1.1.1:... } a 1... ix), 1:59! i. 9465: 9:3,. 9.. 3 . n, 7 a... . ‘n .73... ~. 6.3 . a .....L. 1 I. “.5 t . U. I .m»... .42. ‘ 1L. 1 .p its xi... 3 I. ) I‘l‘, .llchnm. . 5‘47 .7 . ‘04:)... 5.41:. .Ifiit 0“ £13.33? a rut...” . A . N! 3%.... $52.3» MW . .I’ti: 5:47 “I, «I It..." 9... I affix? u... .. of r , rfiondli as figLiL.‘ an." ..... 5...... l . ‘3 -. 1.51.11?! 3.. . a... l .r: . 3:; .r a... . I v.1 . I! 3 5...... = 2. .. in. ! JIO‘ . S... A .34?!) 1. A. 1!. 2 Jr. 3.3:. E...- 5.33.! . . 513.71.. ~ Ar . 6.93”.“ hau.r3..5._ ; UBRARY am Michigan State University This is to certify that the dissertation entitled HIGH-PRECISION MASS MEASUREMENT OF 38Ca AND DEVELOPMENT OF THE LEBIT 9.4-T PENNING TRAP SYSTEM presented by Ryan Ringle has been accepted towards fulfillment of the requirements for the Ph.D. degree in Physics kl M D) W (3 Mfior Professor's Signature \lel17006 Date MSU is an Afiirmative Action/Equal Opportunity Institution PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 2/05 p:lClRC/Dale0ue.indd-p,1 HIGH-PRECISION MASS MEASUREMENT OF 38Ca AND DEVELOPMENT OF THE LEBIT 9.4—T PENNING TRAP SYSTEM By Ryan Ringle A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 2006 ABSTRACT HIGH-PRECISION MASS MEASUREMENT OF 38Ca AND DEVELOPMENT OF THE LEBIT 9.4-T PENNING TRAP SYSTEM By Ryan Ringle The Low-Energy Beam and Ion Ti'ap facility, LEBIT, has been designed to facili— tate a variety of experiments at low energies with rare isotopes produced by fast-beam fragmentation. Gas stopping of the fast-fragment beams and modern ion manipula— tion techniques are used. The first experiments to be performed are high-precision mass measurements made possible with a 9.4 T Penning trap mass spectrometer. LEBIT has been commissioned and first experiments on stable and unstable nuclides have been performed. Here I present the results of a mass measurement on the un- stable 38Ca isotope, measured with a precision of better than 6m/m=1 x 10—8. 38Ca was the first successful radioactive nuclide measured with LEBIT. It is of particular interest as it is a 0+ —> 0+ emitter, which makes it a possible test candidate for the Conserved Vector Current (CVC) hypothesis [1]. Also presented are design and commissioning details of the 9.4 T Penning trap system and mass measurements of the stable krypton isotopes, 83’84Kr, leading to improved mass values. In the interest of pushing the current limits of Penning trap mass spectrometry two new methods will be introduced. The first is a new excitation scheme of the ion motion using an octupolar radiofrequency field. The second, using the Lorentz steerer, is a fast preparation of ions previous to their capture in the Penning trap. Both methods have the potential to reduce the necessary measurement time, thus making high-precision measurements of shorter-lived species possible. ACKNOWLEDGMENTS To begin, I would like to offer my heartfelt gratitude to the N SCL at large. With— out the assistance of all of the employees working here the completion of LEBIT could not have been realized. Making mass measurements of radioactive species is certainly not accomplished by one group, alone. Thank you so very much for running a world-class facility and offering the best support a researcher could ask for. You’ve made my time here pleasant and productive. Next I would like to thank those in the design and machining departments who helped me to bring the LEBIT Penning trap from conception to reality. Thank you Don Lawton, Jim Moskalik, and Keith Leslie. Your exceptional efforts were integral to the success of the LEBIT program. Thank you to the gas cell team, led by Prof. Dave Morrissey, for all of your efforts. Without the close collaboration of this group we would never have made a radioactive mass measurement. I’d like to finish by acknowledging those I’ve worked closest to during my tenure at the NSCL. Although I was the first student on the LEBIT project (by two weeks, Pete!), the first generation LEBIT team expanded to three students (me, Peter Schury and Tao Sun) and a post-doc (Stefan Schwarz). By now you all feel like family to me. Thank you for your assistance and strong efforts. Without you I doubt that LEBIT would have been such a tremendous success in the timeframe we managed to squeeze it all into. I would also like to thank the next generation of LEBIT students (Josh, Amanda, Ania, and Greg) for (somewhat) gently removing me from the lab, thus iii forcing me to write. I would like to offer special thanks to Prof. Georg Bollen, my advisor. Thank you for always making yourself available and involved. Your guidance and encouragement have meant more to me than I have words to express. iv TABLE OF CONTENTS LIST OF TABLES viii LIST OF FIGURES ix 1 Introduction 1 1.1 Significance of nuclear mass measurements ............... 1 1.2 Mass measurement techniques for rare isotopes ............. 4 1.2.1 Production methods ....................... 5 1.2.2 Indirect mass measurements ................... 6 1.2.3 Direct mass measurements .................... 6 1.2.4 LEBIT - the first Penning trap mass measurements on rare isotopes produced by projectile fragmentation ......... 9 1.3 Outline of the dissertation ........................ 10 2 LEBIT Overview 11 2.1 The Coupled Cyclotron Facility ..................... 11 2.2 The gas stopping and ion guides station ................ 13 2.3 The test beam ion source ......................... 14 2.4 The ion cooler/buncher system ..................... 16 2.5 The 9.4 T Penning trap mass spectrometer ............... 19 2.6 Overview of the performance of the LEBIT system .......... 20 3 The LEBIT High-Precision Penning Trap 22 3.1 Basic Penning trap concepts ....................... 22 3.1.1 Ion motion in a Penning trap .................. 24 3.1.2 Ion capture in a Penning trap .................. 25 3.1.3 Excitation of ion motion ..................... 26 3.1.4 Time-of-flight cyclotron resonance detection scheme ...... 28 3.1.5 Mass measurement procedure using the LEBIT Penning trap mass spectrometer ........................ 30 3.2 The LEBIT Penning trap mass spectrometer .............. 32 3.2.1 High-precision electrode system ................. 35 3.3 Minimizing effects due to magnetic field imperfections ......... 41 3.4 Penning trap injection/ ejection optics .................. 45 4 Study of the classical quadrupolar and new octupolar excitation schemes 48 4.1 Introduction ................................ 48 4.2 The quadrupolar excitation revisited .................. 49 4.2.1 Equations of motion ....................... 49 4.2.2 Phase dependence of the quadrupolar excitation ........ 52 4.3 Octupolar excitation ........................... 55 4.3.1 The octupolar field ........................ 56 4.3.2 Single-ion octupolar simulations ................. 58 4.3.3 Realistic multi-ion simulations .................. 66 4.3.4 Experimental procedure and results ............... 69 4.4 Summary and conclusions ........................ 75 5 Development and detailed study of the Lorentz steerer 79 5.1 Charged particle motion in a region of perpendicular electric and mag- netic fields ................................. 80 5.2 Lorentz steerer design .......................... 82 5.3 Lorentz steerer beam calculations .................... 84 5.3.1 Ion cloud deflection simulations ................. 84 5.3.2 How to achieve minimum cyclotron motion? .......... 86 5.4 Lorentz steerer measurements ...................... 87 6 Mass Measurements of Stable Krypton Isotopes 95 7 High-Precision Mass Measurement of 38Ca and its Contribution to CVC Tests 100 7.1 Experimental procedure ......................... 101 7.2 Experimental results ........................... 104 7.3 Uncertainty analysis ........................... 106 7.3.1 Analysis of online results ..................... 106 7.3.2 Independent precision test with 23Na+ and 40Ar2+ ...... 111 7.4 38Ca mass evaluation ........................... 112 7.5 Impact of improved 38Ca mass on the precision of the .7: t value of the 0+ —+0+ decay .............................. 113 8 Summary 116 APPENDICES 119 A Solution to the Electric Potential of an Infinite Quartered Cylinder 119 vi B SOMA documentation B.1 Introduction . . B.2 Getting to Know SOMA ......................... 8.2.1 The main window ......................... 8.2.2 The uncertainty window ..................... B.2.3 The info window ......................... 8.2.4 The plot window ......................... 8.3 Analysis Methodology .......................... C SCM_Qt Documentation C.1 SCM-Qt Intro . C.2 Using SCM_Qt OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO C21 The main window ......................... C22 The isotope list window ..................... C23 The results window ........................ C .3 Additional Tools C31 The mass fragment calculator .................. C32 The frequency separation window ................ D Fit procedure to minimize higher-order electric field terms BIBLIOGRAPHY vii 123 123 124 124 126 128 129 130 133 133 134 134 136 136 137 137 138 141 144 3.1 3.2 3.3 3.4 5.1 6.1 6.2 7.1 B.1 LIST OF TABLES Specifications of the LEBIT 9.4 T magnet ............... Parameters of LEBIT Penning trap .................... Optimum electrode voltages (relative to the ring voltage) and the as- sociated Cl’s calculated using the LEBIT Penning trap SIMION ge- ometry described above. ......................... Frequency shifts, AVG, associated with the octupole and dodecapole components of the electric field using the optimum electrode voltages. Summary of the results highlighting the precision placement of ions in a Penning trap using the Lorentz steerer ................. Cyclotron frequency ratios for stable krypton isotopes obtained in this work ..................................... Mass excess values ME for krypton isotopes with mass number A as obtained from the measured frequency ratios and compared to their AME03 [2] values .............................. 39 90 99 99 Count rate information for each 38Ca2+ measurement of the second run. 111 SOMA constants and values ....................... viii 130 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 LIST OF FIGURES Layout of the Coupled Cyclotron Facility (CCF) at the NSCL ..... Layout of the LEBIT facility at the NSCL ................ Schematic diagram and photographs of select components of the gas stopping station. ............................. The test beam ion source. ........................ Design drawing and photographs of select components of the LEBIT beam cooler and buncher. ........................ Photograph of the LEBIT Penning trap mass spectrometer. ..... Cartoon illustrating the basic Penning trap concept ........... Cartoon illustrating the three independent eigenmotions executed by an ion in a Penning trap .......................... Cartoon illustrating the process of capturing an ion in a Penning trap. Schematic illustration of the driving of ion motions in a Penning trap. Cartoon illustrating the time-of-flight detection principle ........ Chart illustrating the mass measurement cycle used with the LEBIT Penning trap mass spectrometer. .................... Cyclotron resonance curve of 82Kr+ ions with an excitation time of Trf = 200 ms. .............................. Cartoon illustrating the mass measurement process ........... Schematic drawing of the LEBIT Penning trap mass spectrometer. . . Photograph of the LEBIT high-precision Penning trap with one endcap electrode removed. ............................ ix 11 13 15 16 17 19 23 25 27 28 29 30 31 32 33 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 Design drawing of the LEBIT high-precision Penning trap. ...... SIMION geometry of the LEBIT Penning trap. ............ SUSZI geometry of the LEBIT Penning trap ............... Induced magnetic field contribution due to Penning trap materials as a function of axial position. ....................... Quadratic fit to total magnetic field on the axis of the Penning trap. . Relative frequency deviations due to magnetic field imperfections. . . Photograph of the complete injection / ejection optics system with Pen- ning trap. ................................. Schematic drawing of the injection/ ejection ion optical elements with typical voltages ............................... Radial energy gain ion an ion subjected to a quadrupolar excitation. . TOF line shape obtained from a quadrupolar excitation of 40Ar+ with excitation time Trf = 200 ms. The solid line is a fit [3] of the theoret- ical line shape to the data ......................... Single-ion quadrupolar excitation simulations illustrating phase depen- dence. ................................... Measured cyclotron resonance curves showing phase dependence of quadrupolar excitation ........................... Octuoplar RF field configuration. .................... pi of an ion confined in a Penning trap and subjected to an azimuthal octupolar RF field at frequency 21/0 .................... Beat patterns of pi of an ion subject to an octupolar excitation. . . . Beat frequency of an ion subject to an octupolar excitation as a func- tion of p_,o for different values of (15.30 .................. Beat frequency of an ion subject to an octupolar excitation as a func- tion of ¢_,o for a constant Urf ...................... Beat frequency of an ion subject to an octupolar excitation as a func— tion of Urf for three values of d>__,0 .................... 36 38 42 43 44 45 46 47 50 51 53 54 56 58 59 60 61 62 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 5.1 5.2 5.3 Beat frequency due to an octupolar excitation as a function of p_,0 holding p2_,0 - Urf constant. ....................... Phase invariance of the octupolar excitation, part 1 ........... Phase invariance of the octupolar excitation, part 2 ........... Simulated single ion octupolar cyclotron resonances for several values of 45-,0. .................................. Cuts from simulated single ion octupolar cyclotron resonances ..... Multi-ion simulation of radial energy pickup in resonance for an oc- tupolar excitation. ............................ Realistic, multi—ion octupolar time of flight simulation. ........ Schematic drawing of experimental setup used to produce a azimuthal octupolar RF field in the LEBIT Penning trap. ............ Experimental time of flight for ions subject to an octupolar excitation with Vrf = 21/3 ............................... Octupolar TOF curves of 23Na+ as a function of Urf with Vrf = 21/6 for several excitation times. ....................... Product of U0 and Trf of an applied octupolar RF field at which the TOF curve reaches its minimum. .................... In—resonance time-of-flight curves obtained with an octupolar excita- tion for several Lorentz steerer settings .................. Experimental octupolar beat frequency .................. Experimental octupolar cyclotron resonance curves. .......... Mass comparison of 39AilK'l' using octupolar excitation. ....... Cartoon illustrating the Lorentz steerer principle. ........... Radial displacement as a function of time for a charged particle passing through a region of uniform, perpendicular electric and magnetic fields. Square of the radial velocity as a function of wet of a charged particle after it has travelled through a region of uniform, perpendicular electric and magnetic fields ............................. xi 63 64 65 66 67 68 70 71 72 73 74 75 76 77 78 79 81 82 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 6.1 6.2 7.1 7.2 7.3 7.4 7.5 7.6 Isometric section of the Lorentz steerer and nearby optics elements. The dashed line is the axis. Typical operating voltages are listed. Schematic layout of the Lorentz steerer’s electrode configuration. . . . Comparison of Lorentz steerer potentials obtained from a SIMION calculation and the analytical solution. ................. SIMION simulation of mean radial displacement of ions traveling through the Lorentz steerer. ....................... Steering strength of the Lorentz steerer as a function of mass number. Calculated radial displacement as a function of time for an A = 70, singly-charged ion as it travels through, and out of, the Lorentz steerer. Time of flight as a function of voltage applied to Lorentz steerer elec- trodes. ................................... Initial magnetron amplitude as a function of voltages applied to Lorentz steerer electrodes. ........................ Precision ion positioning with the Lorentz steerer ............ The difference between mass values measured with LEBIT and AME03 [2] for the stable krypton isotopes ..................... Deviation of individual mass measurements of 83’84‘Kr from AME03. Activity as a function of the mass—to—charge ratio A / Q of ions extracted from the gas cell and selected with the mass filter ............ Cyclotron resonance of 38Ca2+ measurement taken with an excitation time Trf = 300 ms. ........................... Dev1atlon of 1nd1v1dual frequency ratio measurements, R 2 VC / 116,7.ef, from the mean. .............................. Variations of the magnetic field and atmospheric pressure vs. time. The magnetic field strength nonlinearity parameter, s(AB) / B, as a function of At between reference measurements. ............ Average counts / cycle for each 38Ca2+ measurement made in the sec- ond run. .................................. xii 82 83 84 86 87 88 92 93 94 96 98 103 105 109 7.7 7.8 7.9 A.1 B1 B2 B3 8.4 C.1 C2 C3 C4 C5 Initial magnetron radii, p_,0, for 38Ca2+ and H3O+ as determined from a theoretical fit to the cyclotron resonances. ........... 112 The relative uncertainties in f, (Sf/f, due to the uncertainties in the Q EC value of the 12 most precisely known 0+—+0+ decays [1] and 38Ca using the new LEBIT mass value. ................ 114 The 12 most precise f t values from a recent review [1] along with the 38Ca precision limit due to new mass value. .............. 115 Cartoon illustrating the method used to solve for the electric potential of a quartered cylinder ........................... 120 The SOMA Main Window which is presented upon opening the ap- plication ................................... 125 The SOMA Uncertainty Window accessed through the Tools menu. 127 The SOMA Info Window accessed through the Tools menu ...... 128 The SOMA Plot Window accessed through the Tools menu ...... 129 The SCM_Qt Main Window which is presented when opening SCM_Qt.134 The SCM-Qt Isotope Window which lists the isotopes to be used in the search .................................. 137 The SCM_Qt Results Window which lists the matches generated by the search algorithm using the specified search parameters. ...... 138 The SCM_Qt Mass Fragment Calculator Window calculates which molecules fall within a given mass range. ................ 139 The SCM_Qt Frequency Separation Window which calculates the difference in the cyclotron frequencies between two species, as well as the energy separation. .......................... 140 Images in this dissertation are presented in color. xiii CHAPTER 1 Introduction High-precision mass measurements of short—lived, radioactive nuclides have an impact on various branches of nuclear physics, such as nuclear structure studies, the study of astrophysical nucleosynthesis, and the test of fundamental interactions. One of the most important recent developments in mass spectrometry is the use of Penning traps. Penning traps, used for many years in mass spectrometry of stable charged particles, have proven themselves invaluable for their accuracy, efficiency and reliability [4]. For the study of short-lived, rare isotopes several Penning trap mass spectrometers have been installed at low-to—medium energy radioactive-beam facilities around the world. The Low Energy Beam and Ion Tfap facility (LEBIT) at the National Superconducting Cyclotron Lab (NSCL) is the first to implement Penning trap mass spectrometry at a high—energy, rare-isotope facility using projectile fragmentation. In terms of reach far from the valley of stability this production mechanism provides advantages over lower-energy techniques. 1.1 Significance of nuclear mass measurements One of the most fundamental properties of an atomic nucleus is its mass. Since it is simply defined as the sum of its constituent nucleons minus the binding energy, measuring the mass of a nucleus essentially tells us how tightly the nucleons are bound, i.e., it serves as a probe of the total Hamiltonian of the nucleus. Nuclear structure effects can be explored by the systematic measurement of nu- clear masses. Shell effects distinguish themselves as a break in the smooth trend of nuclear binding energy as a function of the number of protons or neutrons. Near the valley of stability these closures happen at very specific magic numbers. Mass measurements provided the first evidence that nuclei outside of the valley show an evolution of shell structure, such as shell quenching and the emergence of new magic numbers. The first observance of such an effect was in 1975 in mass measurements on neutron rich sodium isotopes [5], giving the first indication of the erosion of the magicity at N = 20. This region of the nuclear chart is still of interest today. As one moves toward the driplines on the chart of nuclides, the nuclei become more weakly bound, resulting in diffuse or halo nuclei [6]. Halo nuclei consist of a tightly-bound core and one or two loosely-bound valence nucleons whose wave functions extend out much further than those of the core. Measurements of neutron- rich lithium isotopes [5] provided the first experimental evidence of the loosely bound nature of these systems. To date, several light, neutron-rich nuclei have been found to possess halo ground states, such as 11Li, 148e, and 6He . Although these halo nuclei are short lived, the first two having half lives of T1/2 < 10 ms, they are not beyond of the reach of precision mass spectrometry [7] and future Penning trap mass measurements of these rare isotopes are planned. Another interesting nuclear structure phenomenon that is currently under inves- tigation is neutron-proton pairing, also associated with the so—called Wigner energy [8]. Even-even and odd-odd nuclei near the N = Z line of the chart of nuclides exhibit a cusp in the trend of binding energies. This effect is associated with an additional binding energy in nuclei in which protons and neutrons occupy the same nuclear shell. Some of the most recent LEBIT mass measurements in the N as Z R: 33 [9] region will contribute to these studies. Isospin is a key concept in nuclear physics. With respect to the strong nuclear force, nuclear systems with the same isospin should have the same wavefunctions and energies. The Isobaric Multiplet Mass Equation (IMME), first introduced by Wigner [10] in 1957, predicts the mass excesses of each isobaric multiplet as a unique isospin-dependent function. High-precision mass measurements have been performed on multiplet isotopes to test the validity of this equation, and the improved 37Ca mass value obtained with LEBIT [11] contributes to such tests. High-precision mass measurements also play an important role in nuclear astro— physics. Stellar nucleosynthesis beyond the iron region, Z = 26, is carried out via neutron [12, 13] and proton [14] capture processes. The r process takes place on the very-neutron-rich side of the nuclear chart and synthesizes a heavier element by the capture of a neutron followed by a 6‘ decay. Proton capture occurs via the rp process and occurs on the proton-rich side of the nuclear chart. Here single protons are captured and followed by a subsequent 6+ decay, synthesizing a heavier isotope. Modeling these processes require neutron and proton separation energies, obtained from mass differences, as input parameters. Ideally the masses are obtained experi- mentally. In particular, in the case of the r process, which involves nuclides far from the valley of stability which beyond the reach of present-day facilities, mass predic- tions are important. Mass models are employed to make mass predictions when the nuclides of interest are beyond experimental reach. Several mass models and semi-empirical mass formulas have been introduced. A general feature of these models is a tendency to agree with one another in the region of known masses. However, their predictions begin to diverge once you enter terra incognita [15]. In order to test the predictive power of these models, and to help to improve their accuracy, more mass measurements are required in these areas of interest. High-precision mass measurements also play an important role in weak interaction studies, such as the mass measurement of 38Ca, presented in this work. Nuclear beta decay provides us with a convenient laboratory for exploration fundamental inter- actions and symmetries. One example is the test of the Conserved Vector Current (CVC) hypothesis which asserts that the vector part of the weak interaction is inde- pendent of the nuclear interaction. This means that the fl-decay strength, or ft value, should be constant in super-allowed 0+—> 0+ transitions. In order to measure the ft values of these decays, three experimental quantities are required: the half life of the species, the branching ratio, and the Q EC of the 0+ ——>0+ decay. Certain corrections to the ft value are also required as the decay takes place within the nucleus, yielding a modified ft, or f t, value which, according to the CVC hypothesis, should be constant. To date there are 12 well-known CVC test candidates from which a mean f t value [1] is calculated, verifying the CVC hypothesis to a level of 3x10‘4. Using the mean .7: t value the vector coupling constant Gv can be calculated. Gv can be used to cal- culate the up—down quark matrix element, Vud, of the Cabibbo-Kobayashi—Maskawa (CKM) matrix. Together with the other two elements, Vus and Vub: the unitarity of the CKM matrix (required by the Standard Model) can be tested. In the interest of increasing the precision of GU, more CVC candidates are desirable. 