An electrostatic decelerator for a high-field radio frequency quadrupole ion guide beam cooler
by Ricardo Larnbo
Department of Physics Mc Gill University Montreal, Canada
August, 2005
A thesis submitted to the faculty of Graduate Studies and Research in partial fulfillment of the requirements of the degree of Master of Science
© Ricardo Larnbo, 2005 Library and Bibliothèque et 1+1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l'édition
395 Wellington Street 395, rue Wellington Ottawa ON K1A ON4 Ottawa ON K1A ON4 Canada Canada
Your file Votre référence ISBN: 978-0-494-24713-6 Our file Notre référence ISBN: 978-0-494-24713-6
NOTICE: AVIS: The author has granted a non L'auteur a accordé une licence non exclusive exclusive license allowing Library permettant à la Bibliothèque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l'Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans loan, distribute and sell th es es le monde, à des fins commerciales ou autres, worldwide, for commercial or non sur support microforme, papier, électronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats.
The author retains copyright L'auteur conserve la propriété du droit d'auteur ownership and moral rights in et des droits moraux qui protège cette thèse. this thesis. Neither the thesis Ni la thèse ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent être imprimés ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission.
ln compliance with the Canadian Conformément à la loi canadienne Privacy Act some supporting sur la protection de la vie privée, forms may have been removed quelques formulaires secondaires from this thesis. ont été enlevés de cette thèse.
While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. ••• Canada Abstract
An electrostatic dece1erator system capable of bringing to rest a continuous 60 ke V De ion-beam was developed and tested as to its suitability as the first part of a high performance RFQ buffer gas beam cooler. The dece1erator is intended to decelerate high energy ions for injection into an RFQ ion-guide capable of producing bunches of ions at a repetition rate up to 100 Hz. Simulations of the ion motion through the decelerator and into the guide were made to test the design, and the dece1erator was then constructed following the parameters used in the simulations. Although problems, such as sparking between the electrode surfaces of the decelerator, prevented it from being operated at potentials high enough to decelerate a 60 ke V beam, its implementation proved, as in the simulations, that high-energy ions could indeed be decelerated to energies that would allow for their injection into an RFQ ion-guide by this design. Résumé
Un décélétateur électrostatique capable d'amener au repos un faisceau d'ion continu de 60 keV a été développé et testé, et constitue la première partie d'un RFQ refroidisseur de faisceau à gaz tampon à haute performance. Le décélérateur est conçu pour décélérer des ions de hautes énergies pour injection subséquente dans un guide d'ion RFQ capable de produire des amas d'ions à une fréquence de 100 Hz. Des simulations du mouvement des ions à travers le décélérateur et le guide ont été faites afin de tester le concept du prototype, et le décélérateur a été construit suivant les paramètres déterminés lors des simulations. Malgré quelques problèmes, tel l'ionisation du gaz résiduel dans le décélérateur, qui ont limité l'utilisation de potentiels suffisament élévés pour décélérer un faisceau ionique de 60 keV, la mise en oeuvre du design à 12 kV démontre que tout comme dans les simulations, des ions d'hautes énergies peuvent en effet être décélérés à des énergies permettant leur injection dans un guide d'ion RFQ.
11 Acknowledgements
1 would like to thank Professor Robert B. Moore for his superVISIOn and assistance at all stages of my thesis project, from its design and simulation to its construction and testing, as well as for his help in the writing and editing of this thesis. 1 also extend my thanks to Dr Leo Nikenen for his advice on the electrical and vacuum systems of the decelerator; to fellow graduate student, Mr Omar Gianfrancesco, for his help in providing a working ion-source for the experiment; to summer student, Mr Mark Antoine Goulart, for his assistance in assembling and testing the apparatus; and to Mr Steve Kekani and Mr Eddie Del Campo for their help in machining its parts.
111 Table of Contents
Abstract i
Résumé ii
Acknowledgements iii
Table of Contents iv
1. Introduction 1
2. Theoretical Background 7
2.1 Bearn Acceptance ...... 7
2.2 The Acceptance of RFQ Ion Guides ...... 9
2.2.1 The motion of a single partic1e in an RFQ field ...... 9
2.2.2 The motion of a collection of partic1es in an RFQ field ...... 14
2.2.3 The acceptance of an RFQ ion guide in terms of bearn emittance ... 17
2.3 Basic Principles of the Decelerator ...... 17
2.4 Scaling of the Decelerator with the Ion Mass ...... 20
2.5 Scaling with the Decelerating Potential ...... 21
3. Computer Simulation 22
3.1 Spreadsheet Calculation for Decelerator System ...... 22
3.2 Higher-Order Simulation ...... 25
3.2.1 Truncation of the Entrance and Ring Electrodes ...... 26
3.2.2 Correction ofthe Ring-Electrode ...... 28
3.2.3 Simulation of Decelerator Operation ...... 30
4. Apparatus, Experiments and Results 34
4.1 The Decelerator ...... 34
4.2 The Experimental Set-up ...... 38
IV 4.3 Measurements and Results ...... 43
5. Summary and Conclusion 48
References 49
Appendix A: Technical Drawings Al
Appendix B: Matrix Optics for Low Energy Beams Bl
Appendix C: The Ion Source Cl
v 1. Introduction
Recently, the cooling of ion beams, particularly radioactive ion beams, has become of great interest in nuclear physics. This is because the reduction in phase space achieved by cooling, as in high-energy particle physics experiments, enables much greater accuracy and sensitivity in experiments on the ions. This is very important for measurements such as those of radionuclide masses at ISOLDE at CERN where manipulations of radioactive ions in a Penning trap has enabled accuracies exceeding one part in 108 on over 100 radionuclides, many of very short haIf-life. Cooling of high-energy particle beams is achieved by either immersing the particles in a collinear bath of cold electrons moving at the same velocity as the particles, referred to as "electron beam cooling", or by feeding electric fields back onto the particle that are opposite to those that are due to the potential that the particle itself induces in the electrode of a detecting circuit, a technique referred to as "stochastic cooling". However, neither of these techniques is applicable to the relatively low velocity of ions that are delivered by on-line mass separators operating on the output of a target at high-energy accelerators, by far the most prolific source of radioactive ions. High-powered lasers are sometimes used for ion cooling in traps by a technique called "side-band cooling". Here an intense laser beam is tuned to a frequency just slightly lower than an optical resonance of the ion. Photons are then only absorbed when the ion is moving against the laser beam, and therefore receiving a retarding momentum. However, this technique is only useful for low temperatures and cannot be easily employed to on-line mass separator beams. The technique that can be used for the cooling of hot ions is deceleration and thermalization of the ions in a thin buffer gas such as helium. The ions then come close to thermal equilibrium with the helium molecules. If the helium is pure enough, the ions remain as ions and can be extracted from the helium at temperatures close to that of the helium itself, i.e., of the order of 1/40th of an electron volt, as compared to a typical temperature of 10 electron volts from the accelerator target. For the ions to be thermalized in a small volume, suitable for subsequent extraction, they must enter the gas at relatively low energies, of the order of several electron volts, and must be radially confined before and after they are thermalized. The
1 confinement can be achieved by a transverse radiofrequency quadrupole (RFQ) electric field. The low energy is achieved by electrostatic deceleration. For example, the ISOL TRAP system used at ISOLDE for mass measurements is shown in fig. 1.1.
60keV ISOLDE Electrostatic RFQ ion guide Penning trap ion-beam RFQ ion trap decelerator cooler cleaner/cooler
Penning trap mass snectrometer
Figure 1.1: Schematic of the ISOLTRAP facility at ISOLDE, CERN
The deceleration of ion beams is a very old subject, and many decelerators have been built for various purposes. The most common use is for ion implantation when it is desired that the ions not go very deep into the target material. Such a system was reported by Freeman et al., [1] in 1976, the basic geometry of which is shown in fig. 1.2
-2 kV +24.5 kV OV
1 Target 1 -+1 1+- 10 mm
Figure 1.2: Cross-section of the decelerator developed by Freedman et al., for implanting ions at 10 to 1000 eV from an initial ion beam of 25 keV. The potentials on the electrodes are shown for 1 keV implantation.
The paper of Freeman et al., [1] also inc1udes a short history of decelerator development before their work, reporting that the first decelerator to be used on accelerated beams was by Oliphant and Yeats in 1938 [2].
2 Another representative decelerator was built by Foo et al., [3] and reported in 1990. This decelerator was designed to take a 3 keV argon-ion beam to 10-200 eV for dry etching and thin film deposition. This was a set of 5 cylindrical electrodes to which potentia1s could be applied to achieve various focusing conditions at various residual ion energies (fig. 1.3). -3000 V -150 V -4500- V -1500 V -10 V ----§ ==
1...... ------100 -mm ------+1-1
Figure 1.3: Cross-section of the decelerator developed by Foo et al., for dry etching and thin film deposition at 10 to 200 eV from an initial ion beam of3 keV.
The first decelerator to be designed for injecting low-velocity ions into an RFQ containment device, actually a Paul trap, was developed by Rouleau and Moore [4]. This device was designed by manipulation of electrode geometries until computer simulations indicated that an acceptable fraction of the ions of a 60 ke V ISOLDE beam would enter the orifice leading into the trap at energies of about 100 eV so as to be captured in the trap. Its geometry is shown in fig. 1.4.
IN..ECTION END ELECTRODE ELECTRODE OFTRAP DECELERATION ELECTRODE
......
INTERIOR o OFTRAP 60 keVBEAM o
59990 V 10CM----t
Figure 1.4: The geometry of the decelerator developed by Rouleau and Moore for injecting a 60 keV ion beam into a Paul trap.
3 This deceleration scheme was also used by Dezfuli [5] to inject ISOLE beams onto a very large Paul trap. This trap was actually used at ISOLDE for mass measurements and was the first instance of the deceleration and capture of an ion beam with no interaction with material walls. This was very important in that it allowed measurement of the masses of rare radioactive mercury isotopes that would have been 10st had there been any such interaction.
Deceleration electrodes vzzzzztZZZZ2~Z-Z--Z-Z-4a-----~~
PZZZZZZZZZZZZZZZZ~
Trap end electrode-- __v
1+~---100 mm ---+,1
Figure 1.5: The geometry of the decelerator developed by Dezfuli., for injecting a 60 keV ion beam into a very large Paul trap.
In aIl of the above decelerators the primary concern was to get the decelerated ions to be confined to as small a spot as possible, there being very little concern about the divergence of the beam at that spot. This is because the angle at which the ions hit the surface of a target, or the angle at which it enters a trap, is of little importance. In fact, for the trap injection, the beam divergence was deliberately made as large as possible so as to spread the ions throughout the volume of the trap and enhance the chance of capturing them. However, in the case of delivering ions to an RFQ ion guide cooler one must take into account the beam divergence as weIl as the beam diameter and there must be a very specific relationship between the two. In technical terms, the incoming beam emittance must match the RFQ ion guide acceptance.
