencies were chirped at a rate of 2 GHz/s. With a chirp rate of 2 GHz/s the 4GHz scan is done in 2 seconds, and as we used a standard delay of 1 second after injection before initiating the chirping, a plateau of constant detuning was reached 3 seconds after injection. At this point a velocity distribution was measured using the LIF technique with the PAT. The diusion rate was measured by measuring the evolution in the velocity distribution as a function of time of the initially cold ion beam after cooling had been stopped (i.e. the lasers 'removed'). However, in order to measure the velocity distribution we need a laser which is resonant with the ions. In these measurements two solutions to this was tried. One solution (a) was to detune the laser away from resonance very rapidly. In these experiments it was done at a rate of 20 GH/z. In order that we could still measure the velocity distribution the lasers were however only detuned -2GHz (this corresponds to a potential change on the PAT of 124V). A dierent approach (b) was also used. In this approach one laser was blocked, and the other was kept at constant detuning. This technique had the advantage that the diusion would not be stopped by the laser when particles diused enough to be resonant with the -2GHz detuned laser, but had the disadvantage that the laser in the distribution might in uence the measurement. Figure 5.3 illustrates the two techniques. Number of particles

a)

l1

l2

b)

v

Number of particles

l1

v

Figure 5.3: Measuring diusion. a) The lasers are detuned several GHz out of resonance very fast. b) One laser is blocked, while the other remains xed in frequency in order for the velocity measurements to be done. With these two techniques for letting the velocity distribution evolve without cooling, we could measure the longitudinal diusion by measuring the velocity distribution evolution as a function of time after the cooling was stopped. Eight velocity distributions were measured in each series. Each velocity distribution measurement was done in 128ms, and with some extra delay the time between the center of each PAT scan was 160ms. In Figure 5.4 the results from every second velocity distribution measurement of a 2A injected beam are shown for measurements using both techniques. The laser power in these measurements was  20mW, and the lifetime of the ion beam  10s. Measurements where carried out with the two techniques with injected currents of

5.1 Coasting Beams

71

0ms

0ms

334ms

334ms

654ms

654ms

974ms

974ms

-1000

-500 0 500 Relative Velocity [m/s]

1000

-1000

-500 0 500 Relative Velocity [m/s]

1000

Figure 5.4: Example measurements of the longitudinal velocity distribution as a function of time after the cooling is stopped. The left plot used method a) and the right used method b). Both measurements were done with 2.0A injected current. 1.0A, 1.5A and 2.0A. From these measurements the velocity spreads was extracted using Gaussian ts. In the measurements using method a) it turned out that as the two lasers were only 4GHz apart, corresponding to about 1100 m/s in velocity, the free evolution of the velocity distributions (as can be seen in Figure 5.4) was in uenced. The measurements using method b) were analyzed by tting with a function consisting of two half Gaussians with dierent widths, and then using the width of the part away from the laser as the velocity spread of a freely evolving distribution. Figure 5.5 shows an example of the evolution as a function of time for a 2A injected beam. The measurement was done using method b). Similar curves were drawn for the ve other measurements conducted and Table 5.1 lists the results. The uncertainties stem from the uncertainties in the Gaussian ts. The longitudinal diusion coecient Dk is given by hk2i = 2Dk t (5.1) where Dk is assumed not to depend on the longitudinal velocity. Thus we should expect that the velocity spread development when the cooling is stopped is q (5.2) k(t) = 2Dk t + k(0) As we saw in Figure 5.5, there is good agreement with the expected square root dependence of the longitudinal velocity spread on time.

72

Longitudinal Dynamics 250

Longitudinal Velocity Spread [m/s]

Measured Spreads + Diffusion Fit 200

150

100

50

0

0

0.2

0.4

0.6

0.8

1

Time after stop of cooling [s]

Figure 5.5: Velocity spreads measured as a function of time with method b). The injected beam current was 2.0A. The diusion coecient corresponding to the t is D = (2.90.1)
104 m2/s3 . Injected current 1.0 A 1.5 A 2.0 A

Diusion Coef. (a) (4.3 0.4)
104 m2/s3 (4.9 0.3)
104 m2/s3 (5.0 0.3)
104 m2/s3

Diusion Coef. (b) (2.1 0.3)
104 m2/s3 (2.5 0.2)
104 m2/s3 (2.9 0.2)
104 m2/s3

Table 5.1: Diusion coecients for various currents measured with method a) and b). The initial velocity spread was  25 m/s. The diusion coecients listed in table 5.1 show the clear tendency that the diusion coecients measured using method b) are smaller (about half) than the diusion coecients measured using method a). This observation is consistent with the sample evolution shown in Figure 5.4, where it is easy to see that the velocity distributions evolve to become much wider in method a) than in method b). It seems therefore that the laser in uence in method b) is signi cant. Another point, which is also observable in Figure 5.4 is that the maximum of the velocity distributions in method b) is moving away from the laser - this means that both the two 'half' Gaussian distributions increase in width, a feature which makes it dicult to decide what velocity spread to extract from the measurements. The measurements using method a) are, as mentioned before, also in uenced by the lasers. The in uence is also seen in Figure 5.4, as a small edge in the velocity distribution at the point where the laser is resonant (zero velocity). The diusion measurements discussed here are a good demonstration of the powerful diagnostics we have available for studying the dynamics of ion beams. However, we have observed that in the exact procedures presented the free evolution of the velocity distributions after the cooling was stopped was in uenced by the lasers. From the presented

5.1 Coasting Beams

73

data it is hard to estimate exactly how much this has in uenced the results, but the large dierence in the diusion coecient extracted using the two methods demonstrate that we should be careful when drawing conclusions. It has later been realized that by exciting the PAT with a voltage to compensate for the detuning of the laser in method a) we can make the laser resonant with the center of the velocity distribution in the PAT only. This means that we can detune both the lasers -4GHz (corresponding to 1100m/s), and thereby bring them far enough away that they do not in uence the measurement. With this improved technique it should be possible to obtain unambiguous measurements of the longitudinal diusion, a powerful tool in the understanding of the dynamics of cold ion beams. One of our initial goals of these measurements was to use the longitudinal diusion to extract information about the transverse dimensions of the ion beam. This would also be a good supplement to actual transverse diagnostics, as the diusion coecient contains information about the density of the ion beam, as well as of the relative kinetic energy. The next section therefore considers how information about the ion beam can be extracted from the longitudinal diusion. Comparison with theory

In order to extract information about the beam density we need an expression for the diusion. In the smooth, non-relativistic approximation, assuming constant beam size around the ring, we can calculate this under the assumption that the transverse temperature is much larger than the longitudinal. The diusion is part of the collision term in the Boltzmann equation mentioned in Section 3.1. The collision term is sometimes called the Landau collision integral [53]. The details of the calculation are shown in Section C.3, and the rst order, velocity-independent, result is 4

Dk = p ne 2LC2 8 20m ?

(5.3)

where n is the ring averaged particle density and ? is the ring averaged transverse velocity spread. A result with a couple of higher order corrections was published in [49]. We have in Section C.3 compared this analytic calculation of the longitudinal diusion to the results from an intra-beam scattering (IBS) code developed by Giannini and Mohl [24] based on Piwinski's IBS-theory [80]. It turns out that the simple approximation dier from the IBS program with about a factor of 5, but the scaling with the transverse emittance is reasonable (appendix C). This deviation is probably due to the rather crude nature of the approximation, furthermore, dispersion, which is not included in the simple model, has an in uence on the coupling. Due to these observations it was decided to compare the measurements to the results of the ibs program instead of the analytical diusion calculation. The analytical study is still reproduced in appendix C as it gives some insight into the physics of the problem. In Figure 5.6 the longitudinal diusion coecient as a function of the emittance is shown for

74

Longitudinal Dynamics

a beam current of 1.0A. The IBS program generates growth times given by !,1 d 1 k  =  dt k

(5.4)

from which we can extract a diusion coecient, by using (5.1)

k2 D=  (5.5) which if smaller than zero indicates cooling instead of heating, in which case the growth time is a cooling time, which is connected to the friction force F by the following equation !,1 @F 1 (5.6) c  , m @v v=v0 where m is the mass of the ion and v0 is the stable point for the cooling (in the transverse plane v0 = 0). 103 Horizontal Vertical 102 106 101

100 105 10-1

Transverse Cooling Time [s]

Longitudinal Diffusion Coefficient [m2/ s3]

107

Longitudinal Diffusion 104 -7 10

10-6

10-2 10-5

Transverse Emittance [π m rad]

Figure 5.6: Diusion coecient and sympathetic cooling time with a current of 1A and a longitudinal velocity spread of 25m/s. The dashed horizontal line indicates the measured value of diusion at 1A (using method a). In Figure 5.6 the horizontal line indicates the diusion coecient measured in a 1 A beam with method a). From the Figure we extract a transverse emittance of order 6.5  mm mrad. This is equal to a beam size at the camera position of about h = 9 mm and v = 4 mm. We did not at the time have any notion of what to expect, but the transverse cooling time of about 150s is rather long. As we will learn in the next chapter the physical overlap is crucial to the sympathetic cooling, a fact which has been realized all along, but the sensitivity was not known, as the cooling in the longitudinal dimension always worked quite well. The reason it is hard to

5.1 Coasting Beams

75

determine the exact dependence on the physical overlap without transverse diagnostics is that improved cooling leads to colder beams and higher densities and therefore increased diusion due to intra-beam scattering. With only longitudinal diagnostics, this will emerge as increased tails in the velocity distribution. However, if the overlap is bad, the bad coupling to the ion beam can lead to fewer particles being collected into the cold part of the beam, which will also emerge as tails in the velocity distribution. It is interesting to calculate the ring averaged density from the results above, as this may give us some insight into which regime we are working. In our 1A beam current example the density becomes 3.5
104 cm,3 which is 100 times smaller than the maximum density in the smooth approximation (the uniform, zero emittance beam) (see Section 6.1). This indicates that the sympathetic cooling rate is low. A careful study of Figure 5.4 also reveals that there seems to be tails on the cold distributions indicating that we have not cooled all particles. This may be due to large-angle intra-beam scattering events which may cause particles to have there longitudinal velocity changed enough that they are sent outside the short range of the laser forces (called the Touschek eect), but it could also be due to very bad laser - ion beam overlap - or a combination of the two. The most likely conclusion is that it is an overlap problem, as the diusion in the beam decreases with decreasing density, and we observe a very low diusion coecient. The discussion in this section shows how critical it is to control all parameters in the experiment, as we will see more evidence of later. The results demonstrate a useful procedure by which the emittance can be determined. This procedure is complementary to nding the emittance by measuring beam sizes, and using these methods together may reveal if other eects play a role in the dynamics. The chirped laser technique is recommended for coasting beams, as leaving one laser in the beam leads to data which are dicult to interpret, and the measurements here seems to indicate that it may have some in uence on the diusion coecients measured. In a bunched beam where cooling can be accomplished with one laser alone, the measurements are much easier, as the cooling laser can just be cut o when necessary.

5.1.4 Touschek Scattering

The dynamics of a laser-cooled ion beam are, as we saw in Section 5.1.2, dierent from standard ion beams, the main dierence being that the velocity distribution may be nonGaussian and have sharp edges due to the very strong laser force. This behavior is due to the narrow range and high strength of the laser-force. If small-angle scattering events, i.e. diusion, were the only heating mechanism in the beam the narrow range of the laser-force would not play a role, only the strength would, as velocity changes of the ions would be small compared to the line width of the cooling transition. However, if the transverse dimensions are much warmer than the longitudinal (like we saw in the previous measurements), energetic collisions transferring transverse momentum into longitudinal momentum, as illustrated in Figure 5.7, are possible. These collisions can lead to particles scattering outside the range of the laser force and generate tails on the distribution, as illustrated in Figure 5.9. This process is called Touschek scattering [5]. This problem

76

Longitudinal Dynamics

can be rather severe, and various experiments to counteract this eect by extending the laser-force capture range have been carried out elsewhere [102, 4]. y

s

Beam Direction

Figure 5.7: A large angle scattering event may transfer energy from the transverse to the longitudinal dimension. When the longitudinal temperature is low, the lasers are close together in longitudinal velocity space, and we expect the losses due to Touschek scattering to increase. The Touschek eect is therefore an important mechanism in laser-cooled ion beams, and we have conducted some experiments to determine the exact behavior of the eect. We have measured how the number of particles in the cold portion of a laser-cooled coasting ion beam decays. This was done using the standard cooling procedure for cooling coasting beams described previously. After the laser chirp to the desired detuning, the detuning was maintained for 4 seconds, during which several velocity distributions where measured at intervals of 0.5 sec. The lifetime of the beam during these experiments where   30s. The decay of the maintained cold distributions in these experiments was analyzed by integrating the cold part of the distribution. Figure 5.8.a shows the number of particles in the cold part of the velocity distribution as a function of time for a 4.0 A beam (injected). Similar measurements were carried out at currents of 1.0 A and 2.0 A. In Figure 5.8.a the number of particles have been normalized to be 1.0 in the initial distribution. Another method would of course be to observe the number of particles in the tails as a function of time, however the statistics when doing this are much worse, and in the measurements here it would have been impossible. The rate of Touschek scattering does of course depend on the relation between the transverse and the longitudinal temperature in the beam. If we do the calculation in the smooth approximation and assume that the transverse temperature is much higher than the longitudinal, the probability for a particle to undergo a velocity change larger than v is given by [49] "p ( ) # 1 dN (v) =  (1 + 2
2) 1 , p2 Z
e,x2 dx ,
e,
2 (5.7) e4 N  2 2 N dt 645=220M 2v2v?R  0 where N is the number of particles in the beam, M the mass of the ion, v? is the rms transverse velocity,  is the rms beam radius, R is the mean radius of the storage ring, and
= v=v?. Details of the calculation of this rate can be found in for instance [60].