38Ca, whose mass was measured to sufficient precision in this work, is such a potential candidate. 1.2 Mass measurement techniques for rare iso- topes Today there are two different techniques for mass measurements of radioactive species. The indirect technique involves reaction and decay measurements, whereas the direct technique usually involves a time-of-fiight measurement, cyclotron frequency deter- mination, or both. The indirect technique is called as such because it yields mass differences, although direct mass measurements are not absolute and must be cali- brated with some reference mass. This naming scheme is a long-standing convention and will be preserved here. 1.2.1 Production methods There are several production methods for rare isotopes. They all possess specific advantages and complement one another. The isotope separation online (ISOL) tech- nique uses a high-energy primary beam of light ions to bombard a thick target of heavier elements to produce rare isotopes via spallation, fission, and fragmentation. The targets are maintained at high temperatures so that the reaction products can be diffused out and into an ion source, where they are ionized. Afterwards they are accelerated to an energy of a few tens of keV, a beam energy well suited for Pen- ning trap mass spectrometry. Very intense beams with excellent beam properties for certain elements are available, but chemical selectivity and decay losses in the target ultimately limit the range of secondary beams available [16]. Rare isotopes can also be produced by fusion or fusion-evaporation by bombarding a thin target with a low-to-medium energy primary beam, typically with energies of a few MeV/u. The heaviest elements are produced by fusion reactions. SHIP [17] at GSI utilizes a thin target of heavy metal, Pb or lBi, to produce heavy elements by fusion of the target and projectile. A primary beam, delivered by the UNILAC facility, is impinged on the target where fusion products exit from the other side at a few 100 keV/ u. The ATLAS facility at Argonne National Lab produces rare isotope beams by fusion evaporation and in—fiight separation. The ion guide isotope separation online (IGISOL) method [18] at JYFL produces rare isotopes by using a primary beam to bombard a thin target located within a gas cell. The reaction products exit the target and are thermalized in the buffer gas and extracted with electric fields. The advantage over the ISOL method is that it is not chemically selective and the extraction times are in the sub—ms range, minimizing decay losses. A powerful method of rare isotope production is the fragmentation of fast, heavy— ion beams after impinging on a light, thin target, which are then mass separated in flight. This is the method employed at the NSCL. The benefits of this method are many fold. The process produces fragments lighter than the projectile with no dependence on chemistry. Also, as there are no delay times associated with diffusion out of a target, shorter-lived nuclei are available from the fragmentation technique than from the ISOL technique. This method is very sensitive and can be used for the detection of rare isotopes with production rates on the order of a few per day. The resulting fragmentation beam still possesses a majority of the energy of the original primary beam. High-energy, high-emittance beams are poorly suited for low-energy experiments, but good for Coulomb excitation and reaction studies. 1.2.2 Indirect mass measurements Indirect mass measurements involve the determination of Q values of nuclear reac- tions or radioactive decays. Various types of reactions have been used to determine masses both near and far from stability. With respect to the latter, reaction Q-value measurements allow for the masses of unbound nuclei to be determined. The Q-value of nuclear decays provide mass differences between parent and daugh- ter nuclei. In order to arrive at a final mass value from a decay the unstable species must be linked to a known mass. For isotopes far from stability the final mass de- termination may involve long chains of decays which can lead to the accumulation of error in the final mass value. 1.2.3 Direct mass measurements The first direct mass measurements of rare isotopes were performed with magnetic spectrographs and spectrometers [19, 20]. In 1996 the last facility with a rare-isotope mass measurement program based on deflection-voltage measurements, Chalk River, was closed. Today, all direct mass measurements are made by employing some com- bination of time of flight, frequency determination, and magnetic rigidity. Several spectrographs and spectrometers exist around the world for making mass measure- ments of rare nuclides at various precisions. SPEG [21], at GAN IL, uses time of flight and magnetic rigidity measurements to achieve a mass resolution of z 10_4. The time of flight is on the order of 1 us, and although the resolving power is relatively low, SPEG’s high sensitivity have allowed it to perform measurements far from the valley of stability. TOFI [22] at LANL was designed to measure the masses of light, neutron-rich nuclei. The S800 spectrograph [23] at the N SCL also uses time of flight and magnetic rigidity for making mass measurements in a similar manner, and regime, as SPEG. First mass measurements with the S800 in the 66Fe region have recently been performed [24]. For a given time resolution the overall mass resolution is limited by the total time of flight of the ions. In an effort to improve the mass resolving power two approaches have been made. The first is to use cyclotrons to prolong the time of flight. In the case of the CSS2 [25] cyclotron, at GANIL, projectile fragments are injected into the cyclotron and radiofrequency fields are applied to accelerate species with a certain mass-over-charge value. The arrival time of the ion at a detector inside of the cyclotron is measured as a function of the phase of the radiofrequency, allowing a mass determination to be made. A similar approach [26] was used with SARA at Grenoble. The second approach is to use a storage ring, such as the Experimental Storage Ring (ESR) at GSI. By operating in isochronous mode [27] the revolution times of ions are recorded as they travel around the ESR, using a thin-foil detection technique for the time-of—flight detection. Ions of different mass create different time— of-flight spectra which can be compared to reference mass spectra to determine a mass value. Mass resolving powers on the order of 105 can be achieved and the method has been employed to measure the masses of many short-lived, neutron-rich nuclides [27]. Even greater mass resolving powers can be obtained through frequency determi- nation. Using the ESR it is possible to use cold electrons to cool the rare isotopes as they travel around the ESR and employ the Schottky method to detect the revolution frequency of the ions [28]. This method increases the resolving power to z 106 and requires a few thousand ions, yet the lengthy cooling times limit the half lives to T1/2 > 5 s. The RF spectrometer MISTRAL, at ISOLDE, accepts the 60 keV ISOLDE beam, imposing a two—turn helicoidal trajectory through a homogenous magnetic field. Using MISTRAL one determines the cyclotron frequency, l/C = qB/(21rm), of the nuclide of interest by the application of a radiofrequency signal applied to electrodes located at the one—half and three—half turn position within the magnetic field. If the phase difference in the applied RF signals is correct then ions experience an equal but opposite deflection when passing through the second electrode structure. An incorrect phase difference will result in a net deflection and the ions will not pass through an exit slit, which allows resolving powers of greater than 105 to be reached. Today, Penning trap mass spectrometry is the most precise method available for measuring stable and unstable nuclides. Relative mass precisions of 6m/ m < 10‘10 [4] for stable species and 6m / m < 10—8 [29] for unstable species have been observed. Penning trap mass spectrometry of unstable nuclides with half lives down to 10 ms are considered to be possible, and the shortest-lived isotope studied so far is 74Rb [30] which has a half life of T1/2 = 65 ms. Penning trap mass spectrometers have been installed at radioactive beam facili- ties around the world. The first Penning trap mass spectrometer for rare isotopes, ISOLTRAP [31], was installed at the ISOLDE facility at CERN. The low-energy, low-emittance ISOL beams are ideal for precision experiments, such as Penning trap mass spectrometry. Following ISOLTRAP’s success, several Penning trap spectrom- eters were installed at other facilities: JYFLTRAP [32] at Jyviiskyla, CPT [33] at Argonne National Lab, and SHIPTRAP [34] at GSI. Although CPT and SHIPTRAP are installed at facilities which produce low-to-mid energy rare isotope beams, an additional step is required to thermalize the beam before it can be used for Penning trap mass spectrometry. Thermalization of the rare isotope beam is accomplished by stopping the beam in a gas cell and extracting it at low energies. 1.2.4 LEBIT - the first Penning trap mass measurements on rare isotopes produced by projectile fragmentation LEBIT is the first facility designed to thermalize high-energy, rare-nuclide beams produced by relativistic in-flight separation for precision experiments. Utilizing gas stopping of the high-energy secondary beam, it has been shown that fragmentation beams can be made amenable to precision low-energy experiments, such as Penning trap mass spectrometry. This demonstration also opens the door to future ISOL-type experiments, such as laser spectroscopy, in-trap decay studies, and experiments with post acceleration of nuclei unavailable at lower-energy facilities. The first experiment on a thermalized secondary beam was the Penning trap mass measurement of 38Ca, discussed in this work, and of 37Ca [11]. In order to take full advantage of the short- lived isotopes produced by produced by projectile fragmentation, new techniques must be developed to reduce the measurement time necessary to achieve a given relative mass uncertainty. In this work two such techniques are introduced. A new device, called a Lorentz steerer, is used for a fast preparation of the ion motion necessary for a cyclotron frequency determination. A new RF excitation scheme at twice the ion’s cyclotron frequency, leading to an increase in resolving power, is studied in detail. 1.3 Outline of the dissertation Due to the scale of the LEBIT project, many people were involved in bringing LEBIT from conception all the way to the first successful radioactive mass measurement of 38Ca. In this document I will first provide an overview of the NSCL and LEBIT followed by more in depth discussion of the aspects that I was primarily responsible for. These topics include the design, construction and commissioning of the 9.4 T Penning trap system, stable Kr isotope measurements and the exploration of a new excitation scheme. Finally, I will present the results of the 38Ca mass measurement, introducing 38Ca as a test candidate of the CVC hypothesis. 10 CHAPTER 2 LEBIT Overview 2.1 The Coupled Cyclotron Facility EcR K590 N1 N2 N3 N4 N5 N6 SRF CLEAN ROOM f'e- E "I: "I ] la \ \ \ \ ’ —= ' K1200 A1900 s1 32 $3 CRYOGENIC PLANT Figure 2.1. Layout of the Coupled Cyclotron Facility (CCF) at the NSCL. The Coupled Cyclotron Facility (CCF) at the NSCL on the campus of Michigan State University is the premier rare isotope beam facility in the US. Fig. 2.1 presents an overview of the CCF. One of two Electron Cyclotron Resonance (ECR) ion sources is used to produce a primary ion beam of highly charged stable isotopes which is in- jected into the smaller K500 cyclotron. The K500 accelerates the beam to about 14 11 MeV/u. The extracted beam is injected into the larger K1200 cyclotron, where their remaining electrons are removed by a stripper foil. The K1200 then accelerates the beam to energies on the order of 140 MeV/ u. The fast primary beam is extracted from the K1200 and focused on the production target, to produce rare isotopes by fragmentation reactions. The secondary beam is then injected into the A1900 frag- ment separator [35], where the beam is purified and a small range of fragments is selected by a two—stage separation. This secondary beam of rare isotopes can be de- livered to the different experimental vaults (N2-N6, Sl-S3). LEBIT is located in the N4 and N5 vaults. The gas stopping station is located in the shielded N4 vault as due to the high-energy beam. The rest of the LEBIT facility is located in the N5 vault as the beam has been thermalized, with an energy of only a few keV. The layout of the LEBIT facility is shown in Fig. 2.2. There are four main compo- nents that are connected via an electrostatic beam transport system [11]. Six beam observation boxes (BOBs) are located along the beam line and are equipped with various detector systems, such as Faraday cups, microchannel plate (MCP) detectors and silicon detectors. The first component is a gas stopping station consisting of a gas cell and mass filtering radiofrequency quadrupole (RFQ) ion guides [9]. Here the fast secondary beam of rare isotopes delivered by the A1900 is stopped and and a low energy beam is produced. After the wall in the N5 vault is another RFQ sys- tem [11] for cooling and bunching the DC beam delivered by the gas cell/ ion guide system. There is also a plasma test beam ion source that provides stable beams for offline testing, system optimization, and for providing reference ions for magnetic field calibration. Finally, at the end of the LEBIT beam line is a 9.4 T Penning trap mass spectrometer designed to make high-precision mass measurements on short-lived radioactive species. 12 Gaull RFQ Buneoohr a) Ibullo ionouido Indbunclnr m I‘VIII 9.4-T Pmninglnp mas sputum Massfilbt Pia-coder Tm . i Ion Em” Mll mm Maura , I ....... ...... , ; .. mamas?» H ] ] 9.4T Penning trap ] b) masss ectrometer ’ beam cooler /\ p and buncher 3035 113086 ;—:/”§¢i|_ \ .z _»_-o,..'-"},_.‘ \tfi WE] '\ A ‘ "‘ i. s EJ- r beam Irom [I f—' _. . “NH”. .« /'{3 \‘ng A1900 a, a a-.. ;,'-‘- /‘ v511511§~"' " ’ "fie-7‘!”- 8081 test beam ion source gas stopping cell and ion guide system Figure 2.2. Layout of the LEBIT facility at the NSCL. BOB denotes beam observation boxes equipped with various detector systems. The two figures are (a) the schematic layout and (b) design drawing of the LEBIT system. 2.2 The gas stopping and ion guides station Fig. 2.3 schematically shows the gas cell and ion guide system together with pictures of selected components. In order to stop the fast rare isotope beam from the A1900, it passes through a system of adjustable glass degraders where it loses most of its energy before passing through an aluminum wedge degrader which drastically reduces the momentum spread of the fragment beam. It then passes through a thin beryllium window into a 51 cm—long gas cell [36] filled with 1 bar of ultra-pure helium gas, where the ions ultimately come to rest. Helium is used because it has the highest 13 first ionization energy of any ion. As highly charged ions are delivered to the gas cell by the A1900 the ions are stopped by collisions with the gas. By charge exchange reactions with the helium gas they lower their charge state until they are, at least, in the Q = +1 state. Inside of the gas cell a system of ring electrodes (Fig. 2.3(b)) is used to create a DC gradient which pulls the ions through the gas towards an extraction nozzle. A series of electrodes (2.3(c)) near the nozzle focus them into a region where gas flow provides the dominant force and sweeps the ions through the nozzle, out of the gas cell, and into an RFQ ion guide system. This ion guide system is contained in three vacuum chambers and divided into sections by a small RFQ (uRFQ) (2.3(d)), which facilitates differential pumping and better beam transmission between the first and second chambers, and a diaphragm. The pressure in the first ion guide can reach as high as 0.2 mbar, while the pressure in the final chamber is on the order of 10—6 mbar. The mass-selective ion guides span the last two chambers. They allow filtering by mass-to—charge ratio, A / Q, and have been designed to achieve a resolving power of z 50 with no loss in efficiency. A biased needle electrode can be inserted into the beam for collection and retracted to measure the collected activity on a nearby fl detector. This method is used to measure the rate at which rare isotopes are extracted from the gas cell. When used in conjunction with the mass filter, the A/ Q values of the radioactive atomic or molecular ions extracted out of the gas cell can be determined. 2.3 The test beam ion source Beams from the test beam ion source are used extensively in system tuning and optimization, and also to provide reference masses during mass measurements of rare nuclides. Fig. 2.4 shows a photograph of the plasma ion source presently used. Gas is introduced through a needle valve into a gas inlet tube. Noble gases like Ar, Kr and Ne are typically used. To ionize these gasses, a tungsten filament is heated so that it 14 wedge degrader uRFQ ion guide l/ l ' \ , ‘. *- - , _ / mi... till RFQ ion guide and \ mass filter collection needle and B detector glass degrader system gas cell Figure 2.3. The gas stopping station. (a) schematic diagram of the gas cell and ion guides system. (b) gas cell ring structure for creating a DC gradient inside of the gas cell. (c) gas cell ”flower” electrode for creating focusing potential. ((1) partial section of the RFQ ion guides connected to the ,uRFQ. produces electrons and is biased at around 100 V to produce a discharge. By changing the polarity of this bias alkali metals can also be produced via surface ionization present as impurities in the filament of the ion source. The test ion source is located perpendicular to the main LEBIT beam line. An electrostatic quadrupole deflector is employed to send the beam downstream to the cooler/buncher and Penning trap or upstream towards the gas cell. Figure 2.4. The test beam ion source. 2.4 The ion cooler/buncher system The LEBIT ion accumulator and buncher accepts the 5keV-Q continuous ion beam from the gas cell and converts it into a low-energy, low-emittance ion pulses. This device is a linear RFQ ion trap filled with a buffer gas at low pressure for ion cooling [11, 37, 38]. It features two separate vacuum sections (2.5(a)), one for beam pre- cooling and one for final cooling, trapping and beam bunching. The first section (2.5(b)) is typically operated with helium or neon at a pressure of z 10‘2 mbar. Neon can be used to increase the efficiency of Collision Induced Dissociation (CID) for the break up of molecular ions delivered from the gas cell. Before entering the system, the ions are electrostatically decelerated to a few tens of eV. Here the ions are transversely cooled and slowed down before they pass through a miniature RFQ ion guide (an efficient differential pumping barrier) into the trap section. The linear trap (2.5(c)) is operated with helium at a pressure 1-2 orders of magnitude lower than that in the first section to minimize beam heating during the pulsed-beam extraction. The ions are typically stored for 20-30 ms for their final cooling before they are extracted as a sub-as ion pulse. 16 RFQ trap ,7 ‘ acceleration opflcs deceleration optics _ uRFQ FIFQ trap ‘ I (RN47) NW roolei Figure 2.5. The LEBIT beam cooler and buncher. (a) design drawing of the LEBIT cooler and buncher. (b) Photograph of the ion guide cooler section together with the uRFQ. (c) Photograph of the RFQ ion trap for accumulation and bunching. Both the pre-cooler and the trap sections have been built as cryogenic devices and can be cooled with LN 2 provided from a stationary cryogenic line, reducing the kinetic energy of the buffer gas. This increases the efficiency of the system by reducing the diameter of the beam in the cooler, making the transport between the two sections via the miniature RFQ more efficient. The low buffer gas temperature also decreases the cooling time and has the potential to reduce the emittance of the resulting pulse which would increase the efficiency of injection into the Penning trap. Cooling the system also results in the significant reduction of residual pressures of gases other than the noble gases used for beam cooling. 17 In contrast to standard RFQ ion coolers [39] used elsewhere the LEBIT system uses a wedged-electrode design which allows the electric drag potential in the cooling section to be created without the need for segmented rods. The LEBIT beam cooler and buncher has been extensively tested and its properties were found to be in very good agreement with beam simulations [40] involving realistic ion-atom interactions. In pulsed-mode operation the overall efficiency was found to exceed 50%, while in continuous mode operation values of up to 80% were observed. In its normal mode of operation, continuous or pulsed ion beams from the gas cell or the test ion source are cooled and accumulated in the cooler/buncher and released as short ion bunches for capture in the precision trap of the LEBIT Penning trap mass spectrometer. For diagnostic purposes and buncher optimization time—of— flight distributions are routinely measured with a MCP detector downstream of the buncher. More details of the design of the buncher system can be found in the Ph.D. thesis of Tao Sun [11]. 18 2.5 The 9.4 T Penning trap mass spectrometer beamline from buncher 9.4 Tesla superconducting . MCP(Daly) and magnet “ Si detectors Figure 2.6. Photograph of the LEBIT Penning trap mass spectrometer. The LEBIT Penning trap mass spectrometer, shown in Fig. 2.6, features a 9.4 T superconducting magnet. Housed within the bore of the magnet is the Penning trap system which was designed to make high—precision mass measurements of rare isotopes. Ion bunches are delivered from the buncher and trapped in the Penning trap. While trapped, the ion’s motion can be driven with applied RF fields. The ion’s response to the driving RF fields can be measured and used to determine its mass via a cyclotron resonance time-of—flight detection technique. Detailed information of the LEBIT Penning trap mass spectrometer will be presented in Chapter 3. 2.6 Overview of the performance of the LEBIT system The efficiency of extracting rare nuclides which were stopped in the gas cell was measured to be about 5% for an implantation rate of z 40 pps for a mixed 38Ca/37K beam [36]. The total efficiency of the transport of rare isotopes from the gas cell to final detector in BOB6 is estimated to be about 15 %. After careful tuning of the potentials of the Penning trap and optimization of parameters for beam transport, ion capture and ejection, excellent line shapes and high resolving powers have been obtained with the LEBIT Penning trap mass spec- trometer. The highest resolving power observed thus far is about 3 million for an excitation time Trf = 1 s of 40Ar+ ions. In order to study the achievable precision and accuracy a large number of test measurements have been performed with stable ions, in particular 23Na+, 39K‘l', 40Ar+, 40Ar2'l', and AKr+. From these tests we conclude that LEBIT has the po- tential to achieve a very high accuracy. For example, the measured frequency ratio of stable 40Ar2+ and 23Na+, both known with sub—ppb precision, is in full agreement with the expected ratio obtained from literature mass values [2], showing an insignif- icant deviation of only 3(5) x 10‘9. This example corresponds to a close-doublet situation. The results obtained for a AKr+ -39K+ mass comparison, discussed in Chapter 6, shows excellent agreement even for large mass differences, indicating that mass-dependent systematic uncertainties are small. In the period from May of 2005 through November of 2006 LEBIT has made high- precision mass measurements of 26 rare nuclides: 37’38Ca, 64’65’66Ge, 40,41,42’43’448, 66,67,68,80AS, 29,341), 63,64Ga, 68,69,70,81m,SIgSe, 333i, 70m,7lBr. The mass mea_ surement of the super-allowed 0+ ——> 0+ emitter 38Ca, presented in this work, with a relative mass precision (Sm/m = 8x10"9 illustrates the precision to which the mass of 20 rare nuclides can be measured with the LEBIT system. LEBIT’s sensitivity is demon- strated by the mass measurement of 37Ca [37] where cyclotron resonance curves were obtained with a total of a: 50 detected ions. The mass measurement of 66As, with T1/2 = 96 ms, indicates the short time scales necessary for the performance of high- precision mass measurements. 