HIGH-PRECISION MASS MEASUREMENT OF 32 Si AND
DEVELOPMENTS AT THE LEBIT FACILITY
By
Anna A. Kwiatkowski

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Physics and Astronomy
2011

ABSTRACT
HIGH-PRECISION MASS MEASUREMENT OF 32 Si AND
DEVELOPMENTS AT THE LEBIT FACILITY
By
Anna A. Kwiatkowski

The mass of an atom, a fundamental property, serves as a unique identifier of the
nucleus like a fingerprint and represents, through the binding energy, the sum of all
atomic and nuclear interactions. Precise mass values are essential in many disciplines
including nuclear structure and fundamental symmetries. The highest precision mass
measurements are obtained with Penning trap mass spectrometers; of those, only
the Low Energy Beam and Ion Trap (LEBIT) facility has been able to measure rare
isotopes produced by projectile fragmentation. In this work, the high precision mass
measurement of unstable 32 Si, δm/m = 1 × 10−8 , is presented and found to deviate
from the literature value by four standard deviations. The measurement is combined
with the mass values for the A = 32, T = 2 quintet to provide the most stringent test
of the isobaric multiplet mass equation (IMME) to date. The experimental results
indicate a dramatic breakdown of the quadratic form of IMME and require a large
cubic term. In addition, the masses of 33 Si, 31 P, 34 P, and 32 S were measured, which
agree with earlier experimental data.
Beam purity is critical to high-precision Penning trap mass spectrometry. This dissertation documents the development of in-trap cleaning of stored ion samples based
on stored waveform inverse Fourier transform (SWIFT). The technique can eliminate
isobaric contaminants without the need to identify individual contaminants unlike
the method used in the present work. SWIFT cleaning minimizes possible systematic
errors in mass determination and will increase LEBIT’s measurement efficiency.

For my parents

iii

ACKNOWLEDGMENTS
My advisor, Georg Bollen, has encouraged and assisted me throughout my graduate career. Georg instructed me in the lab, sent me to several conferences and summer
schools, and provided many interesting research projects. During my first few years in
LEBIT, Stefan Schwarz played a pivotal role in my training, from soldering lessons to
data analysis. In the lab, Stefan was my mentor and answered my every question. In
the latter years, Ryan Ringle oversaw the recommissioning of LEBIT, pushed forward
testing of SWIFT cleaning, and proofed my thesis as quickly as I could write. Josh
Savory facilitated my joining of the group; we often spent weeks working side-by-side
in the lab. To the rest of the LEBIT team, both past and present, I also express my
gratitude. Many thanks to the gas cell team, led by Dave Morrissey who frequently
acted as a second advisor to me. I credit the wonderful scientists and staff of the NSCL
for enhancing my graduate studies. In particular, Dan Morris deserves recognition for
his help in achieving SWIFT beam purification. Dan lent me his spectrum analyzer for
extended periods, explained RF mixers, and advised me on many technical aspects.
Rhiannon Meharchand and I co-founded a student group, WaMPS, and her skill
sets complemented mine. As our faculty advisor, Filomena Nunes counseled and nurtured the group. From its inception, Wolfgang Bauer and Michael Thoennesson have
generously provided financial support from the PA Department and the NSCL respectively. I am gratified to see WaMPS’ success and the unexpected popularity of
my “little” mentoring program.
I would also like to thank the many friends I have made here. With three in
particular, I weathered class work, subject exams, experiments, the highs and lows of
research, and graduate student life: Carol Guess, Dan Olds, and Megan Romanowich.
Last, I thank my family for unwavering support and constant encouragement.

iv

Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Examples of the Importance of Precise Mass Values in Nuclear Physics
1.1.1 Fundamental Interactions and Tests of the Standard Model . .
1.1.2 Nuclear Structure . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Nuclear Astrophysics . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Nuclear Models and Mass Formulae . . . . . . . . . . . . . . .
1.2 Penning Trap Mass Spectrometry of Rare Isotopes . . . . . . . . . . .
1.2.1 Rare Isotope Production . . . . . . . . . . . . . . . . . . . . .
1.2.2 LEBIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 The
2.1
2.2
2.3
2.4

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3 Test of the Isobaric Multiplet Mass Equation with the A = 32, T
= 2 Quintet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Systematic Error Analysis . . . . . . . . . . . . . . . . . . . .
3.2.2 Comparison of Measured Mass Excesses to Literature Values .
3.3 Impact of the present 32 Si mass measurement on IMME . . . . . . .
3.3.1 Possible Reasons for a Non-zero d Coefficient . . . . . . . . . .

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4 Motivation for and Foundation of Stored Waveform Inverse Fourier
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Broadband excitation . . . . . . . . . . . . . . . . . . . . . . . . . . .

47
48

2.5
2.6

LEBIT Facility . . . . . . . . . . . . . . . . . . . .
Beam Production at the NSCL . . . . . . . . . . . . .
Gas Stopping Station . . . . . . . . . . . . . . . . . . .
The Beam Cooler and Buncher . . . . . . . . . . . . .
The Penning Trap Mass Spectrometer . . . . . . . . . .
2.4.1 Ion motion in a Penning trap . . . . . . . . . .
2.4.2 Excitation of ion motion . . . . . . . . . . . . .
2.4.3 Mass Determination . . . . . . . . . . . . . . .
Beam Purification Techniques . . . . . . . . . . . . . .
Overview of LEBIT Performance . . . . . . . . . . . .

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4.2
4.3
4.4

Optimal phase modulation . . . . . . . . . . . . . . . . . . . . . . . .
Signal distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SWIFT-based cleaning at LEBIT . . . . . . . . . . . . . . . . . . . .

5 Implementation of SWIFT Cleaning at LEBIT . . . .
5.1 Signal Synthesis . . . . . . . . . . . . . . . . . . . . . . .
5.2 Preliminary Studies . . . . . . . . . . . . . . . . . . . . .
5.3 First implementation of SWIFT . . . . . . . . . . . . . .
5.4 Second implementation of SWIFT . . . . . . . . . . . . .
5.4.1 Characterizing the SWIFT waveform . . . . . . .
5.4.2 Determining the cleaning amplitude . . . . . . . .
5.4.3 Measurement of the SWIFT waveform with ions .
5.4.4 Isobaric separation . . . . . . . . . . . . . . . . .
5.4.5 Mass measurement with SWIFT . . . . . . . . . .
5.4.6 Measurements with CO and N2 . . . . . . . . . .
5.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . .
6 Summary and Outlook

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80

A Experimental Data of the A = 32, T = 2 Quintet . .
A.1 Mass Value of the Ground State of 32 Si . . . . . . . . .
A.2 Mass Value of Isobaric Analog State of 32 P . . . . . . .
A.3 Mass Value of the Isobaric Analog State of 32 S . . . . .
A.4 Mass Value of the Isobaric Analog State of 32 Cl . . . .
A.5 Mass Value of the Ground State of 32 Ar . . . . . . . .

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B Fourier Transforms and Their Properties . . . . . . .
B.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . .
B.1.1 Basic Properties . . . . . . . . . . . . . . . . . .
B.2 The Discrete Fourier transform . . . . . . . . . . . . .
B.3 Fast Fourier transform . . . . . . . . . . . . . . . . . .
B.4 Sampling theorem . . . . . . . . . . . . . . . . . . . . .

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C Simulations of RFQ Mass Filter: Calculations of Transmission
Emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1 Ion Motion in an RFQ Ion Guide . . . . . . . . . . . . . . . . .
C.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . .
C.3 Transmission through the Mass Filter . . . . . . . . . . . . . . .
C.4 Emittance and Beam Profile of Mass-filtered Beams . . . . . . .

and
. . . 95
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. . . 100
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. . . 104

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

vi

List of Tables

1.1

Applications which utilize precise mass measurements and the typical
precision δm/m required to probe the associated physics. . . . . . . .