4 The first to do this was Kellerbauer [6], who designed a unique electrode shape to accommodate an ISOLDE-type ion beam and prepare it for entrance into an RFQ ion guide of the type developed by Kim [7] for buffer gas cooling of 100 eV ion beams. The geometry of this decelerator is shown in fig. 1.5 and it is, in fact, the decelerator used in the system in fig. 1. It has been shown to decelerate 60 ke V ion-beams at energies down to 50 eV and introduce them into an RFQ ion-guide with a transmission of 15 to 30 percent.
Focusing lens (59,900 V)
RFa entrance (59,950 V)
I~ 100mm~1
Figure 1.5: The geometry of the decelerator developed by Kellerbauer for injecting a 60 keV ion beam into a Paul trap.
Computer simulations of the ion trajectories through this device for ion beams representative of ISOLDE are shown in fig. 1.6.
DisplClY of x vs z
1"'0 œntral p::rticle in a"'t""CJ:I time = 5.121JS
1--______Z9:5.00 non ------1
Figure 1.6: The geometry of the decelerator developed by Kellerbauer for injecting a 60 keV ion beam into a Paul trap.
5 Recent advances at McGill in the generation of very high field RFQ containment by Gianfrancesco [8] have shown that ion beam cooling can be extended to much higher beam intensities and to much tighter confinement than was previously thought possible. However, this puts even more pressure on designing a decelerator system that matches the RFQ ion guide acceptance of these high-power devices. This thesis describes such a design.
6 2. Theoretical Background
2.1 Bearn acceptance
As pointed out in the introduction, the design of a decelerator for injection into an RFQ ion guide must start with the acceptance of the guide, where acceptance means the range of displacements and momenta of the incoming ions that will allow them through the guide without encountering any material walls. This full range defines a volume in 6- dimensional phase space for the incoming ions and, in general, aIl 6 coordinate of a partic1e in this space must be considered if it is to be determined whether it will be accepted by the system. However, in so-called "linear" systems the displacement momentum diagrams, i. e., the action diagrams, are orthogonal, meaning the behaviour of particles in a particular action diagram is independent of its behaviour in the diagrams for orthogonal directions. (Linear systems are those where the components of a force on a particle depend linearly only on the spatial and velocity components that are in the same direction as the force.) The boundaries of the 6-dimensional phase space volume accepted by the device therefore become independent boundaries in the three action diagrams. If the momentum displacement coordinates of a particular partic1e lie within these boundaries in aH three diagrams, then it is accepted by the system no matter where it is in each ofthose diagrams (fig. 2.1).
x y
x action boundary
y action boundary z action boundary
Figure 2.1: A representative set of action acceptance diagrams for an arbitrary collection of particles.
7 The electric field in RFQ ion guides is very close to being linear, deviating only from this condition at the relatively short entrance and exit regions. For the practical purposes of ion injection the action diagrams can be taken to be independent. AIso, the acceptance in the action diagram for the beam axis, usually taken to be z, is not an issue since the purpose of the ion guide as a cooler is to reduce, and finally remove, the axial motion. Only the transverse action diagrams need be considered, other than making sure that the incoming energy is not too high for the guide to remove. Furthermore, the axiperiodic field of the quadrupole means that the ion guide works on the particles in the x-direction in a symmetrical fashion to that in the y-direction, and so only one transverse action diagram need be considered in order to determine the acceptance boundary. The type of system for which it is simplest to determine the acceptance boundary is one in which there is a transverse force on the particle that is towards the center of the system and proportional to the displacement from this center, i.e., is linearly related to it. The transverse motion is any particle is then simple harmonic of angular frequency ru = ..J k / m , where k is the proportionality constant of the force and m is the particle mass. The action diagram of any particle is then the right ellipse shown in fig. 2.2.
...... r m(f)Xmax
x
Figure 2.2 : The action acceptance diagrams for a collection of particles in simple harmonie motion ..
8 The most deviant particle accepted by the system will than be the one that has its maximum displacement equal to the closest distance of approach of any material of the system that produces the force-field. This maximum displacement is, of course, the displacement semi-axis for the acceptance ellipse. Any other particle within this ellipse will automatically be allowed through. The "acceptance" of the system is therefore the area of the ellipse mmr~ax' For ion manipulation, the usual unit of acceptance is the action unit electron-volt-microsecond (eV-ils), and that ofmomentum is the eV-ils/mm, if the displacement is expressed in mm.
2.2 The Acceptance of RFQ Ion Guides
2.2.1 The motion ofa single particle in an RFQ field
Unfortunately, it is impossible to set up an electrostatic field that will confine a charged particle to simple harmonic motion in the two orthogonal coordinates simultaneously. This is because an electric field that is converging onto an axis in one plane must diverge from the axis in the orthogonal plane if there is no charge on the axis to absorb the electric flux. Expressed as Gauss's theorem in terms of the electric potential l/J in a vacuum,
(2.1)
and for a uniform field in the z direction
éil/J éP(jJ _ aE _ aEy x (2.2) iJx2 = - (]y 2 ; iJx (]y
Thus an electric field that results in simple harmonic motion in the x direction would result in an exponentially increasing displacement in the y direction. The y acceptance diagram would have zero area and hence there would be no acceptance of any ions. The first to circumvent this problem was Paul [8], for which he was co-winner of the Nobel Prize for physics in 1989. The technique he used was to simply oscillate an electric field that would be altemately confining and de-confining, the simplest such field being a quadrupole field such as
(2.3)
9 Such a field can be set up by the electrodes at ± ~o shown in fig. 2.3,
2 2 x - electrodes ., x = r0 + y2
Figure 2.3: The geometry of electrodes that will create an axiperiodic quadrupole electric field.
l Oscillating the potential so that, for example ,
~o = A sin(cot) (2.4) 2 would produce the desired altemately confining/de-confining electric field. While at first it might seem that the de-confining field would then simply cancel the effects of the confining field, it should be noted that an oscillating electric field will induce a driven oscillation of a charged particle that is out of phase with the driving force. The electric field is then a maximum toward the center of the driven oscillation when the particle is farthest from that center of the field. On the other hand, it is a maximum away from the center of the motion only when the particle is closest to the center of the field, and this maximum is less than the maximum at the other extreme of the oscillation. Its effect during the half of the cycle when it is pushing the particle inward is therefore greater than its effect during the half-cycle when it is pulling the particle outward. Over the full cycle of the particle motion there is then a net impulse in the direction in which the field is decreasing.
1 The altemation of the electric field does not have to be sinusoidal. It can be any periodic function, providing is spends as much time positive as negative. In fact, in ion confinement a "square-wave" is frequently used.
10 The average force on a particle in a sinusoidally oscillating electric field with a gradient dE/dx can be easily calculated for the case when the amplitude of the driven oscillation is small compared to the distance of the particle from the field center. For a driven oscillation of amplitude A, the equation for the change in momentum of the
particle Px in time t due to the electric field is, to first order,
(2.5)
where e is the charge of the particle, Eo is the amplitude of the oscillating electric field at the center of the motion and (J) is the angular frequency of the motion and of the field. The momentum change of the particle over one period T of the oscillation is then
(2.6)
This results in an average rate of change of momentum of
dfix =_!eldEI A (2.7) dt 2 dx 0
which is equivalent to an average field in the x-direction of
E =_!ldEI A . (2.8) x 2dx o
Of course, because of the non-uniformity of the electric field the driven oscillation will then not be strictly simple harmonic. However, for a small motion in which the electric field over the range of the motion only slightly deviates from its value at the center of the motion, the oscillation is simple harmonic with small higher-order harmonics of the basic frequency. To first order these harmonics can be ignored. The amplitude of the basic simple harmonie motion is
(2.9)
11 where m is the mass of the particle. The average field on the particle is then
E =_!_e_EIdEI (2.10) x 2 moi 0 dx 0
In the special case of a focusing quadrupole field the gradient is uniformly that at the quadrupole center and the electric field at the center of the particle oscillation, expressed as Eo at X center relative to the quadrupole center, is
(2.11)
The average force on the particle directing it to the quadrupole center is then 2 F _ 1 e (dEJ2 (2.12) x - -2" moi dx Xcenter
This force, being in the direction of the quadrupole center and being proportional to the distance of the oscillation center from that center, results in the oscillation center itself oscillating about the quadrupole center at an angular frequency
m =_1 _e (dE) (2.13) M J2 mm dx
This frequency can itself be expressed in terms of the dimensionless parameter q commonly used to indicate the strength of an oscillating quadrupole field when expressing the motion of a charged particle in terms of Mathieu functions2 rather than in terms of sinusoidal oscillations;
q =~(dE) (2.14) mm2 dx
2 The Mathieu functions are the solutions of the Mathieu equation, a dimensionless formulation, developed in the 19th century by Emile Mathieu, of the equation of motion of a partic1e operated on by a spring whose spring constant is itself oscillating sinusoidally (sometimes referred to as a "parametric oscillation"). The aim of this formulation was to de scribe the properties of planetary orbits due to periodic impulses from other celestial objects. The properties of these functions become important at their limit of stability, i. e., at the limit when they resemble sinusoidal functions rather than exponential functions. For a purely sinusoidal variation of the spring constant, with no constant offset, that limit occurs at about q = 0.92. A full discussion of the Mathieu functions can be found in Dawson [10] or March [11]
12 The re1ationship is, simply,
(2.15)
This is the well-established relationship of the so-called macromotion frequency to the field frequency for RF quadrupole ion motion in the Iimit of weak quadrupole fields. It is generally accurate to one percent for values of q up to 0.5, which is weIl above the limit at which RFQ confinement is use fuI for buffer gas cooling. The relationship of the driven oscillation, commonly referred to as the "RF motion" or the "micromotion". to the macromotion is shown in the graphs of fig. 2.4, which were derived by numerical integration of the equation of motion.
5 10 o 5 10 TIME - RF CYCLES TIME - RF CYCLES
q = 0.91
o 5 10 o 5 10 TIME - RF CYCLES TIME - RF CYCLES
q = 0.93 motion unstable
o 5 10 TIME - RF CYCLES
Figure 2.4: The result of numerical simulation of the equation of motion of a cesium ion in an RFQ field ofvarious strengths q.
13 Thus the motion of a single ion under RFQ confinement in a buffer-gas cooler is a driven sinusoidal oscillation superimposed on a larger, but slower, simple harmonic motion, where the amplitude of the driven oscillation at any point is proportional to the point on the macromotion onto which it superimposed. (Again for weak quadrupole fields the proportionality constant can be shown to be q/2.)
2.2.2 The motion ofa collection ofparticles in an RFQ field
The simple harmonic macromotion of a single particle under RFQ confinement will have random phase and random amplitude. The action diagram for macromotion of a collection of particles will therefore be as in fig. 2.2. If this were the only motion that the particles had then the acceptance diagram of an RFQ ion guide would be just an ellipse of displacement semi-axis equal to the closest approach allowed to the RFQ electrodes and with a momentum sem-axis equal to this of displacement times the ion mass and the angular frequency of the macromotion. However, the micromotion disturbs the action diagram of the macromotion, quite significantly. Being a coherent motion, where aIl the particles are oscillating in antiphase to the RF, they do not themselves add any area to the action diagram. This is because in an action diagram the points representing the particles allline up radially at the phase of the RF for which the action diagram is obtained (fig. 2.5).
x
Figure 2.5: The action diagram ofa group ofparticles oscillating at a corn mon phase.