21 CHAPTER 3 The LEBIT High-Precision Penning Trap Over the years a large amount of effort has been expended to optimize both the precision and accuracy of Penning trap systems [3, 41, 42]. Today they are the most precise instruments for making mass measurements on both stable and unstable species. In this chapter I will explore some basic Penning trap concepts dealing with ideal traps and RF excitations, and then move on to properties of realistic traps and discuss the design of the LEBIT Penning trap system. 3.1 Basic Penning trap concepts Three-dimensional ion confinement in a Penning trap is achieved by superimposing an axial, electric quadrupole field in a homogenous, axial magnetic field, as illustrated in Fig. 3.1. The electric quadrupole field is generated by two hyperbolic endcap elec- trodes and one hyperbolic ring electrode. Hyperbolic electrodes are used to minimize higher-order electric field contributions. Other configurations also exist [43], but are not relevant to this discussion. This electric field confines the ions in the axial direction. By placing this electrode 22 Figure 3.1. Cartoon illustrating the basic Penning trap concept. An axial electric quadrupole field is generated by a hyperbolic electrode configuration and embedded in a homogenous magnetic field oriented in the axial direction. system in an axial magnetic field full three-dimensional confinement is achieved. The electric quadrupole potential created by the electrode configuration can be described in cylindrical coordinates (r,0,¢) by ¢2 = 0%(222 - T2) (3.1) This potential can be realized by two hyperboloids of revolution given by 2 22 — 3 = fizz?J (3.2) where zo is half the distance between the two endcap electrodes, as seen in Fig. 3.1. If we specify the quantity U0 as the potential difference between the endcap and ring electrodes, then the coefficient of the quadrupole expansion can be written as _£’2 _2d2 where d is a characteristic trap parameter given by d = (mg/4 + 23/2. This yields the trapping potential a2 (3.3) 23 U0 452:4}? (22.2 — r2) (3-4) and leads to the quadrupole trapping field _ Ur Er- a pa Ezz—ygl‘éé 3.1.1 Ion motion in a Penning trap Charged particles in a magnetic field execute a circular cyclotron motion around the field lines with frequency wzla (w) m where q is the charge of the particle, B is the strength of the magnetic field and m is mass of the particle. Superimposing an azimuthal, electric quadrupole field within a uniform magnetic field and using them to trap a charged particle causes the particle to execute three independent eigenmotions. The electric quadrupole field is responsible for an axial oscillation with a frequency tag. The remaining two eigenmotions are in the radial plane: a slow magnetron motion with frequency w_. due to the ExB drift motion of the particle, and a cyclotron motion with a modified frequency (42+. Fig. 3.2 illustrates the three eigenmotions. In a pure electric quadrupole field the frequencies of the two radial eigenmotions are given by 2 2 C C Z _ L“ :l: L _ _°° 3.7 where the axial oscillation frequency is given by _ / qUo 24 In the general case of a strong magnetic field and a comparatively weak elec- tric field w+ >> (412 > w_. Two other important relationships between the three eigenmotions are (42+ + w~ = we (3.9) and 2 w+w_ = 02-3 (3.10) magnetron (-) reduced cyclotron (+) Figure 3.2. Cartoon illustrating the three independent eigenmotions executed by an ion in a Penning trap. The two radial eigenmotions are the cyclotron and magnetron motion, while the third motion is the axial oscillation. The mass of an ion can be determined from Eq. 3.6, if we know the charge state of the trapped ion and the strength of the magnetic field. This can be determined by a calibration measurement of a well—known stable species. 3.1.2 Ion capture in a Penning trap There are two methods used to capture ions in a Penning trap: a dynamic and a static method. In the static method the Penning trap electrodes are used to create a static potential well within a buffer gas. Ions are injected into the Penning trap, losing enough energy via collisions with the buffer gas such that they cannot escape 25 the trapping potential. The static method is not compatible with high-precision mass measurements due to subsequent interaction of the ion of interest with the buffer gas during the measurement. The dynamic method involves fast switching of voltages applied to the Penning trap electrodes. An ion is ejected from the cooler/buncher and drifts into the Penning trap system. When the trap is open the injection endcap voltage is low while the ejection endcap voltage is kept high. When the ions have drifted into the trap the voltage on the entrance endcap is raised and the ions are trapped. The process is illustrated in Fig. 3.3. In a perfect capture the ion comes to rest at the trap center when the voltage on the entrance endcap is raised which results in an ion with no axial energy. This is achieved by adjusting the slope of the injection potential such that the ion comes to rest at the trap center and immediately switching to the trapping potential. When multiple ions are captured a perfect capture results in the mean axial position of the ion bunch remaining at the trap center with the individual ions oscillating about the axis with some amount of axial energy. To remove ions from the trap the voltage on the ejection endcap is lowered, allowing ions to drift out. 3.1.3 Excitation of ion motion Once the ion is trapped its motion can be excited through the application of multi- polar RF fields. For the mass measurement technique used in LEBIT, RF excitation with multipolar, azimuthal RF fields is important. The most common method of in- troducing such a field is by segmenting the ring electrode and applying RF voltages to the ring segments for different multipolarities. An RF dipole field can be created by applying two RF signals with the same amplitude which are 180° out of phase to two diametrically opposite ring segments. The RF dipole field can drive one eigenmotion. By applying an azimuthal RF dipole field at frequency 11+ or u- it is possible to drive either the cyclotron or magnetron motion of a trapped ion, increasing the amplitude, 26 \V/W \V/ +V‘ open trap “ closed trap E E :2 cf. / axial direction Figure 3.3. Cartoon illustrating the process of capturing an ion in a Penning trap. Shown is the potential along the axis of trap between the endcaps. at or p_, of the eigenmotion. An azimuthal RF quadrupole field couples both radial eigenmotions. An azimuthal RF quadrupole field is created by applying two RF sig- nals with the same amplitude which are 180° out of phase to a pair of diametrically opposite ring segments. With LEBIT the most important frequency driven with an azimuthal quadrupole field is 12+ + l/_ 2 VC. The application of a quadrupole RF field at frequency 116 causes a periodic beating between the magnetron and cyclotron motions of an ion in a Penning trap, illustrated in Fig. 3.4, and is used to pump energy into the system which is detected and used to make a mass determination. The ion must initially be executing some radial motion to make this conversion. Traditionally this is achieved by first applying an azimuthal RF dipole field to drive the ion off axis. The figure on the left represents an ion beginning in a state of pure magnetron motion, p0 = p_, as illustrated by the circle. When the quadrupole RF field is turned on, the cyclotron amplitude begins to grow and the magnetron amplitude begins to shrink. The figure on the right skips forward in time until the magnetron amplitude 27 has decreased to p_ = 0 and p+ = p0. Since 11+ >> 1/_ the conversion of p_ to p+ is accompanied by a drastic increase in radial energy of the stored ion. A 7r—pulse at frequency 11,. f = 110 is defined as the product of the length of time for which the RF excitation is applied, Trf, and RF amplitude, Urf: such that the initial magnetron motion is completely converted into cyclotron motion. In addition to the excitation of ion motion via the application of azimuthal dipole and quadrupole RF fields an octupole RF field can be used drive the radial eigen- motions at a frequency Vrf = 211+ + 21/. = 21/0. This excitation requires eight ring segments and offers an increase in resolving power over the standard quadrupolar excitation. The octupolar excitation will be discussed in detail in Chapter 4 Y (a) Figure 3.4. (a) Ion begins in a state of pure magnetron motion, until quadrupolar RF field is turned on and the reduced cyclotron radius begins to grow. (b) full conversion of magnetron to reduced cyclotron motion. 3.1.4 Time—of-flight cyclotron resonance detection scheme After the application of a quadrupolar excitation at frequency 11,. f, the trapped ion may have gained some amount of radial energy if 11,. f z 14;. In LEBIT, and in other Penning trap experiments, this energy is detected with a time—of—flight resonance detection technique [3, 44, 45]. In order to detect a change in the radial energy of an 28 Z 63 _ E. as 17:- __ #82 Bo 62 Figure 3.5. Cartoon illustrating the time-of-flight detection principle. The ion ex- periences a force in the axial direction due to the magnetic field gradient and gains axial energy. ET is the radial energy of the ion and Bo is the initial magnetic field strength. ion the voltage on the ejection endcap is lowered, as described in Sec. 3.1.2, and the ion drifts along the magnetic field to a detector where the time of flight is measured. As the ion travels through the ejection optics the strength of the magnetic field begins fall off. Since the ion is executing cyclotron motion it has a magnetic moment, and the ion experiences a force in the axial direction due to the gradient in the magnetic field. This process is illustrated in Fig. 3.5. Here ET is the radial energy of the ion and B0 is the initial magnetic field strength. When the ion has left the magnetic field all of the energy gained during the excitation has been converted into axial energy and a reduced time of flight is measured at the end of the system with an MCP detector. The total time of flight of the ion can be calculated by 7'1 m 2 m”) z]... (2iEo—q-V(z>warp-Ben] ‘1’" ‘3'”) where E0 is the total initial energy of the ion, q is the charge of the ion, V(z) is the electric potential along the ion’s path, and B (2) is the magnetic field strength along the ion’s path. A theoretical description of the ion motion during a quadrupolar 29 excitation [3] can be used in conjunction with Eq. 3.11 to determine the initial radial eigenmotion amplitudes, p+,0 and p_,0, if V(z) and 8(2) are known. This has been used extensively in the study of the octupolar excitation (Chapter 4) and in characterizing the Lorentz steerer (Chapter 5). 3.1.5 Mass measurement procedure using the LEBIT Pen- ning trap mass spectrometer Frequency Scan capture ions in Penning trap increment vri and repeat until Vrt = Vcenter+ Avl2 choose scan range eject ions perform RF vn = Vcente, - Av/2 from buncher excitations eject ions and measure TOF x N = Measurement Figure 3.6. Chart illustrating the mass measurement cycle used with the LEBIT Penning trap mass spectrometer. A cyclotron resonance curve measured with the time-of—flight technique outlined in the previous section is obtained in the following manner. First, a frequency scan range, AV is chosen depending on the mass uncertainty of the ion being measured. Next, an ion bunch is delivered from the buncher and trapped in the Penning trap. After purification of the ion bunch by the application of a mass-selective dipolar excitation, a quadrupolar excitation at frequency Vrf = Vcenter — AV / 2 is applied. The ions are ejected from the trap and their time of flight to the MCP is recorded. This process, 30 called a frequency scan, is repeated, incrementing 11,. f until 11,. f = Vcenter + AV/ 2. A measurement consists of one or more frequency scans and is outlined in Fig. 3.6. A sample cyclotron resonance curve for 82KBr ions is shown in Fig. 3.7. For an excitation time of Trf = 200 ms a line width of Au = 5 Hz is obtained for N as 1000 detected ions, resulting in a resolving power of R = V/Au z450,000. Perfect agreement is observed between the data points and the theoretical fit to the data [3]. l 11.30 - [ 9.60 « "a" .3. u. 7.90 4 O t— 6.20 . 4.50 . . . 25.4 31.4 37.4 43.4 49.4 55.4 v" [Hz] - 1760700 Figure 3.7. Cyclotron resonance curve of 82Kr+ ions with an excitation time of Trf = 200 ms with N z 1000 detected ions. Solid line is a theoretical fit [3] to the data. In order to make a Penning trap mass measurement the magnetic field must be known very precisely in order to determine the mass via a determination of the cyclotron frequency, Vc = qB/ (27rm). This is accomplished by making a measurement of a nuclide, usually stable, with a very well-known mass. The mass of the ion of interest can then be determined by M = (Vc,ref /1/c)-M,.ef where M is the mass of the ion of interest, Mref is the mass of the reference ion, :16 is the cyclotron frequency of the ion of interest, and Vc,ref is the cyclotron frequency of the reference ion. In order to determine the atomic mass of the ion of interest the electron mass and 31 electron binding energy are required. Isotopic shifts in the electron binding energies are negligible, and the electron binding energies of stable isotopes can be used. With any real persistent superconducting magnet the field decays over time, and multiple reference measurements are required to determine the magnetic field as a function of time. In order to determine the strength of the magnetic field during a real mass measurement a reference measurement is taken before and after the mea- surement of the ion of interest. The cyclotron frequency of the reference ion is then linearly interpolated to the time at which the ion of interest was measured, as is illustrated in Fig. 3.8. A L reterenoe . measurement >5 3 C... Q) lnterpolated "a, L: reference ° reference Time 3.2 The LEBIT Penning trap mass spectrometer Fig. 3.9 shows the schematic layout of the experimental setup of the LEBIT Penning trap mass spectrometer. The magnetic field is provided by an actively shielded, per— sistent, solenoidal, superconducting 9.4 Tesla magnet system built by Cryomagnetics, Inc. with a room—temperature, horizontal bore. The magnet system has been up- 32 Penning Trap Figure 3.9. Schematic drawing of the LEBIT Penning trap mass spectrometer. graded by the addition of external-field compensation coils [46] to reduce the effect of external field changes that may occur in an accelerator environment. The employ- ment of a 9.4T field, as compared to z 6 T which is typical of current systems, has the advantage that a given precision can be achieved in about half the measurement time. The primary magnet specifications are listed in Table 3.2. A precisely machined vacuum tube, mounted inside of the room-temperature bore of the magnet, serves as an ion optical bench for the trap electrode system. Since bore of the magnet may not be exactly parallel with the magnetic field axis the ends of the bore tube are secured to the magnet using translational mounts. To align the bore tube with the magnetic field an electron gun was used. The electron gun was placed in the center of the bore tube with two detection electrodes separated into quadrants on either side. The electron current on each quadrant can be read while the the position of the bore tube is adjusted with the translational mounts. When the currents on each quadrant are roughly identical the bore tube was declared to be aligned to the magnetic field axis. The magnet rests on a translational mount which allows it to be moved so that the injection and ejection optics systems can be easily removed from the bore tube. On either side of the cryostat are two 300 L/s turbo pumps. After several months of pumping the pressure in the two chambers is on the order of 10—9 mbar. An MCP 33 Table 3.1. Specifications of the LEBIT 9.4 T magnet Parameter Value central field homogeneity i 10 ppm over a cylindrical volume 5.1 cm diameter x 10.2 cm length persistent mode field decay (dB / B) / dt z -1><10‘8 bore diameter 12.8 cm cryostat diameter 116.8 cm cryostat length 101.6 cm LHe capacity 325 L LHe refill rate 3 3 months (250 L) LN2 capacity 150 L LN2 refill rate z 2 weeks (140 L) detector in a Daly [47] configuration is placed in BOB6. Several improvements to the magnet system have also been added. The pressure of the helium bath is stabilized by an electric valve operated on a PID loop to eliminate non-linear magnetic field effects, on top of the natural magnetic field decay, due to variations in the helium boil-off rate. The Penning trap and nearby ion optical elements can be cryogenically cooled, reducing residual gas pressure which can cause frequency shifts while performing a measurement. A pair of insulated copper wires wound around the tube allow for either baking or compensation of the natural decay of the main magnetic field during measurements. The Penning trap and associated injection and ejection optics have been optimized to avoid introducing large uncertainties in the measurements or losses in efficiencies. The electrode structure of the LEBIT Penning trap has been designed such that higher-order contributions to the electric trapping field have been minimized. The materials used to construct the trap have also been distributed such that the magnetic field in the center region of the trap remains homogenous. The injection optics have 34 been designed to minimize transverse energy pickup as the ions are injected into the magnetic field. The ejection optics must transport the ions out of the magnetic field and efficiently focus them onto the MCP detector at the end of the system. Details on the design of these systems will be covered in the following sections. 3.2.1 High-precision electrode system Figure 3.10. The LEBIT high-precision Penning trap with one endcap electrode removed. Note the hole in the endcap for ion injection/ ejection. The final design drawing of the LEBIT high-precision Penning trap is shown in Fig. 3.11. In addition to the ring and endcap electrodes there are two sets of correction electrodes. In order to generate a perfect electric quadrupole field the hyperbolic electrodes would need to extend to infinity. In a real Penning trap the electrodes are truncated, which introduces higher-order field contributions. These higher-order terms can be minimized by adding correction ring electrodes to compensate for the finite nature of the ring and endcap electrodes. The holes in the endcap electrodes also introduce higher—order field contributions and are partially compensated by the addition of correction tube electrodes. The LEBIT Penning trap’s electrodes are constructed of high-conductivity copper and plated with gold to reduce possible patch effects for the creation of a homoge- 35 nous electric field. The shape and dimensions of the electrodes have been optimized to achieve close-to-ideal electric and magnetic fields within the trap, and will be dis- cussed in detail in the following sections. The insulators are made of Alumina. The ring electrode has an eight-fold segmentation. This allows not only for the creation of a quadrupole RF field, as required for the excitation of the ion motion at the ion’s cyclotron frequency V5 [42], but also the application of an octupole RF field, which makes ion motion excitation at 2 116 possible, which is discussed in Chapter 4. Fig. 3.10 shows a photograph of the LEBIT high-precision Penning trap with on endcap removed. The electrodes are gold and the Alumina insulators are white. Correction Tube \ Correction Ring \ V //A . \Kx\\\\\\\\\\\\\\\ [///l///V// [f ///F//// % \\\\ //////////////<10_7 for the copper pieces. Fig. 3.14 displays the results of the SUSZI calculation, plotting the induced mag- netic field contribution, AB/ B, as a function of axial displacement for the copper 42 657'; — Copper 557,3 . - - Alumina 1 ] t, -- Sum 1' 4E-7{ 3 '3 . | . ‘ ' a 3E'7': ‘ 2| i 0 Q 25-7g ' m : <’ 15-7-4 ............ 05+0-j 45-75 --------------- - 257r -15 -1O -5 0 2 [mm] Figure 3.14. Induced magnetic field contribution, AB / B as a function of axial posi- tion, z. The individual contribution from the electrodes (copper) and the insulators (Alumina) are plotted together with their sum. electrodes, the insulators, and their sum. Note that the copper and Alumina curves are constant in the middle of the trap. Since the magnetic field imperfections are symmetric about the z = 0 plane the lowest order component of the magnetic field deviation will be the quadrupole term, 1+ [32 (22 — §)] (3.19) where Bo is the base magnetic field strength. Then, according to [42] the cyclotron which is expressed as BzZBo frequency shift is given by 2 2 2 02+ —w_ 2 02+ —w_ If we use the approximation 02+ — w_ m we, then Awe = [32%(22 — p2_). (3.21) By taking the sum data shown in Fig. 3.14 and performing a least-squares fit to a second order polynomial it is possible to extract the fig term. Fig. 3.15 shows the 43 ~9.00E-08 82 = 1.3x10'10 mm’2 '9-055'08" I -9.1OE-08- <1 -9.15£-oo- 9206-089 '9-255'08 f I I I l l l 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 2 [mm] Figure 3.15. Fit to sum data from Fig. 3.14 from z = 0 mm to z = 3.5 mm. The fit yields a 52 = 1.3x10—10 mm—Z. sum data from z = 0 mm to z = 3.5 mm and least-squares fit, which yields [32 = 1.3x10—10 mm‘g. An overall relative shift of the magnetic field of z 9x10"8 is observed. Fig. 3.16 plots AVG/11¢ as a function of the axial oscillation amplitude for four different values of p_. The maximum p- allowed by the entrance and exit holes of the LEBIT Penning trap is 2 mm. If we assume a p- of 2 mm then the maximum relative frequency shift is Ave/11¢ z :i: 1x10_9. The relative deviation of the measured frequency ratio due to magnetic field im- perfections is found to be AR T 2 where A2 = Zre f - 22 and Ap_ 2 p2_ ref - p2_. The relative deviation of R is zero in R5 182(Az — Ap—)1 (3.22) the case that amplitudes of the axial and magnetron motions are the same for both ion species. Experimentally it is not difficult to ensure that the difference between P—,ref and p- is small, and a 10% difference is a conservative estimate. Similarly, the difference in the axial oscillation amplitude should not be very great as it depends on a well- controlled timing sequence. For 10% differences in amplitudes not exceeding 2 mm 44 1.5509 0:05mm ------ Q=L0mm 1.08-00 - 5.054 0 - s.” > d 0.06100 * 6.054 0 - 0.0 0.5 1.0 1.5 2.0 2.5 3.0 2 [mm] Figure 3.16. AVG/Va as a function of the axial oscillation amplitude, z, for four different values of p_ for 32 = 1.3x10‘10 mm-z. the relative shifts will be below 10‘“). 