2

For each mass measured, the isotope is listed with its mass number, its
molecular ion, the molecular ion’s mass number, its reference ion and
its mass number, the number of measurements N, the mean frequency
¯
ratio R, and the relative uncertainty . . . . . . . . . . . . . . . . . .

35

Experimentally determined mass excesses and a comparison to literature values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Experimentally determined mass excesses and a comparison to mass
values obtained by Penning trap mass spectrometry at FSU . . . . .

38

3.4

The mass excesses used to fit IMME . . . . . . . . . . . . . . . . . .

40

3.5

The four data sets a, b, c, and d used to test IMME . . . . . . . . . .

41

3.6

The coefficients and the χ2 /n of the quadratic and cubic forms of
IMME to the A = 32, T = 2 quintet . . . . . . . . . . . . . . . . . .

44

A list of contaminants identified during different mass measurements
and the separation of their ν+ from the ion of interest . . . . . . . . .

59

The maximum cleaning bandwidth in MHz for an AFG with memory
N points for cleaning times Tclean = 10-100 ms . . . . . . . . . . . .

60

Parameter values used to generate the excitation profiles depicted in
Figure 5.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

A.1 Comparison of neutron separation energies Sn (in keV) for several Si
isotopes from the AVOGADRO project, the IAEA database, and the
preview of the next Atomic Mass Evaluation (AME’11) . . . . . . . .

84

3.1

3.2

3.3

5.1

5.2

5.3

vii

A.2 Recent ground state mass measurements and values of 32 Si . . . . . .

85

A.3 Recent ground state and excited state energy values of 32 P . . . . . .

85

A.4 Recent ground state and excited state energy values of 32 S . . . . . .

86

A.5 Recent ground state and excited state energy values of 32 Cl

. . . . .

88

C.1 Fit parameters for Equations C.6 . . . . . . . . . . . . . . . . . . . .

99

C.2 Typical pressures P measured along the RFQ ion guides . . . . . . .

99

C.3 Transmission from the segmented RFQ to the mass filter using hardsphere collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

C.4 The inter-rod distance r0 and the RF amplitudes for A = 28, 45, 85 of
the RFQ ion guide sections . . . . . . . . . . . . . . . . . . . . . . . . 100
C.5 The electrostatic potentials for the electrodes in the ion guides . . . . 100

viii

List of Figures

2.1

Layout of the LEBIT beamline from 2005-2009 . . . . . . . . . . . . .

11

2.2

Layout of the experimental area at the NSCL from 2007-2009 . . . .

12

2.3

Mechanical design of the gas cell . . . . . . . . . . . . . . . . . . . . .

13

2.4

Mechanical design of the RFQ ion guides . . . . . . . . . . . . . . . .

14

2.5

Photographs of the LEBIT beam cooler and buncher . . . . . . . . .

15

2.6

Cartoon depicting the principles of the LEBIT beam cooler and buncher 16

2.7

Schematic layout of the LEBIT Penning trap mass spectrometer . . .

17

2.8

(a) Illustration of a hyperbolic Penning trap. (b) Photograph of the
LEBIT hyperbolic Penning trap . . . . . . . . . . . . . . . . . . . . .

18

Representation of the three independent eigenmotions of an ion in a
Penning trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.9

2.10 Schematic drawing of the segmented ring electrode in a Penning trap

21

2.11 Illustration of quadrupolar excitation of ion motion . . . . . . . . . .

22

2.12 Cyclotron resonance curve of 39 K+ . . . . . . . . . . . . . . . . . . .

23

2.13 Results from the multi-stage beam purification system of LEBIT . . .

26

3.1

3.2

The difference between experimental and IMME-predicted mass excesses for A = 32 in 2003 and in 2006 . . . . . . . . . . . . . . . . . .

30

Measured β-activity of 33 Si after the RFQ ion guides . . . . . . . . .

31

ix

3.3

The mean time-of-flight as a function of the applied RF frequency in
the Penning trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Fit of the quadratic form of IMME to data with the four different data
sets indicated in Table 3.4 . . . . . . . . . . . . . . . . . . . . . . . .

41

Fit of the cubic form of IMME to data with the four different data sets
indicated in Table 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

A rectangular or uniform amplitude spectrum in frequency space created with 1028 points and its inverse Fourier transform . . . . . . . .

49

4.2

Phase coherence of cosine functions with similar frequencies . . . . .

50

4.3

Comparison of three choices of the phase function φk and the corresponding SWIFT time-domain waveforms . . . . . . . . . . . . . . . .

51

4.4

Gibb’s oscillations and power leakage of the frequency-space signal . .

52

4.5

A comparison of frequency spectrum obtained by the Fourier transform
of a rectangular excitation in the time domain multiplied by different
apodization functions . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.6

The desired excitation profile for beam purification at LEBIT . . . .

55

4.7

The time-domain waveform of the frequency excitation shown in Figure
4.6 with quadratic phase modulation . . . . . . . . . . . . . . . . . .

56

4.8

Comparison in frequency-space of apodized cleaning excitations . . .

56

5.1

A cartoon of the desired SWIFT profile . . . . . . . . . . . . . . . . .

59

5.2

Amplitude modulation of the SWIFT signal . . . . . . . . . . . . . .

61

5.3

Amplitude modulation of the SWIFT signal . . . . . . . . . . . . . .

63

5.4

SWIFT waveform output by Tektronix 3102 AFG and captured on an
Agilent EXA Signal Analyzer N9010A . . . . . . . . . . . . . . . . .

64

Amplitude-modulated SWIFT signals captured on the Agilent spectrum analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Sine waveform captured with the Agilent spectrum analyzer . . . . .

67

3.4

3.5

4.1

5.5

5.6

x

5.7

Effect of the Agilent 33521 AFG built-in filters on the SWIFT signal
as captured on the Agilent spectrum analyzer . . . . . . . . . . . . .

68

5.8

Scan of the SWIFT amplitude determined with 23 Na+ . . . . . . . .

69

5.9

Measured side lobes of different apodization windows . . . . . . . . .

70

5.10 Comparison of expected SWIFT profiles with observed ion profiles . .

72

5.11 A closer look at the gap of the SWIFT excitation profile experimentally
determined with 39 K+ . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5.12 Detected 23 Na+ ions and mean TOF as the gap size was varied from
0 to 500 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

5.13 Deviations from the mean value of the mass measurement of 39 K+ with
SWIFT-based cleaning . . . . . . . . . . . . . . . . . . . . . . . . . .

76

5.14 Quadrupole resonances (TRF = 7 ms) with CO+ and N+ simultane2
ously loaded in the Penning trap . . . . . . . . . . . . . . . . . . . .

78

B.1 Illustration of the sampling theorem . . . . . . . . . . . . . . . . . . .

94

C.1 Mechanical design of the RFQ ion guides . . . . . . . . . . . . . . . .

96

C.2 Stability diagram in the Mathieu parameter space (a,q) . . . . . . . .