5.1 Coasting Beams

77

0.8

0.6

0.4

0.2

1.0 microamps 2.0 microamps 4.0 microamps 4.1e-6*detun^-1.29 1.3e-6*detun^-1.27 2.8e-7*detun^-1.24 Equation (5.8) Equation (5.7)

1e-08 Loss rate, r [1/s]

Normalized number of particles in peak

1.0

1000 MHz 760 MHz 530 MHz 410 MHz 300 MHz 180 MHz 100 MHz

1e-09

1e-10 0.0 0.0

1.0 2.0 3.0 Time after release of distribution [s]

4.0

100

1000 Laser detuning [MHz]

Figure 5.8: a) Relative number of particles in the cold part of the velocity distribution of a 4.0A beam for various laser detunings. The lines are ts with equation (5.9). b) Decay rates (r) as a function of detuning. The calculations were done with ? = 4  mm mrad, the distribution function used in (5.8) was a sum of a square and a parabolic. As the loss of a particle scattering a speci c distance in longitudinal velocity space depends on its initial \position" in velocity space (see Figure 5.9), it is necessary to integrate over the normalized distribution f (v) to get the number of particles lost per unit time " # dNlost = 2
Z v ,l=2 f (v0)
1 dN (v , v0 +  =2) + dN (v + v0 +  =2) dv0 (5.8) l l dt 2 dt  dt  0 where l is the FWHM of the electronic transition and v is the detuning of the laser. From equation (5.8) we can de ne an N independent loss rate r as dNlost =dt = r(v )
N 2. The dependence of r on the longitudinal temperature of the beam (or rather the detuning of the laser v) can then be found by numerical integration of equation (5.8). These losses together with the normal decay of the beam due to electron capture from the rest gas (dN=dt = ,
N , r(v )
N 2), give us the following behavior of the number of particles in the cold distribution: (5.9) N (t , t0) = e(t,t0)= + r(v )NN0  (e(t,t0)= , 1)  0 where  is the beam lifetime, and r(v ) is the loss rate due to Touschek scattering as calculated from equation (5.8). The decays of the cold part of the velocity distributions, of which an example was shown in Figure 5.8, have been tted with equation (5.9), and the results are shown as lines in the gure. The agreement with the expected decay behavior is good. From the ts the decay rate r could be extracted. The decay rate r as a function of the laser detuning is shown in Figure 5.8.b for three dierent injected currents. In Figure 5.8.b some ts to the scaling of r are shown, as well as calculations using the theoretical equations.

78

Longitudinal Dynamics N

a)

kept

-v

δ

N

b)

kept

kept

v

δ

V

-v

δ

N

c)

lost

v

δ

lost

V

-v

δ

lost

v

δ

V

Figure 5.9: a) Two possibilities for a particle scattered by an amount v. b) With the same amount velocity change, particles at a dierent place in the velocity distribution would be able to escape. c) In [60] they assume a thin distribution, and thus uses (5.7) directly obtaining a too low loss rate (i.e. they ignore case b). v is the detuning of the laser. If we assume, as for instance done in [60], that we do not have to integrate over the longitudinal velocity distribution the scaling of the loss rate with longitudinal velocity spread is r  v,2. This model does not include the contribution from particles at the edge of the velocity distribution, which have an increased probability of being lost. The loss rate in the model ignoring the velocity distribution and the loss rate in the modi ed model, which does include the velocity distribution, are both shown next to the measured loss rates in Figure 5.8.b. The measured behavior is closer to the modi ed version integrating over the velocity distribution. The example emittance of 4.0  mm mrad which gives theoretical results quite close to the measured is in good correspondence with the results from the diusion measurements. But the theoretical loss rate also depends critically on exactly how far we assume that a particle must scatter before being lost. As evident from equation (5.8) it has been assumed that a particle must at least jump a transition line width before being lost. This assumption gives the discussed results, and an increase in the minimum scattering length does not alter the results much. Thus the theoretical model seems quite sound. The decrease of the loss rate with increasing current is however not explained, and probably relies on some changes in the emittance with current. Measurements including a direct measurement of the emittance are necessary to clarify this. The relatively large emittance extracted from the Touschek measurements agrees with the results from the diusion measurements in the previous section. At this point the reason for the large emittance could not be determined, as the quality of the physical overlap between the ion beam and the cooling lasers could not be determined, and the overlap is believed to be important for the transverse to longitudinal coupling. Optimizing the relative position of the ion beam and the laser beam is dicult with the setup used here, as optimum overlap and thus sympathetic cooling lead to heavier losses because of the increased intra-beam and Touschek scattering, while non-ideal overlap would also lead to losses due to less laser interaction with the ion beam. In conclusion the Touschek measurements are consistent with the diusion study, and

5.2 Bunched Beams

79

the experimental results suggests that Touschek loss measurements are a good supplement to the density information obtainable from diusion measurements. The density information can be compared to the density measured by the transverse imaging system, and thereby give insight into which processes are important for the transverse to longitudinal coupling and thereby to the limits to laser-cooling of ion beams. The Touschek losses may however also turn out to be a severe limitation to the achievable cooling results in coasting beams, and in that case we may be forced to introduce one of the capture range extending methods discussed in Section 3.3.2.

5.2 Bunched Beams In a bunched beam the longitudinal dynamics is complicated by the presence of synchrotron oscillations and of course the separation of the beam into bunches. When a bunched beam is cooled to a point where the longitudinal energy spread is low, the space charge forces start to dominate the longitudinal dynamics. As the longitudinal and the transverse dimensions are usually only weakly coupled (as we have discussed in the previous sections) the longitudinal relative kinetic energy may, independently of the transverse, become low compared to the inter particle Coulomb repulsion (a high longitudinal plasma parameter) and the bunch lengths become space charge limited. This has been observed in electron cooling experiments [23] and is routinely seen in high energy accelerators, where the longitudinal con nement elds may be very high. Strong con nement will decrease the inter particle distance, and situations arise where the Coulomb forces are more important than the thermal motion. Using laser-cooling we have produced longitudinally space-charge limited beams [30]. Apart from the scaling of the bunch length with particle number the actual pro le of a bunched beam as a function of the temperature (or rather the longitudinal plasma parameter) is interesting because the occurrence of instabilities may depend on the detailed density pro le of the bunch [87]. Furthermore we can measure the current directly in a bunched beam, which makes the results much less ambiguous. As our typical bunches are very long compared to their width, we may be able to use the IBS program (which does not include space charge) by assuming that bunched beams are just coasting beams with higher linear density - this method of course only works when there are no resonant couplings between the synchrotron motion and the transverse motion. If the sympathetic cooling turns out to be too weak, a couple of enhancement schemes have been proposed for bunched beam, which utilize resonant couplings between the oscillations [74, 73], yet another reason to study bunched beams.

5.2.1 Cooling a bunched beam In Section 3.3.3 the principle behind bunched beam cooling was described. There were two fundamentally dierent ways to accomplish the cooling, one was to use one xed laser and use the synchrotron oscillations to bring the particles into resonance and thereby cool

80

Longitudinal Dynamics

them. Another was to chirp the lasers as often done in coasting beams. We have until now only worked with the xed laser procedure, as this procedure only needs one cooling laser, and is technically simpler to accomplish. The time it takes to cool a particle with velocity amplitude
v using a xed laser can be calculated by integrating the laser force on the particle over a synchrotron period, assuming a harmonic potential and thus ignoring the discrete nature of the bunching process. However, in our case the synchrotron frequencies are of order 150 Hz, corresponding to a period of 7 ms. This is large compared to the time it takes to accelerate the beam one transition line-width, which we earlier calculated for a coasting beam to be 35 s. We may therefore assume that all particles with amplitudes larger than the laser detuning are damped by one transition linewidth in one synchrotron period. This means that it would take approximately 0.6 seconds to cool a beam with a longitudinal velocity spread of 2 =1100 m/s, 150 times more than the time it takes to cool a coasting beam of equal velocity spread with chirped lasers. This means that by using two counter propagating chirped lasers we could cool much faster. However the process of using only one laser has the intriguing property that the laser does not have to be tunable if one can adjust the revolution frequency to Doppler-shift the ions to a laser frequency obtainable from a xed laser, it has therefore been the method of choice so far. Figure 5.10 shows how an injected coasting beam into the storage ring with the bunching drift tube active is cooled by one xed laser detuned  -200 MHz. The circulating beam current was  80nA, and the power in the cooling laser  45mW. The time scale for the cooling process is observed to match the estimate of 0.6 seconds rather well. For all experiments discussed in this thesis the standard settings for bunching in ASTRID are those listed in table 5.2. Parameter Symbol Value Drift Tube Length L 1.10 m Typical Harmonic h 16 Typical RF peak-peak voltage Vpp 4.55V Typical Synchrotron frequency s 150 Hz Equivalent synchrotron tune Qk 3.7
10,3 Average vacuum chamber radius b 5.0 cm

Table 5.2: Parameters for bunching in ASTRID. The rather low synchrotron frequency in ASTRID will set a limit to how low temperatures can be reached with one laser. This limit arises because the longitudinal diusion in velocity due to intra-beam scattering may become faster than the synchrotron period, and may cause a constant (asymmetric) tail on the velocity distribution. This problem is often observed in the form of asymmetric velocity distributions when the laser detuning is small.

5.2 Bunched Beams

81 a)

p

AAAA AA AA A A A

730

b)

530

Time

a fte

ctio r inje

n [m

s]

s

330

p

s

130 -500

0

-500

Relative Velocity

Figure 5.10: Left: Longitudinal velocity distributions at dierent times after injection of a coasting beam in a ring with bunching turned on and a xed laser with a detuning of  -200 MHz. Right: The synchrotron oscillations rotate the initially low energy-spread but high 'time' spread beam. Due to the sinusoidal potential the synchrotron frequency will be amplitude dependent and we will observe a relatively warm bunched beam after a short time.

5.2.2 Bunch Pro les

Following the arguments of [87] we nd the longitudinal bunch pro le as a function of the longitudinal temperature of the bunch. The general assumption is that we are in the nonrelativistic limit and can ignore beam-induced magnetic elds. Furthermore, we assume that the transverse temperature is high enough that the space-charge forces are negligible in the transverse dimension, and the coupling to the longitudinal motion therefore weak. In this regime the longitudinal beam pro le can be described entirely by longitudinal force equilibrium. The longitudinal Maxwell-Boltzmann distribution is given by ! H k fk = fk;0 exp , k T (5.10) B k where Hk is the longitudinal part of the Hamiltonian. If we integrate the over the longitudinal velocities and multiply with the charge q, we obtain the longitudinal Boltzmann relation for the line charge density ( ) q [ conf (s) + sc (s)] (s; Tjj) = (0; Tjj) exp , (5.11) k T B jj

82

Longitudinal Dynamics

where the potentials are the con nement and space charge potentials respectively. We assume that the con nement potential is that of equation 3.14. b a ds

Figure 5.11: Integration path for calculation of the longitudinal electric eld Ejj(s). The gure is not to scale. If we assume that the bunches are much longer than they are wide, the electric eld from the bunches is purely transverse. In Figure 5.11 the integration path for calculating the longitudinal electric eld in this situation is shown schematically - the gure is not to scale, as the bunch radius a must ful ll a  L where L is the length of the bunch. Furthermore, image eects from the vacuum chamber must be negligible, i.e. b  a. The longitudinal electric eld Ejjsc(s) can then be found via I 0 = Escdl , Zb Z0 0 = Ejjsc(s)ds + E?sc(r; s + ds)dr + E?sc (r; s)dr (5.12) 0

b

where we use that the transverse electric eld from a uniform cylinder of charge can be extracted using Gauss' law, and nd g0 [(s + ds) , (s)] = 0 ; g = 1 + 2 ln(b=a) Ejjsc(s)ds + 4 (5.13) 0 0 where a and b are the beam and vacuum chamber radii respectively. The longitudinal electric eld is therefore g0 d (5.14) Ejjsc(s) = , 4 0 ds Zero temperature

In the zero temperature limit, which we studied in [30] the longitudinal charge distribution can be found directly as the resulting longitudinal force on any particle must be zero. We nd, using the harmonic approximation for the potential, that the distribution is parabolic (or a cosine if the exact potential is employed) with a bunch length of s s0 = 3 163Qg2h0qCF (5.15) 0 0

5.2 Bunched Beams

83

where Q is the total charge in the bunch, h is the harmonic number, and F0 is the con nep ment force constant de ned in equation (3.13). The FWHM of the bunch is given by 2 s0. It should be emphasized that this equation shows that when the longitudinal temperature is zero the bunch length scale as a cube root in the total charge in the bunch (the number of particles). Finite temperature

With a nite temperature we use the Maxwell-Boltzmann equation (5.11) and with the harmonic potential we nd ( )  2 2 hF qg 0 0 h h h  (s; Tjj) =  (0; Tjj) exp , Ck T s , s0 , 4 k T  (s; Tjj) (5.16) B jj 0 B jj where the constant s0 can be chosen such that ( hh i) qg hF 0 0 2 h h h  (s; Tjj) =  (0; Tjj) exp , Ck T s + 4 k T  (0; Tjj) ,  (s; Tjj) B jj

0 B jj

(5.17)

from which the charge distribution function for a given longitudinal temperature can be found numerically by iteration. Now, it turns out that we may have bunches long enough to be in uenced by the anharmonicity of the longitudinal potential, thus we need to take that into account. This is done by using the sinusoidal con nement potential derived from equation (3.13), and we nd that the longitudinal density distribution is given by

(s; Tjj) =

8 ! # g q (0; T ) , (s; T ) 9 < F0C " 0 jj jj = 2 h (0; Tjj) exp : 2hk T cos C s , 1 + ; (5.18) 40kB Tjj B jj

where we again can nd the density distribution by iteration. Figure 5.12 shows how the longitudinal distribution changes as a function of the number of particles for the harmonic and the sinusoidal potential. The bunch length as a function of the number of particles per bunch is shown for various longitudinal velocity spreads (Tk = mk2=kB ). For high temperatures or large particle numbers the distribution may become longer than the potential (high velocity tails) - these particles will not be con ned to the bunch and will therefore constitute a coasting beam 'background' which is not visible in our measurements. This has been taken into account by only counting the number of particles in the bunch - therefore the particle number 'saturation' tendency in Figure 5.12.b. When space charge forces becomes negligible, the bunch length is (for a harmonic potential) s B Tk ln 2 sFWHM = 2 CkhF (5.19) 0

84

Longitudinal Dynamics

which is independent of the particle number, equivalent to horizontal lines in Figure 5.12.b. In the other extreme, when the beam is space charge limited, the bunch length scales as a cube root in the number of particles, as we observed in the zero temperature limit investigated earlier (equation (5.15)).

c)

-0.50

-0.25 0.0 0.25 Relative Position [Wavelength]

b)

d)

-0.25 0.0 0.25 0.50 Relative Position [Wavelengths]

Length of bunch (FWHM) [m]

a)

1.0

20 m/s 50 m/s 100 m/s 200 m/s 400 m/s 0.1

10

4

5

6

10 10 Number of particles in bunch

10

7

Figure 5.12: Left: Longitudinal charge distributions for the exact case (black lines) and the harmonic approximated case (red lines) for a longitudinal velocity spread of jj = 19 m/s (Tk = 1 K). The longitudinal axis is in units of the potential wavelength. Standard bunching parameters were used (table 5.2). The four dierent plots are for four dierent numbers of particles. a) 8.0
106 , b) 4.0
106 , c) 2.0
106, d) 1.0
106. Right: Bunch lengths versus beam current with sinusoidal con nement, for various velocity spreads. In actual measurements the high temperature / high current situation is hard to investigate, as we need a well de ned zero between bunches in order to determine the background level on the pickup used for measuring bunches. Without the zero level the absolute current and the density distribution cannot be determined unambiguously.