3.4 Penning trap injection/ ejection optics Fig. 3.17 shows a photograph of the complete Penning trap injection/ ejection optics system, including the Penning trap (see also Fig. 3.10). These ion optical systems are inserted into the bore of the magnet. The injection optics is composed of eight drift tube sections and a field termination plate located just before the trap. They are used to decelerate the ion bunch delivered from the cooler / buncher with a kinetic energy of K m 2 keV-Q to K m a few eV when entering the Penning trap. One element of the injection optics is the ”Lorentz” steerer, which is a new technique used to prepare the ion bunch for a quadrupolar excitation during the injection process and is described in Chapter 5. Detailed SIMION simulations have been performed to optimize the voltages applied to the individual ion optical elements such that a space focus is created at the trap center. The ejection optics consists of five drift tubes and their purpose is to guide the 45 Figure 3.17. Photograph of the complete injection/ejection optics system with Pen- ning trap. ion bunch out of the Penning trap, through the magnetic field to the MCP detector located at the end of the system. The ions typically travel through the first drift tube with an axial kinetic energy of K N 30 eV-Q. By the time the ions have reached the end of the ejection optics they have an axial kinetic energy of K m 1.8 keV-Q, plus whatever energy they gained as a result of an RF excitation in the Penning trap. SIMION simulations have been performed to optimize the location of the beam spot on the MCP, and to increase the the time-of—flight separation between ions that have gained radial energy during the quadrupolar excitation and the baseline. Fig. 3.18 shows a schematic drawing of the injection and ejection ion optics with typical voltages listed. 46 Injection Ejection Optics Optics f— -2250 V -200 V CV '50 V -8 V -—L250 V -2000 V ____11.\.L _____ LELZC<_____ I Steerer -200 V :t U1,2 Figure 3.18. Schematic drawing of the injection/ejection ion optical elements with typical voltages. 47 CHAPTER 4 Study of the classical quadrupolar and new octupolar excitation schemes 4. 1 Introduction Within this work detailed aspects of the quadrupolar excitation as presently used for mass determination were investigated. In particular, the phase dependence of the quadrupolar excitation was studied, which has never been studied in detail. In addi- tion, ion motion excitation using an octupole RF field was investigated and demon- strated for the first time. An octupolar excitation in a Penning trap is achieved by applying a octupole RF field at frequency 2116, instead of a quadrupole field at 11¢, and requires an eight-fold segmentation of the electrode to which the RF is applied. Using this new excitation scheme it may be possible to increase the resolving power of Penning trap mass measurements. 48 4.2 The quadrupolar excitation revisited In order to provide some background for the discussion of both quadrupolar and oc- tupolar excitations the major steps leading to the equations of motion for a quadrupo— lar excitation, as derived in [3], will be briefly reviewed. 4.2.1 Equations of motion ——> ——> Begin by introducing the vectors V+ and V‘ [41] such that Viz—pf—wqyjofxez, (4.1) where if is the ion’s position vector. Additionally, (4.2) V+—V_ y = _a;__.r_ where x and y are the ion’s position in Cartesian coordinates. Eq. 4.1 successfully decouples the equations of motion of an ion confined in a Penning trap. Using this coordinate transformation and applying the rotating wave approximation the equa- tions of motion can be solved [3] for an ion subjected to an applied quadrupolar RF field. The solution is - :l: PLOW“) ‘wC)l+PIF,oko Pi(t) 2 [Pin 0030080 q: % rf “B (4.3) x sin(wBt)]e%(wa_wC)t 9 where pi(t) are the magnitudes of the cyclotron and magnetron radii as a function of time and pi“, are the initial cyclotron and magnetron radii. Additionally, 1 023 = 5([00,). — wc)2 + kg (4.4) . U and k3: = koeiZAfb where k0 = fiffiw+lw . Here, q is the ionic charge, m is a _ the ionic mass, Urf is the applied RF amplitude, a is the radius at which the RF is 49 applied and A4 = 4.. f — (4+ + 44 (4.5) qérf is the phase of the applied RF and qbi are the cyclotron and magnetron phases. As stated above, radial energy gained during the excitation process is detected via a reduced time of flight of ions ejected from the trap to the detector. Under normal conditions 01+ > w_, so we can write the radial energy as Er(t) z émwipfijt). b.) :‘5 a.) 5 £5 (Avc)FWHM ~ 0.13/Trf .2. N + O. :5 4 -2 o 2 4 6 AVG [1nd] (0,, = U0) c.) p +2 [arb. units] Urf (AVG = 0) Figure 4.1. (a) surface plot of p_2[_ as a function of Urf and 11,. f for a given excitation time Trf- (b) cut from plot (a) at Urf = U0. Full conversion from magnetron to cyclotron motion is achieved at AVG = 11,. f - Vc = 0. (0) slice from plot (a) at AV 2 0 illustrating the beating of pi as a function of Urf' Fig. 4.1(a) shows a 3D plot of the calculated values of pi, which is proportional 50 to the gained energy, as a function of Urf and frequency detuning for a given Trf for a single ion initially executing pure magnetron motion, p+,0 = 0. Fig. 4.1(b) shows a cut at U1. f = U0, which is the excitation amplitude at which a full conversion of magnetron to cyclotron motion has occurred for Vrf = Vc. The width of the resonance curve is given by theory as AVFW H M m 0.8/ T1. f. Such a response is reflected in the experimental resonance curve shown in Fig. 4.2. Fig. 4.1(c) is a cut along AV = 12,. f - VC = 0 illustrating the change of pi as a function of Urf for a constant excitation time. Calculating the energy gain from the excitation and accounting for the ejection optics and magnetic field it is possible to calculate the theoretical line shape [3] expected for the time-of—flight measurement. This shape has been used to fit the experimental data shown in Fig. 4.2. Due to a nonlinear conversion between radial energy and time of flight, the full width of the time—of—flight curve is 0°9/Trf for LEBIT. TOF [us] ~10-8-6-4-20 2 4 6 810 Vrf - 36066143 [Hz] Figure 4.2. TOF line shape obtained from a quadrupolar excitation of 40Ar+ with excitation time Trf = 200 ms. The solid line is a fit [3] of the theoretical line shape to the data. 51 4.2.2 Phase dependence of the quadrupolar excitation The solution to the equations of motion as derived in [3] contains a phase dependence, the consequence of which has not been previously investigated. The phase dependence appears in Eq. 4.3 through the kg: term. We algebraically expand 2 2 2 2 2 2 2 4p 0) cos a) t) k _ +p w —-w) p+p+*= *“0 B 2 (B 0 ’0 +43; 7 C sin2 40) 440 B B k _ w —w sin A45 ‘ _ 0p ,0p+,0( 7'5 C) ( )Sln2(u}Bt) (46) 4°”B 2k _ A _ 0p ,0p+,0;)B COS( ¢) Sln(2LUBt) 4wB The phase dependence of the radial energy change is located in the third and fourth terms of Eq. 4.6. Both terms scale with the product of p+,0 and p_,0, which means if the ions begin in a state of pure cyclotron or magnetron motion at the beginning of the excitation the effect will not be present. Fig. 4.3 presents the results of simulations of quadrupolar resonances for single 39K+ ions with Trf = 1 s. Both simulations were performed with p_,o = 1 mm and ¢+,0 = ¢_,0 = 0°. An initial cyclotron radius of p+,0 = 0.1 mm was used on the left and p+,0 = 0.4 mm was used on the right. In each case the value of Urf was adjusted such that in resonance and for qSTf = 0° the radial energy was maximized. Looking closely at the contour plot on the left one can see a distortion of the central peak and a shift in the position of the side bands as A05 changes. Resonance profiles for which A05 = 90° and 270° become asymmetric and the position of the minimum time of flight has shifted slightly. Increasing the value of p+,0 to 0.4 mm only exacerbates the situation. The most obvious implication of these phase-dependent effects involve low-statistics measurements where A43 is allowed to vary. If the measurement samples any particular phase range more frequently and the resonance is fit with a line shape with a symmetric center then the analysis will include an additional error. 52 M [deg] TOF Ills] -'2m-'1'.'5m-'1m-0‘.5m 0:5 1 1.5 2 Av [Hz] Figure 4.3. Simulations of quadrupolar resonances for single 39K+ ions with Trf = 1 s. The two plots in the top row are contour plots of the resonances as a function of A05 and Au. Below are cuts at given values of A43. Both simulations were performed with p_,0 = 1 mm and ¢+,0 = ¢_,0 = 0°. An initial cyclotron radius of p+,0 = 0.1 mm was used on the left and p+,0 ---= 0.4 mm was used on the right. In order to see the phase dependence in the radial energy change of the quadrupo- lar excitation, two conditions must be met. The first is that the RF voltage must have a defined phase relation to initial ion motion. A03 must not be random which means, according to Eq. 4.5, that all individual phases must be well defined. If this is not the case then the final line shape will be an average over the line shapes of random Agb’s. The second condition is that the ion ensemble cannot begin in a pure state of cyclotron or magnetron motion. For the test of this phase dependence 39K+ ions were injected into the Penning trap, and an initial magnetron motion was introduced with the Lorentz steerer. The 53 143 q)”: 3600 TOF [11.8] 161 14g TVTV r177 YV’I TTTTT O) I b-i is- 0) Figure 4.4. Three 39K+ quadrupolar resonances for Trf = 500 ms measured for different values of 45,.f. The solid line is the fit of the theoretical line shape. excitation RF was phase locked to the time of ion capture in the trap, which defines the initial phase of the ion motion. Several resonances with Trf = 500 ms were taken with varied values of (1),. f. Fig. 4.4 illustrates the effect of varying ¢.,.f on the resonance shape. Here we see experimentally how the sidebands change shape as the value of A05 is changed. When A¢ is allowed to be a free parameter a good agreement is obtained between the experimental data and the fit. All three fits yield consistent values for p_,0 = 091(2) mm and p+,0 = 0.11(1) mm. For each resonance 45,]. was incremented by 100°. Within 20% this phase dependence was reproduced in the fit parameter A45. 54 As discussed earlier, only the third and fourth terms of Eq. 4.6 contain a A43 term and contain the product of p+,0 and p_,0. While p_,0 is introduced deliberately, a finite p+,0 must be the result of an asymmetric injection of ions into the magnetic field of the Penning trap which leads to a pickup of cyclotron motion. If the amount of radial energy gained during the injection process can be reduced, then p+,o will be reduced, thus reducing the phase-dependent effect. This could be useful in fine-tuning the injection of ions into a Penning trap. 4.3 Octupolar excitation Excitation of ion motion in a Penning trap by application of an octupolar RF field at frequencies near 2V0 have been studied experimentally and in simulations. Singleion simulations were used to explore the resonant response of pi for a variety of initial conditions. Simulations utilizing realistic distributions of multiple ions were used to predict resonance profiles under realistic conditions. Two independent codes were developed and used in these studies: a fourth-order Runge-Kutta routine written in Fortran and compiled on a Microsoft Windows operating system, and a fourth-order Runge-Kutta routine with adaptive step size control written in C++ and compiled on an OSX operating system. Experimental results together with those from simulation are used to make esti- mates of the distribution for initial conditions. In many cases 2D simulation results have been presented to illustrate the scope of the simulations performed. In prac- tice, though, the agreement between experimental and simulation data is more clearly observed by presenting projections of the data. 55 4.3.1 The octupolar field Since much of what we learn from the octupole excitation will be based on simulation it is important to adequately outline the methods employed should anyone care to reproduce the results given here. Bearing this in mind I will present the formulation of the octupole field orientation which I used and the associated equations of motion which can be solved numerically (using a Runge—Kutta routine [49] in the present work). The expansion for the charge distribution, 050, is 450 = arr4 + bx3y + csc2y2 + dxy3 + ey4. (4.7) Since the space is charge free it must satisfy the Laplace Equation, V205 = 0. Applying this condition to Eq. 4.7 yields the following conditions: 6a + c = 0 b + d = 0 (4.8) c + 6e = 0 Figure 4.5. Octuoplar RF field configuration used in simulations shown at t = (2n+1)7r/(2w,.f). Nodes are located at (2n+1)7r/8. Generated with b = -d = 1 anda=c=e=0inEq.4.8. Choosing a nontrivial set of parameters which satisfy Eq. 4.8 yields an equation describing a valid octupole field. Different parameters correspond to a rotation of the 56 field’s orientation in space. For illustration we will choose a=c=e=0 and b=-d=1. This is the orientation used in all of the calculations, and is shown in Fig. 4.5. Calculating the electric field produced by a time-varying octupolar field produced by a set of electrodes at a radius a from the trap center yields the following field components: U Ea: = :3} sin(w,.ft + ¢rf)(y3 — 3:02.11) (4.9) U Ey = ——a-r1£ sin(w,.ft + ¢rf><$3 — 3$y2)' Writing out the full equations of motion of an ion in a Penning trap subjected to an octupolar RF field using the transformation given in Eq. 4.2 yields (VJ—vx“)2i(va?“—V;)2—3(V;44,—)2] ':l: _ _ i_ Va; _ wit/y k (w+_w_)3 (4.10) . v+—V‘ 2 V+—V_ 2—3 V+—V" 2 vyi: in§t+k(y y)l(y y)3(:1: all (w+-w—) where U f9 k = T—sinfiurft + (prf). (4.11) Unfortunately, the transformation does not decouple the equations of motion. The ansatz presented in [3], Vi: Ai(t)eii(wi+¢i), where Ai is an amplitude, does not simplify the problem. In the case of the quadrupolar excitation after the ansatz is inserted the high-frequency components can be neglected and the solution remains a physical description of the system. Following the same procedure in the case of the octupolar excitation yields a non-physical description of the system. Unless a suitable coordinate transformation is found an analytical solution may not be possible. At present, numerical solutions to the equations of motion have to be used. By calculating the vectors 7i and using Eq. 4.1 it is possible to extract p+ and p- as a function of time. Since Er or pi, the radial energy pickup due to the application of an octupolar RF field is directly accessible. 57 4.3.2 Single-ion octupolar simulations Octupolar excitation of the motion of a single ion will be used to illustrate the simi- larities and differences between the octupolar and quadrupolar excitation schemes. Motional beating of single ions with VB F = 2V0 0 ' 012 ’ 0347 Trf [arb. units] 1 Octupolar “2 fl fl _, 0.8{ m 59- : g 06—; {2 0.4-5 3. : N + 0.2% °- 03.11 .I,..i.,..L- 0.6 0.8 i Figure 4.6. pi of an ion confined in a Penning trap and subjected to an azimuthal octupolar RF field at frequency 2116. Fig. 4.6 shows the variations of pi of an ion as a function of time subjected to an azimuthal octupolar RF field. The ion is initially in a state of pure magnetron motion. A periodic beat pattern is observed. Compared to the quadrupolar case (Fig. 4.1c) the beating is no longer harmonic but begins to approach a square wave in shape. According to Eq. 4.4, the beat frequency of an ion induced by a quadrupolar RF field with 11,. f = Vc is proportional to Urf1 and independent of the initial ion motion. This is not the case with the octupolar excitation. Fig. 4.7 displays the pi of a single ion, subjected to an octupolar field with l/rf = 2110, for different initial magnetron radii, p_,0, and applied RF amplitudes, Urf- The panels on the left side display the 58 p_ = po p- = 72/2 00 ..... 1Z- Urf - 2% 203:. t. C '2 : 3.0.6? :- N+O.2E- .- 0. 0 mg. J 1...11.11. -1L..1...1...1.L.J J 1 7 111111111710 E'nflmnr1n“”" H '_ L 'E 0.8E : 3.0.6? :- 3.0-“: ~.°—2siuUUtUUt 31111111111111. 0. Obi...1...1...1...1...1 -1...1...1...1...1nn.1 O 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Trf [arb. units] Trf [arb. units] Figure 4.7. The left hand panels show pi as a function of Trf for an ion with an initial p_,0 = p0 and p+,0 = 0 with applied RF amplitudes of Urf and 2Urf1 respectively. The right hand panels pi for an ion with an initial p_,0 = Aézpo and applied RF amplitudes of 2Urf and 4Urf1 respectively. beat patterns for an ion with an initial magnetron radius of po. Doubling Urf results in an increase in the beat frequency. The panels on the right side illustrate that doubling Urf and scaling the initial magnetron radius by \/2 / 2 preserves the original beat frequency. In the case of octupolar excitation the beat frequency depends not only on the excitation amplitude, but also on the initial motion of the ion in the trap. Fig. 4.8 shows the beat frequency as a function of the initial p_,0 for three different initial magnetron phases, ¢_,0. The figure reveals that the beat frequency depends not only on p_,0, but also on ¢_,0. The dip in the 45-,0 = 0° curve is not a numerical artifact, but has been verified by two independent simulations. By changing the phase one can change the position of this dip, but it quickly moves to a larger value of p_,o as one moves away from ¢_,0 = °. With (15,. f = 0° the dip occurs at p_ z 2.1 59 0.6- "'1 0.5{ 2“ . § . 3.. 0.4“ >. . g . a) 0.34 a o- . g . LL 0.24 a . 03 I m 0.1— 0‘....,..A.j-...,....,-4.+, o 0.5 1 1.5 2 2.5 p_ o [arb. units] Figure 4.8. Beat frequency between magnetron and cyclotron motions as a function of p_,0 for constant Urf and for three initial magnetron phases, (1)-,0 = 0°,22.5° and 45°. and ”beat/V- z 0.26. Fig. 4.7 might lead one to believe that Vbeat cc p_2_,0 ~ Urf, yet taking Fig. 4.8 into account the relation would need to be amended to Vbeat o< f (05-) ~p2_,0 - Urf° The effect of holding Urf and 05-,0 constant and plotting the beat frequency as a function of ¢_,0 can be seen in Fig. 4.9. The beat frequency seems to sample the octupolar RF field’s spatial orientation, and is at a minimum at the anti-nodes, and maximum at the nodes, of the field. According to Eq. 4.4 the beat frequency in the case of 12,. f = uc due to a quadrupo- lar excitaton is linear with respect to Urf and is phase independent. Again, this is different in the case of octupolar excitation. The beat frequency is not linear with respect to Urf, and is dependent upon the 45-,0, as is illustrated in Fig. 4.10 which shows the beat frequency as a function of Urf- As the beat frequency nears u_ the beat pattern begins to lose its periodic nature. Zooming in to low frequencies and 60 constant Urf 0.96] 0.922 0.88! Beat Frequency [arb. units] 0.843 0,3‘..4.,2.4-,....,. 0 20 40 60 80 100 ¢-,o [deg] rfi I I Y I Y Y I Figure 4.9. Beat frequency as a function of 05-,0 for a constant Urf- excitation amplitudes the (15.30 = 0° and ¢_,0 = 45° curves become indistinguish- able, while the ¢_,0 = 22.5° curve remains distinct. Again, similar to Fig. 4.8, a dip is observed in the (0-,0 = 0° case, again located at ”beat /1/_ z 0.26. We are now in a position to investigate a possible invariant of the motion. Fig. 4.11 shows the beat frequency when changing p_,0, while at the same time keeping the 2 product Urf - p_,o constant. It can be seen that the beat frequency is constant for a given initial phase, ¢_,0. This was found to be true for ions in a state of pure magnetron motion, and for mixed initial motions, p0 = \/p3_,0 + p2_,0, as well. Therefore, we arrive at a general relationship ”beat 0( a(pi,o, (1532.01 0)”) - pg - Urf, where a is a scaling factor which depends on all four of the initial p’s and dis. 61 0.004 ,_ 1'4“ 0.003 ‘2' 1.2; 0.002 ¢ =450 ,6 - >3 ,_ 0.001 ¢_=00 2:9 0'8_ ”,th 22.5 ,. g 0.6— C- 9 . U. 0.4— 8 0.2— m /‘_4, 0. I I I fir *I Urf [arb. units] Figure 4.10. Beat frequency as a function of Urf for three initial 05-,0’5. The insert zooms in to low beating frequencies and amplitudes. Invariant phase relation of the octupolar excitation According to Eq. 4.5, A05 = (1),]: — (¢+ + 0L) is a constant of the motion for a quadrupolar excitation. We investigate if a similar phase relation holds for the oc- tupolar excitation. Fig. 4.12 shows scans of pi as a function of ¢+,o and time for a given 03-,0 with qbrf = 0° in all cases. By comparing the patterns it is easy to conclude that the time dependence of pi is the same if ¢+,o + ¢_10 = const. It was verified that this also holds true if (prf is changed. Fig. 4.13 is similar to Fig. 4.12, except ¢+10 is held constant at 0° and the individ— ual panels show p3 as a function of ¢_’o and time. Here we verified that for 03+“, = 0°, érprf - qb_,o = const. the same time dependence of pi is observed. Testing this condition for many values of ¢+,0 yields the same results and leads us to conclude that the invariant phase in the case of octupolar excitation is A05 = %¢rf - (03+ + 62 1 ............................... I _______________________ 0,. [deg] i 3”? 1 _____________________________ _ 0 'E 1 —-- 5.625 3 0.95- D. , —..— 11.25 ‘— g 16.875 >5 . — — 22.5 8 0.9- Urf*(p )2=const. cu ] -.o 3 C. . o . h LL 1 45 035-] <9 co 0,3,24.,....,....,..22, 0.65 0.75 0.85 0.95 1.05 1)-,0 [arb. units] Figure 4.11. Beat frequency due to an octupolar excitation as a function of p_,0 holding p30 - U1. f constant for several values of ¢_,0. 0)-). This differs from the quadrupolar phase relation, Eq. 4.5, only in the factor of 1 / 2 multiplying the qSTf term. This appears plausible as the octupolar field has twice the number of nodes and anti-nodes as the quadrupolar field. The factor of 1/2 reflects that the ions do begin their motion at the same position with respect to the field orientation. Single-ion resonance curves At first, we will examine octupolar resonances of single ions initially in a state where p0 = p_,0. The general observation is a periodic change of pi as a function of Urf- Along the line Vrf = 2V0 a resonant effect is observed. Fig. 4.14 illustrates 63 9 <1, 0 [deg] 15 ’ ' ¢_,0=67.5° ¢_ 0 = 90° Time [arb. units] Figure 4.12. 2D simulations of pi as a function of time and (15+,0 for a given ¢_,0 with (prf = 0°. The grey scale is proportional to pi. the dependence of pi on frequency detuning, AV 2 Vrf — 2V0, and Urf for various values of 05-,0. At 05-,0 = 13.5° a secondary resonant effect becomes visible which sweeps through the primary resonant structure as the value of (1)-,0 is changed. This effect lies on a line with an origin at Urf = 0 and AV = 0 which rotates clockwise for increasing values of ¢_,o. At 05-,0 = 18.0° the structure can be seen across all three sections of the primary resonant structure. At d>_,0 = 22.5° the secondary structure lies on the AV 2 0 line. A close look at the resonances reveals a narrowing of the resonance profiles at Au 64 rr—f/i/E ¢rr=°° kt «11,, [deg] Time [arb. units] Figure 4.13. 2D simulations of pa as a function of time and 05-,0 for a given (1),. f with ¢+,0 = 0°. The grey scale is proportional to p3, = 0 for certain values of U1. f. Fig. 4.15 plots 2D cuts from the 05-,0 = 0° (left) and (15-,0 = 22.5° (right) cases shown in Fig. 4.14. Moving from the top panels to the bottom steps through the first conversion of magnetron to cyclotron motion. The ¢_,0 = 0° case exhibits a larger width in the radial energy distribution at the lower values of Urf'Trf as in (1)-,0 = 22.5° case. The central peak narrows for larger values of Urf'Trf and at Urf'Trf = 2.90 V-s the radial energy gained during the excitation begins to drop. The top three panels of the 45-,0 = 22.5° case show a suppression of radial energy at Au z 2Vc. The three bottom panels no longer exhibit 65 9 14A 6 1.; 41A A. .1. =00 3'1 I - -10 0 I I I I I I I I I fT I I I I I I I f1 91 '*'_——"- l — ——— = 13.5 3-1 3‘ I ¢ 0 0 T I T I I I I I I I I I I I I I I I I I 91 "‘— 7)‘ e. h ‘— l o I: 4—— = 18.0 a. 3- -__.._.L- “1° Pt 0 I l I l fiI I I I r I r I T In FI I I 91— ~ --_._ . 6‘ J \ 0 D 3": + ‘,O 0 I I I I I I I I I I I I I I I I I fI 1 9-, fl 1 6‘ %..— — o 34 -k 5,0 _ 27.0 0 I I I I I I I I 1 I I I r I I I Ifi I I 9‘]_———' I ——— 6" v O 3. _ .- 5'0 - 31.5 0 I In I I—I I I I I T I I I I r I I I I -6.37 -3.18 0 3.18 6.37 Av [1nd] Figure 4.14. Simulation of pi for a single ion with p_,0 = 1 mm and p+,0 = 0 mm as a function of frequency detuning, Au = 12,. f - 2110, and Urf for different values of ¢_,0. The grey scale is proportional to p3. this behavior, but the central peak continues to narrow. In both cases AVFW H M falls below 1/ (100-Trf), which corresponds to a factor of a: 200 increase in resolving power over the quadrupolar excitation for the same excitation time! It remains to be seen how well the ions can be prepared to reproduce these results under realistic conditions. 4.3.3 Realistic multi-ion simulations Simulations involving multiple ions representing a cloud are required to study the ion behavior in the case of an octupolar excitation under realistic conditions. First we will examine the radial energy gain as a function of Urf1 with 14. f = 2110. Gaussian 66 0.5—i WA 1.50 V*S _ 05 0: .. 7 ...... .,,.2-,--...., 0 1j 1‘ 3 1.sB5V* I .—. 0.5- 10.5 ‘gg i iAL 3 c 0‘. 2, ....,222,. ,...,...,2.2,...,..2,2.-,...,...‘o 3 11 11 .Q . >. 01...,2..,J..L.,...,.2-,.¥¥..- .. .. 1N... .JL ...:0 U) 1 '— 1 g : u.1 05‘; E 0‘...,../.,¥...,22.,.2, 'C 1‘3 (U 1 (I 05- l 1 o “2425., ,-,2,, 1 0.5 3.25V*s 0.5 0,...’,~...,...,...J,...,...,...7..., ,...fi..,...,...,..., ..,. .42. .,0 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 Av[1/Tn.] Figure 4.15. 