98

C.3 Simulated transmission as a function of mass filter strength . . . . . . 101
C.4 Ion trajectories of A = 28 ions in the mass filter . . . . . . . . . . . . 102
C.5 Simulated x- and y-emittances as a function of mass filter strength

. 103

C.6 Simulated x- and y-emittance and beam profile for A = 28 ions with
damped initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . 106
C.7 Simulated x- and y-emittance and beam profile for A = 28 ions with
“ideal” initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.8 Simulated x- and y-emittance and beam profile for A = 45 ions with
damped initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . 108
C.9 Simulated x- and y-emittance and beam profile for A = 45 ions with
“ideal” initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 109

xi

C.10 Simulated x- and y-emittance and beam profile for A = 85 ions with
damped initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . 110
C.11 Simulated x- and y-emittance and beam profile for A = 85 ions with
“ideal” initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 111

xii

Chapter 1
Introduction
In the ideal high precision experiment, the scientist probes a pure sample in an isolated
environment with precise tools. While the ideal laboratory has yet to be achieved,
ion traps offer nearly ideal conditions: confinement in a small volume, well-controlled
electromagnetic fields, long observation times, and single-ion sensitivity. These parameters make it a tool of choice for precision measurements and versatile enough to
serve areas of science that range from forensics to nuclear physics.
The most important ion traps are the Paul (or radiofrequency) and Penning traps.
The former has been deployed for laser and radiofrequency spectroscopy [1–3], the
realization of a frequency standard [2,3], and manipulation of rare isotope beams [4,5].
In the field of mass spectrometry, the precision of Paul traps is limited to ≈ 10−6
by the stability of the electric field [6]. In comparison, Penning traps are renowned
for the high precision achieved in many types of measurements. Even the earliest
experiments with a Penning trap, measurements of the free g-factor of the electron by
Hans G. Dehmelt in 1986 [7], yielded a precision of 4 × 10−12 . “For the development
of the ion trap technique,” Dehmelt and Wolfgang Paul, inventor of the Paul trap,
were awarded the 1989 Nobel Prize in Physics [8]. (It was shared with Norman F.
Ramsey “for the invention of the separated oscillatory fields method and its use in
the hydrogen maser and other atomic clocks [8].) Penning traps lend themselves to

1

Application

δm/m

Nuclear structure
≤ 10−6
Nuclear models and formulae ≤ 10−7
Nuclear astrophysics
≤ 10−7
Weak interaction
≤ 10−8
Fundamental constants
≤ 10−9
CPT Violation
≤ 10−10
QED
≤ 10−11
Neutrino physics
≤ 10−11
Table 1.1: Applications which utilize precise mass measurements and the typical precision δm/m required to probe the associated physics.
other high precision measurements such as verification of CPT (charge, parity, time
reversal) symmetry [9], tests of the Standard Model [10], decay studies [11], and mass
spectrometry, which is the subject of this thesis.
Mass is a fundamental and unique characteristic of an atom. The atomic and nuclear binding energy (by way of the atomic mass) represents the sum of all atomic
and nuclear interactions. Mass spectrometry, then, is a powerful probe used in many
areas of physics, some of which are listed in Table 1.1 with the typical precision δm/m
needed [6]. The focus of the present work lies in nuclear physics, where broadly speaking there are four applications: fundamental symmetries, nuclear structure, nuclear
astrophysics, and nuclear models.

1.1

Examples of the Importance of Precise Mass
Values in Nuclear Physics

1.1.1

Fundamental Interactions and Tests of the Standard
Model

Several tests of the Standard Model were featured on the cover of Discover Magazine in 2002 as part of the eleven greatest physics mysteries [12]; number four, “do

2

neutrinos have mass?” In the Standard Model, the mass of the neutrino is zero. The
observation of zero-neutrino, double beta decay [13] would contradict the assumption
that the neutrino is massless and demonstrate that the neutrino is its own antiparticle. Accurate and precise knowledge of the masses of the mother and daughter
nuclei facilitates calculation of the decay rate and narrow the energy window of the
double beta decay spectrum. Another test of the Standard Model is the unitarity
of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix [14]. The up-down
quark matrix element Vud can be extracted from the f t-values of superallowed beta
emitters, a measure of the decay strength. Three quantities are needed to calculate
the f t value: the half life, the branching ratio, and the Q value or mass difference
between the mother and daughter nuclei.

1.1.2

Nuclear Structure

The nucleus has been found to have a higher binding energy, colloquially to be more
stable, at certain numbers of protons and neutrons, the so-called “magic numbers”
(2, 8, 20, 28, 50, ...), which provided the impetus for the nuclear shell model. Shell
structure can be easily seen in double mass differences, i.e. two-neutron (two-proton)
separation energy S2n (S2p ). In general, the S2n decreases smoothly as one moves
away from the valley of stability; deviations from this trend indicate microscopic nuclear structure effects such as a shell closure or nuclear deformation [15]. Far from
stability, the well known magic numbers may vanish and new ones emerge [15]. Another interesting class of interaction that can be studied with masses is the interaction
of the valence neutron(s) and proton(s). The binding energy is enhanced along the
N = Z line because of the isospin T = 0 interaction [16,17]; that is when a valence neutron and proton occupy the same orbital, they have nearly identical wavefunctions
and interact more strongly than valence nucleons in N = Z nuclei. For extremely
loosely bound nuclei, a neutron or proton “halo” may occur [18], one or two valence
nucleons orbiting a tightly bound core. The extent of the halo is related to the neu3

tron separation energy (the mass difference between neighboring isotopes). The first
experimental evidence for halo nuclei was seen in 11 Be [19].

1.1.3

Nuclear Astrophysics

Also on the list of greatest physics mysteries [12] is the origin of elements heavier
than iron. Three main processes are thought to contribute: The slow neutron capture
(s-) process is believed to be responsible for the creation of nuclei close to the valley
of stability. The rapid neutron capture (r-) process, expected to be the source of over
half the abundance of heavy nuclei, is postulated to occur in supernovae. The rapid
proton capture (rp-) process powers type I x-ray bursts and is thought to supply
nuclei close to the proton dripline. The path these processes follow depends on the
neutron and proton separation energies and on other environmental factors, e.g. the
neutron density and the temperature. The mass values are crucial for calculations of
final abundances and luminosity [20,21]. As the r-process involves many masses which
are inaccessible by mass spectrometry, astrophysicists are forced to rely on nuclear
mass models.

1.1.4

Nuclear Models and Mass Formulae

The nucleus is a many-body quantum system under the influence of the electromagnetic, strong, and weak forces. Two problems present themselves: the lack of an exact
description of the strong interaction and the complexity of many-body problems. Nuclear mass models have been developed based on microscopic, macroscopic, or both
approaches in their description of the nucleus. An early model [22, 23] partly resembles a liquid drop, hence the name the liquid drop model (LDM), and predicts the
mass in a macroscopic framework through the volume energy, surface energy, symmetry energy, etc. An example of a microscopic model has already been mentioned,
the nuclear shell model, in which each nucleon is bound by a central force calculated

4

from the average effects of all nucleons [24]. A review of nuclear mass models and
formulae [25] showed that the models tend to agree in the region of known masses
but diverge in the terra incognita.
In addition to global predictions, there are so-called local models intended to predict an unknown mass close to several known masses [25]. An elegant example and
the subject of this dissertation is the isobaric multiplet mass equation (IMME) proposed by Wigner in 1957 [26, 27]. To first order, the neutron and proton are identical
particles excepting the electric charge of the proton. They can be treated as the different projections of the isospin operator T, where the neutron has isospin up and the
proton isospin down. Certain states in different nuclei have the same mass number
(A), angular momentum (spin), parity, and isospin; these states have nearly identical wavefunctions and are called isobaric analog states. The small differences in the
wavefunctions can be attributed to the Coulomb interaction. As a result, the masses
of the multiplet are a quadratic function of the isospin projection Tz = (N-Z)/2:
2
M = a + bTz + cTz

(1.1)

The IMME has been extremely successful [28] and only failed so to speak in light
multiplets with an unbound member. Recent high precision measurements have upheld the validity of IMME [29–32] until a remeasurement in the A = 32, T = 2
quintet [33] found a 3 keV shift in the Tz = 0 member. The authors of [33] debated
whether isospin mixing or an erroneous measurement of the Tz = 2 (32 Si) member
was responsible for the breakdown of the relationship. In the present work the high
precision mass measurement of 32 Si is presented, and the obtained mass value is used
to test IMME.