5.2.3 Pro le Measurements In recent experiments we have measured the longitudinal bunch pro les as well as the transverse beam sizes of laser-cooled bunched beams with dierent longitudinal velocity spreads. Figure 5.13 shows examples of longitudinal velocity and density pro les for dierent longitudinal velocity spreads [56]. The measurements were done using the xed laser cooling technique as described in Section 5.2.1, and then at 7s after injection bunch pro les were measured, 0.1 second after that the longitudinal velocity distribution was measured using the PAT (from 5.5 to 6.5 seconds after injection the CCD was exposed, but the information gained from those measurements will not be discussed in this section). The cooling laser power were for all measurements in the range 35 - 50 mW, whereas the probe laser power was 15 - 35 mW. The lifetime of the ion beam was  25 sec.

5.2 Bunched Beams

1.0

85

55.4 m/s

55.4 m/s

0.5

1.0

164 m/s

164 m/s

Fluorescence [Arb. Units]

Induced Voltage on Pickup [V]

0.0

0.5 0.0 1.0

209 m/s

0.5 0.0 1.0

235 m/s

209 m/s

235 m/s

0.5 0.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 Relative Position [m]

1.5

-750 -500 -250 0 250 500 750 Relative Velocity [m/s]

Figure 5.13: Longitudinal bunch and velocity pro les for bunches of 4
106 particles. The gray (red) lines in the velocity pro les are parabolic ts to the data (folded with the laser line shape), and in the bunch pro les the theoretical pro le is extracted from the parabolic modi cation of the Maxwell-Boltzmann distribution. The velocities in the upper left corners of each curve are the extracted rms velocity spreads. When trying to match these measurements to the theory from the last section we quickly realized that the match was insucient. And if we take a look at Figure 5.13 we can see why. The longitudinal velocity distributions generated by laser-cooling are not Gaussian (as we already discussed in Section 5.1.4), but rather parabolic. This means of course that as the longitudinal bunch shape in a low density (dominated by the kinetic energy of the particles and not by the intra particle Coulomb forces) beam is equivalent to the velocity distribution, as the distribution is 'rotating' in phase space - it will be parabolic. Thus when laser-cooling is applied, the longitudinal density distribution will be parabolic for all temperatures (ignoring anharmonicity in the longitudinal potential). As the Maxwell-Boltzmann distribution assumes a Gaussian velocity distribution it will not properly describe the change in bunch length as a function of particle number and temperature.

86

Longitudinal Dynamics If we assume a parabolic velocity distribution equation (5.18) changes to (s; Tjj) = (0;"Tjj) ( " ! #  )# F C 2 h g q 0 0 1 , 2hk T cos C s , 1 + 4 k T (0; Tjj) , (s; Tjj) (5.20) 0 B jj

B jj

where  = 52 . The  factor stems from a parabolic distribution being (1 , x2=x20) whereas a Gaussian is exp(,x2=2hx2i) and for a parabola hx2i = 5x20 (see appendix A.1). A comparison of the distributions assuming a Gaussian velocity pro le and a parabolic are shown in Figure 5.14 together with a measured pro le. 1.2

Induced Voltage [V]

1

Measurement Parabolic Velocity Distribution Gaussian Velocity Distribution

0.8

0.6

0.4

0.2

0 -1.5

-1

-0.5

0

0.5

1

1.5

Relative Position [m]

Figure 5.14: Measured bunch pro leqtogether with the two discussed model distributions. The longitudinal velocity spread is hv2i = 100 m/s. Note the large dierence in FWHM of the two distributions. The smoothness of the parabola is because it has been folded with the pickup shape. The current was  200nA, corresponding to 3.5
106 particles in the bunch. From equation 5.20 we can calculate the expected bunch length for various velocity spreads as a function of the beam current. Figure 5.15 shows some measurements of the bunch length dependence on the beam current for dierent longitudinal velocity spreads, together with the expected lengths extracted from equation (5.20). The calculations and measurements of how the length change as a function of particle number and velocity spread correspond quite well. The changing transverse beam sizes have not been incorporated in the calculation as the change in the results are negligible (due to the logarithmic dependence of the bunch length on the transverse beam size). However, there is a tendency for the bunches to be shorter than expected, a tendency which becomes more pronounced with decreasing velocity spread and high current. This deviation is believed to be due to the fact that we have ignored any direct in uence of transverse temperature and/or beam size in our model. As mentioned above a simple

5.3 Discussion

87

Bunch Length (FWHM) [m]

1.5

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 28.4 m/s - 50 m/s 128 m/s - 144 m/s 235 m/s - 248 m/s 0.2 5 10

106

35.5 m/s 139 m/s 241 m/s 107

Number of particle in bunch

Figure 5.15: The bunch length versus the number of particles in the bunch. The solid lines are results from the parabolic model. A transverse beam size of 2mm has been assumed in the calculation. inclusion of the measured radii in the simple model here does not alter the results enough. One other possible explanation could be that the bunches are space charge limited in three dimensions. However, this would mean that we would expect the bunches to be longer, as the space charge forces will be stronger [98, 22]. Thus there is no immediate simple explanation to the question without a detailed investigation of the development of the beam size. In the next chapter we will discuss the transverse dynamics of laser-cooling, and return to this question.

5.3 Discussion From our studies of the longitudinal dynamics we have learned several things. First of all laser-cooling has a tendency to suppress Landau damping and create strong charge density waves in a coasting beam. These waves may lead to problems when trying to reach the crystalline state, however the eect depends on a non-Gaussian velocity distribution, and at very low temperatures where the lasers are detuned to be few line-widths from each other, the friction forces become approximately linear, the diusion due to intra-beam scattering strong, and we observe a tendency for the velocity distribution to become Gaussian and the coherent signal to disappear (see Figure 5.2). The diusion and Touschek scattering experiments seemed to indicate inecient sympathetic cooling, but were limited in their usefulness by the missing control of the beam-laser overlap. However, the results demonstrated that it should be possible to use these methods to study what processes dominate the dynamics of the transverse to longitudinal coupling once a transverse diagnostics system is in place. The study of the longitudinal shapes of a laser-cooled bunched beam revealed a good correspondence with a simple model which to a large extent ignored the transverse degrees

88

Longitudinal Dynamics

of freedom. The results also demonstrated that laser-cooled bunched beams have virtually no tails, and have a parabolic velocity distribution (and longitudinal density distribution), this may have an in uence on the occurrence of instabilities at high currents. We learned from the comparison with the simple model that the bunches are not transversely space charge limited (uniform density), as this would mean that the bunch lengths should be larger than the simple model suggests. However, we observed that the bunches in the ultra cold case were in fact shorter than expected, a fact which could not readily be explained - but a possible explanation has been found by studying the transverse dynamics. The results from the studies of the transverse dynamics are discussed in the next chapter.

Chapter VI Transverse Dynamics Understanding the transverse dynamics of stored ion beams is crucial to successful operation of a storage ring, as we discussed in Chapter 2. Especially when laser-cooling, which as we have seen only operates in the longitudinal dimension, the transverse dynamics, and coupling to the longitudinal motion is important for cooling the transverse degrees of freedom. With the new transverse diagnostics system described in Chapter 4 we have been able to study the transverse beam sizes of a stored ion beam. In this chapter the results and implications of these studies are discussed. The discussion starts with a review of a calculation on the transverse beam pro les of nite temperature ion beams discussed in [87], as this calculation is important when trying to understand how strongly the beam is in uenced by space-charge. The main purpose of the experiments discussed in this chapter has been to understand what mechanisms are important for successful sympathetic cooling, and how strong the mechanisms are. We know from the experiments discussed in the previous chapter that the laser - ion beam overlap is important, and some of the discussion is therefore devoted to experiments on the in uence of changing the laser ion beam overlap. A spin-o from these experiments was a procedure to optimize the sympathetic cooling. However the experiments also demonstrated that something seems to be limiting the maximum densities attainable. Most of the discussion in this chapter focuses on experiments which were aimed at understanding how this limitation arises. In the most recent experimental run a rather surprising result arose which indicates a phase transition in low current laser-cooled ion beams. The last section of this chapter is devoted to this recent result.

6.1 Beam Pro les As we discussed in Section 2.3 a beam in ASTRID cannot form a 3-dimensional crystal due to the low periodicity. However it is still interesting to know what will happen to the beam if cooled to very low temperatures in all dimensions. The pro le of a cold ion beam has been studied by Reiser et al. [87], under the simple assumption that an ion beam has

90

Transverse Dynamics

a transverse and a longitudinal temperature, which are mostly independent due to weak coupling between the two degrees of freedom (see also Section 5.2.2). Furthermore it is assumed that the velocity distribution is Gaussian, and for simplicity only space charge and con nement forces (i.e. no image charges and so on) are considered. This implies non-relativistic beams, and beams which are transversely small compared to the vacuum chamber. These assumptions are reasonable for the situation in our experiments; the results may therefore be of some value to us. All calculations assume the smooth approximation, i.e. that the storage ring provides uniform focusing. Furthermore we use an axisymmetric con nement for simplicity. Thus our con nement force is given by

F?conf (r) = ,m!?2 r ; !? = !rev Q?

(6.1)

where m is the ion mass, !rev is the revolution frequency, and Q? is the transverse tune.

6.1.1 Space charge limited In the space charge limited (or zero emittance) case the resulting force on each particle is zero and the system is in equilibrium, thus the space charge forces must be linear inside the beam. This is equivalent to the charge density being uniform, i.e. the beam being cylindrical, thus the electric eld from the beam is purely transverse and can be found via Gauss' law to be given by (  =r ; r  a (6.2) E?(s; r) = 2  1r=a 2 ;r < a 0 where  is the linear charge density, r is the radial distance from the beam center, and a is the beam radius. This gives us an equilibrium space charge limited beam radius of s  (6.3) a0 = m!q 2 2 ?

0

with an equivalent maximum density of

 = 20 m!2 n0 = qa 2 q2 ? 0

(6.4)

where for a coasting beam in a storage ring  = qN=C , where N is the total number of particles and C is the circumference. In ASTRID, using an average transverse tune of 2.5 and 2.78
107 100 keV 24Mg+ particles the equilibrium radius is 0.26 mm. The maximum density for a tune of 2.5 is 3.38
106 cm,3. For comparison the density of lattice sites (nuclei) in a typical solid is of order 1023 cm,3.

6.1 Beam Pro les

91

6.1.2 Finite Temperature

Reiser et al assume a nite temperature [87]. When the temperature increases from zero to temperatures high enough that space charge is negligible, the density distribution must change from the uniform density in the space charge limit to the Gaussian which characterizes a Maxwell-Boltzmann distribution. The transverse particle density distribution is given by " 2 r2 q s (r) # m! n(r) = n(0) exp , 2k ?T , k ?T (6.5) B ? B ? where s? is the transverse space charge potential which has to be found self consistently by using Poisson's equation r2s? (r) = qn(r)=0 (6.6) thus " # Z r q Z r0 2 r2 m! q ? 00 00 00 0 n(r) = n(0) exp , 2k T + k T r n(r ) dr dr (6.7) B ? B ? 0 0 r 0 0 where n(0) is found by requiring that the linear density is constant, i.e. that Z1  = 2 r n(r) dr = a20n0 (6.8) 0 which can be solved numerically as shown in Figure 6.1. 2K

Transverse Density [arb. units]

5K 10 K 20 K 35 K 50 K 100 K 200 K 600 K Zero

0

0.2

0.4

0.6

0.8

1

Radial Position [mm]

Figure 6.1: Transverse Distributions for various transverse temperatures. N = 2.78
108 and Q?=2.5. Now when space charge becomes important for the single particle motion the emittance is no longer a conserved quantity, and thus is no longer a good gure of merit of the beam. This means that we can no longer assume that the beam size scales with the beta function, as we usually do when we calculate the average density around the ring. This problem has not been addressed until very recently [51], and may also be too complicated for analytical calculations, which therefore means that it may have to be addressed by MD simulations including the real lattice.

92

Transverse Dynamics

6.2 Position Dependence Initial measurements with the transverse diagnostics revealed that the eciency of the transverse to longitudinal coupling depended critically on the overlap of the cooling laser and the ion beam. Therefore a series of measurements to investigate this dependence in detail were carried out, these experiments are described in this section. Recently we experienced diculty with the position stability of the laser-beam. As stability is important in order to understand how the cooling depends on position, and the measurements are a good illustration of the importance of the overlap, this problem is discussed rst.

6.2.1 Drift and oscillations During the most recent experimental run we had problems with a slow drift in the laser position which turned out to be due to temperature drift in the external cavity for UV generation. This caused the co-propagating laser to have an oscillatory position variation with a period of 100s and an amplitude of more than 0.2mm. It turned out that this was enough that the cooling eciency changed drastically during the time period. Measurements were done with xed laser cooling of a coasting beam. This implies that the initially cold beam (due to the pre-acceleration) is injected into the ring where two counter propagating lasers are present, each detuned red from being resonant with the center of mass of the beam. The detuning chosen was such that the longitudinal velocity spread maintained was  51 m/s. After injection the cooling proceeded continuously until the next injection. The CCD's were exposed from 5.5 s to 6.5 s after injection, and we used the standard procedure where we alternately exposed the CCD for the horizontal dimension and the CCD for the vertical dimension. The counter-propagating laser was used for cooling only, and the co-propagating laser was used for cooling and probing. Both lasers delivered about 30 mW of UV light, for the co-propagating laser 3 mW of these were directed to the probe section using a polarization beam splitter. The time between injections were about 10 s, thus the movement period of the laser beam was approximately 10 injection cycles. We observed that in one part of the period the uorescence on the PMT in the probe section was low, and in another part the uorescence signal was high. In order to determine whether this eect was caused by changes in the probe measurements, or due to changes in the beam, two sets of beam pro les where measured. Each measurement took four injection cycles, i.e. two exposures of each CCD. The four cycles were chosen to be while the PMT signal was low and while it was high. Figure 6.2 shows the resulting beam pro le measurements. The measurements demonstrate that the periodic change in the uorescence signal on the PMT is due to a periodic change in the transverse beam size, probably caused by a periodic change in the transverse to longitudinal coupling, as the longitudinal temperature was constant. This indicates that one of the laser beams used for cooling had some periodic motion. Periodic motion of the laser is a normal phenomenon in our experiments, as the laser beams propagate a considerable distance in air before entering the storage ring. In order to minimize beam motion due to air motion the laser beams are directed through

6.2 Position Dependence

93

Low fluorescence High fluorescence

0

5

Low fluorescence High fluorescence

10

15

20

25

30

0

5

10

Horizontal Position [mm]

15

20

Vertical Position [mm]

Figure 6.2: Horizontal and vertical beam pro les averaged over 2 injections each. The longitudinal velocity spread was  51 m/s (unaected by the oscillations). The distributions have been normalized to the same area for good and bad cooling situations. The injected current was  400nA. shielding tubes, which reduce the air motion around the laser beams. These shielding tubes do not remove all motion, and some motion also stems from ground motion. We had however not earlier observed any in uence of this residual motion of the laser beam on the cooling results. Therefore, in order to determine the source of the problem we decided to measure how the laser position changed as a function of time. 10.3

Vertical position [mm]

10.2

10.1

10

9.9

9.8

9.7

0

50

100

150

200

Time [s]

Figure 6.3: Vertical position of probe laser measured with the imaging system. The solid line is a harmonic t to the points, with a period of  100 sec, equal to the period of change in the sympathetic cooling eciency. Figure 6.3 shows the relative vertical position of the co-propagating laser measured as a function of time. The measurement was accomplished by doing a series of measurements of the laser beam pro le using the transverse imaging system. The laser-beam pro le can

94

Transverse Dynamics

be measured by imaging the uorescence from an often injected high current beam1 which is uncooled. The beam pro les were analyzed and the centroid in each pro le measurement was found. As the horizontal positions were almost constant, only the vertical positions are shown in Figure 6.3. The excursions observed in this measurement had the same period as the beam size changes. The vertical nature of the oscillation led to the discovery that temperature drift in the KDP crystal used for frequency doubling caused a variation in the horizontal plane of the exit angle of the light from the UV cavity (this would cause horizontal variations, but the beam was rotated 90 degrees before entering the storage ring). The problem will be solved by xing the temperature stabilization. When considering the measurements shown in Figure 6.2 one may realize that, as we usually do the beam size measurements by averaging over some time, a varying cooling laser position may cause us to average between signi cantly dierent transverse distributions and thus get ambiguous data. We did in fact observe this eect, and it was therefore decided to implement an active feedback system to stabilize the laser position, this system will also provide us with a reproducible way of measuring the in uence of o center positioning of the cooling lasers.