2D cuts from the (15.2,0— 2 0° and 050,22— 2 225° 3D profiles in Fig. 4.14 for six different values of Urf'Trf- distributions were used in generating values for the initial ion cloud. Fig. 4.16 shows the results of three simulations for three different values of ap_’0 with p_,o = 0.55 mm. p+,0 = 0.050(5) mm was used, being a conservative estimate from simulated radial energy gain during injection into the Penning trap. This value also agrees with those obtained from fits to quadrupolar resonances obtained under similar conditions. Widths of the 05 distributions seem to have a small effect on the multi-ion response curves, and d)- = 0(5)° and 05+ 2 0(50)° were used. For the largest value of ap_,0 shown in Fig. 4.16 the curve seems to mimic the behavior of a damped oscillator, 67 converging to some average asymptotical value of radial energy. This is the result of the beat frequency being dependent on p0, as discussed in Sec. 4.3.2. The larger the spread in pa, the larger the range of beat frequencies. The distribution of beat frequencies determines how quickly the average beat pattern is damped. 1_ .. -._. op_’0=0.12mm “f \ -006mm ‘ .a' '..'-| \ _ _ Op-,O_ ‘ .— . i '..'- H _ op =0.03 mm .32 0-8" ’I ' ‘. \ .30 c ‘. 3' \‘\\ fl!“ Q < . E. 06— "|\ ". ll\/\\-‘|\ > I |_'-I :‘.,, [‘]|".'|lv..l'7 \" :\ ‘1 9 ' \"./.1‘ I“! I 4'} a) 04 pl ‘\,‘\..l ‘ '_1 l Iv” UCJ '_ J l ‘l I r \l 1 a , / ‘6 .- \ ”I «5 0.2- ,- ,/ “5 .' < l 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Urf*Trf[V*S] Figure 4.16. Simulated radial energy gain for three different values of ap_ o as a function of Urf for Vrf = 2Vc and fi_ = 0.55 mm. Fig. 4.17 shows the results from two octupolar simulations of 23Na+ with Trf = 50 ms. Both simulations were performed with identical phase and p+ distributions as the simulations shown in Fig. 4.16. E_ was also held constant, but the widths of the distributions were changed. The simulation shown in the first column was performed with p_,o = 0.80(3) mm and in the second with p_,0 = 0.80(13) mm. Instead of pi the calculated time of flight was plotted for easier comparison to experimental results 68 covered in the next chapter. The first row shows the dependence of the time of flight as a function of Urf and Au. The grey scale indicates smaller values of time of flight. The remaining rows are cuts through the data at different voltages. Note the resemblance of the top left plot to the single-ion case shown in Fig. 4.14. For the larger ap_,0, shown on the right, only one conversion is seen in the grey-scale plot before settling down to an average value. This is also reflected in Fig. 4.16. The 2D profiles also reveal that there is a greater separation in time of flight between the baseline and the minimum for the profiles on the left. In both cases as one proceeds from smaller to larger values of Urf the resonance narrows. On the right-hand side the narrow peak begins to develop on top of a broader resonant structure, while on the left the time of flight baseline doesn’t change. At Urf = 44.5 V the resonance curve on the left has achieved a maximum change in time of flight and a minimal width of % 0.14/Trf, corresponding to a gain of a factor of 13 in resolving power over the quadrupolar excitation. 4.3.4 Experimental procedure and results Fig. 4.18 presents a schematic drawing of the electronic setup which was used to produce an azimuthal octupolar RF field in the LEBIT Penning trap. An arbitrary function generator (AFG) prodived a signal to a broadband RF amplifier with 65 dB gain. The signal from the amplifier was fed into a phase—splitting coil. The two output signals, Urf,1 and Urf,2, which are 180° out of phase, were then fed into the ring electrode segments. The amplitudes of the two phases agree to within z 15-20 ‘70. From now on, their average will be quoted as the excitation amplitude, Urf- In the experiments 23Na+ ions were used. Ion bunches from the cooler/buncher were injected off-axis, via the Lorentz steerer, described in detail in Chapter 5, and trapped in the LEBIT high-precision Penning trap. 69 urf [V] 75' E. u. 0 l— 19 I I r l I ‘ I x 1 l_" '_'1 I 164 141 12‘ 38.4V -5.2.5 -3'5”-ii.s 0 ”17's 33'5'52'5 65535 V7193 ' o 1'7'.5 35 52.5 Av [1fl'rf] Figure 4.17. (Left): multi—ion simulation of TOF as a function of Urf and AV with three cuts at different RF voltages. (Right): same as the left only the width of the p_,o distribution has been increased by a factor of 4. Experimental octupolar studies in resonance 23Na+ ions were excited with an octupolar excitation with 11,. f = 2116 for Trf = 50 ms and their time of flight was measured as a function of U1. f. The result is shown in Fig. 4.19 together with a simulated curve. The simulated curve corresponds to the aP—,o = 0.13 mm case from Fig. 4.16. In order to get a good agreement between experimental data and the simulation results it was necessary to divide RF voltage values measured at the output of the circuit by a factor of 0.7. Such a factor makes sense as it accounts for RF attenuation and partial shielding due to the geometry of the trap. Using experimental data like that shown in Fig. 4.19 and comparing 70 Figure 4.18. Schematic drawing of experimental setup used to produce a azimuthal octupolar RF field in the LEBIT Penning trap. to the corresponding simulation results allows the values of 7)_,0 and ap_,0 to be determined. The minimum time-of-flight value achieved is a function, primarily, of fi_,0, and the damping of the curve is determined by the ratio of 0p_,0 and ‘p‘__,o. In the case of quadrupolar excitation, the beat frequency of pi (t) is proportional to the product Urf'Trf (see Eq. 4.4). We will experimentally explore if this holds true for the octupolar excitation, as well. Fig. 4.20 shows the time of flight of ions as a function of Urf for five different excitation times. As expected, the minimum time of flight is reached at a lower value of Urf for longer excitation times. The value of Urf where the TOF curve reaches its minimum will be labeled U0. Fig. 4.21 displays the product Uo-Trf for the fives cases shown in Fig. 4.20, along with simulated data using our best-fit parameters determined by the comparison shown in Fig. 4.19. Both the experimental and simulated data agree and are constant within 5%. This value will not be constant for all circumstances, and depends on initial conditions. By adjusting the Lorentz steerer we can control fi_,o. The displacement of the ions as a function of the applied voltage is linear. Fig. 4.22 shows six different octupolar scans of TOF as function of Urf in the case of Vrf = 2110 for six different voltages, V L, applied to the Lorentz steerer. The first minimum in the curves represent the 71 16— « - experiment ‘ — simulation 15~ 75' 3 . u. 14- O . '— 13~ ‘ i 0 40 80 120 160 200 240 280 urf [V] Figure 4.19. Time of flight as a function of Urf for a octupolar excitation of 23Na+ with T1. f = 50 ms at 11,. f = 2120. The solid line shows the results of a simulation. necessary Urf to, on average, bring the ions to a state of maximum radial energy. Fig. 4.23 contains the beat frequencies observed in Fig. 4.22 as a function of p_,0. Two of the curves are simulation results and one is experimental data. The experimental observation does not seem to confirm the nonlinear response illustrated in Fig. 4.8. However, the ap_,0 = 0.13 mm curve represents our best-fit simulation scenario, and matches the data quite well. The third curve reduces the best-fit value of ”IO—,0 by a factor of four to 0.0325 mm. Now the curve begins to recover the nonlinear shape shown in the single ion simulations. Again the variation of individual beating frequencies of ions in the cloud play a dominant role in determining the overall response of the system. 72 structure. o 20 4o 60 a'o1601éo urf [V] Figure 4.20. Octupolar TOF curves of 23Na+ as a function of Urf with Vrf = 211C for several excitation times. Experimental octupolar resonances Fig. 4.24 displays several octupolar resonances of 23Na+ produced with Trf = 50 ms for various values of Urf- Included in the figures are simulated resonances which were produced using the same initial conditions as the simulation in the right column of Fig. 4.17. For Urf S 40 V the width of the resonances are between 1.3/TH and 0.9/Trf. Proceeding towards larger amplitudes reduces this width. At Urf = 72.9 V a narrow resonant peak with a width of z 0-2/Trf forms atop a broader resonant 73 ‘L9— . -< O o O : . 6 5 _ 75' 1.8a 4g . ._>_. . t 1 7L 3 . f t g - experiment '— 1'61 o simulation 'LS .fi...... ....... .....-. .., 20 30 4o 50 so 70 1]1[nns] Figure 4.21. Product of U0 and Trf, where Trf is the duration of the octupolar excitation and U0 is the amplitude of applied octupolar RF field at which the TOF curve reaches its minimum. Mass measurements with octupolar excitation To verify that octupolar resonances can be used for precision mass measurements, a mass measurement of 41K+, using 39K+ as a reference, was performed. An excitation time of Trf = 200 ms was used in each individual measurement. Urf was chosen such that the resonances were Gaussian. Fig. 4.25 shows the results of this mass measurement. As there is no theoretical line shape all resonances were fit with a Gaussian profile. The difference of the mean mass value extracted from the octupolar measurements from the accepted literature values (Atomic Mass Evaluation [2]) is shown. The dashed lines represent the uncertainty in the mean of the experimental results. The solid lines represent the uncertainty in the AME values. As can be seen, there is excellent agreement within the uncertainty. 74 12.5 '5, :“WI-" VL=-4oov I I s..."1...,..,...., , 161,. In H a. 12.53 '1 ‘1, "‘ VL=-35OV 9 ...,....fifi,.fi-fi,.. , 16?. ‘l '67 12.5 ‘- a" VLz-soov 3 3 “'5" LL 9‘...., , fifijr..., 12.5% VI. = '250 V 9‘ .fi, .mr 161W 12.53 VL = -200 V 9‘....,....T....,...WW“, 167”“ l 9‘ .................. uFf [V] Figure 4.22. Time of flight as a function of Urf with AV = 0 using 23Na+ with Trf = 50 ms octupolar excitation for six different Lorentz steerer voltages, V L- The greater the magnitude of applied voltage, the larger the initial average displacement of the ions from the center of the trap. 4.4 Summary and conclusions Although an analytical solution to the octupolar excitation has yet to be found, nu- merical simulations combined with experimental results have offered significant insight into this complicated problem. The most important observations can be summarized as: o The beat frequency of an ion subjected to an octupolar RF excitation applied at UR F = 211+ + 2V- = 2V0 is dependent upon the initial conditions of the ion motion. This is in contrast to the quadrupolar excitation at uc where such a 75 0.03- I experiment '6' —— sim:op_=0.13 mm :2." — - sim: (1 =0.0325 mm c V 3 4 ‘5 0.02- t. .52. > 0 C Q) 8- . 9 0.01~ LL ‘6 Q) m 0 """""" l fl * fl I * Y ' ' Y ' ' V ' I 0.3 0.4 0.5 0.6 0.7 0.8 9-,0 [mm] Figure 4.23. Beat frequency as a function of fi_,0. The experimental curve is extracted from data shown in Fig. 4.22. The 0p_,0 = 0.13 mm curve is a simulation based on the parameters extracted from the simulation shown in Fig. 4.19. The second simulated curve was generated using identical parameters, except 0p_’0 = 0.0325 mm. dependence does not exist as long as p+,0 = 0. o The octupolar resonance profiles have a radically different shape than their quadrupolar counterparts. o For certain initial conditions it is possible to reduce the Width of octupolar res- onances by a factor of 10 or more beyond what is achievable with a comparable quadrupolar resonance performed with the same excitation time. We have verified a factor of 9 gain in resolving power over the standard quadrupolar excitation scheme. According to the simulation results presented in Fig. 4.17, a factor of four reduction in p_,o would yield a factor of 13 gain without the broad resonant structure observed in Fig. 4.24. We have also shown that the initial conditions of the ion cloud determines the ultimate resolving power that can be achieved. Simulations show even higher resolving powers are possible, provided that the ions can be prepared 76 171 TOF [us] 11: .52.;r.“.s1'.;=,' 3675163 3115' 35.5 vrf - 125395725 [Hz] Figure 4.24. Several octupolar resonance profiles of 23Na+ with Trf = 50 ms for various values of Urf (data points) compared with simulated results (solid lines). appropriately. Although more work is required to assess the absolute accuracy of mass measurements performed with octupolar excitations, a promising first step has been taken towards the implementation of octupolar resonances in high-precision mass measurements. Motivated by the phase dependence exhibited by the octupolar excitation, we have also revisited the quadrupolar excitation and confirmed a phase dependence of resonance line shapes. 77 1? - - - Octupole Uncertainty 0.75? —— AMEOS Uncertainty 31‘) . ._. 0.254 1 i' I ('0 Z a ———————— }—————— 2 ? " 1’ fit : :8 0.25? — — — ——————— ‘L_ — — _:|_ O . 2 -0.5{ -0.75-§ '1 l I I l I I O 1 2 3 4 5 Measurement Figure 4.25. Mass comparison of 39"l‘lK'l' using octupolar excitation. 39K+ was used as the reference and all measurements were performed with 200 ms excitation times. 78 CHAPTER 5 Development and detailed study of the Lorentz steerer TUT —-: , Q B’ O O -—I '— AHA. Figure 5.1. Cartoon illustrating a positively-charged ion passing through a region of perpendicular electric and magnetic fields, resulting in an off-axis capture in a Penning trap. MFG As mentioned in Sec. 3.1.3, before the application of a quadrupolar (or octupolar) field it is necessary for a trapped ion to be executing some initial motion. Normally this is achieved by first applying a dipole RF field at frequency 11+ to drive the ion’s magnetron motion. After the ion has been driven out to some radius the quadrupolar excitation will drive a beating between the magnetron and cyclotron motions. This 79 dipolar excitation requires some finite amount of time, on the order of 10 ms, to perform. Additionally, the RF excitation must be phase locked to the time of ion capture to ensure that for each individual measurement that the value of p_ , previous to the application of the quadrupolar RF field, is the same. Since every millisecond is precious when making mass measurements on short- lived species, a method that could prepare the ions in—flight would be very desirable. The Lorentz steerer accomplishes this by creating an electric dipole field before the Penning trap, but well within the strong magnetic field region. The ions experience a net force in the EXB direction which results in an off-axis capture in the Penning trap, resulting in an initial magnetron motion and eliminating the need for an initial dipolar excitation. This process is illustrated schematically in Fig. 5.1 5.1 Charged particle motion in a region of perpen- dicular electric and magnetic fields To begin we will consider the motion of a particle as it travels through a region of perpendicular electric and magnetic fields. Let R = BE and E = Eg’]. The equations of motion in the radial plane which must be solved are 72*: (Eg+f~'x 32). (5.1) Consider a particle initially at the origin traveling in the 5 direction with no initial radial energy. The velocity in the radial plane and the radial displacement is then 7"(t) = 1%ng — cos(wct) (5.2) r(t) = B£%\/2m2[1 — cos(wct)] + [Bth + 2mBqt sin(wct), (5.3) 80 where E is the strength of the electric field, q is the ionic charge, m is the mass, B is the strength of the magnetic field and we = (q/m)B. For large values of t, . E _ tl_1>rr.10'r(t) = E -t = ”drift - t, (5.4) where adrift is the average drift velocity of the ions. The displacement after traveling a distance L is proportional to t = L/vz 0( M. For example, Fig. 5.2 shows the radial displacement as a function of time for an A = 40, singly-charged particle as it travels through a region of perpendicular, uniform electric, E = 28.6 V/ m, and magnetic, B = 9.4 T, fields. This displacement is accompanied by a pickup of cyclotron motion which leads to the observed non- linearity. Fig. 5.3 shows the square of the radial velocity, which is proportional to 0.5 0.4 - 0.3 - 0.2 - 0.1 - Radial Displacement [mm] 0.0 I I 0.0 0.1 0.2 0.3 0.4 Tlme [us] Figure 5.2. Radial displacement as a function of time for a charged particle passing through a region of uniform, perpendicular electric and magnetic fields. the radial energy, of a particle as a function of wet. From Eq. 5.2 it can be seen that the radial velocity as a function of time scales as 1-cos(wct). If we consider the particle to enter and exit the electric field suddenly, then upon exit it would have a radial velocity between 0 and fiE / B, depending upon the time of exit. Having left the electric-field region the ion performs a cyclotron motion with a radius that 81 depends on the time of exit. For the example parameters used here the maximum radial velocity is VT = 4300 m/s, which corresponds to a radial energy KT = 5 eV and a radius of the cyclotron motion of p+,o = 0.17 mm. 2.0 ‘ v3 [252/32] 3 O l l 0 11' 211 311 411 Angle [tuc't] Figure 5.3. Square of the radial velocity as a function of wet of a charged particle after it has travelled through a region of uniform, perpendicular electric and magnetic fields. 5.2 Lorentz steerer design Lorentz Tube 1 Tube 2 Tube 3 Steerer (-2250 V) (~2250 V) (-200 V) Plates /W ('200 V i U1,2) WWI/k. “ ’ ’l/Iu/I/ Wm, I Figure 5.4. Isometric section of the Lorentz steerer and nearby optics elements. The dashed line is the axis. Typical operating voltages are listed. 82 The design of the Lorentz steerer is remarkably simple, being four segments of a cylinder quartered in the radial plane. Fig. 5.4 shows a design drawing of the injection optics of the LEBIT Penning trap mass spectrometer which contains the Lorentz steerer. The Lorentz steerer is located just upstream of a field termination plate, and after a series of drift tubes. The steerer electrode system is 16.5 mm in length and 28 mm in diameter with 3 mm gaps between the four segments. Typical operating voltages are listed. These voltages were also used in SIMION simulations discussed later. AL V X Figure 5.5. Schematic layout of the Lorentz steerer’s electrode configuration. Each pair of diametrically opposing electrodes can be used to create an electric dipole field. Both pairs can be used simultaneously to orient the dipole field in space. Fig. 5.5 shows a simple sketch of the electrode layout of the Lorentz steerer. Each pair of diametrically opposing electrodes can be used to generate an electric field with a dominant dipolar component. Changing the values of U1 and U2 such that ‘/U 12 + U22 remains constant results in a constant magnitude of the dipole field, but changes its orientation in space. This imparts a constant radial displacement with an 83 adjustable angle to ions passing through the steerer. 5.3 Lorentz steerer beam calculations 5.3.1 Ion cloud deflection simulations V [mm] 4440—6 -2 z 61014-14-10-6 -2 2 61014 x[mm] x[mm] a.) b.) Figure 5.6. Contour plots of the potential at the mid-plane of the Lorentz steerer given by a.) the analytical solution and b.) the SIMION calculation for U1 = 0 V and U2 = 1 V (see Fig. 5.5). Within a circle of radius of 6 mm the relative difference of the potentials obtained by both methods is less than 4x10—3. The ion deflection in the Lorentz steerer has been calculated both analytically and numerically. While the magnetic field is assumed to be homogeneous, the electric field inside the steerer is required. One method is to use SIMION [50] to numerically solve the Laplace equation in a geometry based on the design drawing shown in Fig. 5.4 and to trace the ions through the electric and magnetic fields. Another method to obtain the electric field is to analytically solve the electric potential for an infinite quartered cylinder. This potential is given by U(7‘, (15) = 27} [arctan (W) — arctan (W)] + g2 [arctan (W) + arctan (WM , (5.5) 84 where R is the radius of the cylinder and U1 and U2 are the applied voltages. In order to account for the finite length of the Lorentz steerer the analytic potential given by this equation can be multiplied by a function f(z). From a comparison with the potential obtained with SIMION a function f(z) = e(—z/a)4 with a = 1x10—2 m was found to fit the potential along the axis well. Fig 5.6 plots the mid-plane potentials obtained from the SIMION calculation and the analytical solution for U2 = 1 V and U1 2 0 V. Within a 6 mm radius from the center of the steerer the potentials agree within <4x10‘3. A series of simulations was performed, using both the analytical and the SIMION potentials, to study the properties of the actual Lorentz steerer. In the case of the SIMION simulations the geometry and potentials as shown in Fig. 5.4 were used. The initial ion distribution contained 500 ions and was centered on the axis of the Lorentz steerer. The Width of the initial distribution in the radial direction was 0.1 mm. The ions started their flight in Tube 1, shown in Fig. 5.4, with an axial energy of 2 keV and no radial energy. After passing through the Lorentz steerer their radial positions were recorded at the location of the field plate. Fig. 5.7 shows the mean radial displacement as a function of the potential ap— plied to U2 obtained from the SIMION simulations. As expected, the average radial displacement is proportional to U2. The slopes of these lines represent a steering strength which depends on the mass and axial energy of the ion passing through the Lorentz steerer. Fig. 5.8 shows steering strength values obtained from the SIMION simulation (like those shown in Fig. 5.7). The solid line is the result of a calculation using the analytical solution for the potential of the quartered cylinder. There is good agreement between the SIMION simulations and the calculations. 85 __ 1.2- E e A: 10 4}” .z 1.0“ O A=40 XI / é o A=70 ,, ,/ I, / § 0.8- ,<>' 0’ I / 9 / a 0.6“ 9’ 3 “’ x .3 o4_ a” ’l a: - ,Q) I, / é ,o/ g 0.2“ ”1’8" < ’1 / ,/ V 0-0 I I r I 0 so 100 150 200 250 U2 [V] Figure 5.7. Results of SIMION simulation showing mean radial displacement (data points) as a function of Lorentz steerer voltage (U2) for three different masses. Lines are the results of a linear regression analysis. 5.3.2 How to achieve minimum cyclotron motion? When injecting ions into a Penning trap it is important to introduce as little radial energy in the process as possible, as non-zero values of the initial cyclotron radius, p+,0, can introduce asymmetries in the resonance line shape [51] and reduce the mag— nitude of the resonance signal. Fig. 5.9 shows the radial displacement as a function of time of an A = 70, singly-charged ion for two different of axial energies. The steering voltage was adjusted to result in the same final radial displacement. The curves were calculated using the analytical potential. The solid line is the result for an ion with K = 200 eV of axial energy and a steerer voltage U2 = 400 V. The dashed line is the result for an ion with K = 50 eV of axial energy and a steerer voltage U2 = 200 V. The K = 200 eV ion exits the electric field region after z 1.25 as. The oscillations are due to the induced cyclotron motion (see Sec. 5.1), in this case leading to an amplitude p+,0 z 0.1 mm. The slow ion exits the electric field region after z 2.5 as and achieves the same mean radial displacement as the fast ion, but with a cyclotron 86 0| l h l Steering Strength [um/V] 1- K=200eV IIIII 010203040506070 Mass Number Figure 5.8. Steering strength as a function of mass number of a singly-charged ion with K = 200 eV axial energy determined from SIMION simulations. The solid line is obtained using the analytical solution for the potential. amplitude of p+,0 z 2 pm. Qualitatively this can be understood upon closer inspec- tion of Eq. 5.2. Slower ions require more time to pass through the Lorentz steerer and therefore require a smaller electric field strength E to drive them to the same final radial displacement as a faster ion. For the system employed in the LEBIT Penning trap mass spectrometer an axial energy of less than 75 eV reduces the radial energy gain to sub-eV levels for the maximum applied steering voltages of 400 V. 5.4 Lorentz steerer measurements 39K+ ions were created with a plasma ion source and delivered as short pulses using the LEBIT beam cooler and buncher [37]. After passing through the Lorentz steerer the ions were captured in the Penning trap. If the ions enter the trap off axis then upon being trapped they perform an initial magnetron motion. Application of a 87 2.5 E E. 1.5- , E - 200 eV 0. ,l u2 - 400 v .1 1.0- ,t O ,I _______ E - 50 eV 3 l.’ u2 - 200 v I E 0.5+ / ’I ’I ’I 0'0 l I I I l I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Tlme [us] Figure 5.9. Calculated radial displacement as a function of time for an A = 70, singly- charged ion as it travels through, and out of, the Lorentz steerer. The solid line is for a beam with K = 200 eV axial energy. The dashed line is for a beam with K = 50 eV axial energy. The steering voltages have been adjusted to result in the same final radial displacement. 