5

1.2

Penning Trap Mass Spectrometry of Rare Isotopes

The precision of mass measurement is critical for tests like IMME of isospin symmetry;
and, precision is the hallmark of Penning trap mass spectrometry. The technique has
proven itself to be highly reliable; no statistically significant disagreements between
Penning trap mass measurements have been found. On the other hand, Penning trap
mass spectrometry has exposed erroneous mass values determined with other techniques, including β-decay, nuclear reactions, Q-values, and direct mass measurements
some of which were well established and highly regarded. Deviations from the accepted mass values of stable isotopes have been observed, such as 31 P and 32 S, which
were measured for this work. These discrepancies have far-reaching consequences:
stable nuclei are often used to anchor network calculations and serve as calibrations.
Penning trap mass measurements have obtained precisions that range from 10−6 to
10−11 for stable isotopes [34] and to 10−9 for unstable nuclei [30], which matches the
range of required δm/m listed in Table 1.1. For short-lived isotopes, the precision is
typically limited by the half-life; the shortest T1/2 of an isotope measured by Penning
trap mass spectrometry is on the order of 10 ms [35].
Compared to measurements of stable isotopes, additional challenges are faced
when trapping radioactive isotopes. First, the ions have to be created in nuclear reactions; depending on the production mechanism, the beam energy can range from
tens of keV to several GeV. The more exotic and more interesting species are generally produced with a small yield, as low as a few ions per hour. Therefore, highly
efficient methods are required to manipulate these beams to achieve the beam properties needed for Penning trap mass spectrometry, namely high quality, very low-energy
ion bunches. Second, as one approaches the most exotic nuclei at the driplines, the
half lives grow shorter, decreasing to the order of tens of milliseconds or less. Consequently, the beam handling and the mass measurement must be executed quickly.

6

The solutions to these challenges have been tailored to the beam properties, which
are largely determined by the production mechanism.

1.2.1

Rare Isotope Production

The three predominant methods to produce rare isotopes are isotope separation online (ISOL), fusion-evaporation reactions, and fast projectile fragmentation at higher
energies. The ISOL technique impinges a high-energy primary beam of light ions
on a thick target of a heavier element; products diffuse out of the target, which is
held at a high temperature, and into an ion source. Next they are accelerated to an
energy of a few tens of keV, mass separated, and delivered to experiments like the
pioneer of rare isotope Penning trap mass spectrometry, ISOLTRAP [36], at ISOLDE
at CERN. The related IGISOL technique uses heavy ions which impinge on a thin
target; the fragments are thermalized in a small gas volume, swept out, and re-ionized
if necessary, and then extracted as low-energy ions; this method is currently being
exploited by JYFLTRAP [37]. For fusion and fusion evaporation reactions, low- to
medium-energy ions bombard a thin target; the reaction products have significant
forward momentum which allows a fraction of them to be mass separated in-flight with
magnetic separators. Typical beam energies are 100 keV to a few MeV per nucleon and
requires re-thermalization in a low-pressure gas catcher to be coupled to Penning trap
mass spectrometers. SHIP at GSI [38] employs this technique to study the heaviest
elements, and its resident Penning trap mass spectrometer, SHIPTRAP [39], made the
first direct mass measurements in transuranic elements. Fast projectile fragmentation
on the other hand relies on accelerating a fast, heavy primary beam that reacts
with a thin, light target and in-flight mass separation. Several rare isotope facilities
worldwide use of projectile fragmentation and in-flight separation, including SPIRAL
at GANIL [40], the FRS at GSI [41], RIPS at RIKEN [42], and the NSCL in the
United States [43], where the present work was performed. Statistical in nature and
without decay losses, projectile fragmentation can access isotopes far from the valley
7

of stability on either the proton- or neutron-rich side. However, the beam properties
are somewhat opposite to those needed for trapped ion studies. The challenge is to
convert a high-energy, high-emittance, continuous, cocktail beam into a low-energy,
low-emittance, pulsed, pure beam.

1.2.2

LEBIT

The Low Energy Beam and Ion Trap (LEBIT) facility at the NSCL [44] was the
first facility to trap ions produced by fast projectile fragmentation and in the process
developed several advanced ion manipulation techniques. A persistent problem of
using projectile fragments is beam purity as impurities may shift the measurement or
entirely mask the signal in the Penning trap. The beam thermalization process ionizes
residual impurities in the buffer gas, creating various stable molecular ions, which are
transported out of the gas cell along with the rare isotopes. For his dissertation [45],
Pete Schury implemented a multi-stage beam purification system, which filters ions
by their mass-to-charge ratio and eliminates all contaminants except isobars (ions
with the same mass number, A).
Currently, two techniques are employed to clean isobars. At LEBIT, a dipole
excitation [46, 47] drives the ion from the trap but requires an exact identification
of each contaminant species, which is a time-consuming and involved process. At
ISOLTRAP [36] and JYFLTRAP [37], a buffer-gas filled (preparation) Penning trap
accumulates and purifies ions [48–50]. Although the contaminant ions need not be
identified, the process is slow and better suited for long-lived species. A faster and
more efficient technique was explored, and the results are presented in the second
half of this dissertation. Stored waveform inverse Fourier transform (SWIFT) [51]
eliminates the need for individual contaminant identification like dipole cleaning and
does not require cooling like mass-selective buffer gas cooling.

8

1.3

Outline

This dissertation describes the LEBIT system and the principles of Penning trap mass
spectrometry before presenting the results of mass measurements of 32,33 Si, 31,34 P,
and 32 S. The nuclides 32 Si and 32 S are members of the A = 32, T = 2 quintet,
and their mass values were used for the most stringent test of IMME. Next the
discussion turns to reducing the systematic uncertainties in such mass measurements
due to beam impurities, which posed a significant challenge in the measurement of
32 S.

The SWIFT cleaning technique is described, its implementation chronicled, and

the ultimate test of SWIFT cleaning, separating the carbon monoxide and nitrogen
doublet, is presented.

9

Chapter 2
The LEBIT Facility
Located at the center of the Michigan State University campus is the National Superconducting Cyclotron Laboratory (NSCL), home to the nation’s foremost rare isotope
facility and the LEBIT Penning trap mass spectrometer. LEBIT was the first and remains the only Penning trap mass spectrometer to study rare isotopes produced by
fast beam fragmentation. This chapter presents the facility and the advanced ion manipulation techniques used to convert the relativistic beam into brilliant ion bunches.
Figure 2.1 shows an overview of the LEBIT beamline. Three major components
are connected by an electrostatic beam transport system [52]. Rare isotope beams are
delivered from the Coupled Cyclotron Facility (CCF) to the gas stopping station [53,
54], where the beam is thermalized in the gas cell and transported by radiofrequency
quadrupole (RFQ) ion guides [55]. After being transported out of the high energy
vault, the continuous beam is cooled and bunched in the beam cooler and buncher
[5, 56], another RFQ device. Finally, the 9.4 T Penning trap mass spectrometer [57,
58] receives the ion pulses for high-precision mass measurements. Beam Observation
Boxes (BOB) are situated along the beam line and contain diagnostic equipment
such as Faraday cups, microchannel plate (MCP) detectors, and silicon detectors.
Subsequent to the present work, the LEBIT facility was decommissioned and moved
to a new location in the NSCL. These physical changes and their impact on LEBIT

10

Figure 2.1: Layout of the LEBIT beamline from 2005-2009. For interpretation of the references to color in this and all other figures,
the reader is referred to the electronic version of this dissertation.

11

will be discussed in Chapter 6. The mass measurements described in this thesis
were made in the configuration described in this chapter and shown in Figure 2.1.