6.2.2 Sympathetic cooling The problems described in the previous section can be very severe, and are at the heart of our ability to sympathetically cool the transverse degrees of freedom in an ion beam. Another problem is that the laser beam positions cannot be reproduced with an accuracy better than about 0.5mm. The reason for this is that only 'human-mechanical' means are available when moving the laser, and that the laser positions are read by eye on a millimeter grid. As the laser beam is very non-Gaussian it depends much on the actual person doing the job what the read out is. This is undesirable and a system for doing these things reproducibly is currently being implemented. In spite of this, experiments have been carried out which study the dependence of the eciency of the sympathetic cooling on the beam-laser overlap. In order to minimize the ambiguity in the results these experiments were carried out with bunched beams, as this meant that only one laser was used for the cooling. Before a better system for positioning is implemented, and the laser intensity pro le is improved any dierence in the dependence on position in bunched and coasting beams can not be determined unambiguously. In Figure 6.4 measurements of the horizontal and vertical beam size as a function of the horizontal position of the cooling laser are shown. These measurements were done with xed laser bunched beam cooling, i.e. the ion beam was injected into the ring where a (counter-propagating) cooling laser, detuned to give a nal velocity spread of  40 m/s, 1 This is referred to as fast injection (rate about 10 Hz). The method discussed here is the standard

way of measuring the laser beam pro le and position in relation to the ion beam. At a high current the ion beam is much wider than the highly focused probe laser, and by measuring the uorescence in a repeatedly injected ion beam we measure the laser pro le. The fast injection is necessary to avoid the laser depopulating the velocity class on resonance.

6.2 Position Dependence

95

was present. The cooling laser power was  40 mW. The PAT was excited to 400V, and the probe laser detuned to be resonant with the beam inside the PAT. The circulating beam current was  90nA. The power in the probe laser was  25mW. The CCD's were exposed for one second each injection, from 5.7 to 6.7 seconds after injection. The bunch pro les where measured at ten points during each injection, one of the measurements while the CCD's were exposed. The longitudinal velocity spread was measured 7 sec. after injection. Each transverse pro le is the sum of 10 exposures, thus each measurement took 20 injection cycles. 2.5 Vertical Sigma Longitudinal FWHM

2

5 1.5

4 3

1

2 0.5 1 0

-4

-2

0

2

4

6

Horizontal Position of Cooling Laser [mm]

8

0

Horizontal Distribution (sigma) [mm]

6

2

8 7

1.5

6 5

1

4 3

0.5

2 Horizontal Sigma 1 0

Vertical Sigma Longitudinal FWHM -4

-2

0

2

4

6

8

Vertical and Longitudinal width [mm]/[m]

Horizontal Sigma

Vertical and Longitudinal width [mm]/[m]

Gaussfit to Horizontal Distribution [mm]

7

0

Vertical Position of Cooling Laser [mm]

Figure 6.4: Bunch dimensions as a function of horizontal and vertical laser displacement respectively. The bunch lengths (diamond) are unaected by the movement. Measurements were taken with approximately 2.5
107 particles (90nA). Transverse sizes are extracted from Gaussian ts to the distributions, in order to give an estimate of the beam size, as some distributions were non-Gaussian. The positions given in Figure 6.4 are somewhat relative, as no means existed by which the absolute position of the laser-beam with respect to the ion beam could be determined. The zero point in the gure has been chosen as the point where the highest density was reached. In Figure 6.4, positive is the outwards direction. The beam sizes are determined by tting a Gaussian to the actual distribution, which may (see Figure 6.5) be very nonGaussian, in order to obtain an estimate of the rms beam size. We nd that both for horizontal and vertical displacement of the laser-beam with respect to the ion beam the longitudinal cooling is rather unaected. This agrees with earlier experience where high reproducibility of longitudinal dynamics results have been obtained without detailed control of the overlap of the laser beam with the ion beam. The drop in horizontal size at the right of the left plot is due to bad ts, as the beam blows up more than the imaging system could measure (see Figure 6.5). The rms spreads in Figure 6.4 are slightly deceiving, but give an impression of the general behavior. The reason is that the distributions become far from Gaussian when the laser is displaced. In Figure 6.5 the actual distributions corresponding to the horizontal displacement data are shown. As we can see the horizontal distribution is heated when the cooling laser is displaced outwards. The horizontal dimension is heated so much that

-10.0

-4.5 -2.5 0.0 0.0 1.5 5.0 7.5 -5.0 0.0 5.0 Horizontal Position [mm]

Cooling Laser - Horizontal Position [mm]

Transverse Dynamics

Cooling Laser - Horizontal Position [mm]

96

-4.5 -2.5

0.0 0.0 1.5 5.0 7.5 -5.0 0.0 5.0 Vertical Position [mm]

Figure 6.5: Measurements of the horizontal and vertical beam density distribution for various horizontal displacements of the cooling laser. The rms spread extracted from Gaussian ts to these distributions were shown in Figure 6.4. The distributions have been normalized to the same area. we cannot measure the beam size properly - and the Gaussian ts in Figure 6.4 become meaningless in this region. The measurements show that when we change the horizontal position of the laser beam we observe an asymmetric behavior of the horizontal beam size, whereas the vertical is more symmetric. The dierence is believed to be due to the dispersion via a mechanism rst proposed by Wolf in 1991 [111] (the dispersion is zero in the vertical dimension). The explanation goes as follows (Figure 6.6); In a ring section with positive dispersion a particle with a momentum deviation
p0 will have a deviation in its equilibrium orbit of
x0 around which the particle will perform betatron oscillations (Figure 6.6.a). If the particle momentum is decreased by an amount p when the betatron phase of the particle is  (Figure 6.6.b) the horizontal position of the equilibrium orbit is changed by an amount x such that the betatron amplitude is suddenly smaller, and thus has been damped. In the opposite case where the phase was zero (Figure 6.6.c) the betatron amplitude would be increased and the beam heated. Horizontal cooling is accomplished by this mechanism if the process causing decreased amplitudes is more probable than the other. If the opposite is the case the laser will cause horizontal heating. This eect of dispersion in a cooling section has been observed in electron cooling experiments [9] and very recently in laser-cooling experiments at the TSR in Heidelberg

6.2 Position Dependence a)

97

x’

b)

x’

c)

Laser kick

Laser kick

x ∆x

0

x’

x δx

x δx

Figure 6.6: Illustration of how dispersion can cause horizontal cooling from longitudinal cooling. The orange curves in b and c indicates the phase space orbit after the laser kick. Further explanation in text. [46]. In the experiments whose results were presented in Figure 6.4 we used a counter propagating laser (as in the dispersion discussion), thus, as dispersion is positive in ASTRID, we would expect an increased horizontal cooling when the laser is displaced horizontally inwards. As the horizontal size explodes when the laser is moved outwards this complies well with the expectations, however measurements where the overlap is known with good precision (for instance via the camera) are needed to quantify the eect. The asymmetry in the horizontal beam size as a function of the vertical cooling laser position is not explained as there is no vertical dispersion. However, the laser beam is very inhomogeneous and it is not unlikely that the centroid of the intensity distribution depends on the vertical position, thus causing dispersive heating when moved in one direction but cooling when moved in the other direction. The change in the vertical distribution indicates less cooling with less overlap as expected. That we are still overlapping with (at least some of) the ion beam at +7 mm is seen from the fact that the bunch lengths are independent of the position. This explains how it was possible to obtain rather large emittances while performing ecient longitudinal cooling in the Touschek and diusion measurements discussed in the previous chapter. Thus it seems that the explanation for large emittances observed in those measurements is that the transverse to longitudinal coupling was not optimized. As a consequence of these investigations we have adapted the following alignment procedure for optimizing overlap.  Optimize probe laser overlap with ion beam at the camera/PMT position.  Optimize the signal on the PMT, to obtain max. signal with max. overlap.  Optimize cooling laser position by optimizing the uorescence in the PMT while cooling for a short time. This procedure will overlap one cooling laser with the ion beam. The other cooling laser (for coasting beam cooling) is then overlapped with the rst (note that the dispersive

98

Transverse Dynamics

coupling actually implies that the co- and counter propagating lasers should be displaced horizontally relative to each other to obtain the best horizontal cooling, however our mechanical alignment is not precise enough for this to have been tested presently). The procedure works very well, and gives, reproducible results. The reason is that once optimum overlap is achieved in the probe section, any improvement in cooling, which will cause a decreasing beam size will lead to more uorescence, whereas a slightly misaligned probe may cause the beam to shrink out of overlap. The measurements discussed in this section demonstrates that it is crucial to have a stable laser position, and to have control over the laser beam overlap with the ion beam in order to have ecient transverse cooling. As discussed we are presently not able to position the lasers accurately with respect to the ion beam, and we still have motion of the laser beam due to ground motion and air motion. The problem is compensated somewhat if a laser beam with a large cross section is chosen, however, the beam used for the probe section must be small for optimum performance of the imaging system. Presently a system for stabilizing the laser position, as well as moving the laser with more precision is being implemented. With this system it will be possible to investigate the dispersive coupling in detail, as the laser beam position can be controlled precisely. The imaging system can be used to determine the exact position of the laser beam relative to the ion beam.

6.3 Time Scale In Section 3.3 we discussed the longitudinal cooling time, which with chirped laser cooling was very short (down to about 4 ms total time). In Chapter 5 we used the IBS program intrabtc to calculate the expected cooling rates from the longitudinal velocity spread and beam current, however in the previous section we saw that dispersion is very important when considering the coupling to the horizontal dimension from the longitudinal. It is therefore interesting to try to measure the time-scale of the equilibration of the transverse dimensions during longitudinal laser cooling. This time scale is also important when considering the minimum temperatures reachable, and the possibility for crystallization, as it is a measure of the transverse cooling force, which is to overcome the heating from IBS and other sources. Furthermore, it is important to know how fast the beam dimensions stabilize when cooling from injection, as this determines when equilibrium measurements can be done.

6.3.1 Measurements We have done some preliminary studies of the time to cool the transverse dimensions of a coasting ion beam which is longitudinally cooled with laser-cooling. The discussion will be kept brief, as the measurements presented here were done before the optimization procedure from the previous section was adapted. Figure 6.7 shows how the transverse beam size and the density develops as a function of time after injection with laser-cooling active from injection. The measurement was carried

6.3 Time Scale

99

out by injecting a coasting beam into the ring, while two counter propagating cooling lasers were present, and detuned slightly red from resonance to obtain a nal velocity spread of  28 m/s. The cooling laser powers were  25mW. The dierent times after injection were measured with dierent camera exposure times relative to injection. The duration of each exposure were adjusted to be short compared to the time scale of the change in the transverse beam size. Each point is the sum of several injections (varied to adjust for the change in light ux on the CCD originating from the change in transverse beam size). The lifetime of the ion beam was  30 s. 4 105

2.5

3 105

2

2 105

1.5

1

Particle Density [cm- 3]

Transverse Beam Size [mm]

3

Horizontal Spread (sigma) Vertical Spread (sigma)

0.5

Particle Density 0

0

5

10

15

20

25

30

35

105 40

Time after Injection [s]

Figure 6.7: Change in beam size as a function of time after injection. The longitudinal velocity spread is kept at 28 m/s from the time of injection. The current was  840nA (2.3
108 particles). The density given is the peak density. The longitudinal velocity spread, not shown in the gure, reached the constant value of  28 m/s within the rst few milliseconds, the longitudinal spread of the injected beam was less than 28 m/s. From Figure 6.7 we learn that the transverse beam sizes stabilizes after about one second, where the density is observed to reach a constant level. As the density is constant as a function of time, the natural decay of the ion beam emerges as a decreasing beam size as a function of time. The 1 second total transverse cooling time corresponds to e-folding times of order h = 4.3 sec and v = 2.5 sec. The calculated cooling times from the IBS program are hibs = 3.2s and vibs = 0.9 sec. Thus, as expected, the vertical cooling is faster than the horizontal, and we observe the correct order of magnitude. The constant density observed in the experiment was determined not to depend on whether the laser was chirped for cooling or xed at injection. Some measurements were done on the time development with chirped cooling, the chirping was however done rather slowly compared to the expected sympathetic cooling times, thus an actual coupling strength was hard to determine, as the transverse dimensions shrunk at approximately the same rate as the longitudinal (i.e. as the chirping).

100

Transverse Dynamics

6.3.2 Discussion

In order to determine if laser-cooling could be used for instability damping in high intensity applications, as for instance accumulation in an injector for heavy ion fusion [39], the sympathetic cooling rate is important. Presently the time scales seem to be of order a few seconds, which is most likely not enough. With the dispersion coupling mentioned in the previous section this rate is improved in the horizontal dimension. This improvement in the horizontal cooling can aect the vertical either indirectly through intra beam scattering or directly by introducing resonant coupling between the vertical and horizontal motion. Another possibility would be to introduce skew quadrupoles (or a longitudinal magnetic eld) which would mix the horizontal and vertical degree of freedom. In bunched beams increased transverse to longitudinal coupling can also be introduced through a resonant coupling of the betatron motion to the synchrotron motion [74, 73]. Resonant coupling of the motion in the dierent degrees of freedom is not possible if the beam freezes into the crystalline ground state. Furthermore, the crystalline ground state requires a constant angular velocity while the laser-force gives the ion beam a constant linear velocity, as discussed in section 2.3.3. The laser force will therefore cause heating of the crystalline ground state. This problem could be resolved by introducing tapered cooling as proposed in reference [105]. With the available diagnostics it should be possible to investigate in detail the in uence of these dierent schemes for improved transverse to longitudinal coupling, as well as the eects of introducing tapered cooling.