7r-pulse with frequency Vrf % VC can be used to completely convert the magnetron motion into cyclotron motions as discussed above. The accompanied increase in radial energy which can be detected via measurement of the time-of-flight of the ions [44] from the trap to a detector located outside of the magnetic field. By taking the radial energy gain and the known electric and magnetic fields traversed by the ion on its path to the detector, the value for the initial magnetron amplitude can be determined. Fig. 5.10 (top) shows the time of flight of ejected 39K+ ions as a function of voltage applied to the Lorentz steerer electrodes after capture in the trap and subsequent quadrupolar excitation for a time Trf = 50 ms. U2 was held at a constant 84 V for the following reasons. Due to imperfections in the injection of the ions into the magnetic field of the Penning trap spectrometer steering voltages of U1 = U2 = 0 do not necessarily correspond to injection on the trap axis, and an initial steering by the Lorentz steerer may be required. The values U13 2 (~24 V, 84 V) correspond to 88 the maximum time of flight (and minimum initial magnetron amplitude). For ease of discussion we introduce offset-corrected steering voltages such that U1,2’ = (0 V, 0 V) corresponds to the maximum time of flight observed for 39K+ ions. Fig. 5.10 (bottom) shows the values of the initial magnetron amplitude calculated from the time-of-flight data in Fig. 5.10 (top) as a function of Ul’. The solid lines il- lustrate that the size of the magnetron radius p_,0 of the captured ions is proportional to U1’, as expected from the simulation results shown in Fig. 5.7. Fig. 5.11 shows the results for the initial magnetron radius p_,o as a function of the true voltage U1 obtained from measurements with three different ion species. The solid lines are linear fits to the data. The numbers in parentheses are the slopes of the lines which correspond to steering strength values (in pm/V). According to Eq. 5.4, the displacement respectively steering strength is proportional to «772 for ions with the same axial velocity. This is reflected in the experimental data and the measured steering strength values for 23Na and 39K also agree with the calculated values shown in Fig. 5.8. The Lorentz steerer should allow for precision control not only of the amplitude of the magnetron motion, but of the initial phase, as well. In order to investigate how well this control works, an experiment was performed according to the following procedure. 39K+ ions were placed off axis into the trap using the Lorentz steerer, resulting in an initial magnetron radius. Next, a dipolar RF field, at frequency V- which is phase-locked to the time of ion capture, was applied. Depending on the phase of the RF field the excitation of the magnetron motion will result in a larger or smaller final magnetron radius. Using a 7r-pulse with 11,. f = VC and the time-of- flight measurement allows the amplitude of the magnetron motion after the dipolar excitation to be determined. Fig. 5.12 presents the results of such a measurement. The magnetron amplitude p- of 39K+ ions is plotted as a function of the initial phase of the dipolar RF field for four different Lorentz steerer settings. Each setting 89 corresponds to a magnetron phase change of 90° from the previous setting and should provide the same steering strength. The data were fit with a sinusoidal function (solid line), p_(q$) = p_,0 + Ap_osin(¢,.f - 05-,0), where p_,0 is the initial magnetron amplitude introduced by the Lorentz steered, Ap- is the maximum amplitude change due to dipolar excitation, (1),. f is the initial phase of the RF, and ¢_,0 is the initial magnetron phase. The results, given in Table 5.1, show that the initial magnetron amplitudes, p_,0, due to the effect of the Lorentz steerer alone, agree to within a few percent. The phase change of 90° from one case to the next is confirmed to within i2°. Both of these results together confirm that the ions were moved on a circle in the radial plane of the Penning trap and demonstrate the precision control of the initial ion placement in the trap available with the Lorentz steerer. Table 5.1. Summary of the results of a fit of a sinusoidal function to the data presented in Fig. 5.12. ULQ’ are the offset-corrected voltages applied to the Lorentz steerer, p_,0 is the initial magnetron amplitude given by the fit, ¢_,0 is the initial magnetron phase given by the fit, and Ad) = [052,0 — 052:3] is the phase advance from one setting to the other. U1’ [V] U2’ [V] p—,o [mm] d>—,o [deg] A2 [deg] 1 -100 0 094(2) 102(1) 87(2.2) 2 0 +100 096(2) 189(2) 93(22) 3 +100 0 093(2) 282(1) 91(2.2) 4 0 -100 096(1) 13(2) 89(2.2) A new technique for the precise manipulation of ion injection into a Penning trap has been developed and tested. It is used routinely in LEBIT’s Penning trap mass spectrometer. Compared to the alternative method of magnetron preparation via dipolar excitation, it is less complicated and requires no additional preparation time. The Lorentz steerer offers complete control over the injection process and 360° placement of an ion cloud in the radial plane of a Penning trap. We expect that such 90 a Lorentz steerer is a likely successor to dipolar excitation for magnetron preparation in future Penning trap systems. It is already planned to be used in the TITAN [52] Penning trap mass spectrometer, presently under construction. 91 E : §§§ 27- : é : '5" J 5 i 5:} a; 25 I : 3 E : fr . s 0.0 [ I I 400 -200 0 200 400 U1' [V] Figure 5.10. Top: Time of flight of 39K+ ions ejected from the trap after being subjected to a quadrupolar RF field with a frequency Vc as a function of U1 with U2 = 84 V. Bottom: Initial magnetron amplitude, p_,0, as calculated from the time-of-flight data shown above. U1,2’ are offset-corrected values of U13 such that for U1,2’ = (0 V, 0 V) the value of p_,0 is minimized. Lines are to guide the eye and to illustrate the linearity of steering with applied voltage. 92 x e 23Na(2.4) + 39K (3.4) x x 8590(49) Figure 5.11. Initial p- as a function of U1 for three different ion species. The numbers in parentheses are the steering strength values in um / V. 93 0. [mm] 1.60' 1.201 0.80 ‘ 0.40 . 1.60‘ 1.20‘ 0.80 . 0.40 1.00‘ 1.20‘ 0.40 1.60‘ 1.20‘ ' 200 300 (Pd [deg] U1.= -100 V, UZI=OV 1 331!\ u,‘ = +100 V,U2' = 0V 3 I u,' = 0v. U," = -100V 4 l I 0 100 400 Figure 5.12. Magnetron amplitude p- as a function of ¢rf of 39K+ ions subjected to a dipolar RF field at frequency 11. for an excitation time of Trf = 50 ms. The choice of steering voltages is such that the ions should be captured on a circle at angles 90° apart. Solid lines are sinusoidal fits to the data. The fit results are summarized in Table 5.1. 94 CHAPTER 6 Mass Measurements of Stable Krypton Isotopes Krypton beams from the test ion source have been used extensively to characterize the performance of the LEBIT Penning trap mass spectrometer in the regime of medium heavy masses. Mass measurements utilizing 86Kr as a reference mass, measured to high precision by SMILETRAP [53], were performed on 78’80’82’83’84‘Kr, revealing somewhat surprising results [54]. Initial LEBIT measurements had indicated that the AME03 [2] mass values of 83’84’Kr were too large. To explore this discrepancy in greater detail an additional series of measurement were performed in which 4-5 cyclotron frequency measurements were made of 78’80’82Kr'l' and 11 cyclotron frequency measurements were made of 83’84Kr'l'. Each of these individual measurements was bracketed by a cyclotron fre- quency measurement of the reference ion, 86Krl‘. The time separating each reference measurement was approximately half an hour. During the measurement of any spe- cific Kr isotope, possible contamination from the other isotopes was cleaned away via dipolar excitation as discussed in Section 3.1.3. Table 6.1 provides the frequency ratios obtained in the run with their statistical uncertainties in parentheses, and an 95 additional uncertainty of 1x10—8, added in quadrature, in curled brackets. This additional uncertainty accounts for any additional systematic uncertainties that we have not been able to rule out. Using the known mass value for 86Kr and the frequency ratios obtained here, mass values for 78’80’82’83’84‘Kr were obtained using an analysis application, SOMA, discussed in Appendix B, which I created to calculate final mass values from fits of the cyclotron resonances. The final mass values are listed in Table 6.2 together with the AME03 [2] value and the difference between these values. This difference is shown as a function of mass number in Fig. 6.1. 4 2 - I’f" _____ I,_._/’ i I. 0— [ [ -21 “DELETE—“L \___ MLEBlT—MAMEOS (keV) L 78 80 82 83 84 Figure 6.1. The difference between mass values measured with LEBIT and AME03 [2] for the stable krypton isotopes. The error band corresponds to the uncertainty of the literature values, the error bars to the uncertainty of the mass values determined in this work. Excellent agreement is observed for the three isotopes 78’80’82Kr within measure— ment uncertainties. The mass values for these isotopes are known with very high precision and are dominated by data from other Penning traps. A significant but 96 equally large deviation is observed for 83Kr and 84 Kr for which the masses have been determined by many different and partially inconsistent data (discussed below). The observed averaged relative deviation of the LEBIT results from the AME03 values is less than 6 X 10‘9, if 83 Kr and 84’Kr are excluded. As a part of this measurement se- ries, mass comparison of 86Kr+ and 39K+ yields a mass-dependent systematic effect of less than 10‘9/u. An analysis of the data used in the Atomic Mass Evaluation [2] shows that mass values for 83 Kr and 84Kr are determined via a network of Q-values from decays and reactions combined with data from doublet mass measurements. The mass values for 83Kr and 84'Kr are linked strongly by an (my) reaction [55] which has an uncer- tainty of 0.3 keV. The Q-value agrees within 1.5 a with the value calculated from the LEBIT data. The absolute mass of 84Kr is determined primarily by the Q-value of the fl-decay 84Rb(fl'l')84‘Kr [56, 57], and the result of a doublet mass spectrometer measurement, C6H12—84Kr [58]. The mass of 83Kr is primarily determined by the link to 8‘l‘Kr, but also by a doublet mass measurement C6H11—83Kr [58]. Two recent measurements of 84Kr are not included in the AME03. One was made at Florida State University, FSU [59], and another [60] at ISOLTRAP [61]. Fig. 6.2 displays the deviation of mass excesses of 83’84Kr, as determined by individual measurements, from AME03 values. The three measurements made using Penning traps, FSU, ISOLTRAP and LEBIT, of 84“Kr agree very well. The doublet measurements of both krypton isotopes in question also agree with the Penning trap measurements, but may have a slight systematic bias towards heavier masses. The mass excesses calculated from the results of the 6 decay measurements are obtained by using the AME03 values of the parent nuclei. The fact that they do not agree well with the Penning trap measurements could indicate that there is one or more incorrect input data used in the AME03 in the region of 84Rb and 83Br. Assuming that the AME03 mass values for 83Kr and84Kr are suspect, this provides great confidence in 97 + :Br(p‘)83Kr +3Rb(p*)8‘Kr 2 _ Kr—CeH ‘ 1 _ Kr-CSH1 2 —e— LEBIT —e— ISOLTRAP o- - —v— FSU —e— LEBIT _2 . -4 - Mmas-MAM E03 (keV) _ l . 1 l I Figure 6.2. Deviation of individual mass measurements of 83’84Kr from AME03. Values obtained from 6 decays use AME03 values for the masses of the parent nuclei. the data obtained with LEBIT. 98 Table 6.1. Cyclotron frequency ratios R = V(AK7+)/u(86Kr+) obtained in this work. Column 1 contains the atomic number of the krypton isotope. Column two contains the weighted average of the measured ratios with the statistical uncertainties in parentheses and an additional 1 x 10’8 uncertainty, added in quadrature, in curled brackets. A R 78 1.102 544 461(9){14} 80 1.075 006 812(9){14} 82 1.048 797 277(9){ 14} 83 1.036 139 857(9){14} 84 1.023 824 21g8){13} Table 6.2. Mass excess values ME for krypton isotopes with mass number A as obtained from the measured frequency ratios and compared to their AME03 [2] values. A ME L E B IT keV ME AM 1903 keV AME keV 78 —74179.4(0.9) —74179.7(1.1) 0.3(1.4) 80 -77892.4(1.0) -77892.5(1.5) 0.1(1.8) 82 -80590.4(1.1) -80589.5(1.8) -0.9(2.1) 83 -79991.2(1.0) -79981.7(2.8) -9.5(3.0) 84 -82438.8(1.0) —82431.0(2.8) -7.8(3.0) 99 CHAPTER 7 High-Precision Mass Measurement of 38Ca and its Contribution to CVC Tests The Conserved Vector Current (CVC) hypothesis asserts that the vector part of the weak interaction is independent of the nuclear interaction. This means that the vec- tor coupling constant, G1), is truly a constant and does not require renormalization. This constant, when combined with the purely leptonic muon decay constant, G F, determines the up—down matrix element, Vud: of the CKM quark-mixing matrix. Measuring the ft values of super-allowed 0+—+ 0+ transitions allows Cu to be deter- mined. A precise determination of Vud has been, and continues to be, an important component in the test of the unitarity CKM quark-mixing matrix and the search for physics beyond the Standard Model. In order to determine the ft values the decay half life, the branching ratio for the 0+ —> 0+ transition, and the Q EC values need to be known. Small radiative and isospin-breaking corrections have to be applied to determine a corrected f t value that can then be used for the test of CVC and the determination of vari- 100 To date there are 12 well-known CVC test candidates that provide an average .7 t value with a relative uncertainty of 3.7x 10—4, as presented in a recent survey [1]. Nine of the candidates are close to the valley of stability and their Q EC values were measured in reaction experiments. I will refer to these as classical candidates. Penning trap measurements of 0+ ——2 0+ emitters now allow us to determine high-precision Q EC for isotopes which were inaccessible in the past. Three such species, 22Mg [62, 63], 34Ar [64] and 74Rb [65], have recently been included in [1]. The addition of more candidates is important for testing the CVC hypothesis and for benchmarking the calculation of the theoretical corrections. Assuming CVC is true, the accuracy of the theoretical corrections to the ft values can be tested and their calculation can be improved. N uclides with large theoretical corrections include the even-Z, Tz = —1 nuclei with 18 g A g 42 and odd-Z, Tz = 0 nuclei with A 2 62. 22Mg, 34Ar, 74Rb, and 38Ca, are among them. Penning trap mass measurements can also be used with advantage to revisit the classical cases. A Penning trap measurement of 46V [66] with the CPT spectrometer at AN L found a significant deviation from the literature Q EC value of the 0+ —> 0+ decay of this nuclide, previously determined by an average of reaction measurements. A recent measurement with JYFLTRAP [67] validated CPT’s result for 46V. The JYFLTRAP group also measured the classical candidates 42Sc, 26Alm and showed that speculations made in [66], that more reaction experiments might be wrong, could not be substantiated. 7 .1 Experimental procedure The data presented in this thesis were taken during two separate experiments in which the same primary beam was utilized. The first experiment was dedicated to studying the stopping and extraction of Ca ions from the gas cell, mass separation techniques, and identification of 38Ca2+ ions in the Penning trap. A primary beam of 40Ca at 101 140 MeV/u was produced by the CCF and reacted with a 460 mg/cm2 beryllium target. The secondary beam was purified by the A1900 fragment separator [35] and delivered to the gas cell as a cocktail beam consisting of 50% 38Ca, 35% 37K and 15% 36Ar. A series of range measurements were performed to study the stopping distribution of ions in the gas cell. A Si detector which can be positioned in the beam path was used to identify the ions passing through the gas cell by time of flight and energy loss. By varying the angle of the glass degrader the energy loss experienced by the ions passing through the degrader changes. Scanning the degrader angle and measuring the activity collected on the Si detector with and without buffer gas in the gas cell provided information on the fraction of the beam which was brought to rest in the gas cell. After the optimal degrader angle was found the next step was to determine the chemical form of the 38Ca activity extracted from the gas cell. Using the mass filter the activity was collected on the needle electrode as a function of the mass- to—charge ratio, A/ Q, and the results are shown in Fig. 7.1. Peaks in the measured activity appeared at A/Q values of 28, 37, 46 and 55. This pattern is consistent with water molecules attached to 38Ca2+ ions to form [38Ca(H20)n]2+, with n = {1,2,3,4}. The appearance of water adducts was not particularly surprising due to the relatively-high water partial pressure in the gas cell resulting from an earlier venting and cleaning of the gas cell electrodes. The Q = +2 charge state is also to be expected as calcium has a second ionization energy of only 12 eV, while the first ionization energy of helium is 25 eV [68]. We chose the ions with A/ Q = 28, corresponding to 38Ca(H20)2+, to be transported to the cooler/buncher. It was found that collisions of the 38Ca(H20)2+ ions with the helium gas in the cooler were energetic enough to dissociate the 38Ca(H20)2+ ions into its constituent components, 38Ca2+, A/ Q = 19, and H20. This breakup allows for a very efficient suppression of undesired stable molecular ions with A / Q = 28 which were transported along with the 38Ca(H2())2+ 102 1.2 - 38Ca(H20)22+ ‘ r 1-0- 3°Ca(H20)2+ [f]- ' l L. °'°‘ ‘. : 38Ca(H20)42+ 1 . 0.6. .‘ Activity (arb. units) 0.4 - g 02- fi [E 0.0 . . . . . . , . , . . . , . , 25 30 35 4o 45 50 55 60 NO Figure 7.1. Activity as a function of the mass-to—charge ratio A/ Q of ions extracted from the gas cell and selected with the mass filter. A 38Ca secondary beam was stopped in the gas cell. Lines are to guide the eye. The radioactive molecular ions assigned to these peaks are indicated. ions. The probability of a breakup of the contaminants into fragments with A/ Q = 19 was found to be small. Therefore, the time-of-flight mass separation using the fast beam gate between the Penning trap and the cooler/buncher was very efficient. A few cyclotron resonance measurements of 38Ca2+ were performed in the Penning trap during the first run to evaluate if a precision mass measurement of 38Ca would be possible. The second run was dedicated to the high-precision mass measurements of 3738 Ca. Fig. 7.2 shows one of the 38Ca2+ measurements taken during the second run. In this measurement an RF excitation time Trf = 300 ms was used, which resulted in a resolving power of R = 2x106. A total of twenty-one resonance curves were ob- tained for 38Ca2+. High-level contaminating molecules were identified by measuring 103 cyclotron frequencies over a broad frequency range and using SCM_Qt, described in Appendix C, to determine the molecules. In all of the 38Ca2+ measurements, with the exception of the first seven measurements in this run, all conceivable contami- nants were cleaned via dipolar excitation and the fast beam gate was used to suppress possible contaminating species with different A/ Q values. Additionally, the number of ions simultaneously stored in the trap was kept at about 1 ion/cycle to avoid frequency shifts due to interactions with possible contaminants that were not being cleaned. Taking the MCP efliciency into consideration this was achieved by keeping the average number of ions detected per cycle to S 0.4. H3O+ ions from the gas cell were chosen as the reference species to calibrate the magnetic field. Alternating measurements of 38Ca2+ and H3O+ were made. Because H3O+ has the same A / Q value as 38Ca2+ it is an ideal reference species since it minimizes mass-dependent systematic effects. The same precautions were taken with for H3O+ cyclotron measurements as with the 38Ca2+ measurements, such as cleaning of contaminants, single ion storage, etc. Reference measurements were taken on the order of every 20 minutes. The twenty one 38Ca2+ measurements taken over a period of 22 hours were necessary to achieve the mass uncertainty required for a CVC test candidate. Each measurement consisted of the detection of approximately 300 ions. 7 .2 Experimental results As mentioned in Sec. 3.1.5, the primary experimental result of a mass measurement using the LEBIT mass spectrometer is the ratio, R = Vc/Vc,.,.ef, of the measured cyclotron frequency, Vc, of the ion of interest and that of the reference ion, 11¢,"3 f. Each 116 measurement should be bracketed by two 116,”3 f measurements which can be used to make a linear interpolation of the magnetic field. The cyclotron frequencies 104 24- _22- .3 L<5 20- I'- ”Ca2+ 18- I -10'"'-5""0""5'”'1b VRF [Hz] - 7595522 Figure 7.2. Cyclotron resonance of 38Ca2+ measurement taken with an excitation time Trf = 300 ms. The solid line is a fit of the theoretical line shape [3] to the data. and their statistical uncertainties are determined by fitting the measured resonance curves using a theoretical line shape [3], as shown in Fig. 7.2. Other sources of uncertainty to be considered will be discussed later. 8.0x10‘- 33 2 « + g» 4.0x1o°- [ i f Ca (0 m M—eriwfi—ri— ' 1 i E f [I -4.0x10“‘- f '8-0x10—B ' I ' I ' l ' l ' l ' l ' l ' l ' I T" I fl Figure 7.3. Deviation of individual frequency ratio measurements, R = Vc/Vc,ref, from the mean. The error bars correspond to the statistical uncertainty. In the first run, which served as a feasibility test, 38Ca2+ was identified in four cyclotron frequency measurements during the last 16 hours of the run. These data are not used in the final mass analysis since contaminants were not identified and cleaned. The fast beam gate had also not yet been installed, and the time separating 105 the reference measurements was on the order of hours. An RF excitation time Trf = 200 ms was used only on the first resonance taken during the second run. The remaining resonances were obtained using an RF exci- tation time Trf = 300 ms. The first two resonances were also taken with the fast beam gate off and therefore exhibit a higher average rate of detected ions per cycle, _ = 0.27, 0.37 compared to a rate of 71‘ = 0.07 — 0.22. During the 14 final measure- ments the cleaning scenario was in place and nothing was changed. Fig. 7.3 displays all of the frequency ratios obtained during the second 38Ca2+ run. The mean value of all data taken during the second run, including those with larger count rates and incomplete cleaning is R = 1.001 592 097(3). 7 .3 Uncertainty analysis Experimental data obtained in various online and offline experiments were used to determine possible sources of uncertainty. The online data were analyzed for uncer- tainties associated with nonlinear magnetic field effects, systematic frequency shifts due to ion-ion interactions, and relativistic effects. The offline results include a mass measurement of 40Ar2+ using 23Na+ as a reference (both available from the test ion source), to test for any unforeseen uncertainties associated with mass measurements of doubly charged ions. 7.3.1 Analysis of online results Magnetic field strength calibration The linear interpolation of the magnetic field assumes that nonlinear magnetic field changes are insignificant. The greater the time span between reference measurements the greater the probability that nonlinear changes in the magnetic field become sig— nificant. Fig. 7.4 shows the magnetic field as a function of time, as determined 106 6E-07 994 4E-07 J - 990 T.‘ Q 3E-07 - g m - 988 E < - _ ._. 