2.1

Beam Production at the NSCL

The layout of the experimental area at the NSCL at the time of this work is shown in
Figure 2.2. An Electron Cyclotron Resonance (ECR) ion source produces a primary
ion beam of highly charged stable isotopes, which is injected into the smaller K500
cyclotron, which accelerates the beam to on the order of 14 MeV/u, depending on the
ion of interest. The ions are ejected from the K500 and subsequently pass through
a stripper foil, which generally removes the remaining electrons, near the center of
the K1200. The second cyclotron accelerates the beam to 80-150 MeV/u. When the
stable beam impinges on a (thin, hundreds of mg/cm2 ) target, fragmentation occurs
and rare isotopes are created. The ions of interest are separated in-flight from other
fragmentation products in the A1900 fragment separator [43] and then delivered to
the different experimental vaults (N2-N5, S1-S3). LEBIT is located in the N4 and N5
vaults. The high energy section of LEBIT, the gas stopping station, is located in the
N4 vault while the low energy sections are found in N5, where the beam energy has
a few keV.

Figure 2.2: Layout of the experimental area at the NSCL from 2007-2009.

12

2.2

Gas Stopping Station

The A1900 fragment separator delivers the beam to the gas stopping station with an
energy of approximately 100 MeV/u. The beam loses most of its energy as it passes
through a system of adjustable glass degraders. A pair of plates are rotated to adjust
their effective thickness and to control the stopping distribution of the ions in the
gas. An aluminum wedge degrader is used to reduce the momentum spread of the
secondary beam, thereby minimizing the range distribution of the ions in the gas.
The beams enters the 51 cm-long gas cell through a thin beryllium window. The
mechanical design of the gas cell is pictured in Figure 2.3. The chamber is filled
with 200-500 mbar of ultra-pure (ppb) helium gas. Since helium has the highest first
ionization energy of any atom, the rare isotopes do not neutralize. Collisions with the
buffer gas thermalize the incoming ions while charge exchange reactions reduce the
charge state to Q = +1 or +2.
Inside the gas cell, a system of ring electrodes creates an electrostatic field to
guide the ion towards the extraction region. Four spherical electrodes (called the
“flower”) are used to approximate the field of a point charge in order to focus the
beam into the opening of a supersonic nozzle. Finally, gas flow sweeps the ions into
the radiofrequency quadrupole (RFQ) ion guides.

Figure 2.3: Mechanical design of the gas cell. The ring electrodes guide the ion to
the extraction region. The “flower” electrode creates a focusing potential. Gas flow
through the supersonic jet ejects the ions into the RFQ ion guides.

13

Figure 2.4: Mechanical design of the RFQ ion guides. The three sections are followed
by a 5 keV acceleration region and end at BOB1.
RFQ ion guides provide radial confinement with an oscillating voltage applied
to four rods, each phase-shifted 180◦ with respect to their nearest neighbor, as explained in [59,60]. The ion guides were part of the thesis work of Pete Schury [45] and
a schematic representation is shown in Figure 2.4. They are housed in three vacuum
chambers and can be decomposed into three functional sections. The first section,
a segmented RFQ, includes an electrostatic gradient to drag the ions through the
residual helium gas as the pressure in the first vacuum chamber can be as high as
0.2 mbar. A small RFQ, called the µRFQ, provides efficient transport through a small
diaphragm for differential pumping. The third section can be operated as a mass filter to select the mass-to-charge ratio of the rare isotope and to suppress unwanted
contaminants, principally stable molecular ions produced in the gas cell by ionization
of residual contamination. Beam purification was essential for this work, and suppression of these contaminants will be explored in Chapters 4 and 5. Radioactive species
can be detected by their decay after being stopped in a retractable silicon detector in
BOB1. Mass scans are performed by measuring activity as a function of the chargeto-mass ratio transported through the mass filter. In this manner, the form of the
radioactive species (e.g. atomic, molecular, charge state) is identified.
After the mass filter, the ions are accelerated to 5 keV·Q and transported through
the shielding wall into the low energy area, where steering and focusing elements
guide the beam to the beam cooler and buncher. Located between BOB2 and BOB3
is a plasma-type ion source, which can provide stable ion beams for system optimiza-

14

(a) The cooler and µRFQ

(b) The buncher and ejection
optics

Figure 2.5: Photographs of the LEBIT beam cooler and buncher.
tion, magnetic field calibration, and offline tests. Being placed perpendicular to the
main beam line, the stable ion beam is merged with the rare ion beam with a 90◦
electrostatic bender.

2.3

The Beam Cooler and Buncher

The LEBIT beam cooler and buncher accepts the continuous 5 keV·Q beam from the
gas cell (or the stable ion source) and converts it into low-energy, low-emittance ion
pulses for efficient ion capture in the Penning trap. Photographs of the device are
shown in Figure 2.5. It is a two-stage linear RFQ ion trap filled with a low-pressure
buffer gas for ion cooling and capture [5, 56]; it was studied extensively in an earlier
thesis work [52]. The first stage is a pre-cooler filled with helium or neon at pressures
on the order of 2 × 10−2 mbar. Neon is sometimes used to increase the likelihood
of collision-induced dissociation (CID) to break up molecular ions. The second stage
performs the final cooling, trapping, and bunching. The two are connected by a µRFQ
identical to the one in the gas stopping station also for differential pumping.
The ions are electrostatically decelerated to ≈10 eV before entering the precooler, where they are slowed and the transverse emittance is reduced. They are
transported through the µRFQ. The linear RFQ trap is operated with helium at
pressures ≈10−4 mbar to minimize beam heating via collisions with the buffer gas

15

Figure 2.6: Cartoon depicting the principles of the LEBIT beam cooler and buncher.
The solid line corresponds to the electrostatic potential used to trap, cool, and accumulate ions while the dashed line is that to eject ions from the buncher. Ions are
cooled by collisions with the low pressure buffer gas, typically helium.
during extraction. The electrostatic trap is created by applying suitable potentials
on cylindrical electrodes surrounding the RFQ structure. Ions are accumulated and
cooled for 20-30 ms before they are extracted as an ion pulse with a FWHM ≈100 ns.
The ejected ion pulse has a reduced transverse emittance. A cartoon of the process is
shown in Figure 2.6. The cooler and buncher has a cryogenic heat shield with thermal
contacts to the electrodes. Operation of the device at lower temperatures reduces the
kinetic energy of the buffer gas and improves the brilliance of the ejected ion pulses.
The system cooling also reduces the residual pressure of gases other than the noble
gases needed for beam cooling.
A second beam transport system, at -2 kV, connects the beam cooler and buncher
to the Penning trap. It includes a drift tube, steering elements, Einzel lenses, and ion
detectors. The section is also used as a time-of-flight (TOF) mass filter by quickly
switching the potential applied to a deflector in BOB5 as will be described in Section
2.5.

16

Figure 2.7: Schematic layout of the LEBIT Penning trap mass spectrometer. The
inset shows the mechanical design of the Penning trap itself.