The overall action diagram will therefore have the same area as without the macromotion but its shape will be distorted, as shown in fig. 2.6.
14 Figure 2.6: Numerical simulation of the action diagram of a collection of particles in RFQ confinement at a q of 0.5. The action diagram for the macromotion alone is shown as a dotted line.
This shows that, at a q of 0.5, the maximum excursion of the particles is 1.5 times that of the macromotion alone. Worse, for ions entering the RFQ confinement the actual acceptance ellipse depends on the RF phase at which they enter. For example, for those arriving at an RF phase of 90 degrees will find that the acceptance ellipse has only about 70% of the displacement allowed for the macromotion alone, while for those entering at an RF phase of 270 degrees has only about 70% of the momentum allowed for the macromotion. Thus to match even the reduced acceptance due to the micromotion would require the incoming beam to have its action diagram modified in phase with the RF. This is impossible for a De decelerator, which is the only practical decelerator for the ion energies of on-line mass separators for radioactive beams. The requirement that the incoming beam have a fixed action diagram renders the acceptance of an RFQ ion guide even less than ideal. In essence, the action diagram for the incoming ions must be such that, no matter what the phase of the RF at the time, the resulting action diagram within the RFQ confinement must never have a displacement that impinges the ions onto the RFQ electrodes. This will typically result in an action diagram for the incoming beam that may have an aspect ratio (momentum to displacement ratio) similar to that of the macromotion of the confinement but that will be as little as 1/3 rd of the size, or about 1/10 of the area of the macromotion ellipse that
15 would fit inside the RFQ electrodes. This severe effect of the rotation of the action diagrams within the RFQ confinement region is clearly seen in the computer simulation of a typical beam profile shown in fig. 2.7.
Figure 2.7: A typical profile of an ion beam under RFQ confinement. The q for this confinement is 0.32.
As a rule of thumb, it therefore seems that an RFQ ion guide will have an action acceptance ellipse that is a right ellipse with the same aspect ratio as that of its macromotion, but with a maximum displacement that is only about 1I3rd of the minimum distance of the electrodes from the beam axis. If this is expressed as ro then the action acceptance diagram ofthe device is as shown in fig. 2.8.
x
Figure 2.8: The expected action acceptance diagram of a typical RFQ ion guide.
16 2.2.3 The acceptance ofan RFQ ion guide in terms ofbeam emittance
Bearn emittance is a plot of the angular divergence of a particle's trajectory with the beam axis against its displacement from that axis. The angular divergence is
x'= Px (2.16) Po where Px is the momentum of the particle excursion and Po is the axial momentum of the particle. The area of the emittance plot of a collection of beam particles is referred to as the "beam emittance". A beam which has an action area equal to the acceptance Sx of an RFQ device therefore has an emittance
(2.17)
Altemately, to accommodate a beam of emittance C;x requires an RFQ ion guide with the parameters
(2.18)
For example, a typical heam from an on-line isotope separator for radioactive nuclei will he of 60 ke V energy and have an emittance of 251t-mm-rnrad. If an ion of amu
2 133 is to be collected then the product wM ro should be about 70 mm2/lls. For an ro of 4 mm the radian frequency of the macro motion should be about 4.3 radians/ils. This would require a relatively powerful RFQ electric field, and the decelerator must be capable of matching it.
2.3 Basic Principles of the Decelerator
The most straightforward method of achieving an action diagram that is a right ellipse of the correct aspect ratio is to have the heam directed along the central axis of an axially symmetric quadrupole electric field that focuses in the transverse (radial) direction. This is, in fact, the electric field that is used in most electromagnetic traps, the geometry ofwhich is shown in fig. 2.9.
17 2 End electrodes ; r = 2( Z2 - z~)
Figure 2.9: The geometry of a typical electric quadrupole field used as an electromagnetic trap.
Its only difference from the field of a Paul trap is that the field is static, and its only difference from that of a Penning trap is that the field is repulsive in the axially direction, thereby enabling deceleration of the ion as it approaches the center of the field from one of the end electrodes. This electric field is derived from the usual expression of the electric potential of such a field;
(2.19) where q>o is the potential of the ring relative to the center of the field. The electric field along the axis is then
(2.20)
Used for deceleration, such a field would operate at negative z, with the particle entering the field through an orifice in the negative z end-electrode and proceeding to the center of the field. The full decelerating potential would then be simply q>o' which is the integralof Ez from z=-zo to z=O. Of course, the remainder of the quadrupole field is unused. The electrodes setting up the field can then be truncated as shown in fig. 2.10.
18 Ground Decelerating potential
Figure 2.10: The electrode geometry of a static electric quadrupole used as a decelerator.
Here the positive z end-electrode is replaced by a cone with the half-angle corresponding to that of the zero potential surface between the ring electrode that end electrode. This half angle is given by
(2.21)
The ring electrode itself has only to be of sufficient aperture to encompass the region occupied by the ions as they approach the entrance into the cone. Taking this to be
2 ara results in the required potential of the ring above the field center being l/>aa • Setting the actual potential of the negative z end-electrode as ground, the potential of the cone then becomes the decelerating potential V and the potential of the ring electrode becomes
2 V(1 + a ). The radial electric field of this electrode configuration is
(2.22)
19 which, for an ion of mass m and charge e, causes simple harmonic motion in the r coordinate of angular frequency
(2.23)
From the previous section of the motion of ions under RFQ confinement, the shape of the transverse action diagram of the decelerating ion beam is roughly matched to the action of the ion beam in the RFQ confinement if this frequency is made to be the same as that of the macro-oscillation of the confinement. Since lPo must be set to approximately the decelerating voltage, the only adjustable parameter is Zo' which must then be adjusted so as to make the angular frequency calculated from (2.23) to be roughly equal to that of the macromotion. Again for a 60 keV beam of mass 133-amu ions this would make it about 50 mm. The entrance aperture of the grounded end electrode of the decelerator, and the exit aperture in the cone leading into the RFQ chamber, must be at least as large as the diameter of the beam, which for the above parameters would be about 3.2 mm. For safety this was set to 4 mm. This is the basic geometry for the design of the decelerator that is the subject of this work.
2.4 Scaling of the Decelerator with the Ion Mass
An on-line mass separator will deliver a variety of ions with the same ion energy and approximately the same beam emittance. A decelerator for a working RFQ beam cooler should therefore accommodate a wide variety of ion masses. This turns out to be relatively easy since the transverse oscillation frequency of an ion in a decelerator with a fixed geometry and a fixed deceleration voltage will, according to (2.23), be in inverse proportion to the square root of the ion mass. From (2.15) the angular frequency of the macromotion in the RFQ region is proportional to q and to the RF. However, to maintain the same q at different ion masses, (2.14) shows that the RF must also be in inverse proportion to the square root of the ion mass. The decelerator and the RFQ will therefore remain matched for an masses, provided the
20 frequency of the RFQ can be changed accordingly. This is relatively easy to do by tuning the RF circuit powering the RFQ electrodes.
2.5 Scaling with The Decelerating Potential
For various reasons, on-line mass separators are sometimes operated at reduced accelerating potential. The decelerator of the RFQ cooler must therefore also operate at reduced potential. The frequency of transverse oscillation of the ions in the decelerator will decrease in proportion to this potential and so will no longer match the frequency of the RFQ macromotion, unless the parameters of the RFQ are changed. However, this is again easy since a reduced macromotion frequency can be obtained by simply increasing the frequency of the RF in inverse proportion to the square root of the deceleration potential. Now the value of q will be in inverse proportion to the potential leaving the macro motion frequency, like the transverse oscillation frequency in the decelerator, in proportion to the square root of the decelerating potential . As for the ion trajectories, and hence the transmission properties of the decelerator, operating at reduced decelerating potentialleaves them unchanged. This is because the transverse momenta and the axial momenta both change in proportion to the square root of the deceleration potential. This is extremely important in testing a prototype decelerator since operating at high potential, in any vacuum that is achievable without heroic effort, results in extraneous currents of electrons and ions produced by interactions of the decelerating beam with the background gas. These charged particles are of no consequence when operating the decelerator as an injector in an RFQ ion guide since they are rapidly filtered from the beam by the RFQ field. However, they can easily smother the ion beam exiting from the decelerator onto a simple CUITent detector. Testing the transmission of a decelerator without the benefit of an actual RFQ ion guide therefore requires operation at retardation potentials considerable less than 60 kV.
21 3. Computer Simulation
Before building the decelerator system, two simulations were made to test its viability. The first simulation took a linear approach to the partic1e dynamics, which meant that the calculations were carried out by simple two-dimensional matrix manipulation of the elliptical action diagram representing the collection of partic1es on an Excel spread sheet. Once it was established that the decelerator worked to first order approximation, a second, higher order simulation using the relaxation method was made using the commercial pro gram Simion 3D [12] developed by Idaho National Engineering and Environmental Laboratory. The advantage of doing the first order calculation before the higher-order simulation is that it saves a considerable amount of time, since the first order calculation is done in seconds compared to the hours required for a higher order simulation. A brief account ofboth methods is given below.
3.1 Spreadsheet Calculation for Decelerator System
The first-order code used to calculate and display beam envelopes through axiperiodic electric systems was developed by R.B. Moore [13]. It uses 3-dimensional matrices to calculate the Twiss parameters of the action ellipse of the partic1es passing through the system, for which accurate knowledge of the axial coordinate of the partic1e collection at particular times is required. A fifth order Runge-Kutta integration [14] of the equations of motion of the partic1e travelling along the axis of the system, with spline interpolation of the axial electric fields, provided the ion trajectories-inc1uding the axial coordinate of the partic1es. The matrix algebra was then performed on the Twiss parameters and the beam envelopes were calculated from these parameters. (For a description of the use of Twiss parameters and linear matrix algebra to determine beam envelopes see Appendix B.) The inputs required for this linear calculation are the axial field and the gradient of the transverse field along the system axis, and maps of these two fields were created in Simion and imported into the Excel spreadsheet, which could accommodate 10 sets of field maps, each one containing a column of De fields and a column of transverse field
22 gradients. The output of the calculation is the envelope of aIl the particles as a function of time and the axial coordinate. The general overall parameters of a decelerator system for an RFQ ion guide, suggested by the theory presented in chap. 2, are shown in fig. 3.1. This requires only four maps: one for the decelerator system, consisting of the hyperboloidal electrodes, and three maps for the three segments of the RFQ ion-guide.