2E 07 a - 986 15-07 - OE-I-OO - i' 984 (AB/B)ldp = 4.5x10‘8 mbar'l ‘1 5'07 T I I I I 982 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time [days] Figure 7.4. Variations of the magnetic field and atmospheric pressure vs. time. Data points are the relative change in the magnetic field as obtained from the reference measurements taken over the course of the second run. The Wunder line are pressure readings (obtained from a local weather station) near East Lansing, MI. The Setra line is pressure data taken with a precision barometer in the lab. from cyclotron resonance measurements of the reference molecule, H3O+, over the course of the second run. The solid lines show the measured atmospheric pressure as a function of time obtained from a weather station and from a high-precision, absolute—pressure gauge brought into operation during the run. One can easily see that the magnetic field strength and the atmospheric pressure are correlated, an effect common to the type of superconducting magnet system used. Atmospheric pressure changes affect the internal pressure of the superconducting magnet’s liquid helium bath. These pressure changes within the cryostat alter the boil-off rate of the helium, changing the temperature of the bath. It is suspected that this change in tempera- ture changes the magnetic susceptibility of materials used in the construction of the 107 superconducting coils [69, 70], thus changing the total magnetic field strength. The atmospheric pressure dependence has since been removed by maintaining a constant pressure in the helium cryostat using an absolute-pressure gauge and electropneu- matic valve operated on a PID loop. The internal pressure is now stable to about 10 ppm. The uncertainty of the ratio of measured cyclotron frequencies due to the appar- ent nonlinearity in the temporal change of the magnetic field has been analyzed. The measured reference data were fitted with a polynomial function over the time span in which the 38Ca2+ measurements were performed. Reference points were then gen- erated for every minute along the fit line. Using these data points the deviation of the interpolated magnetic field strength Bint from the actual magnetic field strength B were determined. Choosing a time At between two reference measurements and stepping through the points the deviation AB = B - Bint was recorded. From the distribution of AB and including the statistical uncertainty of the cyclotron frequency determination a standard deviation s(AB) was obtained. Fig. 7.5 shows the variation of s(AB) / B, as a function of the time At between any two reference measurements. For times S 1 hr, s(AB) / B was found to be constant and is dominated by the statis- tical uncertainty of the cyclotron frequency measurement. In order to obtain an estimate of the effect of non-linearities in the magnetic field changes a maximum time separation At = 1.5 h was chosen (note that 38Ca reference measurements were performed every 20 min). A linear fit to the data at times less than 1.5 hr resulted in a slope of 2(1) x 10— 10 hr‘l. The largest uncertainty yields a value of 3x10‘10 as an estimate for the maximum systematic error, which is negligibly small. 108 205-08 1.5E-oe- . ' In . ' q o 3' , ' 1.05-08- , ' 5.0E‘09 l l I l o 1 2 3 4 5 AT [hr] Figure 7.5. The magnetic field strength nonlinearity parameter, s(AB) / B, as a func- tion of At between reference measurements. Effects of contaminating ions and background events Even though the number of detected ions per cycle was less than one for all of the 38Ca2+ measurements in the second run, the first two 38Ca2+ measurements in the second run did exhibit higher average counts per cycle. Fig. 7.6 displays the count rate information for the second 38Ca run. The average number of counts / cycle for all of the measurements of 38Ca2+ z 0.15, while the maximum was % 0.37. These measurements were sorted into count rate bins and used to calculate the mean frequency ratios. This procedure allowed identification of frequency shifts associated with residual contamination. Table 7.1 presents the mean frequency ratios obtained for the four bins. The frequency ratios in the first three bins, containing 19 of the 21 measurements, agree perfectly. The bin containing the measurements with the highest count rate exhibits a possible shift towards lower frequencies. A shift towards lower frequencies, i.e. heavier masses, has been found previously to be indicative of the presence of contaminating masses [71]. These two measurements with the highest count rate are among the first seven measurements of the second run. Evidence 109 0.40 z x incomplete cleaning 0 0.35- - used in analysis T: g 0.304 "" i g 0.25- 8 31 0.20- E I I E 0.154 I III II " I 3111 > < 0.10- 82 I I X 0-05 r I w I o 5 1o 15 20 25 Measurement Number Figure 7.6. Average counts/ cycle for each 38Ca2+ measurement made in the second run. The first seven measurements have been excluded from the final analysis due to possible contamination and high count rates (see text). of contamination disappeared after cleaning the possible contaminants via dipolar excitation was initiated. Therefore, we excluded the first seven measurements from the final mass analysis. A linear regression of the mean frequency ratios obtained from the three bins with the lowest count rates yields an additional uncertainty of 4x10“9 due to possible contamination. Relativistic efl'ects Relativistic mass shifts will always occur to some degree in the cyclotron frequencies. As long as both the species of interest and the reference mass are a mass doublet and have the same initial magnetron radius, p_,0, then the relativistic effects cancel in the determination of the frequency ratio. This should be the case as both 38Ca“?+ and H30+ have the same A/ Q = 19. The ion optics for the injection of the ions into the trap should be the same, and the ions should be captured into a magnetron orbit with the same amplitude. The initial magnetron radius is one of the parameters of the 110 Table 7.1. Count rate information for each 38Ca2+ measurement of the second run. The bins are average counts / cycle. N is the number of measurements included in the bin. Bin N R = Bc—S'eLf x3011 5 1.001 592 097(11) 0.11< 10—9 from the ratio obtained from literature [2] values. 7 .4 38Ca mass evaluation A theoretical line shape [3] was fit to the 38Ca2+ and H30+ cyclotron resonances, and the resulting fit files were entered into SOMA. Discarding the first seven mea- 112 surements of 38Ca2+ of the second run and using the remaining 14 we obtain R = 1.001 592 097(3). Within rounding the same value obtained using all of the 38Ca measurements. As an additional systematic error we quadratically add the sum of 3x10‘9 for uncertainties in the magnetron radii and 4x10‘9 for the presence of con- taminating ions to the the statistical uncertainty and get a mean ratio of R = 1.001 592 097(8). The atomic mass value of 38Ca was then calculated by m(38Ca)=2-R—1(m(H30)- me+b1+2me-b2), where bl = 5 eV [72] is the first ionization energy of H30, b2 = 17 eV is the sum of the first and second ionization energies of 38Ca, and me is the mass of the electron. The mass of H30 is known with a precision of 6m/m< 10—9 and its uncertainty is negligible. The result is a mass excess for 38Ca ME = -22058.53(28). This value agrees with the current literature value, ME = -22059.2(4.6) [2], but is over an order of magnitude more precise. 7 .5 Impact of improved 38Ca mass on the precision of the .7: t value of the 0+ ——>0+ decay Using the mass excess of the 38Km 0+ state from [1] an improved Q EC = 6611.7(4) for the 0+—> 0+ transition of 38Ca to 38Km can be calculated. The fvalue depends on the Q EC value to the fifth power, so that the uncertainty is (6f/ f) z 5(6Q EC / Q EC)- The uncertainty in f introduced by the Q EC using the improved 38Ca mass value is only (Sf/f = 3x10‘4, which is a factor of 12 smaller than the previous uncertainty. Fig. 7.8 shows the relative uncertainties in f, (if/f, of the 12 most precisely known 0+—> 0+ decays from Hardy and Towner’s most recent survey [1], along with the relative uncertainty in f of the 38Ca decay using the new LEBIT mass value. The new 38Ca mass value reduces the relative uncertainty in f of the 38Ca decay to the same order as that of the most precisely known super-allowed 0+——> 0+ decays. 113 2.5E-03 4. 205.032 1.5E-03- S 10 1.0E-03- + 5.05-04J + + + + + + + + + + + 0.05100 IIIIIfiTfiIfiII 00 56‘s g>¢ G 0 V Mother Figure 7.8. The relative uncertainties in f, 6f/ f, due to the uncertainties in the Q EC value of the 12 most precisely known 0+—»0+ decays [1] and 38Ca using the new LEBIT mass value. Fig. 7.9 shows the .7: t values for the 12 most precisely known 0+—+ 0+ decays. The solid lines represent the uncertainty in the mean f t value calculated from these results. The precision limit of the f t value for 38Ca due to the improved mass value from this work is also shown. The absolute f t value for 380a is not yet determined as the branching ratio is unknown and the uncertainty in the half life is large. Results of measurements recently performed at Texas A&M [73] and ISOLDE [74] may reduce the half life uncertainty and determine the branching ratio. In the meantime, the mass of 38Ca has been measured at ISOLTRAP with less precision [75], but the value agrees with the mass value obtained with LEBIT. 114 3080 ,. 3085-1 QEC precision limit " )l f38ca .———. 30304 0 .fl. «13 l" [ 14., 1‘ 3075‘i 1 V “ )r K i 4 in it 30705 [ .. J~ [ 3035 r I I I I I I I I I I . '90 \I10 (0% 'f figégg W9? ”Wasn‘t-Q? Figure 7.9. The 12 most precise .7: t values from a recent review [1] along with the 38Ca precision limit due to new mass value (see the text). Solid lines represent the uncertainty in the mean .7 t value. 115 CHAPTER 8 Summary In 2000 the initial design work on the Low Energy Beam and Ion Trap project at the Coupled Cyclotron Facility of the NSCL was begun. By 2004 the system was complete and commissioning of the entire apparatus was begun. In May of 2005 the first mass measurement of a radioactive species, 38Ca, was performed. Thus it was proven that radioactive beams created via fast-beam fragmentation can be prepared with excellent beam properties and high purity which are required for high-precision mass measurements. In the future it is envisioned to make these low-energy purified beams available for other experiments such as laser spectroscopy for the study charge radii and moments of rare isotopes or for post-acceleration for nuclear astrophysics experiments. In the five years required to bring LEBIT to its fully operational state, many students, post-docs and faculty have contributed to its success. My Ph.D. work emphasized the design, construction and commissioning of the LEBIT 9.4 T Penning trap system, the exploration of the octupolar RF excitation, the surprising 83’84Kr mass measurement and the mass measurement of the short-lived 0+ —> 0+ 6 emitter, 38Ca. The design of the LEBIT high-precision Penning trap required extensive simula- 116 tions to reduce the electric field imperfections while simultaneously minimizing the effects of magnetic field inhomogeneities introduced by the materials used to construct the trap. Several iterations and a close collaboration with the design department was required. Many systematic studies of the 9.4 T Penning trap system were performed with stable species to assess the absolute accuracy achievable, and has been reduced to 3 10‘9. Improvements were also made along the way to actively compensate the natural magnetic field decay during long measurements a counter-current introduced via a coil wrapped around the bore tube. Magnetic field changes were further reduced by stabilizing the pressure of the magnet’s liquid helium bath. The first radioactive mass measurement of 38Ca was performed in May of 2005, followed up with another experiment in July 2005 where 37Ca was also measured. The analysis of the 38Ca data was presented in this thesis and a final mass excess of ME = —22058.52(17) was obtained, which agrees with the current literature value of ME = -22059.2(4.6), but is about 30 times more precise. An experiment at Texas A&M university, led by John Hardy, was recently performed to determine the half life and branching ratio of the 38Ca ,8 decay, which is necessary to calculate the .7: t value. The results are forthcoming. Since those first experiments the LEBIT group has been working hard to extend the mass measurement program. In a year and a half of operation a total of 26 radioac- tive masses have been measured: 37’38Ca, 64,65,66Ge, 40’4‘1’42’43’4’4S, 66’67’68’80As, 29’34P, 63’64Ga, 68,69,70’817’1’819 Se, 33Si, 70m,7lBr. This is a very succeszul begin- ning of a mass measurement program which is expected to contribute not only to our understanding of nuclear physics, but also to make important contributions to other fields, such as nucleosynthesis and tests of fundamental interactions and symmetries. 117 APPENDICES 118 APPENDIX A Solution to the Electric Potential of an Infinite Quartered Cylinder Here I present the solution of the electric potential of the infinite quartered cylinder. Using the solution an analytic expression for the electric field can be calculated and used as an approximation for the electric field produced by the Lorentz steerer in- troduced in Chapter 5. The easiest method for solving the electric potential of the quartered cylinder is to first solve for the halved-cylinder potential and then sum over four different orientations to arrive at the solution to the quartered cylinder. The process is illustrated in Fig. A.1 To begin, the general solution to the Laplace equation in polar coordinates is given by V(r, (1)) = [a + b - ln(r)][A + B - (25] + Z [anrn + bar—n] [An cos(n¢) + Bn sin(nq§)] . n=1 (A1) The following list of boundary conditions must be met in order to physically describe the system: 0 V is finite at the origin —> b = bu = 0. 119 We: r'-.- r. - a. . ‘1'. -v./2<>+v./2 + O = ':'<>o +V112 +v' -V1 +V1 , = r \ +V2/2 0 «“16 -V2\ 4v. -V2/2< >+Vzlz + m = f \ v _v\ J, -sz2 2 Figure A.1. Cartoon illustrating the method used to solve for the electric potential of a quartered cylinder. 0 V is an odd function of 05 ——> A = An = 0. o V(r,0) = V(r,27r) = 0 ——2 B=0 and n is integer. The application of these boundary conditions reduce Eq. A.1 to V(r‘, 4)) = 2 Burn sin(n¢). (A.2) n=1 By using the orthogonality of sin(n¢)sin(m¢) over the range of 0 to 27r the Eu coefficients can be solved for on the boundary of radius R at a potential sz1. 371. = f Vsin(n¢)dq§ z f(jr V1 sin(n¢)d¢ — 7?“ V1 sin(n¢)d¢ : 2V1[1 — cos(n¢)] R” f Sin2(’n¢)d¢ R" f027’ sin2(n¢)d¢ anrn (A.3) 120 The first few terms of the expansion are 4V 31 mi 32 _ 0 4V] BB _ 37rR B4 _ 04V B = A4 5 5M ( ) 8271. = O, n >42V B = 1 2 2n+1 (2n+1)7rR2n+T’ n > Now use (rei¢)n - (re—w)" = 2irnsin(n¢) and rewrite Eq. A.2 as 4V1 °° zn—(z'n)* V = —— 7rz' Z n (A 5) n=1 where z = (rem). Using the sum 1 00 1 +11 n3 n5 — = — — A. 2:;In(1_n) n+3+5+ (6) allows us to rewrite Eq. A.5 as 2V1 1+2 Tei¢ V——7r—Im[ln(1_z)],z— R . (A.7) By summing over all of the configurations show in Fig. A.1 the final solution for the potential of a quartered cylinder is V = 121m [ln(1]—i—:-) +ln(a];%)] + (A.8) £21711. [In (ll—1%) +ln(1}%)] wherez=x=flzé£,y=flgW—/22,w=y=tfl—gfl. 121 Expanding the logarithm terms and converting back into cylindrical coordinates, the potential can be written as __1 2Rr Sinlibi _1 2Rr cosgqb} [tan ( R -—r ) _ tan ( R -r ):l + V _1 2Rr sinfjd)? _1 2Rr cosy)! 772 [tan ( R —r ) +tan ( R -r )] =45 V(Ta d» : (A.9) 122 APPENDIX B SOMA documentation B. 1 Introduction SOMA, Simple Online Mass Analysis, is a program designed for online analysis of Penning trap mass measurement data. Ease of use, accuracy of results and easy in- tegration with software currently used at LEBIT, i.e., Mass Measure and Eva (main- tained by Stefan Schwarz), were the primary design considerations. SOMA can be used independently of these pieces of software if the measurement data follows the Eva data formats. With LEBIT there are three steps in the data acquisition and analysis processes: 0 Use Mass Measure to collect time-of-flight spectra as a function of applied RF frequency and write the results to a data file. 0 Use Eva to process the data files from Mass Measure and make fits of the time- of-flight spectra and write the results to a fit file. 0 Use SOMA to process fit files from Eva to calculate a final mass value. Eva outputs two ascii files from fits generated from data collected with Mass Measure, and use the extensions .log and .ft2. Either of these files can be used with 123 SOMA. The .log file contains more information, but none of it is used in the SOMA evaluation. SOMA was written in Qt and has been compiled for Windows, OSX and Linux. Qt is a C++ development framework which is free for open source development and can be downloaded from www.trolltech.com. A folder named massdata must be located in the same folder as the SOMA bi- nary file. Within this folder are two text files which SOMA requires for operation. The awm03.txt file is the AME03 [2] data file and e1ement.names.txt contains the elemental abbreviations and proton number. Both files must be present or an error will be presented upon opening SOMA. B.2 Getting to Know SOMA B.2.1 The main window Upon opening SOMA the user is presented with the window shown in Fig. B.1. The main window is separated into two sections, the calibration menu on top and the measurement menu on the bottom. Valid calibration files, *.log and *.ft2 as provided by Eva, can be dragged and dropped into the Calibration Menu, or the Load File button can be used. When two or more calibration files have been loaded SOMA will display magnetic field decay information to the right of the data file list. Any number of measurement species can be analyzed at one time. In order to load a new measurement species a new mass tab must be created. In the measurement menu use the Add Tab button to create a blank tab. Now drag the files corresponding to a single species into the newly created tab. Do this for each measurement species. Fig. B.1 shows that 25 86Kr+ files have been added to the Calibration Menu and 5 mass tabs have been added for the other stable krypton isotope measurements. Tabs can be removed using the Remove Tab button. Highlighting rows in either 124 _e Q j: sou - sample Online Mass Analysis v.2 ‘ Calibrationllem ..--_ ___ _ -_ _.-_ -__- _-.-_ _ -__- _ _ ,_ ‘____-_\___ l h TWM __ 159“” °L ""“"“‘.JL£“T""‘r _-- W i @ s-neldoocay 3060121 422- men ms 1 29-Jan-2006 14: 32: 24 -7.S66160-06 2 060129- 426- krfiRZ W 1 29-Jan-20M 15: 06: 33l Chi’ 3 060129- 426- Irr66. R2 066 1 29-Jan—2006 15:25:10 2.760330-09 4 060129- 430- kr66.R2 Kr“ 1 29-Jan-2w6 15:37:59 v D! ( Loadruoj (0.1m MD __--_-- Insults man man man Kr631+$~kl41+3w~ rm ll noon- ‘9' norm] Iaoo(moaslrc0 fl 7 04.6 (mm m] “0129- 465- kr64. R2 29—Jan-2006 21: 44: 47 1923624210112. 1976-06); -62436.497(1.7560) ll “3060129- 467-kr64.f12'29—jan-2006b 21:56: 31 1. 02362419640. 273.46) —62437. 457(1. 6193) la] 060129- 469- kr64. R2 29481-2006 22: 09: 01 1. 0236241656t2. 3462-06) —62436. 629(1. 6777) 34.060129- 471- kr64. R2 29—Jan-2006 22:16:17 1. 0236242134(2. 3559-06) 62-436.750(l. 6640i “359,115:- .. . _ ( Addle ) (Delete WsD (WTab) Luna File ) ( Calculate \) Figure 8.1. The SOMA Main Window which is presented upon opening the appli- cation. menu section and pushing the corresponding Remove Row(s) button removes the highlighted files. Pressing the Calculate button will calculate the results of the mass measurements which have been loaded into SOMA. Each mass measurement must fall between two calibration measurements. If any mass measurement lays outside the range of the loaded calibration files then a warning will be displayed and that measurement will not be included in the final analysis. Due to the manner in which SOMA calculates the final mass, all measurements in a mass tab must have the same calibration species for the first calibration measurement. The second calibration measurement does not need to be the same element / molecule as the first measurement as the ratio of 125 frequencies is only calculated with the first calibration measurement. The first mass measurement in a mass tab determines the first calibration species. For any additional mass measurements in the tab which do not share the same first calibration species a warning will be displayed and that measurement will not be included in the final analysis. If the user is interested in seeing how the final analysis changes by excluding certain calibrations or mass measurements, simply highlighting the rows containing the unwanted measurements and pressing the Calculate button will recalculate the results without having to remove the files from SOMA. To undo simply removing the highlighting and press the Calculate button again. The results of the mass analyses are presented in the results tab in the Measurement Menu. There are two menus in the menu bar, File and Tools, each containing several actions. First I will cover the actions in the File menu. The New action clears out all files that are currently loaded into SOMA. Save will save an ascii file with a .par extension which contains all the files, uncertainties and electron binding energies that are currently loaded into SOMA. Load loads a .par file. Export will export data in an ascii format for importing into another program. Upon selection Export a dialog box with three check boxes is displayed. The check boxes available are Calibrations, Measurements and Results. Checking these boxes will choose what is to be exported. Pressing the Save button will opens another dialog where a base filename and location is to be chosen for the exported data. The Tools menu contains three actions, Uncertainties, Info and Plots. Each of these actions open a separate dialog box which will be covered in the next sections. B.2.2 The uncertainty window The uncertainty window (Fig. 82) contains two tabs, Uncertainties and Electron Binding Energies. The Uncertainties tab contains a table with a row 126 00" 7 mummmm f 00" Waning-mac... WW ”M Thomas - 7 Jinn-RE W 7,“... - -_._. _,,-.-- .“- .__.- . - ., . III-6mm Club-boson: _ noun-o. 04-1-. woo/n, mm.? , , I. {linen 0 o 0 . fl m“""“‘ l I ”nob—um) our. 0 o o _},""'°.:'° fl'fi-"ro - *' "not. 0 th 0 0 9 .. .1 00314.0 0 0 fwx°l° ”unto o o 3"'“°-° ‘ ‘ 5' [c.4160 “so...“ 0 fag—m , wow» '0'“ ”T—fl _ fl " "' ' - — — — ' (Ia —— Sums: 0 l ) :0. '7 l__ °fi_) Figure 8.2. The SOMA Uncertainty Window accessed through the Tools menu. for each mass tab in the main window. There are three columns, Mass Dep. U nc., (dB/dt)/hr and Systematic U nc.. Each column represents a possible uncertainty which can be added into the final mass analysis. The Mass Dep. U no. is in units of relative deviation per mass unit. This shifts the final mass value and adds an ad- ditional uncertainty to correct for identified mass—dependent effects. (dB / dt) / hr ac- counts for magnetic field fluctuations over time as described in [30]. Systematic U nc. is an additional systematic uncertainty, such as the known precision limit of the spec- trometer, applied to the final analysis and won’t be reduced by additional statistics. Three line edits appear below the table and are used to apply the same values for each of the three types of uncertainties to each measurement species. The Clear button zeros out all uncertainties. The Electron Binding Energies tab contains two separate tables, one for the calibrations and one for the mass measurements. Here you can enter the total electron binding energies for inclusion in the final mass analysis. Two Clear buttons zero out the entered values. 127 A A A Measurement and Calibrat-on Info ~— --—-——— -— --—{ calibrations measurements?- -—-—--— .-_.________-,-__, w- Calibration File Number ”7*”? 2 File Name 060129.436Juaan2 2 End Date 29-Jan-2005 :1 Species ms :3 End Time 16:31:20 1: Mass Number (A) as 2 Low Freq (Hz) 1678215327 '3 Charge (e) 1 '2 High Freq (Hz) 1678220327 :3 Start Date 29-Jan-2005 'j SpF 5 '3 Start Time 16:26:25 ‘2 Mom Ions 959 2 Mid Date 29-Jan-2006 ‘2' Frequency (Hz) 16782le :3 Mid Time 16:28:52 '2' Uncertainty (Hz) 0.0174 F "“"”"E'EA'H M 'F _——-—;_e; “will": "WWW l 060129_422_kr86.ft2,16782186432 0.0135, 0 2 060129_426_kr86.ft2l1678218454600203 3 060129_428.kr86.ft2f16782184270 0.0189- 4 O60129_430_kr86.f(2f 16782183857 0.0183 5 060129_432_kr86.ft2f16782183287 0.0194 pr 6 060129_,434_kr86.ft2f1678218306400171 Figure 8.3. The SOMA Info Window accessed through the Tools menu. B.2.3 The info window The info window (Fig. 8.3) provides detailed information of the calibration and mass measurement files loaded into SOMA. There are two tabs at the top of the window, Calibrations and Measurements, which allow you to switch between the two file types. In order to view the measurement files the proper mass tab must be selected in the main window. In both cases, selecting a number in the combo box which corresponds to the number of the calibration or mass measurement as shown in their respective tables displays all the information for that given file. By checking the check boxes next to the parameter descriptions you can toggle if the information for all files is shown in the table. 