2.4

The Penning Trap Mass Spectrometer

Mass measurements are performed in the LEBIT Penning trap mass spectrometer,
shown schematically in Figure 2.7. Details on the design and construction of the Penning trap can be found in the dissertation of Ryan Ringle [61]. The main components
of a Penning trap are two hyperbolic end caps and a ring electrode placed in a homogenous magnetic field, B. A moving ion of mass m and charge q in a magnetic
field undergoes cyclotron motion, ωc = q/m · B. While the magnetic field supplies
radial confinement, the axial confinement is provided by the electrostatic quadrupole
potential created by a voltage difference between the ring and end cap electrodes.
Figure 2.8 shows a cartoon of a Penning trap alongside a photograph of the LEBIT
Penning trap.
LEBIT utilizes an actively shielded, persistent, solenoidal, superconducting magnet with a room-temperature horizontal bore and with a field strength of 9.4 T.
Gabrielse coils [62] were included to counteract external field changes. A pair of insulated copper wires are wound around the bore tube to permit either baking of the
bore tube or compensation of the natural decay of the magnetic field [57, 58]. The
pressure of the helium bath is stabilized with a precision barometer and an electro-

17

magnetic valve controlled by a PID loop; variations in the ambient pressure were
found to affect the evaporation of liquid helium that caused non-linear changes in
magnetic field strength.
The injection and ejection optics consist of a series of cylindrical electrodes [61].
The electrode directly before the Penning trap is used as a “Lorentz” steerer [61, 63]
to position the ions in the trap. The cylinder is quartered and biased to produce an
electric dipole field, resulting in a force in an E × B force on the ion and off-axis
injection into the Penning trap. The center of each end cap electrode has a 4-mm
diameter hole to allow injection and ejection of the ions from the Penning trap. A
dynamic capture process where voltages on the end caps are switched allows ions to
enter, to be trapped in, and later to be ejected from the trap. Correction ring and
tube electrodes minimize deviations from the desired hyperbolic electric field. After
ejection from the trap, ions pass through several drift tubes to reach BOB6, where
they are detected with an MCP detector in a Daly configuration [64] or directly with
a channeltron. In the Daly configuration, a metal plate is biased to a high voltage and
mounted parallel to the MCP, both of which are positioned off-axis. Ions strike the
plate and release a fusilade of secondary electrons, which are collected on the MCP.

(a)

(b)

Figure 2.8: (a) Illustration of a hyperbolic Penning trap. A homogenous magnetic field
provides radial confinement while a static electric field provides axial confinement. (b)
Photograph of the LEBIT hyperbolic Penning trap. The end cap has been removed
to reveal the eight-fold segmented ring electrode. A 4-mm diameter hole in the end
caps permits injection and ejection of ions.

18

The LEBIT Penning trap mass spectrometer was designed for use with the time-offlight ion-cyclotron-resonance (TOF-ICR) technique [48, 49, 65], which is particularly
well suited to mass determination of short-lived rare isotopes. The technique will be
described in section 2.4.3 after the basic concepts of Penning traps are explained.

2.4.1

Ion motion in a Penning trap

Detailed discussions on ion motion in a Penning trap can be found in [49,66]; as such,
only a brief overview of the ion trajectory is presented. The applied electrostatic
potential modifies the ion motion from pure cyclotron motion into three independent
eigenmotions. In the axial direction, the potential can be described in cylindrical
coordinates (r, θ, z) by the expression:
V
V (r, z) = DC (2z 2 − r2 ).
4d2

(2.1)

where VDC is the electrostatic potential difference between the ring and end cap
electrodes and d =

2
r2 /4 + z0 /2 is the characteristic size of the trap. The quadrupole

trapping fields then are given by:
VDC
r
2d2
V
Ez = − DC z.
d2
Er =

(2.2)

In the axial direction, the only force arises from the quadrupole electric field, Fz = q · Ez ,
which causes axial oscillations with a frequency:

ωz =

qVDC
.
md2

19

(2.3)

Figure 2.9: Representation of the three independent eigenmotions of an ion in a Penning trap. The two radial motions are the magnetron (-) and the reduced cyclotron
(+) oscillations; the third is the axial oscillation (z).
The remaining two eigenmotions, in the radial plane, are a slow magnetron motion
ω− due to the E × B drift ion motion and a reduced cyclotron motion ω+ :

ω± =

ωc
±
2

2
ωc
ω2
− z.
4
2

(2.4)

The trajectory of the ion is made up from the combination of the axial, magnetron,
and reduced cyclotron motions; the three eigenmotions and their composite are pictured in Figure 2.9.
For an ion with m = 100 u in a Penning trap of size d = 1 cm, in a magnetic
field B = 9 T, with VDC = 10 V, the eigenfrequencies are of the order ω− /2π ≈ 1
kHz, ωz /2π ≈ 100 kHz, and ω+ /2π ≈ 1 MHz. In general, the eigenfrequencies follow
the relation ω− < ωz < ω+ . There are three other important relations among the
frequencies, the first of which is called the Invariance Theorem [67].
2
2
2
2
ωc = ω+ + ω− + ωz

(2.5)

ωc = ω+ + ω−
2
ωz
= ω+ ω−
2

(2.6)
(2.7)

The Invariance Theorem is true even for small (of order 10−3 ) misalignments and

20

Figure 2.10: Schematic drawing of the segmented ring electrode in a Penning trap.
The RF voltage VRF is applied 180◦ out of phase between the red (+VRF ) and blue
(-VRF ) segments. (a) Dipolar excitation. (b) Quadrupolar excitation.
distortions of the trapping potential [66, 68]. The effect of trap anharmonicities on
the mass determination will be addressed again in Section 2.4.3.

2.4.2

Excitation of ion motion

The trapped ion’s radial eigenmotion can be excited by applying multipolar azimuthal
RF fields [48] to the trapping electrodes. The most common method to introduce such
fields is to segment the ring electrode and to apply RF voltages such that the potentials on adjacent electrodes are 180◦ out of phase, as illustrated in Figure 2.10.
An n-pole field can be generated by 2n segments and is capable of exciting eigenmotions at frequencies equal to the sum of 2n−1 eigenfrequencies. For example, a
dipole excitation at ω± drives either the reduced cyclotron or the magnetron motion
and increases the eigenmotion amplitude, ρ± , [46, 47]. Two common uses for dipole
excitations are to place the ion on a magnetron orbit in preparation for mass determination (ωRF = ω− ) or to increase the reduced cyclotron radius of a particular
species of ion and drive it out of the Penning trap (ωRF = ω+ ). A quadrupole RF
field may couple both radial eigenmotions; for mass determination, one applies an
RF field at ω+ + ω− = ωc , which causes periodic beating between magnetron and

21

Figure 2.11: (a) The ion begins with pure magnetron motion, indicated by the red
circle. When the quadrupole RF field at ωc = ω+ + ω− is turned on, the reduced
cyclotron radius grows while the magnetron radius decreases. (b) Full conversion of
magnetron to reduced cyclotron motion is shown
reduced cyclotron motion, illustrated in Figure 2.11. Since ω+ > ω− , i.e. the energy
associated with the reduced cyclotron motion is greater than that associated with the
magnetron motion, the conversion of ρ− to ρ+ is accompanied by a significant increase
in radial energy, which can be detected by the ion’s time-of-flight out of the trap and
used to identify the cyclotron frequency. At LEBIT, the ring electrode is divided into
eight segments to permit dipolar, quadrupolar, and octupolar excitations, the last
excitation offers the potential of increased resolving power and was investigated in an
earlier dissertation by Ringle [61].