Z =50 mm ° Figure 3.1: Diagram of decelerator system and tirst few segments of an RFQ ion-guide
The map for the decelerator was created for electrodes that extended far enough to pro duce practically a pure quadrupole field, except for effect of the holes of 5 mm diameter that were provided for entrance and exit of the ion beam. The map for the RFQ quadrupole segments was made for hyperbolic electrodes with a closest approach to the beam axis of 3.5 mm. (Both these maps were taken from the work of A. Michailopoulus, a summer students who did preliminary calculations on the concept of injecting ions into RFQ devices.) The emittance ellipse for the beam on entering the decelerator was taken in the simulation to be is 25 7t-mm-mrad. A representative ion mass was taken to be 100 amu and a representative energy was taken to be 60 keV. From (2.23) a decelerator length, Zo' of 50 mm results in an angular frequency of the transverse oscillation of the ions in the decelerator of 4.9 rad/f.ls. The transverse action diagram is then as shown in fig. 3.2, showing that the diameter of the beam in the decelerator is 2.66 mm.
23 6.52 eV- s/mm
x
1.33 mm
Figure 3.2: Perimeter of the action diagram of IOO-amu particles in the decelerator.
After a great deal of manipulation of the incoming beam emittance and the RFQ operating parameters the beam envelopes shown in fig. 3.3 were obtained. (The varying beam envelopes within the RFQ section are due to ions arriving in that section at different RF phases.)
15 , mm ' 62000
i 10 i BEAM PROFILE
1 5 1
o
·5
·10
·15 o 10 20 30 40 50 60 70 80 90 100110120130 mm
Figure 3.3: Simulation resuIts for a decelerator system with the ring electrode
24 For this diagram the RFQ was operated at an RF of 5 MHz and an amplitude of 2800 V between adjacent electrodes. The incoming 60 ke V beam, at 10 mm from the plane of the injection hole had a diameter of 3.3 mm and a divergence of 43 mrad. (The beam was being focused so that it became a right ellipse only at the entrance into the decelerator.) In this arrangement the angular frequency of the macromotion in the RFQ ion guide was 4.81 rad/Jls, compared to 4.81 rad/Jls for the transverse oscillations in the decelerator, showing that the two are roughly matched for proper injection from a decelerator into RFQ confinement. In the figure it can be seen that the diameter of the beam remains fairly constant at near the expected value as it travels through the decelerator system and is focused satisfactorily into the RFQ ion-guide. This focusing was achieved by operating the first RFQ section at -700 VDC relative to the cone electrode, giving the ions an axial kinetic energy of about 750 inside that section. This was reduced to 250 eV in the next section by operating that section at -200 VDC • (This remaining axial energy could be easily removed in subsequent sections of the RFQ). Thus the decelerator was shown, at least to first order, to be capable of decelerating a 60 ke V ion beam of 25 n-mm-mrad into an ion-guide suitable for buffer gas cooling of the ions. Nevertheless, one expects there to be aberrations at the entrance and exit of the decelerator due to the truncation of the hyperboloidal electrodes. But because the beam is very rigid at the entrance and the fields are weak at the exit, these are small effects and the first-order simulation cannot take them into account. A higher-order simulation was therefore required to investigate them and to determine the final decelerator geometry. Because of the small diameter of the beam these simulations were carried out with entrance and exit holes of 4 mm diameter so as to decrease the aberrations that they would cause.
3.2 Higher-Order Simulation
In order to create a perfect quadrupole field, the entrance, ring and cone electrodes of the decelerator would have to extend to infinity. Since this is not possible, the electrodes are truncated to a practical length, which causes aberrations in the quadrupole field. In order to investigate these aberrations, various simulations of the decelerator were made on Simion 3D [12], version 7.0, from which a map of the field
25 inside the decelerator could be obtained. Graphs were then plotted of the gradient of the axial field dEz/dz (or d_V/dz-> against the axial distance 'z' (at radial distance, r = 0), which showed up first order aberrations in the path along which the ions travel. The idea was then to eliminate these aberrations, to the greatest extent possible, by using correcting electrodes placed on the entrance and ring electrodes.
3.2.1 Truncation ofthe Entrance and Ring Electrodes
Figure 3.4 shows the plot of the gradient of the axial field for the quadrupole with extensive electrodes used for the calculations leading to fig. 3.3.
dEz/dz vs z for Perfect Quadrupol,
20
10
0 ~ ~ 1 o 1 110 120 130 140 1SP 1 0 ~ -10 ..... ~ -20 N '1:J ~ -30 '1:J -40
-50
-60 z (mm)
Figure 3.4: Gradient of axial field for a perfeet quadrupole
The entry point of the decelerator for this field map is at z = 100 mm and the exit point is at z = 150 mm. Between these two points, the plot is almost completely flat. The curvature at the ends of the flat line (at z = 110 mm and z = 150 mm) is due to the entrance and exit holes and to the limitations of Simion in being able to calculate fields close to electrode edges. The actual decelerator was to have an entrance electrode with a radius of 45 mm (Figure 3.5), so that it would easily fit inside a standard 4-way cross of inner diameter 100 mm. A simulation of this truncated decelerator produced the first-order aberrations shown in Figure 3.6. As one can see, the line between entrance and exit is no longer fiat
26 and has a slight lift near the entry point, indicating that the field inside the decelerator is leaking out.
" ,, dEz/dz vs z for Truncated Decelerata 1. 10
'.1 0 1. 1 0 110 120 130 140 1 0 ~ -10 E Il .!: -20 ~ -S -30 ';;j UJ l' oc -40
1 .1 Il "" -50 l' :' 1 " -60 1 " " 1 z (mm)
Figure 3.5: Decelerator with truncated Figure 3.6: Gradient of axial field for electrodes. truncated quadrupole.
The first step in correcting the field in the decelerator was to screen the who le apparatus by placing it inside a cylindrical shield of radius 48 mm attached to the cone electrode (as shown in fig. 3.7). Figure 3.8 shows the effect ofthis cylindrical shield.
'1 dEzldz vs z for Shielded Electrod, 1 1 '. '.1 20 1 10 1 (\ 1. ~ 0 1 0 110 120 130 140 15 1 o 1 .. ~ .10 ~ -20 1 \!r--- il , ::l -30 ..-.-.- .. , .., r ". ~ 1 ".'- -40 :" J 1 .' -50 ,1 .\.... 1 .>, -60 1 .... .1 .\ z (mm)
Figure 3.7: Decelerator with Figure 3.8: Gradient of axial field with shielded electrodes. shielded electrodes.
Although it prevents the field from leaking out, it depresses the curve near the entry point by shining a new potential into the decelerator. In order to remove this effect, a short flange was attached to the entrance electrode (fig. 3.9), with an inner radius of 43 mm and an outer radius of 45 mm. The length of the flange was then adjusted in the
27 simulation until the field inside the decelerator was correctly shielded and the plot of the gradient of the axial field showed the fiat line characteristic of the quadrupole field. As shown in fig. 3.10 and 3.11, this occurred with a fiange length of 10 mm-roughly at the mid-point between entrance and ring electrode.
&jIIIIIIIII 1 :rt' dEzldz vs z for Deceleratol 1 ", ,:;. l, ,';.' with Smm Flange l, .~,.I 1 .:.:f 20 1 ...•. _...... r' .r 1. --=- j' 10 j' (\ , ::;- 0 1 ~ -101 0 110 120 130 140 1sP 1 o 1 ~ -20 N ~ -30 r w 1 .., -40 1 ) r -50 r 1 -60 - 1 1 aImam l...... ~~~~~~z_(_mm_)~~ ____J
Figure 3.9: Decelerator with flange on the Figure 3.10: Gradient of axial field for ground electrode. decelerator with 5 mm flange.
dEz/dz vs z for Decelerator witt 10mm Flange
30 20 ,.... 10 Ë 0 f\. E ~ -101 0 110 120 130 140 15~ 1 0
--N -20 "C -30 'i::lw "C -40 ,/ -50 -60 z (mm)
Figure 3.11: Gradient of axial field for decelerator with a 10 mm flange.
3,2,2 Correction ofthe Ring Electrode
The addition of the cylindrical shield increases the size and complexity of the decelerator. A simpler method for shielding the quadrupole field is to attach another
28 flange-similar to that of the entrance electrode-to the ring electrode (fig. 3.12). This had the double effect of preventing the field inside the decelerator near the ring electrode from leaking out and also screening it from external sources. The method for finding the correct length of the ring electrode flange was basically the same as that for finding the correct length of the entrance flange. In the Simion simulation, a flange with an inner radius of 43 mm and outer radius of 45 mm was attached to the ring electrode and its length adjusted until the plot of the gradient of axial field showed a straight line between the entry and exit points of the decelerator. The flange on the entrance electrode also had to be adjusted and was lengthened in order close the gap between the two electrodes and compensate for the removal of the cylindrical shield. Figure 3.12 shows the flange configuration, with an entrance flange of 14.5 mm and a ring electrode one of Il mm. Figure 3.13 shows the result - a near-perfect quadrupole field.
lIjIIIBIIIIIIII !IIiIIIIIIIl. , 1 ,:t/ 1 " ':.' dEzldz vs z for Decelerator with end l, " ,y' .~ cap and ring electrode flanges 1 o~ 1 " ,o.r 1 .,.'~ 20 .,-,_,-,. ,.' .r ~ 1. .r 10 f\ i' ~ 0 1 :J 0 110 120 130 140 15 1 o '1, l-lOl ~ 1 ~ -20 .-,-,--'" .... " r '"'., ~ ~ -30 1 --:~~ 'tI -40
1 .~ 1 '. ~ -50 1" ' .... 1 .:>, -60 1 z (mm) aImimœœ IIIIIIIIIBI'" 'P\.
Figure 3.12: Decelerator with flange on the Figure 3.13: Gradient of axial field for ground and ring electrodes. decelerator with ground and ring flanges.
(In the actual construction of the decelerator, these two flanges were simply bolted to the electrodes of the decelerator and therefore easily adjustable, providing an elegant solution to the problem of correcting the quadrupole field.) The results of a final first-order check of the beam envelope inside the decelerator using the accurate field map of the corrected electrodes is shown in fig. 3.14.
29 Figure 3.14: First-order simulation of the beam envolope of a 25 1t-mm-mrad ion beam entering the decelerator at 60 keV and exiting at 100 ev.
For this simulation the beam entered the decelerator at 60,000 eV and exited at 100 eV. The figure shows that indeed the beam can be expected to preserve its diameter throughout the deceleration.
3.2.3 Simulation ofDecelerator Operation
Once the design parameters of the decelerator had been set, simulations of the trajectories of the ions passing through it at various decelerating potentials were made using the higher order calculation to check for the possible effect of aberrations. Here the main point of interest was the transmission rate of the beam i. e., the fraction of the beam that survives deceleration as a function of deceleration voltage. For this test, the beam was taken to be 60 keV 133Cs+ ions of emittance 25 7t-mm-mrad and diameter 4 mm (to match the entrance hole of the decelerator). The results are shown in fig. 3.15. The entrance emittance diagram was created by a sample of 1000 ions randomly distributed within the circle of the beam cross-section and randomly distributed within the limits of the divergence corresponding to the beam emittance.