6.4 Density Limitations In Figure 6.7 we saw that the transverse dimensions reach a limiting size after about one second of cooling, or rather the particle density in the beam reached a limiting value. Limitations to the obtainable beam sizes / densities give important information about the eciency of the applied cooling. In the following series of experiments we have investigated what causes the limiting particle density.

6.4.1 Longitudinal Velocity Spread

The particle density limit was rst observed in a study of how the transverse beam sizes changed as a function of the longitudinal velocity spread. If space charge is negligible we expect the transverse beam size to be given by the temperature and to be independent of the number of particles. IBS calculations using the program intrabtc, discussed in appendix C, which neglect space charge eects, show that the transverse temperature is proportional to the longitudinal. In Figure 6.8 a measurement of the beam size of a coasting beam as a function of the longitudinal velocity spread obtained by laser-cooling is shown. Each measurement were done by injecting the ion beam into the ring where two counter propagating lasers were present. Before injection the laser detunings had been set to the desired nal detuning, which then caused the beam to attain the shown longitudinal velocity spreads. The ion

6.4 Density Limitations

101 1 106 Horizontal Beam Size Vertical Beam Size

Density

2

8 105

1.5

6 105

1

4 105

0.5

2 105

0 400

Peak particle density [cm- 3]

Transverse Beam Size (sigma) [mm]

2.5

0 350

300

250

200

150

100

50

0

Longitudinal Velocity Spread (fwhm) [m/s]

Figure 6.8: Transverse beam size and peak density in a 278nA (7.8
107 particles) coasting beam, laser-cooled to dierent velocity spreads. The dashed lines are calculated using the program intrabsc (see Section C.4.1). The emittance at the plateau is approximately 8
10,2  mm mrad (assuming the lattice is unperturbed by space-charge). beam lifetime was  20s, and the beam pro les were measured 4 sec. after injection. The injected current was constant throughout the measurements. The cooling laser powers were both  30 mW. The co-propagating cooling laser also delivered beam in the probe section for the diagnostics. The laser power in the probe section was  6 mW. The smallest detuning (and thereby longitudinal velocity spread) was chosen as the point where the laser-force was no longer strong enough to overcome the increased diusion due to intra beam scattering caused by the decreased longitudinal velocity spread. The peak particle density calculated using the density de ned in equation (2.21) is also shown in the gure. This density will be called the local peak density, as it represents the particle density at the position of the imaging system, and not the ring-averaged particle density. Figure 6.8 also shows results from calculating the equilibrium emittances and longitudinal velocity spreads for varying longitudinal cooling force using the IBS program intrabsc. The results from the IBS program show that the beam size collapse to zero when the longitudinal velocity spread is zero. This is because the program does not include bulk space charge eects, thus the program results in the low temperature region are not representing a physical reality. The cooling force in the IBS program was expressed by an e-folding time, which for a longitudinal velocity spread of vFWHM = 100 m/s was 300 s, a reasonable cooling time for laser-cooling. The cooling time constant necessary to achieve the various longitudinal velocity spreads with the IBS program are shown in Appendix C. They have not been shown here, as the e-folding times are not directly comparable to the laser-cooling force, as the longitudinal velocity spread obtained with laser-cooling is determined by the laser detuning, and whether the laser-force is strong enough to overcome the diusion of particles past the laser in velocity space. Contrary to the cooling force in the IBS program the laser

102

Transverse Dynamics

only interacts with particles at the edge of the longitudinal velocity distribution (close to the lasers). Thus if more laser-power is applied the longitudinal velocity spread is not decreased, but it is possible to decrease the detuning to achieve lower velocity spreads. Two features are evident in Figure 6.8. First of all we see that the measured horizontal beam size is smaller than the theoretical for the large velocity spreads, while the vertical seems to agree quite well with the calculations. The cooling-optimization procedure of section 6.2.2 was used for these experiments. We would therefore expect the dispersive coupling to contribute to the horizontal cooling. This agrees with the observation that the horizontal beam size is smaller in the experiment than the beam size calculated using the IBS code, which does not take the dispersive coupling into account. The second distinct feature is that there is a plateau in the measured beam sizes, which corresponds to a plateau in the particle density. This could mean, as we have earlier seen in the longitudinal dimension, that the transverse emittance is zero, the particle density uniform, and the beam size therefore determined purely by the transverse con nement. If we calculate the ring-averaged particle density, we nd that, neglecting space charge eects and assuming knowledge of the beta function, the ring-averaged particle density is about 15% higher than the local particle density. Thus a ring-averaged particle density of  7
105 cm,3 can be extracted from these measurements. This is about ve times less that the zero emittance particle density calculated in Section 6.1.1. The rst observation of the plateau phenomenon discussed above was actually in a bunched beam, where we additionally carried out measurements at dierent currents (see Figure 6.9). These measurements where done before we implemented the optimization procedure for overlapping the beam and the laser. Therefore the transverse to longitudinal coupling is weaker in these measurements than in the previous, which may account for lower peak particle density. In connection with the previous measurements, a bunched beam measurement with the same laser beam ion beam alignment as for the coasting beam measurements was carried out (we just switched on bunching in one of the measurements). In this measurement the same peak density as in Figure 6.8 was observed. Figure 6.9 shows how the peak particle density change as a function of the detuning and beam current in a bunched beam. The measurement procedure was similar the procedure used for the coasting beams. The beam was cooled by one xed counter-propagating laser, set to the desired detuning before injection. The co-propagating laser was used to probe the beam in the probe section of ASTRID. The ion beam lifetime was  25s, and the CCD's were exposed from 5.5 to 6.5 seconds after injection. The cooling laser power was 35 - 50 mW, while the probe laser power was 15 - 35 mW. The measurements were carried out over several days, during which motion in the cooling laser position caused the cooling conditions to change slightly. The beam current was varied by injecting dierent currents. In these measurements, we observe as in the coasting beam that the beam particle density reaches a limiting value. The limiting density seems to be slowly varying with beam current, or, if we take into account that the transverse to longitudinal coupling may have varied between the measurements, independent of beam current. In a beam with zero transverse emittance the density is uniform and independent of current, thus the measurements indicate that we have a beam of zero transverse emittance. Furthermore, we

Peak Particle Density [cm- 3]

6.4 Density Limitations

103

Beam Current : 8.6 nA Beam Current : 25 nA Beam Current : 41 nA Beam Current : 77 nA Beam Current : 125 nA Beam Current : 220 nA 105

104

100

10

Longitudinal Velocity Spread (sigma) [m/s]

Figure 6.9: Peak densities extracted from laser-cooling bunched beams of dierent currents to varying longitudinal velocity spreads. The lines are made to guide the eye. observe a dependence of the longitudinal velocity at which the limiting density is attained on the beam current. However, as mentioned, the overlap between the laser and the ion beam varied in these measurements, and this might account for the variation in the longitudinal velocity. The general observation in both bunched and coasting beams is that below some longitudinal velocity spread the beam density attains a limiting value. The measurements indicates that this density is independent of beam current, which indicates that we have reached zero transverse emittance, where the beam density will depend only on the transverse con nement forces. As the densities observed are rather low compared to the expected maximum particle density for the applied con nement, it seems however, that the beams have not reached zero transverse emittance. Before going into more details, there are other aspects of the tendency of a limiting density to be presented.

6.4.2 Laser power As discussed above the limiting peak density seems to be constant as a function of current. However, the density varies somewhat between bunched and coasting beams and between measurements taken at dierent times. As the only things we change between experiments is the overlap of the laser beam and the ion beam and the laser power, one of these must be responsible for the observed variations. In section 6.2 we discussed the in uence of the ion beam laser beam overlap on the transverse to longitudinal coupling, and we saw that due to dispersion the in uence of the overlap on the coupling was very important. This may account for all the variations in the limiting particle density. However, the laser power also changes between experiments, and in order to clarify which eect is most important we have conducted a series of experiments on what in uence the cooling laser power has on the limiting particle density.

104

Transverse Dynamics

2.5

50

2

40

1.5

30

1

20

0.5

Beam Current [nA]

Peak Particle Density [105 cm- 3]

In Figure 6.10 the results of a measurement on laser-cooling of a bunched beam is shown. The laser detuning was kept constant during the experiment, and the ion beam was cooled using the standard xed laser technique, where the desired nal detuning is set before injection. The detuning was set to give a longitudinal velocity spread of  50 m/s. The CCD's where exposed from 4.7 to 5.7 seconds after injection. The cooling laser power was varied by directing only a fraction of the laser power from the UV cavities into the cooling section. The fraction was chosen by rotating the polarization of the laser beam via a /2 retarder and directing the rotated beam through a polarization beam splitter.

10 Peak Density Beam Current

0

0

10

20

30

40

50

0 60

Cooling Power [mW]

Figure 6.10: Peak density and circulating current in a bunched beam as a function of the applied laser cooling power. The velocity spread was  50 m/s, except for the uncooled beam which had a velocity spread of  420 m/s. The measurements shows that the amount of particles captured into the bunched varies with laser cooling power, on the other hand the peak density reaches a constant level when the laser cooling power is increased above a certain level ( 10 mW). The dependence on cooling power for low powers is explained by investigating the simultaneously measured longitudinal velocity distributions. Some longitudinal velocity spreads corresponding to the data shown in Figure 6.10 are shown in Figure 6.11. Figure 6.11 shows the longitudinal velocity distribution for some of the laser powers in Figure 6.10. We observe that at the low powers the longitudinal velocity distribution is both asymmetric and wider than the laser detuning should imply. This is to be expected. At the low laser powers the cooling force is smaller, and thus the cooling time is decreased. As all the measurements are done at the same time after injection, the longitudinal temperature therefore varies when the laser power becomes low enough that all the particles have not been cooled at the time of the measurement. The variation in the beam current is probably caused because the initial heating of the beam is faster than the laser-cooling, and some particles therefore become too energetic to be bunched and later cooled. In the measurements we only measure the part of the beam collected in the bunches. The variation in density at the low powers is therefore due to a lower longitudinal

Fluorescence [arb. units]

6.4 Density Limitations 51mW

-500 0 500

105 9mW

-500 0 500

6mW

-500 0 500

3mW

-500 0 500

Relative Longitudinal Velocity [m/s]

Figure 6.11: Longitudinal velocity distributions corresponding to the densities shown in Figure 6.10. The applied cooling laser power is in the upper left corner of each plot. The distributions have been normalized to the same area. temperature causing a lower transverse temperature. The measurements thus leaves no reason to believe that the laser power has a direct in uence on the transverse to longitudinal coupling. However, we did observe that below a certain laser cooling power, the cooling was too slow to collect all particles into the bunches. In a coasting beam where the particles are kept between the cooling lasers from injection this eect should not arise. As we have noted earlier the laser power does however in uence what detuning the lasers can have and still be able to maintain a cold distribution. This measurement does not give information about that eect, but it would be interesting to know how the necessary laser power depends on the detuning and the beam current. This could be investigated by a measurement where the laser power was high at injection and then, after the beam was cooled, decreased until intra beam scattering causes the longitudinal velocity spread to grow. The discussed measurements where done before we implemented the optimization procedure for overlapping the laser beam and the ion beam. Thus the dispersive coupling has not been strong in these measurements. This explains the low maximum peak density. As the dispersive coupling is a single particle mechanism, it is likely that the strength depends on the laser intensity, as more power would increase the probability for a scattering event. It would therefore be interesting to redo the discussed measurement, as well as the discussed variation of it, with a setup where the beam alignment is such that the dispersive coupling is important.

6.4.3 Beam Current As we saw before it seemed that the limiting particle density did not depend on the circulating beam current. The magnitude of the limiting density in the experiments suggested that this was not due to zero transverse emittance, which would cause the density to depend only on the con nement forces. In order to nd an explanation to the observed tendency we have conducted some more experiments in both coasting and bunched beams.

106

Transverse Dynamics

Coasting Beams

Figure 6.12 shows the results of measurements of the dimensions of a coasting beam while varying the beam current, but keeping the laser detuning constant. The measurements were done using two counter propagating cooling lasers, whose detuning was xed before injection to the desired value. The detuning was constant throughout the experiment. The longitudinal velocity spread obtained was  25 m/s, which corresponds to
p=p  5
10,5 . The cooling laser powers were 25 - 35 mW. The current was varied both by injecting dierent currents, and by measuring at dierent times after injection. The measurements were conducted at times between 10 and 80 seconds after injection. The exposure time of the CCD each injection were increased from 1.0 second at the short times to 5.0 seconds at the late times to increase the photon statistics of the low current beams. The lifetime of the ion beam was  48 s. The measurements were conducted before the overlap optimization procedure was implemented. 2

Transverse Beam Size [mm]

Horizontal Size (sigma) Vertical Size (sigma) Zero emittance calculation 1.5

1

0.5

Longitudinal Velocity Spread 0

0

5 107

1 108

= 25.5 m/s

1.5 108

2 108

Number of particles in the beam

Figure 6.12: Beam dimensions for a coasting, laser-cooled ion beam. The longitudinal momentum spread was
p/p  5
10,5 . The dashed curve is the predicted beam size in the axisymmetric smooth approximation (section 6.1) with zero transverse emittance (Q=2.5). The solid curves are square root ts to the scaling. In Figure 6.12 we observe that the beam sizes scale as a square root in the beam current. A square root scaling of the beam sizes means that the transverse beam cross section scales linearly with the current, thus the particle density is constant as a function of current. This is exactly the behavior we expect if we have a beam of zero transverse emittance. The average local peak particle density we extract from these measurements is 4.3
105 cm,3. The local peak particle density in an uncooled beam is about an order of magnitude smaller. The ring averaged density in the measurements is thus 5
105 cm,3. This density is about seven times smaller than the expected density in a beam with zero transverse emittance (3.38
106 cm,3). Thus the actual density does not indicate that we have reached zero transverse emit-