2.4.3

Mass Determination

At LEBIT, the ωc = ω+ + ω− cyclotron frequency is probed. For a given product of
the excitation time and amplitude and ωRF = ωc , the initial magnetron motion of the
trapped ion is completely converted into cyclotron motion. Since ω+ > ω− , the energy
associated with the reduced cyclotron motion is higher than that associated with the
magnetron motion, and the ion gains radial energy. Changes in the radial energy
are detected with the time-of-flight ion-cyclotron-resonance (TOF-ICR) technique
[48, 49, 65]. After ejection from the trap, the ion travels through ejection optics to a
detector, where its time of flight (TOF) relative to the ejection pulse is recorded. The
magnetic field strength generally decreases along the ion’s path. Due to its reduced
22

M e a n T i m e o f F l i g h t [ µs ]

5 2
5 0
4 8
4 6
4 4
4 2
3 9

4 0
-1 .5

+

K

-1 .0

-0 .5

0 .0

0 .5

1 .0

1 .5

νR F - 3 6 9 8 1 5 7 [ H z ]

Figure 2.12: Cyclotron resonance curve of 39 K+ ions with an excitation time of
TRF = 500 ms. The solid curve is a theoretical fit [49] to the data.
cyclotron motion, the moving ion possesses a magnetic moment µ = Er /B z which
ˆ
interacts with the field gradient, and it experiences an axial force, F = −µ ∂B =
∂z
E
− Br · ∂B , where B0 is the maximum field strength. By the time the ion exits the
0 ∂z

main magnetic field, all of its radial energy has been converted into axial energy and
a reduced TOF is measured. The total TOF can be calculated by the expression:
z1

T OF (ωRF ) =
z0

m
1
dz.
2 E0 − q V (z) − µ(ωRF ) B(z)

(2.8)

where E0 is the total initial energy of the ion, q is its charge, and V (z) is the electric
potential and B(z) is the magnetic field strength, both functions of the ion trajectory
from the trap (z0 ) to the detector (z1 ).
Using the TOF-ICR method, a cyclotron resonance curve can be obtained as a
function of the RF frequency similar to the one shown in Figure 2.12. The resonance
is obtained by applying a quadrupolar RF field over a range of frequencies ωRF close
to ωc and calculating the average measured TOF as a function of ωRF . The actual
cyclotron frequency can be determined by fitting the theoretical line shape to the
data [49].
The magnetic field strength is measured by applying the TOF-ICR technique to

23

a stable ion with a well known mass mref . To obtain the highest precision, each measurement of the radioactive ion is interleaved with a measurement of the reference ion,
whose mass is well known. The primary experimental result is thus the frequency ratio
i
int
int
R = ωc /(ωref )i , where (ωref )i is linearly interpolated from the reference measurei
ments which bracket the measurement of ωc . The ratio of the cyclotron frequencies

yields the mass of the radioactive ion:

m=

q
qref

·

ωc,ref
· mref
ωc

(2.9)

where q/qref is the charge ratio of the rare isotope and the reference ion. Higher
order multipoles in the electric potential and magnetic field inhomogeneities can lead
to frequency shifts in the sideband ωc = ω+ + ω− [68], and a correction must be
added to the ratio R = ωc /ωc,ref (1 + δ) [68]. At LEBIT, the contribution to δ
from various sources of possible trap anharmonicities has been studied carefully [58]:
5 × 10−11 ∆A/Q due to deviations from the quadrupole field and ≤1 × 10−10 ∆A/Q
due to the trap eccentricity, where ∆A/Q is the difference in the mass-number-tocharge-state ratio of the rare and reference ions. The effect of the magnetic field
inhomogeneity shifts the ∆ωc /ωc = ±1 × 10−9 [58].
This correction [68] depends on the eccentricity of the trapping potential ( ) and
the misalignment of the trap axis and the magnetic field (θ).

δ=

At LEBIT, θ and

9 2 1 2
θ −
4
2

ω+,ref − ω+
ω+

ω−
ω+,ref

+ ...

(2.10)

have not been measured directly; however, a highly conservative

estimate is on the order of 10−2 for either anharmonicity. The second and third
factors in Equation 2.10 are of order 10−3 each. A conservative estimate of the shift
in the cyclotron frequency ratio then is 10−10 , an order of magnitude below the lowest
statistical uncertainty of any measurement in this work.
The resolving power, R, achievable with the TOF-ICR technique is given by the
24

expression:
R=

m
ωc
=
∆m
∆ωF W HM

(2.11)

where ∆ωF W HM is the line-width of the ωc frequency determination. For a given
−1
RF excitation time, TRF , the line-width will be Fourier-limited to ∆ω ≈ TRF . The

resolving power then goes as ωc · TRF . The statistical precision can be estimated [69]
as:
δm
γ
2πγ
δωc
√
=
≈ √ ≈
ωc
m
R N
ωc TRF N

(2.12)

where N is the number of detected ions and γ, a system-dependent coefficient, is ≈0.3
at LEBIT [58]. γ was determined from high-statistics measurements of the cyclotron
frequency of 39 K+ from the stable ion source.

2.5

Beam Purification Techniques

The simultaneous storage of more than one ion species in the Penning trap can lead
to systematic errors in the mass determination [70] from the Coulomb interaction of
the radioactive ion with the contaminant ions and must be suppressed. Unfortunately,
the gas cell which enables Penning trap mass spectrometry at the NSCL is also a large
source contaminant ions. Every incoming fast ion produces about one million He+ -e−
pairs in the gas cell; the ionized helium rapidly charge exchanges with residual impurities in the buffer gas. Or, occasionally the rare isotope itself may ionize an impurity.
These contaminant ions will be transported from the gas cell through the beam line.
During the course of many mass measurements, about 700 molecular ion species have
been unambiguously identified at LEBIT. Consequently, the previous work of Pete
Schury [45] included the implementation of a multi-stage beam purification system:
the RFQ mass filter, collision-induced dissociation in the cooler, a time-of-flight mass
filter, and in-Penning trap cleaning. This system was used in the present work and
will be briefly described.

25

1 2 0

C o u n ts

+

O
2

3 0
+

H O

2 5

H

8 0

2

+

O
2

4 0
0

5 0 0

5 2 0

5 4 0

5 6 0

2 0

3 6

1 .2

+

3 8

A rH

+

+

4 0

3 9

+

A r

K

A r

4 0

+

2 5 .5

7

+

H

C
5

3 0

4 0

5 0

6 0

7 0

7

+

O
5

+

H
4

C H

3 9

6 0

0 .4
2 5 .0

+

H
6

H

8 0
+

A rH

2

C

3

2
+

O
8 0

(b) Collision-induced dissociation (CID)

0 .8

0 .0
2 4 .5

C

+

+

O
5

O
7

M a s s F ilte r S e ttin g ( u )

C o u n ts

N o r m a liz e d C o u n ts

A r

H
2

1 0

(a) RFQ mass filter
3 6

C

H
3

1 5

M a s s F ilte r U R F [V p p ]

1 .6

C

2

T O F [ µs ]

1 6 0

2 6 .5 2 7 .0
G a t e T r i g g e r T i m e [ µs ]

(c) Time-of-flight (TOF) mass filter

+

4 0
2 0
0
3 .6 9 4

2 6 .0

K

3 .6 9 6

3 .6 9 8 3 .7 0 0
νR F [ M H z ]

3 .7 0 2

(d) Dipole cleaning in the Penning trap

Figure 2.13: Results from the multi-stage beam purification system of LEBIT. (a)
Mass scan at resolving power ≈50 demonstrates the single mass resolution of the
RFQ mass filter. The mass peaks were identified unambiguously with the Penning
trap mass spectrometer. (b) Time of flight after the cooler and buncher (CID) versus
mass scan of the RFQ ion guides. Helium was used as the buffer gas in the ion guides.
The solid curve indicates ions which did not dissociate, and the two dashed curves
indicate the loss of CH4 and H2 O by the initial ion. (c) Transmission for several ion
species as a function of their TOF from the buncher. A fast kicker prevents unwanted
ions from reaching the Penning trap. (d) Counting rate of detected 39 K+ following a
dipole excitation in the Penning trap.
The mass filter, the last section of the RFQ ion guides (Figure 2.4), selects ions
by their charge-to-mass ratio m/q. A four-stage Brubaker or delayed DC ramp [71]
was applied to the ion guide to maximize the mass filter acceptance and transmission.
The resolving power of ≈50 was found to be sufficient for single mass unit separation
as can be seen in 2.13a.
The molecular isobaric ions are then delivered to the beam cooler and buncher,