128 B.2.4 The plot window ‘ 80”: W WW WW ,, W WW WWP‘otsW W W WW WW , W WW WW A 7 ,. "TCalibratiorTs l mum-1 guilt?- ‘1 2—1 0 ‘ AME Ur‘t’. (keV) ‘ 2.805 2 i 6 _2_ M a _ Meas. Unc. (keV) 3 _4_ 0.922 6 - l Mus—AME03 (keV) _‘ ’ I -6.857 - D 4., . Birge Ratlo to ‘ 0.5522(02384) Figure BA. The SOMA Plot Window accessed through the Tools menu. The plot window contains three tabs, Calibrations, Measurements and Results, each of which displays a different plot. The calibration plot uses the calibration measurements to plot the decay of the magnetic field and fits it to a linear function. The measurement plot plots the individual measurements from each mass tab, shown in Fig. B4. There is a combo box on the right hand side which can be used to switch between the different measurement species. The y-axis is the energy difference, in keV, of the measurement value minus the AME03 [2] value. The solid red lines represent the uncertainty in the AME03 value while the solid blue lines represent the uncertainty in the measurement mean value. A box to the right of the plot displays some relevant information, such as the AME uncertainty, the measurement uncertainty, the difference of the means and the Birge ratio [76] of the measured data. The results plot plots the final mass value for each mass tab along with the AME03 129 uncertainty. The y—axis is again the energy difference. B.3 Analysis Methodology In this section I will cover the methods SOMA uses to arrive at a final mass value given a set of measurement and calibration data. The mass calculation routine is a loop over all entries in an individual mass tab. If there are multiple mass tabs then the loop repeats until they have all been calculated. Table 31 lists the constants and their hard-coded values which are used by SOMA. Table B.1. SOMA constants and values Constant Value 7r 3. 14159265358979 me (keV) 51099898565154 U (keV/amu) 931494.013 The first step in the analysis is to read the date and time of the measurement and find the two nearest reference measurements with respect to time, one before and one after. If the measurement does not lie between two references it is ignored and a warning is issued. Once the two references have been determined a linear interpolation of first reference’s value of Vc is performed based on the magnetic field defined by the Vc’s of the two references. The uncertainty associated with this interpolation is given by the uncertainty in the x2 fit [77] and will be labeled org-”tarp. Next the ratio of the measurement and calibration frequencies calculated and defined as u R = M, (13.1) ”C(Tef) The uncertainty in this ratio is defined as 2 2 OR = R. m + m (B 2) V0 (interp) l’C(meas) 130 An additional term must be taken into account when considering the uncertainty in the ratio. The nonlinear magnetic field term, 03“), described in Sec. B.2.2, is introduced in the corrected ratio uncertainty as oR(CO,.,r)=\/o2 oR+ +(At aw) m2, (B.3) where At is the time separation between the two reference measurements and (IBM is the mass-dependent uncertainty specified in the uncertainty window. Once this has been calculated for all measurements in a mass tab the mean ratio is calculated by Zmeas —_2R—_ (corr) Zmeas ‘7— (corr) R: (13.4) In order to determine if the measurement data scatter is statistical, the Birge ratio is calculated, defined by ratio of the outer over inner uncertainties. The outer and inner uncertainties are defined by 2 23mm (-——— R R 1) ”out = R(corr) (3.5) (N _1) Zmeas __21—— \ (corr) a,” = 1 . (8.6) Zmeas —_%—) The uncertainty in the Birge ratio as given in [76] is 0.4769/\/T_l. A Birge ratio of 1 means that the measurement fluctuations are purely statistical. While if the ratio is greater than one it could indicate that there are additional systematic uncertainties which aren’t accounted for. A Birge ratio of less than one could indicate that the uncertainties of the measurements have been overestimated. The mean ratio, R, needs to be corrected for any mass-dependent shifts inherent in the system, and is defined by Rcorr = R + 0M ' 57' (Ameas — Aref): (B7) 131 where o M is the mass—dependent uncertainty. Using Reorr the corrected mass of the measurement species can be calculated as (Imeas qre f where qref is the charge state of the reference, bref is the total electron binding energy Mmeas = 'Rcorr ‘ (Mref ‘ qref ' me + bref) + Qmeas ' me — bmeas, (B8) of the reference ion’s missing electrons, qmeas is the charge state of the measurement species and bmeas is the total electron binding energy of the measurement ion’s missing electrons. The absolute final uncertainty is given by ”abs = 02'2” "l" (Usys ' RP, (3.9) where osys is the additional systematic uncertainty specified in the uncertainty win— dow. The final mass uncertainty is then ”M = \/(Mref ' Jabslz + (Rcor?~ ' 0M,ref)2a ' (310) where o M ref is the uncertainty in the mass of the reference species in amu. ’ 132 APPENDIX C SCM_Qt Documentation C.1 SCM_Qt Intro SCM_Qt (Search for Contaminant Masses) is a program designed to identify contam- inants in a Penning trap. It is based on a similar command-line—based program first written by Stefan Schwarz. By measuring the cyclotron frequency uc of the contami- nant ion and comparing it to a reference ion it is possible to generate a list of possible contaminants. In order to run the data- files folder must be located in the same directory as the SCM binary. This directory contains four ascii files which SCM uses during normal operation. The awm03.txt is the AME03 [2] data, elementJist.txt is a list of chosen elements and will be described later, e1ement.names.txt contains the elemental abbreviations and proton number and nubtab03.txt which contains the NUBASE03 nuclear data compilation [78]. SCM_Qt was written in Qt and has been compiled for Windows, OSX and Linux. Qt is a C++ development framework which is free for open source development and can be downloaded from www.trolltech.com. 133 0.2 Using SCM_Qt 0.2.1 The main window __6 O n SCM - Search for Contaminant Masses v.2.1 — —— ---------- —— --------- ~91“ ltesulrs Isotope List few fi- . __«" Frequency (Hz) 3590456 Uncertainty (Hz) 5 l l l ,..._.___R¢f“m_.___¢ m... {in ____,_ W“ _W __‘ raements H Ge Eu Species 1K39 He As Gd Li Se Tb Frequency (Hz) 3600000 Br , Dy m Kr “ Ho CW 1 7; - Rb Er & Tm Y IQKL'U ’ Zr ' Nb ' Mo immmmmms “R _ m , ‘ Ru ' Rh Pd All n n. fl‘fiQWL" ..._ Element Max 5 ‘ ‘UNI‘I‘NHUHlfil 'NUI .‘Ul ‘HNUI'HI .‘UI' Q secs N :‘ .3; .PNrd ..‘ Zn .. Pm ' U ‘: Multi Charge 2 Ca 2 Sm Np 1 ;, 1 : :3“:qu Figure C.1. The SCM_Qt Main Window which is presented when opening SCM_Qt. The main window, as shown in Fig. 0.1, is the first thing the user sees when launching SCM_Qt, and where most of the work will be done. The Main Window consists of 4 sub-areas: the Reference Mass area, the Contaminant Mass area, the Search Options area and the Elements area. Each of these sub-areas will be 134 described in full detail in the following sections. The Reference Mass area is where the user enters information concerning the reference mass calibration which was in use when the contaminant mass frequency was determined. By comparing the ratio of the frequencies of the contaminant ion and the reference ion it is possible to determine the mass of the contaminant using the mass of the reference species. The reference species is entered using the Mass Measure convention N(E1)A, where N is the number of atoms of one specific element, E1 is the elemental abbreviation (caps sensitive) and A is the atomic number. Elemental entries are separated by a colon. For example, if H20 was the reference then in the species edit the user would enter 2H121016. If the entry is valid then a formatted string will appear in the bottom of the Reference Mass sub-area displaying the nuclides’ information. In the frequency edit the user enters the cyclotron frequency of the reference species. The charge box is used to select the charge state of the reference species. The Contaminant Mass sub—area is used to enter information concerning the contaminant ion that the user would like to determine. In the frequency edit the user enters the cyclotron frequency of the unknown ion. The uncertainty in the frequency is then entered into the uncertainty edit. The Search Options sub-area is where the search parameters are set. The ele- ment max box sets the maximum number of unique elements to be searched for and currently has a 5 element hard-coded limit. The multi—max box is used to the the maximum N value for each element in the search. If the user would like to add unsta- ble isotopes to the search then they would check the unstable check box and enter the minimum half life for the isotope that they would like included. If the user would also like to search for multiply charged isotopes then they would check the multi-charge check box and set the minimum and maximum charge states to be searched. Note that the more options which are used will increase the time to complete the search. 135 The Elements sub-area determines the elemental species which will be included in the search. Upon opening the program all elements are chosen. Removing a check next to the element will remove it from the search. This applies to any unstable isotopes if they are included. The clear button removes all checks and the select all button applies checks to all elements. It is possible to save and load a set of elements. The Save Elements action in the File menu will write the checked elements to the e1ement.names.txt file mentioned previously. The Load Elements action will reload the saved elements. At the bottom of the main window are two buttons, Generate List and RU N l. The Generate List button list populates a list of searchable isotopes based on the search options settings and the checked elements. The RUN! button starts the search. One does not need to use the Generate List button previous to running the search, but doing so allows manual exclusion of individual isotopes and will be described in the next subsection. C.2.2 The isotope list window After the Generate List button in the Main Window has been pressed the Isotope List Window (Fig. C2) displays all the isotopes to be used in the search, along with its half life and mass excess. To exclude individual isotopes in the list from the search simply highlight the row which contains its information. To include it in the search again, deselect the row. C.2.3 The results window After the RUN! button on the M ainWindow has been pressed the matches appear in the Results Window (Fig. 0.3). Since the search algorithm and the GUI run on separate threads the GUI is able to sort the matches as they’re found. They are listed in descending order of distance from the cyclotron frequency of the contaminant ion 136 i0 9 "‘ SCM - Search for Conurnlnent Masses 17.2.1 :1 :l : 11.31.95 TI :U'S‘F'E 7: Till..':i'”:§§£“4" 1 1 H mu 72.0.9705 2 7 2 H stbl 13135.7216 3W 3 He stbl 7149312147 7 4 He 5th 24243150 f; f5 Li srbl 814086.7930 is ‘7 11 stbl 1490a1410 f7 9 Be at» 113476430 7 l 10 B stbl 12050.7310 fat: 11 a stbl 06673310 11012 c 5th 0.0000 £11 .13 c stbl 0125.011} Fr , . 112 14 N stbl 28614170 “:1 15 N 5th 101.4331 1 u“ 16 o 111» 4737.00“ fIS 17 o ’5th ,_“‘1m 2 Generate Ust l RUN! l Figure 0.2. The SCM_Qt Isotope Window which lists the isotopes to be used in the search. specified in the Contaminant Mass sub-area on the Main Window. The matches are named using the Eva/ Mass Measure convention so they can be copied and pasted into Mass Measure for cleaning, if necessary. C.3 Additional Tools 0.3.1 The mass fragment calculator The Mass Fragment Calculator (Fig. 0.4) is used to determine what molecules are possible within a specified mass region. The user enters in a mass and tolerance, in amu, and presses the RUN! button on the Mass Fragment Calculator Window to 137 16, O f‘ __ ___._-5°':_§ezsh_'etm!sggrlé_--- - —— ----- ’ Main M—l IsotooeList’~~———- - - —- r— _. '*‘.‘.__. _..._*". "—2'2;mu4 _ lg, .5085“ To: E f 4" E411. settle; ; 1'; 3H111H222He312fle421Ne20 1 -0.4ss1 Eislrlauzallerzlns _ l 1 l1.0991 — 31H1:3He3:1l110;1r19 l ’1 72.1790“ 7 fr 3H1:1He3:6He4:ille9l ’1 f-2.6es§ if is 1H1:1H2:2us:1u7:1o17 1 1342.9106 3; 1H122He3z2le9zlfll4 i 1 l-usos ‘21, 1511121141117; 1010 i 1 i 4.166; f. 7H1:1He4:1Li7:1Ne21 1 fuses rezg..m.‘-.;..-+...........‘_..‘..'...-...e......_..;.‘.:.. ....... if??? (Generate List) (' RUN! ) Figure C.3. The SCM-Qt Results Window which lists the matches generated by the search algorithm using the specified search parameters. generate a list of molecules. The search options are specified in the Main Window, as covered previously. The same rules apply to isotope list generation and exclusion. It is a good idea to keep the isotope list as small as possible to reduce the total number of combinations returned. (3.3.2 The frequency separation window The Frequency Separation Window (Fig. O5) is used to calculate the difference in the cyclotron frequencies, and mass in keV, of two species. The first species used is that which appears in the Reference Mass sub-area in the Main Window, and is also shown in the Reference Species sub—area in the Frequency Separation Window. 138 '6 01.1w fie:‘r99:¢.9'ww_- __ Mass (Iran) 60 Tolerance (emu) :1 Mung- _ _ _. -_ e e T :l:-- “We“: . _:___ L 3135 ,1 f5c12 0 .2 ’ 2H2:2C12:1$32 i 4.0002741 :3. 3H1:1c12:1r114;1p31 (40.000310 25:“ 41115 i-o.oo04351 If: 1c13;1n14:1016:1017 7 10.000475; ism 102-211151012 7 (0.0006212 :7: 2011016401: H H Off-0.000705. 11H 3n1:1c12:1c13:1532 ‘ 0001099 V sill—Jimzxuaomalrlg lifli-ol001153 : E — .. ........ w... ,_ 7 -- ——(--— a ‘3. e Figure 0.4. The SCM_Qt Mass Fragment Calculator Window calculates which molecules fall within a given mass range. The other species is entered in the line edit in the Measurement Species sub-area. If a valid string is entered then the Calculate button will be enabled and pressing it will calculate the results. 139 I e O 0 Frequency Separation __Be!e_r_snss_$_2e5l¢s___ _ Measurement Species Species ‘1 1C1311N 15 .l... H m- Charge 1 g _3e_ference - Measurement -383.428 Hz 2500.385 keV (Celeelere) Figure OS. The SCM_Qt Frequency Separation Window which calculates the differ- ence in the cyclotron frequencies between two species, as well as the energy separation. 140 APPENDIX D Fit procedure to minimize higher-order electric field terms The optimum voltages to be applied to the Penning trap electrodes are found by a fitting routine. It is a variant of the traditional X2 fitting routine [77] where a variable function is fit to a data set. Here we have data sets which correspond to the potentials along the trap axis from each electrode pair with 1 V applied. Each data set can be multiplied by an arbitrary factor corresponding to an applied voltage. The total potential along the trap axis for any given point is 4 v (B.3) i=1 where y,- are curve fit points. The first sum is over each value of z, and the second sum is over the unknown potentials. The best fit is that which minimizes Eq. D.3. This is found by solving (‘3 N 4 0=— a. ”£23. -2” 12.) X1011) (DA) k i=1 j1= Now we let Aij = Xj(zz-), b,- = yz- and a = (a1,a2,a3,a4). Substituting these back into Eq. D.4 yields 4 0: Z ink( 2i) —ZaJ-X j(Zi )Xk (2i) 2 AT — (AT - A) -a (OS) We can then solve for a and obtain —1 a = (AT - A) . (AT - b) (D.6) Provided that the inverse exists we have a method for calculating the optimum elec- trode potentials. 142 F BIBLIOGRAPHY 143 BIBLIOGRAPHY [1] J. Hardy, I. Towner, Phys. Rev. C. 71 (2005) 055501. [2] G. Audi, A. H. Wapstra, C. Thibault, Nucl. Phys. A 729 (2003) 129. [3] M. Kénig, G. Bollen, H.-J. Kluge, T. Otto, J. Szerypo, Int. J. Mass Spec. Ion. Proc. 142 (1995) 95. [4] M. Bradley, J. Porto, S. Rainville, J. Thompson, D. Pritchard, Phys. Rev. Lett. 83 (1999) 4510. [5] C. Thibault, R. Klapisch, C. Rigaud, A. M. Poskanzer, R. Prieels, W. Reisdorf, Phys. Rev. C 12 (1975) 644. [6] P. Hansen, A. Jensen, B. Jonson, Annu. Rev. Nucl. Part. Sci. 45 (591). [7] C. Bachelet, G. Audi, C. Gaulard, C. Guénaut, F. Herfurth, D. Lunney, M. De Saint Simon, C. Thibault, Eur. Phys. J. A 81 (2005) 31. [8] W. Satula, D. J. Dean, J. Gary, S. Mizutori, W. Nazarewicz, Phys. Lett. B 407(2) (2000) 103. [9] P. Schury, Ph.D dissertation, Ph.D. thesis, Michigan State University (2007). [10] W. O. Milligan (Ed.), Proceedings of the Robert A. Welch Foundation Confer- ence on Chemical Research, 1957. [11] T. Sun, High precision mass measurement of 37Ca and developments for LEBIT, Ph.D. thesis, Michigan State University (2006). [12] E. Burbidge, G. Burbidge, F. Fowler, F. Hoyle, Rev. Mod. Phys. 29 (1957) 547. [13] A. Cameron, Annu. Rev. Nucl. Sci. 8 (1957) 299. [14] R. K. Wallace, S. E. Woosley, Astrophys. Jour. Supp. 45 (1981) 389. [15] D. Lunney, J. Pearson, C. Thibault, Rev. Mod. Phys. 75 (2003) 1021. 144 [16] A. Kjellberg, G. Rudstam (Eds), The ISOLDE isotope separator facility at CERN, CERN 70-3, 1970. [17] S. Hofmann, G. Miinzenberg, Rev. Mod. Phys. 72 (2000) 733. [18] P. Dendooven, Nucl. Instr. and Meth. in Phys. Res. B 126 (1997) 182. [19] R. Klapisch, R. Prieels, C. Thibault, A. M. Poskanzer, C. Rigaud, E. Roeckl, Phys. Rev. Lett. 31 (1973) 118. [20] D. M. Moltz, K. S. Toth, R. E. Tribble, R. E. Neese, J. P. Sullivan, Phys. Rev. C 26. [21] H. Savajols, Hyperfine Int. 132 (2001) 243. [22] J. M. Wouters, D. J. Vieira, H. Wollnick, G. W. Butler, R. H. Kraus, K. Vaziri, Nucl. Instr. and Meth. B 26 (1987) 286. [23] D. Bazin, J. A. Caggiano, B. M. Sherrill, J. Yurkon, A. Zeller, Nucl. Instr. and Meth. in Phys. Res. B 204 (2000) 629. [24] M. Matos, in: A. Woehr, A. Aprahamian (Eds), Proceedings of the 12th Inter- national Symposium on Capture Gamma-Ray Spectroscopy and Related Topics, AIP, 2006, p. 164. [25] A. S. Lalleman, G. Auger, W. Mittig, M. Chabert, M. Chartier, J. Fermé, A. Gill- bert, A. Lépine—Szily, M. Lewitowicz, M. H. Moscatello, N. A. Orr, G. Politi, F. Sarazin, H. Savajols, P. Van Isacker, A. C. C. Villari, Hyperfine Int. 132 (2001) 313. [26] S. Issmer, M. Fruneau, J. A. Pinston, M. Ashgar, D. Barnéoud, J. Genevey, T. Kerscher, K. E. G. L5bner, Eur. Phys. J. A 2 (1998) 173. [27] J. Stadlmann, M. Hausmann, F. Attallah, K. Beckert, P. Beller, F. Bosch, 11. Eickhoff, M. Falch, B. Hanczak, B. Franzke, Phys. Lett. B 586 (2004) 27. [28] B. Schlitt, K. Beckert, T. Beha, H. Eickhoff, B. Franzke, H. Geissel, H. Irnich, H. C. Jung, T. F. Kerscher, O. Klepper, K. E. G. L6bner, G. Miinzenberg, F. Nolden, Y. Novikov, T. Radon, H. Reick, A. Schwinn, M. Steck, T. Siimmerer, Winkler, H. Wollnick, Hyperfine Int. 99 (1996) 117. [29] G. Bollen, D. Davies, M. Facina, J. Huikari, E. Kwan, P. A. Lofy, D. J. Morrissey, A. Prinke, R. Ringle, J. Savory, P. Schury, S. Schwarz, C. Sumithrarachchi, T. Sun, L. Weissman, Phys. Rev. Lett. 96 (2006) 152501. [30] A. Kellerbauer, K. Blaum, G. Bollen, F. Herfurth, H.-J. Kluge, M. Kuckein, E. Sauvan, C. Scheidenberger, L. Schweikhard, Eur. Phys. J. D 22 (2003) 53. 145 [31] G. Bollen, S. Becker, H.-J. Kluge, M. K5nig, R. Moore, T. Otto, H. Raimbault- Hartmann, G. Savard, L. Schweikhard, H. Stolzenberg, Nucl. Instrum. Methods A 368 (1996) 675. [32] V. S. Kolhinen, T. Eronen, J. Hakala, A. Jokinen, S. Kopecky, S. Rinta-Antila, J. Szerypo, J. Aysté, Nucl. Instrum and Methods in Phys. Res. B 204 (2003) 502. [33] G. Savard, R. C. Barber, D. Beeching, F. Buchinger, J. E. Crawford, S. Gulick, X. Feng, E. Hagberg, J. Hardy, V. T. Koslowsky, J. K. P. Lee, R. Moore, K. S. Sharma, M. Watson, Nucl. Phys. A 368 (1997) 353. [34] G. Sikler, D. Ackermann, G. Bollen, F. Attallah, D. Beck, J. Dilling, S. A. Eliseev, H. Geissel, D. Habs, S. Heinz, F. Herfurth, Nucl. Instr. and Meth. B 204 (2002) 482. [35] D. Morrissey, B. Sherrill, A. Stolz, I. Wiedenhoever, Nucl. Instr. and Meth. B204 (2003) 90. [36] L. Weissman, D. J. Morrissey, G. Bollen, D. Davies, E. Kwan, P. A. Lofy, P. Schury, S. Schwarz, C. Sumithrarachchi, T. Sun, R. Ringle, Nucl. Instr. and Meth. A540 (2005) 245. [37] T. Sun, S. Schwarz, G. Bollen, D. Lawton, R. Ringle, P. Schury, Eur. Phys. J. A 25(31) (2005) 61. [38] G. Bollen, S. Schwarz, D. Davies, P. Lofy, D. Morrissey, R. Ringle, P. Schury, T. Sun, L. Weissman, Nucl. Instr. and Meth. A 532 (2004) 203. [39] R. Moore, G. Rouleau, J. Mod. Optics 39 (1992) 361. [40] S. Schwarz, Nucl. Instr. and Meth. A 566 (2006) 233. [41] L. Brown, G. Gabrielse, Rev. Mod. Phys. 58 (1986) 233. [42] G. Bollen, R. Moore, G. Savard, H. Stolzenberg, J. Appl. Phys. 68 (1990) 4355. [43] H. Raimbault-Hartmann, D. Beck, G. Bollen, M. K5nig, H.-J. Kluge, E. Schark, J. Stein, S. Schwarz, J. Szerypo, Nucl. Instr. and Meth. B 126 (1998) 378. [44] G. Graff, H. Kalinowsky, J. Traut, Z. Phys. A297 (1980) 35. [45] G. Bollen, H.-J. Kluge, T. Otto, G. Savard, L. Schweikhard, H. Stolzenberg, G. Audi, R. Moore, G. Rouleau, J. Mod. Optics 39 (1992) 257. [46] G. Gabrielse, J. Tan, J. Appl. Phys. 63 (1988) 5143. 146 [47] N. Daly, Rev. Sci. Instrum. 31 (1960) 264. [48] S. Schwarz, Aufbau und test einer laserdesorptionsquelle fiir das cern- massenexperiment, Ph.D. thesis, University of Mainz (1993). [49] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing, Cambridge University Press, 2002. [50] D. Dahl, Int. J. Mass Spec. 200 (2000) 3. [51] R. Ringle, G. Bollen, P. Schury, S. Schwarz, T. Sun, Int. J. Mass Spec. in publi- cation. [52] J. Dilling, R. Baartman, P. Bricault, M. Brodeur, L. Blomeley, F. Buchinger, J. Crawford, J. Crespo Lépez-Urrutia, P. Delheij, M. Froese, G. Gwinner, Z. Ke, J. K. P. Lee, R. Moore, V. Ryjkov, G. Sikler, M. Smith, J. Ullrich, J. Vaz, Int. J. Mass Spec. 251 (2006) 198. [53] I. Bergstrém, C. Carlberg, T. F‘ritioff, G. Douysset, J. Schonfelder, R. Schuch, Nucl. Instrum. Methods A 487 (2002) 618. [54] R. Ringle, P. Schury, T. Sun, G. Bollen, D. Davies, J. Huikari, E. Kwan, D. Mor- rissey, A. Prinke, J . Savory, S. Schwarz, C. Sumithrarachchi, Int. J. Mass Spec. 251 (2—3) (2006) 300. [55] R. Firestone, R. Lindstrom, G. Molnar, S. Mughabghab, A. Reddy, Z. Revay, V. Tan, C. Zhou, R. Paviotti-Corcuera, IAEA-Tecdoc. [56] L. Langer, E. Spejewski, D. Wortman, Phys. Rev. B 133 (1964) 1145. [57] H. Booij, E. Van Hoek, H. Van der Molen, W. Slot, J. Blok, J. Borggreen, E. Hyde, Nuc. Phys. A 160 (1971) 337. [58] R. Ries, R. Damerow, W. Johnson Jr., Phys. Rev. 132 (1963) 1662. [59] W. Shi, M. Redshaw, E. Meyers, Phys. Rev. A 72 (2005) 022510. [60] P. Delahaye et al., ISOLTRAP Collaboration, (2006) in preparation. [61] K. Blaum, G. Audi, D. Beck, G. Bollen, F. Herfurth, A. Kellerbauer, H.-J. Kluge, E. Sauvan, S. Schwarz, Phys. Rev. Lett. 91 (260801). [62] M. Mukherjee, A. Kellerbauer, D. Beck, K. Blaum, G. Bollen, F. Carrel, P. De- lahaye, J. Dilling, S. George, C. Guénaut, F. Herfurth, A. Herlert, H.-J. Kluge, U. Késter, D. Lunney, S. Schwarz, L. Schweikhard, C. Yazidjian, Phys. Rev. Lett. 93 (2004) 150801. 147 [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] G. Savard, J. A. Clark, F. Buchinger, J. E. Crawford, S. Gulick, J. C. Hardy, A. A. Hecht, V. E. Iacob, J. K. P. Lee, A. F. Levand, F. Lundgren, N. D. Scielzo, K. S. Sharma, 1. Tanihata, I. S. Towner, W. Trimble, J. C. Wang, Y. Wang, Z. Zhou, Phys. Rev. C 70 (2004) 042501. F. Herfurth, J. Dilling, A. Kellerbauer, G. Audi, D. Beck, G. Bollen, H.-J. Kluge, D. Lunney, R. Moore, C. Scheidenberger, S. Schwarz, G. Sikler, J. Szerypo, Phys. Rev. Lett. 87 (2001) 142501. A. Kellerbauer, G. Audi, D. Beck, K. Blaum, G. Bollen, B. A. Brown, P. Delahaye, C. Guénaut, F. Herfurth, H.-J. Kluge, D. Lunney, S. Schwarz, L. Schweikhard, C. Yazidjian, Phys. Rev. Lett. 93 (2004) 072502. G. Savard, F. Buchinger, J. A. Clark, J. E. Crawford, S. Gulick, J. C. Hardy, A. A. Hecht, J. K. P. Lee, A. F. Levand, N. D. Scielzo, H. Sharma, K. S. Sharma, I. Tanihata, A. C. C. Villari, Y. Wang, Phys. Rev. Lett. 95 (2005) 102501. T. Eronen, V. Elomaa, U. Hager, J. Hakala, A. Jokinen, A. Kankainen, I. Moore, H. Penttila, S. Rahaman, J. Rissanen, A. Saastamoinen, J. Aystii, J. C. Hardy, V. S. Kolhinen, Phys. Rev. Lett. D. R. Lide (Ed.), CRC Handbook of Chemistry and Physics, CRC Press, 2006. S. Brunner, T. Engel, G. Werth, Meas. Sci. Technol. 6 (1995) 222. R. S. Van Dyck Jr., D. L. Farnham, S. L. Zafonte, P. B. Schwinberg, Rev. Sci. Instrum. 70 (3) (1999) 1665. G. Bollen, H.-J. Kluge, M. K6nig, T. Otto, G. Savard, H. Stolzenberg, R. Moore, G. Rouleau, G. Audi, Phys. Rev. C 46 (R2140). J. Melin, J. V. Ortiz, I. Martin, A. M. Velasco, C. Lavin, J. Chem. Phys. 122 (2005) 234417. J. C. Hardy, private communication. B. Blank, private communication. S. George, private communication. R. T. Birge, Phys. Rev. 40 (1932) 207. P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, McGraw Hill, 2003. G. Audi, O. Bersillon, J. Blachot, A. H. Wapstra, Nucl. Phys. A 729 (2003) 3. 148 l[lllgllglllll[gill