6.4 Density Limitations

107

tance. In section 6.1 we calculated the change in the transverse density pro le as a function of transverse temperature. Zero transverse emittance implied a uniform density distribution in the beam. However, the imaging system does not show the transverse density distribution directly, but rather a projection of it. The image of the light intensity distribution from a spatially uniform beam excited uniformly by the laser can be calculated by integrating the distribution in the plane perpendicular to the imaged plane. In a circular beam of radius a the projected intensity distribution from a spatially uniform beam of density n is given by Z ymax (x) p I (x) = 2 (x; y) dy = 2nC a2 , x2 (6.9) 0

where C is the circumference of the ring and the surface particle density of an in nite slice of beam is (x; y) = nC , where p x2 and2 y are constrained to the beam, i.e. x2 + y2  a2. ymax(x) is therefore equal to a , x . Thus the beam pro le we would observe if the beam particle density was uniform is the square root of a parabola. Now, the laser has a small width in the dimension perpendicular to the imaged dimension, thus in very large beams the uorescing beam may not be uniformly excited, and we'll thus see distributions which are closer to the square distribution. 2.0

1.0

2.0 1.0 0.0 12.0

-1

-1

Linear Density [10 mm ]

0.0 8.0 Linear Density [10 mm ]

3.0

7

7

-1

Linear Density [10 mm ]

4.0

-1

Linear Density [10 mm ]

3.0

6

6

6.0 4.0 2.0 0.0 -10

-5 0 5 Horizontal Position [mm]

10

8.0

4.0

0.0 -10

-5 0 5 Vertical Position [mm]

10

Figure 6.13: Horizontal and vertical particle distributions for uncooled (bottom) and cooled (top) beams of  5.8
107 particles. The distributions have been normalized to have an area of 5.8
107. The dashed lines are Gaussian ts to the distributions. Figure 6.13 shows examples of the transverse beam pro les measured during the cooling experiment discussed above. Furthermore, two measurements of uncooled beams are shown for reference. None of the measured distributions corresponding to the transverse beam

108

Transverse Dynamics

sizes given in Figure 6.12 showed any signs of being non-Gaussian. The measured beams do therefore not have uniform particle density, and the transverse emittance is non-zero. Bunched Beams

In section 5.2 we discussed the development of bunch length as a function of beam current and longitudinal velocity spread. As mentioned the transverse beam pro les were also measured during these experiments. The experiments were done with xed laser bunched beam cooling. The current was varied by varying the injected beam current. In Figure 6.14 the part of the measurements done with a detuning leading to a longitudinal velocity spread of  60 m/s are shown. At this longitudinal velocity spread the bunches were longitudinally space charge limited. This is evident from the cube root scaling in current of the bunch lengths. 2 RMS Velocity Spread ~ 59 m/s

2 1 0.9 0.8 0.7 0.6

1 0.9 0.8 0.7

0.5 Horizontal Size Vertical Size

0.6 0.5 0.4

Bunch Length 106

Bunch Length [m]

Transverse Beam Size [mm]

3

0.4 0.3 107

Number of particles pr. bunch

Figure 6.14: The transverse and longitudinal dimensions of a laser-cooled bunched beam as a function of the current. The colored lines are cube root ts to the data, whereas the black line is a parallel translation of a cube root t to guide the eye. Figure 6.14 shows that the bunch lengths as well as the horizontal beam size scales as a cube root in the number of particles. The measurements also indicate that the vertical beam size scales as a cube root at low currents (below  106 particles/bunch). If all dimensions are scaling as a cube root in the current the peak density is constant as a function of current. Thus at low currents we observe that the limiting density is independent of current. For higher currents the vertical beam size scale such that the limiting density is smaller at high currents than at low currents. The ring averaged limiting density at low currents is  2.3
105 cm,3, about an order of magnitude below the particle density in a beam with zero transverse emittance. The density found here is also lower than some of the previously discussed limiting densities, but these experiments were conducted before the procedure to optimize the transverse to longitudinal coupling was implemented.

6.4 Density Limitations

109

Figure 6.15 shows some transverse beam pro les corresponding to the transverse beam sizes shown in Figure 6.14.

0.0

544nA

544nA

192nA

192nA

65nA

65nA

22nA

22nA

5.9nA

5.9nA

5.0 10.0 15.0 Horizontal Position [mm]

20.0

0.0

5.0 10.0 15.0 Vertical Position [mm]

20.0

Figure 6.15: Horizontal and vertical beam pro les for a laser-cooled bunched beam for various beam currents. The distributions vertical scale varies for easier comparison of the shapes. The dashed lines are gaussian ts to the distributions. We observe in Figure 6.15 that the horizontal beam pro les are all Gaussian. The vertical beam pro les are Gaussian at low currents, but deviate from being Gaussian at high currents. In bunched beams we therefore again see indications that the limiting density is independent of current, however, the limiting density is too low to indicate zero transverse emittance, and the transverse beam pro les are Gaussian. Thus the particle density in the beam is not uniform, and we do not have beams with zero transverse emittance. However, we also observe that at high currents, the vertical beam size increases such that the limiting density is no longer independent of current. Origin of the density limit

We have in the presented measurements observed a general tendency for the cold beams to acquire a density which is independent of beam current, and which below a certain longitudinal velocity spread is independent of the longitudinal velocity spread. This indicated that the cold beams had zero transverse emittance, however, this would mean that the density distribution should be uniform in the beam, and the density should be  3.3
106 cm,3. The particle density distributions were Gaussian, thus not uniform, and the observed densities were between 5 and 10 times smaller than the expected density at zero

110

Transverse Dynamics

transverse emittance. Thus the beams did not have zero transverse emittance. In order to explain the observed limits we need a mechanism which can explain how the observed density can be independent of current, and why the slightly dierent laser beam ion beam overlaps in the dierent experiments alters the limiting density, while varying the longitudinal velocity spread does not alter the density. We have earlier considered that the ground state of a beam in a storage ring has a constant angular velocity, while laser-cooling induces a constant linear velocity. This means that there must be a limit to how low transverse temperatures can be reached with 'conventional' laser-cooling. At the center of each straight section in ASTRID the dispersion is approximately 2.74 m, this means that if a beam has a size of one millimeter the dierence in linear velocity between particles in the left side of the beam and particles in the right side of the beam should be  330 m/s for the beam to have constant angular velocity. However, as the heating of the beam due to the induced constant linear velocity depends on the transverse size of the beam, we would expect this eect to depend on the current, as the beam size is observed to increase with current. The transverse to longitudinal coupling on the other hand should not depend much on the beam size, as we seem to be in a low density regime where the coupling is dominated by intra beam scattering, which only depends on the density which is observed to be constant. Therefore we do not believe that the limitation is caused by the constant linear velocity the laser-force induces. We know from the discussion in Section 2.2 that when the density increases the space charge tune shift increases. If the tune shift is large enough to bring us close to a resonance the beam is heated, as has been demonstrated in simulations done by Machida [54] and experiments by Chanel [13]. The rst order space charge tune shift calculated in Section 2.2 depended only on the density of the beam, and may therefore account for our observed constant density as a function of beam current. Figure 6.16 shows the betatron tune as a function of density in a beam with a Gaussian transverse density distribution calculated using equation (2.26). In the gure the tunes corresponding to the three dierent limiting peak particle densities observed in the experiments in this chapter are shown. As marked in Figure 2.5 the betatron tunes for these experiments where Qh=2.27 and Qv = 2.83. When the density is increased both are aected. We observe in the working diagram that the distance to the closest second order resonance (a coupling resonance: Qh + Qv = 5) is about 0.05, and we are 0.25 - 0.3 away from an integer resonance in the horizontal and a half integer in the vertical. These values are of the same order of magnitude as the tune shifts shown in Figure 6.16. A tune shift of 0.2 - 0.25, as we calculate for the two coasting beam measurements discussed in this chapter, seems to comply with the integer and half integer resonances at 2 and 2.5 for the horizontal and vertical tune respectively. The tune shift dierence between the measurements with a shift of 0.07 and 0.2 - 0.25 could be due to increased sympathetic cooling from the better laser ion beam overlap, as the increased sympathetic cooling may be enough to overcome the coupling stop band in the distance 0.05 from the

6.4 Density Limitations

111

2.5

Betatron Tune

2

1.5

1

0.5

0 0 100

5 105

1 106

1.5 106

2 106

2.5 106

3 106

3.5 106

4 106

Peak Particle Density [cm- 3]

Figure 6.16: Tune for a particle with an amplitude equal to the rms width in a Gaussian beam in the smooth approximation as a function of the peak particle density (see section 2.2). The short dashed line marks the ring-averaged limiting peak particle densities observed in the measurements in Figure 6.9, 6.10 and 6.14. The dot-dashed line marks the measurements in Figure 6.12 and the long-dashed line marks the measurements in Figure 6.8. bare tune (also called single particle tune)2, and then bring us close to the presumably stronger stop bands at 2 and 2.5. Any expectations as to whether it will be possible to pass the stop band depend on the precise knowledge of the maximum reachable density, and thus the beam current. This information would give insight into the exact strength of the cooling, and into how much the dispersive coupling in uences this strength. It is very possible that it turns out that the cooling power needed to pass the resonance requires smaller laser detuning than can be sustained without severe Touschek losses or diusion - in that case one of the discussed possible systems for extending the range of the laser force must be implemented. However, as the transverse cooling force has been observed not to increase very much with decreasing detuning, it may not be possible to cross the resonance by brute force alone, other tricks may be needed. As bunched beams are transversely larger than coasting beams of equivalent current (due to the increased linear density) they may behave dierently in the sense that they are in uenced more by amplitude-dependent non-linear resonances. This may be the reason for the observed attening of the distributions for higher currents shown in Figure 6.15. The attening was also described by Machida as a side eect of approaching a resonance 2 The indicated tune shift of 0.07 is calculated from the peak density in the bunch - in a bunched beam

it may be more appropriate to use the average density along the bunch. In the experiments presented here the bunches are parabolic. This causes the average density to be 70% of the peak density - thus it changes the tune shift estimate from 0.07 to 0.05, which directly corresponds to the distance to the stop band proposed to be responsible for the density limit.

112

Transverse Dynamics

[54]. The eect has been observed experimentally by Chanel [13], and the tendency of our bunched beams to expand at higher currents may be explained by this phenomenon. That the beam size increases in the vertical dimension and not the horizontal is probably because the cooling is more ecient in the horizontal due to the dispersive coupling. This increase in beam size seems to explain the deviation from constant density with high currents in a bunched beam, and may also account for the shorter than expected bunch length observed at high currents (Figure 5.15), as increased transverse size increases the average distance between particles, and thereby reduces their Coulomb repulsion3.

6.4.4 Discussion The results presented here indicated that the sympathetic cooling of the transverse dimensions of a longitudinally laser-cooled ion beam, is not strong enough, given the current experimental conditions, to cross the tune stop bands which are met due to the space charge tune shift caused by the increasing particle density during cooling. However, there is some variation in the limiting density we can obtain, a variation which seems to be due to the exact overlap of the cooling lasers with the ion beam, as that is the main parameter which changes between experimental runs, and has been observed to have large implications for the sympathetic cooling. Therefore a better positioning system is crucial to gain a better understanding of the observed density limits. The positioning accuracy is also limited by the homogeneity of the laser beams. For these reasons an active position stabilization system for the lasers, as well as a new external cavity which hopefully generates more UV and a better beam pro le are being prepared [44]. As an increased density means increased heating due to intra beam scattering, we may also need more laser-power. If it turns out that the density limit cannot be overcome by the present cooling technique (which includes the dispersive coupling), it may be necessary to use the tapered cooling scheme discussed in the next section, or introduce one of the proposed coupling schemes in bunched beams. Other means would be to introduce compensating focusing elements in ASTRID to remove the resonance, which would imply more magnets. Another way to 'cheat' the resonance may be to increase the focusing in the lattice momentarily until the beam is dense enough to be on the other 'side' of the resonance.

6.5 Ribbon Beams An intriguing result we got recently, and which has not been studied completely yet, is what we call a ribbon beam. In gure 6.17 is shown a plot of the transverse beam sizes as a function of the beam current in a coasting beam. The measurement was conducted with 3 It should be noted that there is an absolute uncertainty on the current measurements of  10%. This

uncertainty stems from the calibration of the beam current transformer. An error in the current measurement could alter the results somewhat, however the scaling would be the same, thus there would still be a discrepancy between low and high currents, and therefore the conclusions referring to the transverse blowup would still be of relevance

6.5 Ribbon Beams

113

xed laser cooling of a coasting beam. The particles were injected into a ring where the co- and counter-propagating cooling lasers were both detuned slightly red from resonance. The cooling laser powers were  25 mW, and the cameras were exposed at 4.0 seconds after injection. The beam lifetime was  25 sec. The current was varied by injecting dierent currents. We observe that the local peak density is constant at a level of about 5
105 cm,3 consistent with the measurement shown in Figure 6.8 taken with the same experimental setup and laser beam alignment. However below 4.0
106 particles (14nA) the vertical dimension blows up, thus the beam becomes a long vertical ribbon of particles. The longitudinal velocity spreads shown in the gure are observed to decrease for decreasing number of particles. As the laser detuning was constant this indicates that the longitudinal heating decreases with decreasing beam current. The sharp drop at low currents happens at the same current as the sudden decrease in the peak particle density. This drop in longitudinal velocity spread is to be expected as the decreased particle density causes decreased intra beam scattering. The longitudinal velocity spread of 12 m/s the lower limit of our velocimetry resolution, thus the beam may be much colder.

Density

105

1.5

1 104 0.5

0 0

1 107

2 107

3 107

4 107

5 107

Circulating number of particles

6 107

103 7 107

Longitudinal Velocity Spread [m/s]

Horizontal Size Vertical Size

2

Peak Particle Density [cm- 3]

Transverse Beam Size [mm]

70

106

2.5

60

50

40

30

20

10 0

1 107

2 107

3 107

4 107

5 107

6 107

7 107

Circulating number of particles

Figure 6.17: Left: The rms beam size and the peak density as a function of the number of circulating particles in a coasting beam. Note the sudden rise in the vertical size when the particle number becomes very low. Right: The corresponding longitudinal velocity spreads. The reason for this sudden blow up in the vertical beam size is not understood at the moment, and more experiments are needed to characterize the phenomenon. At the moment we believe that the reason the beam becomes ribbon-like is that the laser, as discussed previously, tends to force the particles to have the same linear velocity. Particles with the same linear velocity will have the same ideal orbit to oscillate about, thus the laser tries to put all particles onto the same horizontal position. If the particle number is large the space charge forces will prevent this from happening, but if the number of particles is low enough one might imagine that the particles all acquire the same horizontal position, but in that case space charge would force them to spread out vertically. Of course this explanation does not account for the exact point at which this happens, and is so far rather

114

Transverse Dynamics V

-V

Figure 6.18: Illustration of our scheme for tapered cooling. qualitative, but at least it gives an idea of why at all this kind of beam may exist. In order to clarify this behavior experiments are needed which determines the exact dependence on current and longitudinal velocity spread, and further on how much the phenomenon depends on the procedure used for cooling the beam.