30 1.2 ,--_._--
.... 0 t5 0.8 CO LL c 0 0.6 'inen 'E Emittances en <> 2mrad c 0.4 o 25mrad ~ " 75mrad 0.2
o 1 1 59 59.5 60 Decelerating Potential
Figure 3.15: Plot of simulated transmission factors for a 60keV cesium ion beam of various emittances.
The caesium ion beam available for the test was from a precise source designed to give an emittance of about 2 n-mm-mrad at 60 keV. This was incapable of providing a beam as broad as 25 n-mm-mrad, and so a transmission simulation was carried out for such a beam as weIl, so as to pro vide a better comparison with the experiments. In addition, a transmission test was carried out for the largest possible beam that could be imaged for an on-line isotope separator, i.e., 75 n-mm-mrad. The results of these simulations are also shown in fig. 3.15. These results do show that the transmission factors for a 25 n-mm-mrad beam would not be appreciably different from those for a 2 n-mm-mrad beam but that for a 75 n-mm-mrad would be considerably lower. This confirms that the decelerator would be capable of handling an emimance up to that for which it was designed, but for not much more. Figure 3.15 also shows that the effects of aberrations are expected to show up in the last 400 eV of deceleration from 60 keV. However, there is a curious dip that shows up at about 200 eV ofresidual energy for the 25 n-mm-mrad beam and that is even more pronounced for 2 n-mm-mrad beam. A possible explanation for this is a focusing effect
31 near the exit from the decelerator that can be seen in the first-order simulations. When carried out for the 2 7t-mm-mrad beam, the graph of fig. 3.14 becomes that shown in fig. 3.16. (The size of the beam at entrance to the decelerator was taken to be the full 4 mm ofthe aperture, causing the divergence at the exit to be slightly greater.) ,
Figure 3.16: First-order simulation of the beam envelope of a 2 x-mm-mrad ion beam ente ring the decelerator at 60 keV and exiting at 100 eV.
This shows a dramatic cross-over of the beam just before its exit from the decelerator. With a slightly greater deceleration potential, so that the ions exit with only 70 eV, this pattern becomes that shown on the left in fig. 3.17, where the ions are intercepted by the cone electrode. However, at an even higher deceleration potential, and hence lower residual energy, the ions return to the axis before exiting (right of fig. 3.17).
- -,
Figure 3.17: Simulation of the beam envolope of a 2 x-mm-mrad beam 60 keV ion beam entering the decelerator at 60 keV and exiting at 70 ev (left) and 50 eV (right).
32 Thus the first-order calculations show that a focusing just before the exit from the decelerator could cause a dip in the transmission at a deceleration potential that leaves about 70 eV in the residual energy of the ions. That the higher-order calculations show the dip to occur at about 200 eV residual energy could be due to aberrations in the system. In any case, it appears that the decelerator would work at full transmission to decelerations such that only 400 eV of axial kinetic energy remain in the ions. With modifications to the extraction field by electrodes inside the cone electrode it is possible that it could be made to produce even lower residual energies. It was therefore decided to design and build a decelerator structure based on these considerations.
33 4. Apparatus. Experiments and Results
4.1 The Decelerator
Although it was realized that, for reasons to be outlined later, the decelerator could not be tested at 60 kV, it was nevertheless designed for operation at that potential. Consequently, it has a distance from the ground electrode entrance to the field center, i.e., zo' of 50 mm, and spacings and materials that would allow that voltage. The decelerator, shown in its enclosure in fig. 4.1, was designed and machined out of stainless steel at Mc Gill University (detailed technical drawings can be seen in Appendix A), according to the specifications of the computer simulation described in chapter 3.
Faraday cage ground shield
Decelerator mounting 4·waycross insulator
Ceramic HV insulator Bellows
Beamshield Ring potential Faraday cup current Cone potential
Decelerator assembly Faraday cup & connector mounting plate
Figure 4.1: The assembled decelerator in its vacuum chamber (a 4-way cross) and associated connections. The vacuum cham ber is at the cage potentiaI.
It consists of three separate electrodes: a hyperboloidal entrance-electrode of radius 45 mm with a 4 mm hole in its front; a ring electrode that is a hyperboloid with inner radius 12 mm and outer radius 45 mm; and a conical end-electrode. The ground and ring electrodes were machined on a CNC milling machine with a baIl cutter, and the cone was made using a simple lathe. The electrodes were then polished using fine aluminium
34 oxide grit, finished with a buffing compound and finaHy cleaned with isopropanol. After cleaning, a short flange 14.5 mm in length was attached to the entrance electrode and another flange Il mm in length was attached to the ring electrode in order to correct the field inside the decelerator. The potentials were supplied to the ring electrode and the cone electrode by two stiff stainless steel wires that were wired to vacuum-sealed BNe connectors in the same plate that held the Faraday cup. The entrance electrode was held at ground by being mounted on a tube that connected it to the left flange of the vacuum bream. The cone electrode, meanwhile, was mounted on a plastic disk (as shown in Figure 4.1) and the ring electrode attached to it by three plastic screws that he Id them apart at the correct distance. The first priority, once the decelerator had been assembled, was to determine its breakdown potential. This was done by raising the potential of the system slowly until a spark occurred, which could be detected by a brief and sudden faH in the supply voltage. The potential was he Id at this value in order to condition the electrodes and then raised again; the break down voltage was reached when persistent sparking occurred. At a pressure of 1 mPa, maintained by one mechanical pump and one turbo pump, it was measured to be 35 kV. While this was considerably less than that for which the basic geometry of the decelerator was set, the discharge occurred between the thin shields of the ground and ring electrode, which had a separation of only about 3 mm. On the basis of the work of Gianfrancesco [8] on high-voltage dc discharges between small radius of curvature electrodes across smaH gaps, this was expected. To make the decelerator a truly 60 kV design would require flanges that had more rounded edges, but for the tests to be done here it was more important to have a uniform electrical field gradient along the decelerator axis than it was to have a 60 kV breakdown potential. That design and implementation could be left to later work with beams to be injected into an actual RFQ ion guide. Photographs of the ground electrode are shown in fig. 4.2. Photographs of the ring electrode are shown in fig. 4.3. Photographs of the cone and ring assembly are shown in fig. 4.4 and a photograph of the ring-cone-mount assembly is shown in fig, 4.5.
35 Figure 4.2: The decelerator ground electrode. The view on the left is of the high electric field face. The edge view on the right shows the shield ring.
Figure 4.3: The ring electrode, side view and front view.
36 Figure 4.4: The front face of the co ne electrode (Ieft). A view on the aperture of the cone electrode as seen through the aperture of the ring electrode wh en the cone electrode is mounted to it.
Figure 4.5: Assembly of the ring, cone and insulated mount that supports the assembly in the vacuum cross.
37 Figure 4.6 presents a view of the ground electrode looking backwards along the beam towards the ion source. It shows the relatively tight geometry of the ground electrode fitted into the vacuum cross.
Figure 4.6: Photograph showing the position of the decelerator ground electrode relative to the vacuum cross that is to be raised to potentials of 60 kV relative to it.
4.2 The Experimental Set-Up
The set up to test the decelerator system was similar to that used by Kellerbauer [6] in his pioneering work on cooling ISOLDE-type beams. A schematic of the set-up as used in the present work is shown in fig, 4.7.
Ion source ground cage
Ion source Faraday cage
FARADAY CAGE HV vacuum break Decelerator chamber, vacuum pumps and associated electronics Deflection plates
Ion source-+---*+t=--~=~~~3==-=-=~~~~~------t--
Decelerator
~ 500 mm ----+1 Outer ground cage ~
Figure 4.7: Schematic of the test set-up.
38 The set-up inc1udes the ion source and the faraday cage used by Kellerbauer, with the associated isolation transforrners to power them. The princip le difference in the two set-ups is that the one of Kellerbauer was designed around investigating the properties of the decelerated beam as it entered an RFQ confinement region within the Faraday cage, while the present set-up was designed around investigating the properties of the decelerator designed in the present work to inject beams into a possible high-field RFQ ion guide. A very crucial component of any decelerator system is a high-voltage ceramic vacuum break, as shown in Figure 4.1. It is crucial because it is much more difficult to prevent electrical breakdown along insulating surfaces in a vacuum than it is to prevent discharges across gaps between electrodes. This insulator was therefore a commercial item from CVC vacuum technologies, made of high-purity components and specifically designed for allowing potential differences of up to 60 kV between its vacuum flanges. One of these flanges, the one to the left in the diagram, was connected to the beam line from the ion source, and the other to the vacuum system in the Faraday cage that contained the decelerator. This Faraday cage had been designed and built by Kellerbauer, and tested to 65 kV before electrical breakdown to ground, even though there is only a gap of 80 mm between the Faraday cage and its outer ground shield. To shield the incoming ion beam from the high fields in the ceramic insulator the ground electrode of the decelerator was attached to the beam line by a 190-mm long stainless steel tube, shown in fig. 4.1. The ion-source used to test the decelerator was a zeolite surface-ionisation type, developed by Dezfuli [15], as used by Kellerbauer [6] and shown in fig. 4.8. It consists of an lnconel tube of caesium zeolite powder [16], which is heated by passing a high CUITent (approximately 60 amperes) through it to liberate caesium atoms from the zeolite. Because of its high work function, platinum wire is inserted into the front end of the tube to ionize the caesium atoms as they pass through it. (More details conceming the ion source are presented in Appendix C.) An electrode with a negative DC potential relative to the source and physically near it extracts the ions and a ground electrode shortly after that accelerates them to the kinetic energy equal to the potential of the source. This ion source was capable of
39 producing a beam of caesium ions (133es+) at energy 60ke V at a CUITent of up to about one nanoampere. A photograph of the ion source installation is shown in fig. 4.9.
Platinum wire Caesium Extraction zeolite 1electrode 1
Current Current out in Ground eleclrode 1 ~ 10 mm---+I
Figure 4.8: Schematic of the ion source.
Figure 4.9: Photograph of the ion source installation. (The Faraday cage and the outer ground cage shield of the source are removed.)
40 Since the properties of the beam extracted from this source could not be predicted adequately beforehand, an einsellens was provided in the beam line, as shown in fig. 4.7, in case the beam had to be focused onto the entrance of the decelerator. This lens had an inner diameter of 42 mm and was 50 mm in length overall. The main purpose of the deflectors, also indicated in fig. 4.7, was to be able to direct the beam onto the entrance orifice of the decelerator. (This is why they were mounted after the einsel lens so as avoid having the lens interfere with their operation.) These plates were 40 mm (axially) by 50 mm (transversely) and mounted at a transverse separation of 20 mm between partners. The pairs were mounted orthogonally in x-z and y-z planes with an axial separation between pairs of 10 mm. However, before the decelerator tests, these deflectors were used to determine the actual properties of the beam by using them to sweep the beam across a 2-mm diameter orifice placed at a point in the beam line near where the entrance to the Faraday cage would be (at 800 mm from the source). The ion CUITent passing through the orifice was measured by having it enter a Faraday cup of diameter and depth 15 mm. This cup was connected through a vacuum electrical feed-through to a sensitive picoarnmeter. A plot of the CUITent passing though this orifice as a function of transverse deflection of the beam at the orifice is shown in fig. 4.9. This showed that, at the entrance to the decelerator the beam would be, effectively, about 15 mm in diameter and would have a uniform distribution over a 4-mm diameter at its center.
8 ~------7 <' 6 S 5 c:: -Q) .... 4 :::::1 Ü 3 c:: 0 2 1 0 n 2 4 6 8 10 12 14 Distance tram axis (mm)
Figure 4.9: Profile of the beam as determined by measurement of ion currents traversing a 2-mm diameter orifice placed in the beam Hne 900 mm from the source.
41 This meant that the einsellens would not have to be used for the decelerator test, thereby considerably easing the process of setting up the ion beam for the tests. In the actual tests of the decelerator this same Faraday cup was placed 10 mm behind the exit orifice of the cone electrode. (The sensitive picoammeter was then mounted inside the Faraday cage since the Faraday cup was now at the Faraday cage potential. ) As pointed out earlier, although the decelerator system was designed for up to 60 k V operation, it was not expected that anywhere near that potential could be used for the test of this work. That is because high velocity ions cause a considerable amount of ionization of the residual gas in the decelerator region. The electrons liberated by this ionization are then pulled up onto the decelerating ring electrode, and to a lesser extent the cone, producing themselves ev en more ionization. This avalanche effect causes a considerable amount of electrical CUITent in the Faraday cup, of both electrons sucked into it (negative cUITent) and positive ions that can fall into it or be repelled from it (positive or negative cUITent). To render this effect as small as possible the turbomolecular pump for the decelerator region was mounted in the Faraday cage directly on the 4-way cross containing the decelerator. (Because of the high electric fields that would occur in a vacuum line leading to a mechanical pump at ground, the mechanical pump for the turbopump was also located in the Faraday cage.) Even with the most effective pumping system that could be mounted in the Faraday cage, the residual pressure in the decelerator could not be reduced to below 0.5 mPa (5 xlO-6 mbar). The ionization CUITent produced by the beam was then such that ion beam energies of above 12 kV were not usable. (In the actual usage of a decelerator for injection into an RFQ ion guide the ionization current produced by the beam is of little consequence. This is because it remains weIl below what can be provided by a high-voltage supply to the decelerator, i.e., millamperes, and the ions and electrons produced have no effect on the RFQ confinement of the desired ions. That is provided the buffer gas is ultra-pure helium so that the bulk of the ions produced by the beam are helium ions that are not contained by the RFQ field.) In any case, because of the scaling roles outlined in chapter 2, the results obtained for a 12 ke V ion beam are directly applicable to a 60 ke V ion beam. The transmission
42 tests of the decelerator were therefore undertaken at this beam energy. To achieve a reasonably long life-time for the ion source, it was set to provide ion currents into the decelerator in the range of 20 to 50 pA, corresponding to ion currents of about 200 to 500 pA being extracted from the source.
4.3 Measurements and ResuUs
The first successful scan of the transmission of the decelerator as a function of deceleration potential is shown in fig. 4.10. 40 r------,
« 30 c..
ë 20 @ ..... :::J o c.. 10 :::Jo .· '1 >. as 0 "0 1 1 as~ LL -10
-20 o 2 4 6 8 10 12 14 Potential on cage - kV Figure 4.10: Transmission of a 12 keV beam of caesium ions as a function of deceleration potential.
The scan shows a detector current that lowers as expected with increased ion deceleration due to the production of electrons by residual gas ionization, an effect that increased with reduced velocity of the ions due to the larger number of collisions the ions make with the gas molecules. The effect here is seen to be about 60% of the ion current. (There is ionization when there is no deceleration but the electrons are not pulled up into the Faraday cup.) The transmission current, taking into account the ionization current, is indeed se en to be flat up to when the deceleration potential is near the ion energy and is then seen to fall dramatically.
43 Expanding the plot to show the region near the ion beam initial energy gives the plot shown in fig, 4.11.
1.2 j _1.0 "'C Q) • .~ • CO • E 0.8 ~ • 0 • • c • ---c 0.6 Q) ~ ~ j ::J (.) 1 Cl. 0.4 ~ ::J (.) \ ~ \ "'C 0.2 ~ \ ~ \ \ 0 11.7 11.8 11.9 12 Cage potential
Figure 4.11: Transmission factor (normalized transmission current) for a 12 keV beam of caesium ions as a function of deceleration potential, emphasizing the region where the residual energy of the ions is small. The curve on the left is the experimental data. The curve on the right is the result of computer simulations.
This plot includes a computer simulation similar to that for the 60 ke V ion beams described in chapter 3. Also, in this plot the trend of the data has been extended to include the region of negative CUITent that was not available experimentally. The experimental data shows the same trend as the computer model, except for two things; the offset of the potential for the data compared to the computer simulations, and absence of the bump that appears in the computer simulations in the middle of the down-slope at the end of the transmission curve.
44 The absence of the bump in the experimental transmission curve could possibly be waved off as being due to difficulties caused by the negative electron current in viewing that region of the transmission curve. Or it could be surmised as being due to aberrations near the extraction cone that were not adequately simulated because of limitations of the field map or inadequate handling of the ion trajectories in that region. However, the offset in the potential cannot be so easily disposed of. For this reason tests were carried out for a variety of deceleration conditions. This was easily achieved by varying the ring potential from the supposed ideal of -400 V (for 12 kV deceleration potential). The experimental results for various ring voltages are shown in fig. 4.12
Ring Voltage <> 200V c 300V '0 -Q) 400V .!::! " ct! x 500V E 0.5 .... 0 600V o c: ---c: ....~ 0 :J (,) a. :J \ \ \ \ \ (,) \ \\ \ \ ~ -0.5 \ \\ \ \ '0 ~ \ \\ \ \ If \ \\ \ \ "-"-' , \. -1 +------_r------.------~------_r=-~~~------~ 11.7 11.75 11.8 11.85 11.9 11.95 12
Cage potential
Figure 4.12: Experimental transmission factors for a 12 keV beam of caesium ions as a function of deceleration potential, for various ring potentials relative to that deceleration potential.
During each set of measurements, the potential of the cone electrode varied with the cage voltage, V cage, while the ring electrode floated at a fixed voltage above this, V R, so that its absolute potential was Vcage+VR. The transmission factor was plotted against the cage voltage. Since the main point of interest was the behaviour of the beam near the
45 deceleration voltage, the transmission factor was measured at every 1 kV on the cage until the last 500 V before deceleration (not shown in the plot), after which it was measured at intervals of only 3 V. This was repeated for five different values of VR: 200 V, 300 V, 400 V, 500 V and 600 V. The trend lines for the different ring potentials follow what is expected. The lower ring potentials require that the cone potential be raised so as to achieve the same decelerating effect. However, the raising of the cone potential is expected to be less than the lowering of the ring potential since the ions, in the sensitive region near the decelerator exit, are much more influenced by the cone than the ring. Transmission curves simulated for the fields that would result from the different ring potentials are shown in fig. 4.13.
1.2 -.,------
1 - 10.-
+-'o (J ~ 0.8- oC Potential on ring en 0.6- relative to cone en • 200V E • 300V ~ 0.4- o 400V ~ • 500V 1- x 600V 0.2-
O-~----.----~-~~-~'---~~---~ 11.8 11.9 12 Deceleration Potential (kV)
Figure 4.13: Transmission factors for a 12 keV beam of caesium ions as a function of deceleration potential, for various ring potentials relative to that deceleration potential as simulated by the methods outlined in chapter 3.
These curves show a trend similar to those for the experiments except that the variation of 400 V in the ring potential here causes about an 80 V swing of the curves compared to less than a 50 volt swing in the actual experiments.
46 This stillleaves the offset of the potentials in the experimental results from those of the simulations unexplained. This strongly suggests that there was an error in the measurements of the ion source potential and the cage potential. These measurements were obtained by high-voltage leads from the separate cages bought to a common measuring box where the same high-voltage potentiometer was used to bring the voltages down to a level where they could be measured by the same accurate 5-digit voltmeter. (The difference in potential of the ion source and the cage could not be measured directly by connecting the voltmeter across the two high-voltage leads because high-voltage power supplies present practically infinite impedance to negative currents. And this does not ev en take into account the possible destruction of the meter by accidentally having too high a potential difference. ) This discrepancy between the experimental and simulated transmission curves leaves an uncertainty in the energies of the ion beam exiting the decelerator when operating at full transmission. The most conservative approach is to take the experimental measurements as the most certain. From fig. 4.12 the minimum kinetic energy the ions can have after deceleration from 12 keV would then be about 250 eV, i.e., about 2% of the original. Translated to a beam of60 keV, this would mean entrance at about 1250 eV. This is easily within the range that can be handled by further electric deceleration within the RFQ confinement region itself. Although the experiments were carried out with a beam of only 2 7t-mm-mrad emittance the results of the simulations described in chapter 3 give confidence that the same results could be obtained with a beam ofup to 25 7t-mm-mrad emittance.
47 5. SummaIT and Conclusion
The results obtained from this experiment were highly encouraging in that they showed that a De axially symmetric quadrupole decelerator is capable of decelerating a 12 keV beam of ions with an original emittance of 25 1t-mm-rnrad to 250 eV or less for delivery into an RFQ ion guide. By the scaling laws of deceleration the same decelerator could be used to decelerate a 60 keV beam ofthe same emittance to 1250 eV for injection into an RFQ ion guide. This would accommodate most on-line isotope separator beams at nuc1ear research facilities. The next stage for the implementation of this decelerator for injecting into an RFQ ion guide for beam cooling would be to modify the shield flanges of the ring and ground electrodes of the decelerator so that a potential of up to 60 kV could be maintained between them. However, before actually implementing the decelerator in this fashion the inconsistence between the experimental and simulated transmission as a function of decelerating potential should be investigated. Although the experimental results appear to be more believable that those of the simulation, it could still be possible that there was an experimental error. If there was, and the simulations were, in fact, the more accurate of the two, the energy of the ions delivered to the RFQ ion guide would be even less than determined experimentally. Simulations using different software are suggested. AIso, it is suggested that the design of the decelerator be extended to handle beams of up to 75 1t-mm-mrad, about the large st that is expected for any reasonable on line isotope separator. This would require more careful consideration of the exit region of the decelerator, since the aperture in the extraction cone would no doubt have to be
enlarged relative to the Zo of the decelerator. In particular, an electrode may have to be installed just inside the cone so as to reform the electric field in the extraction region back to more like that of a pure quadrupole. In testing any such design it would also seem advisable to enable the incoming beam to be directed into the decelerator at the large divergences that would occur at such large beam emittance.