6.5.1 Tapered Cooling The eect discussed above may be connected to the laser cooling the particles to a constant linear velocity. As was already mentioned in Section 2.3.3 the crystalline ground state in a storage ring has a constant angular velocity, hence standard laser-cooling will generate shear stresses in the beam, and prevent the beam from reaching its ground state. In [105] it was proposed to solve this by introducing tapered cooling, which can be described as follows [75] " ! # p p x (6.10) Ftap( p ; x) = ,fs p , Cxs D where p=p and x are the momentum deviation and the horizontal displacement respectively, D the local dispersion function, fs is a positive constant cooling strength, and Cxs is called the tapering factor. To realize this the authors proposed to use a prism to introduce a frequency gradient in the cooling laser. This would of course mean that dierent frequencies would have to be part of the laser beam, something which could be accomplished by the frequency comb technique [43]. However the dispersion varies through a straight section, and thus the frequency gradient would have to change or the laser should only be active in part of the section. In ASTRID with a dispersion of 2.7 m in section centers (over a distance of roughly 2 m), the frequency gradient needed would be  1GHz/mm - which will be rather dicult to accomplish when we bear in mind the positioning precision needed and the natural divergence of the laser beam. The suggested method may be possible, but we are working on a much simpler method to introduce tapered cooling. In Figure 6.18 is shown a section of the beam pipe. Two vertical parallel plates have been introduced around the beam in order to introduce a linearly

6.5 Ribbon Beams

115

varying potential across the beam. When a particle enters the section it will experience a velocity change (as in the PAT) which is dependent on its horizontal displacement. If the whole of the setup is given a DC oset all particle velocities can be shifted such that the laser is only resonant in this part of the section. A vertical dipole magnetic eld should be used to compensate for the de ection caused by the electric eld. The linearly varying potential across the beam can be set to compensate for the linear velocity gradient in the beam due to dispersion. With standard ASTRID settings the electric eld needed is 77 kV/m, and the compensating magnetic eld is 86 mT. As both of these values are within reach experimentally the method is feasible. Whether this scheme will actually work in practice depends highly upon what the eects of the stray elds from the capacitor plates will be and on what in uence the system will have on the beam motion. This is currently being studied.

116

Transverse Dynamics

Chapter VII Conclusions and outlook This thesis was a presentation of the work done by myself and coworkers in our attempts to learn about the dynamics of laser-cooled stored ion beams. We have developed a novel system for measuring transverse beam sizes of circulating ion beams and used this system to conduct studies of ion beams during laser-cooling. The system is based on imaging the uorescent light from the laser excited ion beam, and has a better resolution and sensitivity than other similar non-destructive systems. Furthermore it only measures the emittance of a speci c velocity class given by the laser detuning, which makes the transverse emittance measurements direct without need to compensate for the increase in beam size due to dispersion. With this system and laser facilitated velocity measurements as well as the more generally applicable Schottky analysis we have studied the behavior of laser-cooled ion beams. The main focus has been on the coupling between the transverse and longitudinal degrees of freedom (sympathetic cooling). The reason for this is that laser-cooling in a storage ring is only applicable in the longitudinal dimension, and thus cooling of all three dimensions relies on the successful coupling of energy from the transverse dimensions to the longitudinal, sometimes called sympathetic cooling. We have learned that the intra-beam scattering causes rather slow transverse cooling, as one might expect, which means relaxation times of order few seconds, while the longitudinal cooling times are of order some milliseconds. It has been noted, but not studied thoroughly, that an enhancement in the horizontal to longitudinal coupling can be achieved in dispersive sections. However, even with this enhancement a limitation in the maximum particle density which is acquired has been observed. This limiting particle density is independent of the beam current, and we believe that it is due to the beam being space-charge tune shifted into a betatron resonance/stop band. This problem may be severe, as the ground state of a stored ion beam, which is crystalline, has an incoherent tune of zero, and we therefore need to cross several resonances during the cooling. The sympathetic cooling may not be strong enough to overcome these stop bands, however the experiments seems to indicate that an increase in cooling power and improvement of alignment increases the limiting particle density, thus the question is whether this will improve it enough or other methods needs to be applied. In order to learn of the strength of the coupling the diusion and Touschek losses in the beams were studied, but so far only in a preliminary

118

Conclusions and outlook

way, as it was done without the transverse diagnostics operational. However these studies demonstrated that diusion and Touschek measurements may be an important tool for understanding what mechanisms limits laser-cooling of stored beams. One such limit was the above observed stop band limit, but other limits, such as Touschek losses, may become severe and thus have to be countered by extending the range of the laser-force.

7.1 Outlook I think the future of laser-cooling in storage rings is bright. This in spite that it seems that it will take a couple of more years before beam crystallization is a reality, because the only two rings with laser-cooling are unsuited [105]. The reason is that I still think there is much interesting accelerator and beam physics which can be studied without crystallization. Our own work has brought up the question of whether it is possible without any extraneous means to overcome the resonance barrier or stop bands which has to be crossed while cooling to the space-charge limit. This question needs an answer, and it may become necessary to employ schemes to enhance the strength of the transverse to longitudinal coupling. Furthermore the whole question of the ground state and the tapered cooling needs to be investigated. Currently we are in Aarhus working to implement a position stabilization system which will enable us to measure the coupling as a function of cooling laser position with high precision and thus gain more insight into the problem. The next step is to implement a tapered cooling scheme, which if working should be able to suppress the phase transition to the ribbon beam, and teach us whether heating due to shear is an important factor at the present stage. On route to these improvements it will also be necessary to conduct more detailed studies of the transverse to longitudinal coupling by measuring the cooling and heating rates in the beam using the newly implemented transverse diagnostics. These investigations should reveal whether laser-cooling will become a player in storage rings in the long term, or whether seriously new ideas are needed for it to become interesting. If laser-cooling is strong enough it seems likely that it can be used for cooling at facilities for nuclear waste transmutation and/or inertial con nement fusion where high currents and densities are needed, and the accompanying instabilities therefore need to be countered [39]. The possible attainment of a crystalline beam would open up new areas of physics, and increase the understanding of the fundamental limitations of storage rings. A crystalline beam in it self might also be interesting in collision experiments where it has been shown to increase the luminosity [106], and of course it represents the ultimate in energy and time resolution. A more drastic step in the experiment in Aarhus would be to alter the ASTRID storage ring itself into an eight fold symmetry ring in which the attainment of a crystalline state should be possible. This would open up more long term possibilities for continuing the laser-cooling experiments which I think should otherwise take a dierent path in the long term. A dierent path for laser-cooling and laser ion beam interaction in general could be \only" to use the strong tailoring eects of laser-cooling on the beam and the powerful beam diagnostics available, these could for instance be used to enter into studies of non-linear

7.1 Outlook

119

dynamics and chaos as done at the IUCF [50, 96, 12]. With all these options for the future I think there will be plenty of opportunities for innovators and researchers who are intrigued by the wide range of opportunities into dierent areas of physics which laser-cooling in a storage ring opens up.

120

Conclusions and outlook

Chapter VIII Dansk Resume Nrvrende afhandling er en fremlggelse af resultatet af 4 ars arbejde ved Institut for Fysik og Astronomi, Aarhus Universitet. Afhandlingen beskftiger sig med dynamik af ttte laserklede ionstraler. Interessen for ttte ionstraler bunder i at en meget stor del af den moderne fysik benytter sig af partikelstraler af forskellig art. Man benytter partikelstraler til implantering af urenheder i forskellige materialer, noget man f.eks. gr i halvlederindustrien, og man benytter partikelstraler i hjenergifysikken, hvor man kolliderer meget energirige partikler for at kunne studere universets mindste byggesten, og de krfter der holder dem sammen. I eksperimenter som nogle af dem der foregar ved lagerringen ASTRID i Aarhus benytter man at fragmenter fra processer der sker i en partikelstrale ofte vil have et andet forhold mellem ladning og masse og derfor vil blive afbjet anderledes (maske endda slet ikke) nar der kommer et sving1. Man kan saledes detektere fragmenterne og lre noget om de fysiske processer involveret. En nyere anvendelse af partikelstraler, der dog endnu er pa tegnebrttet, er som energileverandr til transmutering af radioaktivt aald2 , og til inertielt sammenholdt fusion3. I mange af disse eksperimenter er det vigtigt at ttheden i stralen er hj idet dette giver en strre begivenhedssandsynlighed. I ere af anvendelserne (som f.eks. de to sidst nvnte) vil man ogsa gerne have en meget hj intensitet, da det er vigtigt at fa mange begivenheder/hj energitilfrsel (man nsker i fusions jemed at levere tndingsenergien med en partikelstrale). Hj tthed i en ionstrale opnaes ved at kle denne. Grunden til dette er simpel. Tykkelsen af en ionstrale er nemlig bestemt ved at denne holdes fanget i et potential, samt at den har en vis temperatur. Jo hjere temperatur, des mere fylder stralen (tnk pa kugler i en skal, hjere temperatur betyder mere bevgelsesenergi, d.v.s. kuglerne kommer hjere 1 Det typiske er at partikler bliver accelereret/lagret i en s akaldt lagerring, som ASTRID er et ek-

sempel pa. Her holder man partiklerne fanget ved at lade dem cirkulerer. For at kunne dette har man afbjningsfelter med jvne mellemrum. Disse felter er enten elektriske eller magnetiske, og er derfor tilpasset de cirkulerende partiklers ladning, saledes vil f.eks. en neutral partikel ikke blive berrt. 2 Ved transmutering af radioaktivt aald forst aes at man ved stor energitilfrsel, f.eks. v.h.a. en partikel strale, fremskynder aaldets henfalds process og derved laver mindre radioaktivt aald. 3 I inertielt sammenholdt fusion tnker man fusions materialet holdt samlet og tilfrt energi ved at bombardere det fra alle sider med f.eks. partikelstraler.

122

Dansk Resume

op ad skalkanten, og dkker saledes mere af skalens areal). Det viser sig desuden, som man maske kunne forvente, at stabiliten af et sadant system falder med stigende strm, ustabiliteter frer i frste omgang til opvarmning, hvilket saledes kan dmpes med kling. Stralekling er derfor et middel til at opna prcis de ting der ofte er interessante i ionstrale eksperimenter. Der ndes 3 mader hvormed man kan kle tungpartikelstraler (elektroner beskftiger vi os ikke med her). Den ldste er elektronkling der foregar ved at man blander sin partikelstrale med klede elektroner med samme middelhastighed som partikelstralen. Partikelstralen bliver herved klet ved kollisioner med den kolde elektrongas. Denne metode er den mest universelle, og ganske eektiv med kletider ned til hundrededele af et sekund. Desvrre er den ikke god (og uforholdsmssigt teknologisk kompliceret) ved hje energier, hvor man i stedet kan anvende en metode der kaldes stokastisk kling, der tilgengld er forholdsvist langsom, og derfor kun benyttes ved hjenergi acceleratorer, hvor den har vist sig at vre srdeles anvendelig (Nobelprisen i 1984). Den nyeste metode er laserkling som egentligt blev udviklet til flder, men som, grundet de meget korte kletider (millisekunder) nu bliver brugt til acceleratorer. Laserkling har to ulemper i forbindelse med partikelstraler. Den frste er at laserkling kun kan anvendes pa ioner der har en bestemt fordelagtig elektronstruktur, den anden at laserkling kun har direkte ind ydelse pa energispredningen i stralens bevgelsesretning, da laserkling krver godt overlap mellem ioner og laser i den dimension der nskes klet, og derfor kun vanskeligt kan bringes til at virke vinkelret pa stralens bevgelsesretning (hvilket vi kalder det transversale plan). At vi alligevel har valgt at studere laserkling skyldes at laserkling er den metode der har kunnet opna de laveste temperaturer i ionstraler, og derfor giver hab om de ttteste straler. I sin yderste konsekvens forventer man at nar en ionstrale bliver kold nok, vil ionernes relative bevgelsesenergi ikke lngere vre nok til at de kan overvinde deres elektriske frastdning, saledes vil de blive fastlast i deres position i forhold til hinanden, og en slags spgelseskrystal er dannet. Denne nye form for stof lover i sig selv nye muligheder, og er blevet observeret, og bliver studeret i ionflder hvor man dog har det noget nemmere idet man ikke har et stort energireservoir i form af stralens kollektive bevgelse der kobler energi ind i krystallen (d.v.s. varmer den op). Derudover giver laserkling, og anvendelsen af lasere til diagnostik nogle unikke muligheder for at studere og lre mere om dynamikken af ttte ionstraler, informationer der er ndvendige for at man kan akkumulere de hje strmme og ttheder man ofte er interesseret i. I det flgende vil jeg i en smule detalje gennemga fysikken bag laserkling og diskutere de resultater der er prsenteret i afhandlingen. Laserkling er en forholdsvist ny op ndelse, der udnytter at fotoner brer bevgelsesenergi (impuls). Hvis en foton bliver absorberet af en ion vil dennes bevgelsesenergi blive overfrt til ionen, der sa vil ge sin hastighed tilsvarende, og nu be nde sig i en eksiteret tilstand. Hastighedsforgelsen per absorberet foton er meget lille, men ved at gentage denne process mange gange, kan man overfre en betydelig mngde bevgelsesenergi, og derved skubbe til ionen. Dette kan selvflgelig kun lade sig gre hvis ionen henfalder til samme tilstand som den startede fra inden for en rimelig tid, da den ellers ikke kan absorbere ere fotoner. Nar ionen henfalder fra den eksiterede tilstand udsender den en foton