48 References
[1] J. Freeman et al., "The retardation of ion beam to very low energies in an Implantation Accelerator", Nuclear. Instruments and Methods 135 (1976) 1. [2] E. Yates, Proc. Royal Soc. A148 (1938) 143; see also M. Oliphant et al., Proc. Royal Soc. A146 (1934) 922. [3] K. Foo and R. Lawson, "Deceleration and Ion Bearn Optics in the regime of 10- 200eV", J. Vac. Sci. Technol. A9(2) (1991) 312-316. [4] R. Moore and G. Rouleau and ISOLDE Collaboration, "In-flight Capture of an Ion Bearn in a Paut Trap", J. Mod. Optics, Vol. 39, No. 2 (1992), 361-371. [5] A. M. Ghalambor-Dezfuli, Injection, Cooling, and Extraction ofIons from a very Large Paul Trap, Ph.D. thesis, McGill University, 1996. [6] A. Kellerbauer, Production of a cooled ion-beam by manipulation of 60-ke V ions into a radio-frequency quadrupole ion guide. Ph.D. thesis, McGill University, 1999. [7] T. Kim, Buffer gas cooling of ions in a radio frequency quadrupole ion guide. Ph.D. thesis, McGill University, 1997. [8] O. Gianfrancesco, Design Principles ofa High Field RFQ Deviee for Ion Cooling and Confinement. M.Sc. thesis, Mc Gill University, 1997. [9] W. Paul and H. Steinwedel, Z. Naturforsch, A, 8 (1953) 448. [10] P. H. Dawson, Quadrupole Mass Spectrometry and Its Applications, Amsterdam, Elsevier Scientific Publishing Company, 1976. [11] R. March and R. Hughes, Quadrupole Storage Mass Spectrometry, New York, Wiley, 1989. [12] D. Dahl, SIMION 3D 7.0 User 's Manual, USA, Lockheed Martin Idaho Technologies, 2000. [13] R. B. Moore, RFQ Beam Cooler Injection Simulation, Lecture Notes, McGill University,2003. [14] W. Press et al. Numerical Recipes in C: The Art of Scientific Computing, Cambridge, Cambridge University Press, 1992.
49 [15] A. M. Ghalambor-Dezfuli et al., 'A compact 65keV stable ion gun for radioactive beam experiments,' Nuclear Instruments and Methods in Physics Research A 368, 611-616, 1996. [16] H. Robson, Verified Syntheses ofZeolitic Materials, Amsterdam, Elsevier, 2001. [17] D. Breck, Zeolite Molecular Sieves-Structure, Chemistry and Use, USA, Wiley,
50 Appendix A: Technical Drawings
Machine surface using tabled values of axial and radial displacements of cutting tool
M3 x 0.5 tap two holes directly opposite
Material - 304 Stainless steel
GROUND ELECTRODE Round corners - 0.5 mm radius
M4tap 3 holes equally spaced 58 B.C. (on
1 43.5 47 70 JJ
CONE ELECTRODE
Al M2.5 tap 5 hales - O~ 86°501,173°401,266°151353°51 5 deep M4 drill 3 hales equally spaced
i
1 ------1----__ 94 93.4 86 (ref.) 1 52 58 78 (ref.) 1 86
1
1 i~5.260 ~26 RING ELECTRODE
1+--,1 --55
A2 Appendix B : Matrix opties for low energy ion beamsl
BI: Introduction
Simulating the behaviour of a beam of particles entering a Radiofrequency Quadrupole (RFQ) beam cooler can be a frustrating experience. This is not only because of the large number of parameters involved in the system design but also because of the large sample of particles required to coyer the range of behaviour that they will experience. As in any beam system, there should be enough particles to represent the action diagram in one transverse displacement-momentum plane, for at least the initial trial designs. More detailed investigation of a trial design might even require an investigation of the particle behaviour in the other two displacement-momentum planes. In the case of beam injection into an RFQ system this complexity is compounded by the fact that particles entering the geometry at different RF phases will experience very different fields. The approach taken here, an approach that is commonly taken in high-energy particle transport design, is to start with a linear approximation to the particle dynamics. This allows the simulation of particle dynamics to be carried out by simple two dimensional matrix manipulation of not only the action points of individual particles but also of elliptical action diagrams representing whole collections of particles. With modem computers this procedure can lead to system evaluation in seconds compared to hours for a higher order calculation.
B2 Action Diagrams
The power of action diagrams derives from the fact that they are projections of the momentum-displacement coordinate pairs for each degree of freedom of the particle motions. Thus to first order, where the motion in each degree of freedom is independent of the motion in the others, the preservation of the particle density in phase space, and hence the preservation of the phase space volume of a collection (Liouville's Theorem), leads to preservation of the density and the area of the action diagrams. (This property of
1 The material presented here is adapted from notes provided by R.B. Moore.
BI particle collections in conservative systems is sometimes expressed as "the incompressibility ofpartic1e collections in phase space".) Since the electric field of a quadrupole is linearly related to partic1e displacement in that field, the forces have always first-order dependence on displacement and so the action diagrams of a partic1e collection retain their local density and overall area. The simplicity of the action diagram is most apparent for the case of a partic1e collection all undergoing simple harmonic motion in the same force field and so at the same oscillating frequency, but at different amplitudes and phases. Each partic1e then follows an elliptical trajectory of the same ellipticity as the rest and at the same orbital frequency, thereby retaining its relative position to all the rest (fig. BI). The perimeter of the diagram is just the trajectory of the most energetic partic1e in the collection.
/-:~: .... ~~, r r comA comA max
1 ?li&~H~'~~), x x
Fig. BI The action diagram for particles undergoing simple harmonie motion in a force field. The diagram on the left is for a single particle. The diagram on the right is for a collection of many particles ail in the same force field and therefore having the same oscillation frequency but with different amplitudes and phases.
The significance of the action diagram is enhanced when the collection of partic1es is in thermodynamic equilibrium. The density of partic1es at a particular point is phase space is then given by the expression
d 6n E --=n e kT (BI) dS 0
B2 where no is the phase space density at the center of the distribution, E is the energy of a particle at the particular point in phase space under consideration, k is Boltzman' s constant and T is the temperature of the collection. For simple harmonic motion, where the energy is divided between the potential energy of displacement and the kinetic energy of the momentum, the projection of this density distribution into an action diagram results in a gaussian density distribution within that diagram given by
(B2)
where nAo is the density distribution at the center of the action diagram and the standard deviations of the distribution are
CI =~JkT (B3) x ru m
The action diagram for the distribution will then be elliptically symmetrical in that points anywhere on the ellipse representing a particular amplitude of oscillation will have a uniform density.
B3: Matrix Algebra ofAction Diagrams
The transformation of a point in an action diagram as the diagram evolves under linear transformations is described by the 2-dimensional matrix M
(B4)
A consequence of the linearity of the transformation is that the determinant of this matrix is unity. AIso, the elements of the inverse transfer matrix
(B5)
can be easily determined from the simple requirement
B3 to be (B6)
where the elements ml!' m12 , m21 and m22 are those of the forward transform. In the case of an ion in an axisymmetric quadrupole field with axis of symmetry along the z axis the electric potential has the form
(B7)
This potential has a radial electric field
a E =--r (B8) r 2 which results, of course, in a radial oscillation of frequency
m= lJ2;rea (B9) where e is the charge and m is the mass of the ion. The radial displacement and radial momentum are then
r=Asin(ca+lfJ) , Pr=mmAcos(mt+lfJ) (BIO)
In the case of the quadrupole being negative, resulting in a radial field that is away from the z axis, the motion becomes
r= A sinh(ca + lfJ) , Pr = mmAcosh(mt+ lfJ) (B11)
where now the magnitude of the quadrupole field is used in evaluating m. These solutions result in the transformation of the displacement-momentum coordinates during an interval t being expressed by the matrices
1 _f cos(ca) _1_ sin(ca)l M M_ = f cosh(ca) mm sinh(mt)lJ (BI2) + mm J -l -mmsin(ca) cos(mt) lmmsinh(ca) cosh(ca) For these matrices the determinants are easily se en to be unity and the inverse matrices, describing a transformation in negative t, are easily seen to follow eqn. (6).
B4 For a field that is not purely quadrupolar the linear approach can be used by taking small steps through the system. In the linear approach the field over a small step can be taken as quadrupolar. This means that ifthere is an axial field gradient in this step then there is also a transverse field gradient
(B13)
The elemental transformation matrice for a step that takes time dt is then
_ f cos( (àft) _t_sin( (àft) l dM (Bt4) + mm -l-mmsin(mdt) cos((àft) J when the axial field gradient is positive and
dM_ =f cosh((àft) ~m Sinh((àft)lJ (Bt5) lmmsinh((àft) cosh((àft) when the axial field gradient is positive, and where
(Bt6)
Elliptical action diagrams are of particular significance in linear transformations since they remain ellipses of the same area. Linear transformations of ellipses are most conveniently expressed in terms of the Twiss parameters A, B, C and e, by which the equation for an ellipse, in terms of the coordinates of its points relative to its center, has the general form
(B17)
For an action diagram it has the specifie form
(Bt8)
The parameter e determines the overall size of the ellipse. Specifically, it is the product of the semi-axes, or Area of ellipse /1(; Thus, defining the action as the area of the action ellipse, it is Action /1(; . The parameters Band C express the ellipticity of the
B5 ellipse. The parameter A expresses the inclination of the ellipse axis with the axis of the coordinate system and is zero when the ellipse is a right ellipse. Because it takes only 3 parameters to specify any ellipse there is a necessary relationship between the Twiss parameters.Itis
(BI9)
The relationship of the Twiss parameters to the cardinal points of an ellipse is shown in fig. B2.
2 Cx +2Axy + Bl= E BC _A 2 = 1
1 JI
-2 x
A =-0.5 B = 0.8 C = 1.5625 E=4
~--~---.!
f----!Œ---I
-3
The semiaxes and the angle of the ellipse axes to the coordinate axis are for diagrams in which the x unit and the y unit have the same geometricallength.
Fig. B2: The relationship of the Twiss parameters of an ellipse to the various cardinal points of the ellipse and its orientation with the coordinate axes.
As an example, the Twiss parameters for the action ellipse of simple harmonie motion are
B6 1 2 A=O, B= -, C=mw, E= mWXmax (17) mW
Because a linear transformation preserves the area of an ellipse, only the parameters A, Band C are transformed. The 3 x 3 matrix that describes this transformation, and its inverse, for an action point transformation described by the 2 x 2 matrix of (4) are
-2mllm12 m llm22 + m 12 m 21 (BI8) -2m21 m 22
2m12~2
m llm 22 + m 12 m 21
2mll~1 (BI9)
If the electric field is static then the envelope of a beam traveling along the axis of the system can be obtained by determining the step by step transformation of the B action parameter and using the relationship shown in fig. B2:
rmax = Jëjj (B20)
B4: Action Diagrams of RFQ Confinement The case of the oscillating quadrupole field that provided radial confinement is
more complicated. Here the frequency W is itself varying and so, strictly speaking, the transformation matrices (14,15) are only valid for infinitesimal time intervals. The
transfer matrix for a finite time t is then the product sum
(B21)
In principle this product sum should be evaluated over infinitesimal time steps, or an infinite set of dM. In practice, for quadrupole field strengths that are oscillating sinusoidally, time steps of 1 degree of oscillation will give accuracies of several parts in 5 3 10 and steps of 5 degrees will give accuracies of about 1 part in 10 .
B7 Thus, even in the linear approach sufficient accuracy is achieved only with many steps per field oscillation. To circumvent this problem the combination of many small steps into larger steps can be carried out beforehand and the results compiled for interpolation during the actual beam profile ca1culations. This is facilitated by describing the sinusoidal variation of the field in terms of the angle of the variation rather than the time;
-=dEr (dEr)- sm. (0 RF ) (B22) dr dr max whereupon the equation of motion becomes
2 -2-d r = (e 2 (dEr)-- sm. (0RF )J r (B23) dORF mCùRF dr max
Using the dimensionless Mathieu parameter
(B24)
(B23) takes the simple form
(B25)
The elemental transfer matrix for a small step in 0RF is then
r 1 1 COSh{isin( 8RF ) Jd8 RF) 1 (B26) dM+=1
1 1 1 l(!Sin(8RF )Y Sinh{!Sin(8RF) yd8 RF ) for 0 ~ °RF ~ 1C, and
B8 r 1 1 COS{-i Sin( 0RF ) JdO RF ) 1 (B27 dM_=1
1 l .!. 1 li-!sin( °RF) Jsin( -~ sin(ORF) JdO RF ) COS«-~Sin(ORF) JdO RF ) for n '5, ()RF '5, 2n These elemental transfer matrices can then be combined to yield the transfer matrices for larger steps. This allows an evaluation of the matrix elements for an 18° step through the RFQ confinement regions by determining the average q during that step and using interpolation to find the ()RF matrix elements for that step. The matrix elements for the transformation according to the equation of motion in time can then be obtained from
mIl = mIle ; m22 = m2'2-e
ml2 (B28) ml2 = e; ~ 1 = ( mmRF )m21e mmRF
B5: Combining the Axial Gradient and the RFQ Transformations The axial field transformation (14,15) and the transverse field transformation derived from (26,27) both include the effect of the axial drift during a step. In combining the se transformations one then has to unwind the effect of one of the se drifts. For example, if the axial field transformation is taken to be the first then the simple drift part of its transformation must be undone before the transverse matrix is applied, as in (29).
ml2axiall (B29) m22axiaJ
Axial Energy Effects of the RFQ Fringe Field Region One of the concems when injecting a beam of ions into an RFQ confinement region is the effect of the fringing field. This is because of the unavoidable axial component of this field, shown schematically in fig. B3.
B9 Fig. 6 A schematic view of the field Iines at the entrance to an RFQ field
An estimate of this effect can be gained by considering the principal multipole of the field that is associated with it. Of the complete multipole set
v = Lalmeim1J RI pt cose (B30) I,m this is the multipole with m = 2, 1 = 3. Expanding the Legendre polynomial of this multipole gives
(B31)
Taking the maximum ofthis function (at l/> = 0) the coefficient a32 is given by
a32=----1 cPEr l ; (Er=---1#321 ) (B32) 30 dZdr =0 a- r=O
whereupon the axial component of the multipole can be obtained from
(B33)
Since the energy change due to this field component depends on the distance of the particle from the axis the overall spread in energy caused by the field can only be
BIO determined by sampling a range of initial action points, each point being sampled over a range of initial phases for the RFQ field. In the present spread sheet the action sampI es are taken every 10 degrees, from 5 to 85° along the perimeter of the initial beam emittance diagram. The RF phases are the same 18° steps as used in the calculation of the beam envelopes. This calculation takes about 10 times as long as the beam envelope ca1culation so that, in practice, it is only run once a seemingly appropriate beam envelope has been achieved from the beam envelope ca1culation. Since the effect on the energy is proportional to the square of the distance of the partic1e from the axis it is important to get as small a beam diameter as possible in the RFQ entrance region. This effect on the axial energy will, of course, render the RK integration based on the axial values of the field inaccurate. Therefore. if the energy calculations indicate an unacceptable effect the RK ca1culations on which they were based cannot be trusted and so the iterations must be repeated until the energy spread is indeed acceptable.
B6: Spectrum of the Axial Energy Spread Due to the RFQ Gradient An estimate of the spectrum of the energy spread due to the axial gradient of the transverse field gradient can be obtained by assuming that the density of the initial action diagrams of the incoming beam corresponds to thermal equilibrium. If the emittance used in the calculation encompasses 90% of the total thermally equilibriated beam (a common practice in beam optics ) then the action density will be
dN =[dN] e -2* (B34) dA dAo where the "radius" parameter r is the distance of the action point from the action diagram center when the momentum coordinate is given the same dimensions as the displacement coordinate (i.e. the ellipse has been scaled into a circ1e) and rois the radius of the "emittance" ellipse.
For a pie shaped slice of angle !1() of this emittance diagram the density of partic1es as a function of radius will be
r dN [dN] _23 : -= - r!1() e ro (B35) dr dA 0
B11 For any given initial action point, the energy deviation caused by the fringe field of the quadrupole will be proportional to the square of the initial r parameter of the action. For the parameter value ro let the energy deviation be designated Eo. The spectrum of the energy deviation for the particles contained within the slice then becomes
E dN N -2.3E" -=-e 0 (B36) dE Eo where N is the number of particles in the slice represented by the segment at the perimeter of the emittance ellipse. For thermal equilibrium and uniform slice angles this number is the same for all segments around the perimeter. AIso, the density of the initial action diagram will not depend on the phase of the RF at which it enters the system. The total spectrum of all the incoming particles can then be obtained by summing
E dN ~ N -HE" [- -nL..-e 0 (B37) dE lotal Eo where the sum is taken over all the initial action points and all the RF phase for each action point. Hence n is the product of the number of action points and the number of phase samples. However, this method of sampling the initial action points assumes that the initial action ellipse is a right ellipse that can be rendered into a circle by a simple scale change of its displacement and momentum coordinates. The actual incoming beam will in general have an ellipse that is inclined to the displacement-momentum axis. The application of this method of sampling action points therefore requires that the right ellipse from which they are sampled be transformed into the actual emittance ellipse of the incoming beam. This requires the Twiss parameters for the starting ellipse. These can be obtained from the beam divergence tlpx and the beam transverse extent Llx using the relationships shown in fig. 5:
BStart- ---el(LlxJ 2 (B38)
B 12 and that the parameter e for the action ellipses is related to the beam emittance through
Emittance e= Po (B39) le where Po is the central momentum of the beam. Note that for a converging beam A is positive while for a diverging beam it is negative. The right ellipse that is taken for the initial azimuthally uniform sampling can be taken to be the starting beam emittance ellipse after a drift to the focus, or in the case of a diverging beam the emittance ellipse at the focus before it drifted to the starting ellipse. In such a transformation the extent of the momentum deviation remains constant, Le.
2JeeStart' The semiaxes of the right ellipse at the focus are then Je/Cstart for the displacement and JeeStart for the momentum (corresponding to the fact that for a right ellipse B = 1IC). Being a simple drift the transfer matrix for the action points is of the form
(B40)
The matrix element ml2 can be obtained from (18) in the form
-2 ml 2 ml2 B [~J =(l~~'""J= l~ 1 -~221 A J (B41) Focus Start o 1 C Start resulting in (B42)
Once the representative action points have been established in the right ellipse of the focus then they can be transformed back to the actual starting ellipse using the inverse of(B38):
x J II -AStar~/ Cstartl x J (B43) lPx Start = 0 Px Focus Again, note that for a beam that is diverging at the start A will be negative.
B13 Appendix C: The Ion-Source
As mentioned in section 4.1, a zeolite surface ionization type source was used to provide ions for the experiment. The ion-source was the same as that used by Kellerbauer [4] to test his decelerator, and its main components are identical the one built by his group for the ISOLTRAP experiment in 1995 [14]. These consist of a cylinder made of Inconel, into which the zeolite powder is packed, and a heater, which evaporates the alkali metal ions from the front of the cylinder by passing a CUITent through it. The basic principle behind such an ion-source is that ions are produced whenever atoms or molecules are brought into contact with a hot metal surface, and the degree of ionization is characterized by a in the following equation:
n. a=-' na (C.l) for which ni represents the number of ions leaving the surface and na the number of neutral partic1es. In the case of positive ions, the degree of ionization is dependent on the i temperature of the ionizing surface, T, its work function, cp, and the ionizing potential, E , of the incident atoms or molecules in the equation found in Breck [16]:
(C.2) for which A =g/ga, the statistical weight ratio of the ionic to the atomic state of the ions leaving the surface (for alkali metals, gj=l and ga=2), Zi is the charge of the ion and k is the Boltzmann constant. Therefore, for maximum ionization, the metallic surface should have a high work function and the alkali metal being ionized should have a low ionization energy. According to Equation C.2, a low surface temperature will also aid ionization, but this is countered by the fact that the rate of release of ions from an aluminosilicate is also temperature dependent, so that if the tempe rature is too low, the ions will not leave the ionizing surface. In this experiment, platinum was chosen as the surface metal because of its high work function (cp =5.32eV) and caesium (133 Cs+) was chosen as the sample ion because ofits low ionization energy (Ei =3.8geV). Another advantage ofusing caesium
Cl ions is that they are very stable and unlikely to exchange its charge state with any residual gas in the vacuum system containing the decelerator, meaning that the ions remain charged for enough time to perform the experiment. Furthermore, the caesium zeolite is very easy to synthesize. In this process, 99.5% pure aqueous caesium hydroxide is added to aluminum silicate and the resulting gel crystallized in an oven at 95°C for 96 hours. This method is a simple adaptation of the recipe for making potassium zeolite to be found in Robson [15], and provides a beam considerably purer than that to be had by the usual ion-exchange method [16] of synthesizing caesium zeolite, which typically has a relative abundance of caesium ions of only 72%. The caesium zeolite is packed into a cylinder made of the alloy Inconel, which is the chosen material for the source holder because of its good machining properties, high electrical resistivity and high melting point. Platinum wire 99.99% pure is then inserted into it to provide the surface for ionization and the cylinder heated by using a low-voltage 2 high-current transformer to pass an altemating CUITent of 60 to 90A through it • A variac controls the input to the transformer and therefore the CUITent through the heater and the beam current. The last part of the ion-gun is the extraction electrode located just in front of the cylinder which extracts the ions at the appropriate energy and also helps to focus them, while removing any variation is the ion energy that could arise from the AC voltage across the cylinder. This simple setup is capable of producing a beam of caesium ions at 60ke V, with a diameter of 30 mm at 500 mm from the source, a maximum current of a few nanoamperes and a lifetime of approximately 1000 hours. The beam is also quite stable and fluctuated by no more than ±5% of the measured current during each experiment.
2 Care must be taken that the CUITent does not go above 90A, since the heater is tapered to be thinnest and hottest near the front, where the ions are produced, and can easily bum out here there if the current is too high.
C2