123 af samme type som den tidligere modtog, heldigvis er retningen af denne foton tilfldig, og i middel (efter mange absorptioner og emissioner) bidrager den spontane emission ikke til nogen hastighedsndring. Herved har vi opnaet at laseren kan skubbe til ionen. Nu er det jo sadan at nar man bevger sig i forhold til en lyskilde, sa skifter lyset farve, dette kaldes Doppler forskydning (det samme som at en ambulances frekvens lyder forskelligt nar den krer mod en, i forhold til nar den krer vk). Dette giver anledning til at de skubbede ioner, der nu har en anden hastighed pa et tidspunkt ikke lngere er resonante med laserlyset, d.v.s. de kan ikke lngere absorbere fotonerne. Men det gr ikke noget for vi kan jo blot ndre laserens frekvens sa den igen passer. Pa denne made kan laseren virke som en sneplov for ioner. Denne metode benytter vi i ASTRID til at kle ionstraler med - princippet er at vi har en laserstale til at bremse ionerne og en til at accelerere dem og vi samler sa ionerne 'mellem' de to hastigheder hvortil de bliver skubbet af laserne, og ved at ndre lasernes frekvenser og stoppe pa passende vrdier kan vi skubbe ionerne sa de alle har nsten samme hastighed. Dette betyder at deres relative (indbyrdes) bevgelse er meget lille, d.v.s de er kolde. Da der skal absorberes mange fotoner for at gre en forskel pa en ions hastighed er denne metode kun god i bevgelsesretningen for stralen, idet vi her kan fa laserstralen og ionstralen til at overlappe et langt stykke. Men en ionstrale der kun er 'kold' i bevgelsesretningen er ikke helt sa spndende (og anvendelig) som hvis man kunne kle de transversale retninger ogsa, da disse, som nvnt, har betydning for ttheden i stralen. Heldigvis falder temperaturen i de to transversale dimensioner (retninger (vandret og lodret)) efterhanden idet partiklerne stder ind i hinanden og pa den made kobler bevgelsesenergi ind i den longitudinale dimension, hvor vi jo kler hele tiden (lidt som hvis man blander to gasser med forskellig temperatur, efterhanden opstar der en ligevgt - her er det 'blot' en gas med to temperaturer). Eektiviteten af denne process, som er vigtig for anvendeligheden af laserkling, har vi studeret i ASTRID i Aarhus. Vi har fundet at denne form for 'vente' kling (til tider kaldet sympatisk kling) er ganske ineektiv, og ikke strk nok til at overkomme forskellige begrnsninger som lagerringsstrukturen stter. Den vigtigste begrnsning som den ikke ser ud til at kunne overkomme er de sakaldte stopband. Hvis man forestiller sig en partikel alene i ringen, sa kan en sadan partikel have en lille fejl i dens bane i forhold til den ideelle bane som er de neret af de magneter der sidder i ringen. Denne fejl, samt de fokuserende elementer i ringen gr at partiklen vil oscillere omkring den ideelle bane pa sin vej rundt i ringen (denne oscillation svarer prcist til de oscillationer jeg omtalte i skalforklaringen tidligere) . Hvis nu denne partikel har en bane der bringer den tilbage til prcis den samme position hver gang den har lavet en omgang i ringen, sa kan man nok forestille sig, at hvis den hver gang den har lavet en runde, far et lille skub ud ad, sa vil alle disse skub addere og fejlen i forhold til ideal banen vil vokse indtil partiklen ramler ind i vacuumkammeret. Hvis partiklen derimod aldrig kommer tilbage til helt den samme position sa vil alle disse skub midle ud og partiklen kan cirkulere i princippet for evigt. Hvis nu der er en masse partikler, men ttheden er lille, sa opfrer de sig nogenlunde som om de hver isr var alene i ringen, men nar vi nu kler stralen sa vil deres oscillationer om idealbanen blive mindre hele tiden, d.v.s. partiklerne vil komme tttere pa hinanden og blive udsat for en slags gnidning, der far

124

Dansk Resume

dem til at oscillere langsommere. Men, hvis de begynder at oscillere langsommere, kunne det jo vre, at de pa et tidspunkt udfrte netop en oscillation per omgang, og derfor ville blive tabt. En oscillationsfrekvens, hvor partiklerne bliver tabt, kaldes et stopband, og kan skyldes, at banen lukker pa sig selv hver gang som i eksemplet her, men ogsa hvis det er efter 2 omgange eller ere. Jo ere omgange partiklen skal lave, fr banen lukker pa sig selv, des strre sandsynlighed er der for, at den har ramt en anden partikel undervejs og sa alligevel ikke lukker banen. Som det fremgar, sa gr kling, at vi kan komme til at ramme et sadant stopband, og hvis dmpningen fra klingen er langsommere end opvarmningen fra stopbandet, sa vokser stralens strrelse - og partiklernes transversale oscillation vil ndres vk fra stopbandet. Denne eekt er netop hvad vi har observeret, nar vi laserkler ionstraler i ASTRID. Jeg beskriver i afhandlingen de forskellige studier, vi har gjort for at undersge hvilken eekt, der er tale om, og i hvor hj en grad den er et problem. Indtil nu har det vist sig, at vi ikke har kunnet nde mader at forstrke dmpningen af de transversale oscillationer nok til, at vi kan komme forbi dette stopband. Dette har stor betydning for laserklings anvendelighed, idet det jo stter en vre grnse pa de ttheder, man kan opna med laserkling. Det er nu op til fortsatte studier at fastsla om denne grnse er de nitiv, eller kan overvindes, og derved afgre om laserkling er kraftig nok til at kunne generere en krystalstrale i en lagerring, eller dmpe instabiliteter i intense ttte straler til f.eks. inertielt sammenholdt fusion. Der er naturligvis mange andre aspekter af det ovenstaende fnomen, som det vil fre for vidt at komme ind pa her. Jeg skal dog lige til sidst nvne, at laser-kling i virkeligheden frst blev udnyttet til at studere kolde atom og ion skyer, og i 1997 blev Nobelprisen givet til fdrerne bag denne teknologi, der har haft og har stor betydning for den fundamentale fysik, specielt kvantemekanikken. Vores arbejde er i periferien af hvad laserkling ellers bliver anvendt til af fysikere, men anvendelsen af laserkling i en lagerring abner mange muligheder som traditionelt ikke har vret tilgngelige for acceleratorfysikken, bade grundet den strke kleeekt, men ogsa fordi laservekselvirkning med ioner giver anledning til uhrt eektive metoder til at studere, hvordan ionstralerne opfrer sig i lagerringen. En af disse metoder har vret en stor del af mit projekt, og bestar kort fortalt i, at man udnytter, at laseren far ionerne til at lyse i alle retninger (spontan emission), og at man ved at tage et billede af dette lys, kan fa et billede af ionernes fordeling i stralen. Metoden har givet meget gode resultater, og leverer meget hjere oplsning end andre metoder, samt giver mulighed for at male pa meget partikelfattige straler. Uden denne metode ville det arbejde, der her er udfrt, ikke have vret muligt.

Appendix A Calculations This appendix includes various calculations which have been considered too long to be included in the main text, or are of a supplementary nature to the main text.

A.1 Velocity and Temperature As the velocity distributions in the laser-cooled ion beams are more often than not parabolic in their shape we need to nd a general term for what we mean by the temperature of the beam. The standard de nition, which is used throughout the thesis, is to de ne the temperature from the mean kinetic energy as kB T = mhv2i (A.1) where m is the mass of the ions, kB the Boltzmann constant, and hv2i is the rms velocity spread of the velocity distribution, which does not have to be Gaussian. A normalized Gaussian distribution can be written as " 2# 1 g(v) = p exp , 2v2 where hv2i = 2 (A.2) 2 and where the full width at half max is given by p vfwhm = 8 ln 2 (A.3) A normalized parabolic distribution can be written as " 2# 1 v p(v) = 4=3v 1 , v2 ; ,v0  v  v0 (A.4) 0 0 where the integral for the root mean square gives Z v0 hv2i = ,v v2p(v) dv = 15 v02 (A.5) 0

126

Calculations

and where the full width at half max is given by

p

vfwhm = 2v0

(A.6)

The parabolic distribution can be written in terms of the root mean square velocity used for the temperature de nition " 2# p p 3 v p(v) = p 1 , 5 2 ; , 5  v  5 (A.7) 4 5 q where we have chosen  = hv2i. The full width at half max is given by

p

vfwhm = 10

(A.8)

q Thus for equal temperature (rms size) the FWHM of a parabolic distribution is 5=4 ln 2, larger than the FWHM of the Gaussian ( 34%). Thus a rather large deviation, as discussed in Section 5.2.

A.2 Bunch Shapes As discussed in Section 5.2.2 we measure the longitudinal charge density distribution of the bunches via an electrostatic pickup. As the pickup has a nite size comparable to the length of the bunches we need to take the shape of the pickup into account when extracting the length of the bunches. This work was earlier done by J.S. Nielsen in [70], and is reproduced here with some additional comments. As discussed in [70] the pickup is a box, but as we need an analytical expression to facilitate the tting of the (many) bunches, we have chosen to approximate the square pickup by a cylindrical pickup with radius dc . The induced voltage on a cylindrical pickup from a point charge q is 2 3 Vind;cyl (q; c) = 2Cq 4 q l=2 , 2c 2 + q l=2 + 2c 2 5 : (A.9) p (l=2 , c) + dc (l=2 + c) + dc The radius dc can be chosen by two criteria. Either we can choose dc so that the cross section area is the same as for the square pickup, which will produce complete agreement when the point charge is far away from the pickup, or we can choose dc=1.122
d which will produce complete agreement when c=0. By integrating equation (A.9) over the parabolic bunch shape, we obtain the induced voltage as a function of the position of the bunch, Vind (pos) = p3Qb [V1(pos) + V2(pos)] ; (A.10) 4 2Cp

A.2 Bunch Shapes where

127

2 !2 1 w 2 4 V1(pos) = 12w3 ,(,8dc + k1 + p , 92 w2)kv1 2 3 !2 " # w 9 1 k + k 3 v 2 2 2 2 5 +(,8dc + k1 , p , 2 w )kv2 + w3 dc k1 ln k + k (A.11) 2 2 v1 2 !2 w 1 2 4 V2(pos) = 12w3 ,(8dc + k4 + p + 29 w2)kv3 2 3 " # !2 1 k + k w 9 5 v 3 2 2 2 5 +(,8dc + k4 , p + 2 w )kv4 + w3 dc k4 ln k + k (A.12) 2 6 v4

and

q q kv1 = q4d2c + k22 kv2 = q4d2c + k32 (A.13) kv3 = 4d2c + k52 kv4 = 4d2c + k62 p p k1 = 2pos , l k2 = k1 , p2w k3 = k1 + p2w (A.14) k4 = 2pos + l k5 = k4 , 2w k6 = k4 + 2w The actual output voltage is given by Vout = gVind , where g is the gain in the ampli er. Figure 4.6 showed how the fwhm of the measured distributions develops as a function of the length (fwhm) of the bunches. From this graph we concluded that it was not necessary to use the rather elaborate scheme above for analyzing the data measured for this report. As it furthermore turned out that the bunches are very parabolic at all temperatures, due to the parabolic velocity distribution, the bunches could be analyzed by tting with parabolas. A calibration using a beam current transformer (BCT) for reference gave us the following relation between the measured area of the induced signal on the pickup and the charge/current in each bunch  = 14:52nA/Vm  0:08nA/Vm

(A.15)

from this calibration constant the capacitance can be found by

gCp = 
TC = (2:60  0:01)
10,11 C=V  10% where T is the revolution time and C the circumference of the ring.

(A.16)

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Calculations

Appendix B Technical Speci cations In this appendix some details are given on important parts of the equipment used for the experiments.

B.1 CCD Cameras The general experimental con guration was shown in Figure 4.11. There the dewar con guration inside the camera was also shown, as extracted from the design drawings from the manufacturer, Photometrics. The cameras are generally identical, however the product name changed, and the CCD chips are slightly dierent. Parameter 'Old' camera (I) 'New' camera (II) Product Name CH260 CH360 CCD Chip TK1024AB SI003AB Resolution 16 bit 16 bit Readout Rate 40 kHz 40 kHz Operation Temperature < -90C < -90C Dark Current 0.26 e, /pixel/hour 2.88 e, /pixel/hour Bias Mean Level  400 ADU  400 ADU Full Well Capacity 272 ke, 371 ke, Conversion factor at 1X Gain 4.16 e, /ADU 5.80 e, /ADU , Noise at 1X Gain 5.3e RMS 7.03e, RMS Linearity at 1X Gain 0.10% 0.36% Conversion factor at 4X Gain 1.02 e, /ADU 1.42 e, /ADU Noise at 4X (2X) Gain 3.3e, RMS 4.90e, RMS Linearity at 4X (2X) Gain 0.11% 0.17% Quantum eciency at 280nm 26.1% 19% Cold time with full N2 dewar 14h 14h

Table B.1: CCD Camera Speci cations. The gains in parentheses are for the right column.

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Technical Speci cations

In table B.1 we use the term ADU when talking about for example the bias level. This is an acronym for Analog to Digital Unit, which is the manufacturers name for the number which is the end result in each pixel in the image on the computer. Thus for camera I, 4.16 electrons of charge in a channel on the CCD chip would cause a 1 to appear in the corresponding position in the computer image. The noise is the readout noise of the system, i.e. the noise in converting the charge in a CCD channel into a number in the computer. Figure B.1 shows an image of the camera with electronics unit and of the camera systems mounted on the storage ring during operation.

Figure B.1: CCD Camera system used for beam imaging. The photo on the right shows the cameras mounted on the storage ring. Note the black bellows between the camera head and the (hidden) lens mounts connected to the ring. To protect the sensitive CCD's and minimize noise the camera's are on a dierent ground than the storage ring.

B.2 Optical Scanners For creating a homogeneous laser intensity distribution over the width of the beam we use two galvo scanners, one for each dimension (vertical and horizontal). The necessary width of the homogenous distribution is approximately 3cm, which, with a distance from the earliest point where the scanners could be positioned (right after the SHG cavities) of  14m, corresponds to a de ection angle of approx. 2 mrad. In tabel B.2 the speci cations for the scanners we currently use are given. As we only have two laser systems (next section) to generate UV and we need three laser beams for coasting beam cooling (two for cooling and one for probing) the beam used for diagnostics is split into two parts, one is sent through the scanner system and

B.3 Lasers

131 Parameter Value Manufacturer Cambridge Technology Model number 6350 Zero Drift (Max) 15 rad/C Repeatability, Short Term (Typical) 3 rad Recommended Load 0.05 - 3.0 gram
cm2 Rated Excursion, Rotor 40 degrees Calibration to ext. input 0.2 degrees/10V Scanner oset setting  10 degrees

Table B.2: Galvanometer Optical Scanner Speci cations. one directly into the cooling section. Figure B.2 shows the setup up for this part of the experiment.

Figure B.2: Path of the beam used for the diagnostics section. The primary beam enters in the lower right corner where part of it is de ected to vertical direction by a beam splitter. In the right part of the picture it passes the two scanners (mounted on aluminium blocks to absorb heat), each set to scan in one dimension, and is guided into the ring hall2(on the left).

B.3 Lasers As mentioned earlier the laser system consists of two parallel systems, each with an Ar+ laser, a dye laser and an external UV-cavity. In Table B.3 relevant speci cations for the ion lasers are presented, in Table B.4 for the dye lasers and in Table B.5 for the UV cavities. 2 The shown mounts are from Thorlabs and Newport, but other possibilities exist [83].

132

Technical Speci cations

The details of the locking system from the UV-cavities, as well as an improvement in the frequency drift of the ring dye lasers are discussed in Section 4.2. Parameter System 1 System 2 Manufacturer Coherent Coherent Model Innova 306 Innova 70-5 Wavelength Range 457.9 - 514.5 nm 457.9 - 514.5 nm Multiline Visible Power (Max) 6.0 W 6.0 W Power Stability 1.0% 0.5%

Table B.3: Argon-Ion lasers used for dye laser pumping [17, 16] Parameter Value Manufacturer Coherent Model CR699-21 Dye Pyrromethene 556 Dye Manufacturer Exciton Operating Wavelength 560 nm Output power @ 6W 1200 mW Frequency Jitter