26

where collision-induced dissociation (CID) can occur. In this widely applied process
[72–74], a molecular ion collides with the buffer gas and generally emits a neutral
molecule. The energy of the collision must be high enough to break the molecular
bond (>10 eV) which can be achieved by either a heavy buffer gas or by increasing
the incident beam energy. The former risks larger recoil energies and RF heating;
however, the latter is easily achieved at LEBIT by increasing all of the gas cell and ion
guide voltages by several tens of volts. The effect of CID is easily analyzed using timeof-flight (TOF) mass spectrometry of the ions exiting the cooler and buncher after
CID as was performed for another recent experiment [45]. At LEBIT, ion pulses are
ejected from the beam cooler and buncher and detected 2.5 m downstream at BOB5.
The time of flight is measured and plotted in Figure 2.13b; the upper diagonal line
of peaks indicate molecules which did not dissociate while the peaks below represent
those which did.
The combination of TOF mass spectroscopy and a fast-switching deflector amounts
to a TOF mass filter, which was thoroughly investigated by Josh Savory [75]. The
deflector consists of a pair of plates in the horizontal and vertical directions; an applied voltage prevents ions from reaching the Penning trap. When the rare isotopes of
interest should pass, a fast switch (≈300 ns) drops the deflector voltage to a near-zero
value for optimal transport. The FWHM of the ion pulse is ≈100 ns; the resolving
power of this TOF mass filter was found to be in excess of 400. The time of flight
separation of masses A = 36 - 41 can be seen in Figure 2.13c.
Any remaining isobaric contaminant can be individually removed from the Penning trap by a dipole excitation of its reduced cyclotron motion [46,47]. For sufficiently
large RF amplitudes, the ion is driven to a large orbit, where it neither interacts with
the rare isotope nor is extracted from the Penning trap. This highly selective process
was exploited in this thesis work but requires the exact identification of the contaminant ion, a somewhat time-consuming process. In Figure 2.13d, the counting rate is
constant except at νRF = νc , where the counting rate falls to zero.

27

2.6

Overview of LEBIT Performance

The extraction efficiency of thermalized, rare isotopes from the gas cell was measured
during the first commissioning runs of the gas cell to be ≈5% at an implantation
rate of ≈40 pps of a 38 Ca/37 K cocktail beam [54]. Studies of the beam cooler and
buncher were performed offline using the plasma-type ion source. Transmission from
BOB3 to BOB6 was 70% for currents up to 10 nA in continuous mode [52]. Losses in
either the -5 kV and -2 kV beam transport systems or the Penning trap are negligible
with optimization of beam transport elements. The total efficiency, including detector
efficiency, of rare isotopes from BOB1 through the Penning trap is ≈15%.
Excellent cyclotron resonance curves require on-axis beam, precision injection into
the Penning trap, well tuned potentials, and proper ejection from the trap. The precision and accuracy achievable at LEBIT has been studied with stable ions, in particular 39 K+ , 40 Ar1+,2+ , and several isotopes of Kr+ . The measured frequency ratio
of 23 Na+ and 40 Ar2+ , both known to sub-ppb precision, was found to agree fully
with the literature values [76] with a deviation of only 3(5) × 10−9 . For ion pairs
who have a larger mass difference, the effect of mass-dependent systematic effects has
been measured to be less than 5(5) × 10−10 /u [30].
During the first running phase at LEBIT, from May 2005 to July 2009, the
masses of 11 elements and roughly 40 isotopes have been measured; the rare isotopes
measured were 26,32,33 Si, 29,34 P, 37−38 Ca, 40−44 S, 63−66 Fe, 63−67 Co, 66−68,80 As,
68−70,81g,81m Se,

and 70,70m Br. With T1/2 = 96 ms, 66 As was as the shortest-lived

nuclide measured at LEBIT [20]. The mass measurement of 37 Ca [77] testifies to the
sensitivity at LEBIT, where cyclotron resonance curves were obtained with a total of
only ≈50 detected ions. 32 Si is a member of the A = 32, T = 2 quintet; and their
mass measurements contribute to the test of the isobaric multiplet mass equation and
is the focus of the present work.

28

Chapter 3
Test of the Isobaric Multiplet Mass
Equation with the A = 32, T = 2
Quintet
To first order, the strong interaction is the same for protons and neutrons. To capitalize on this phenomenon, Heisenberg introduced the concept of isospin [78], which
considers the proton and the neutron as opposite (spin) projections of the isospin
operator T . Although isospin is not an exact quantum number, it is conserved with
high precision. Wigner exploited isospin to predict masses with the isobaric multiplet
mass equation (IMME) [26, 27]; within an isobaric multiplet (states with the same
mass number A, angular momentum, parity, and isospin), the masses of the states
follow the parabolic expression:
2
M (Tz ) = a + bTz + cTz .

(3.1)

where Tz = (N-Z)/2. Higher order terms, dT3 and eT4 , require second order Coulomb
z
z
effects, charge-dependent nucleon-nucleon interaction, or many body forces [79–82].
Numerous studies of isospin multiplets have confirmed the validity of the quadratic

29

4

[k e V ]

3
2

- M E

IM M E

1
-1
-2
-3

E x p

M E

0

-4
-5

(b ) 2 0 0 6

(a ) 2 0 0 3
S i

P

S

C l

A r

S i

P

S

C l

A r

Figure 3.1: The difference between experimental and IMME-predicted mass excesses
for A = 32. (a) The validation of the quadratic form of IMME in 2003 with the
Penning trap mass measurement of 32 Ar [29]. (b) A remeasurement of the excitation
energy of the first T = 2 state in 32 S [33] revealed a breakdown of the quadratic form.
A quartic term e = 0.53(15) keV was required to restore the fit to data.
form of IMME [28], and significant deviations have only been found for in light multiplets with unbound states (such as A = 9, T = 3/2). In addition to the fundamental
importance of isospin symmetry, IMME is employed to predict unmeasured (or insufficiently precise) masses for e.g. mapping the proton-dripline for the rp-process [83]
and to assist in the assignment of nuclear spin [84]. Providing the most precise test of
IMME [28], the A = 32, T = 2 quintet has been the subject of considerable interest
for the past decade. In 1999, IMME was applied to reduce the mass uncertainty of
32 Ar

and enable a search of scalar currents [85]. Four years later, ISOLTRAP val-

idated IMME and its prediction with a Penning trap mass measurement of 32 Ar;
the residuals of the quadratic fit are displayed in Figure 3.1a. Shortly thereafter, a
remeasurement of the energy of the first T = 2 excited state of 32 S [33] disagreed
by 3 keV with the previously accepted value [86, 87]. This deviation led to a violation of the quadratic form of IMME (see Figure 3.1b) and required a quartic term

30

e = 0.53(15) keV. The authors speculated on the importance of isospin mixing in 32 S
or an erroneous 32 Si mass. An analysis of the data indicated that if 32 Si were more
bound by 3 keV then the quadratic form would be restored. To clarify the value of
the mass of 32 Si and to investigate the possible re-validation of IMME was the goal
of the present mass measurement at LEBIT. In addition the masses of 33 Si, 31,34 P,
and 32 S were measured at the same time.

3.1

Experimental Procedure

To produce the radioactive isotopes, a primary beam of 40 Ar at 140 MeV/u was
reacted with a beryllium target. The A1900 fragment separator [43] separated out
32,33 Si

and 34 P ions, and the resulting 94.5 MeV/u secondary beam was delivered

to the gas stopping station. The gas cell was operated at 600 mbar with ultra highpurity (≈1 ppb) helium gas. Recall that the incident ions in the gas cell may charge
exchange with residual impurities, thereby changing the final chemical form.
To determine what, if any, chemistry had occurred with the radioactive isotope, the

3 3
8

3 3

S iO H ( H

S i

+

3 3

7

S iO H ( H

2

O )

+
2

C o u n ts [a .u .]

6
5
4

2

O )

+

3
2
1
3 0

6 0

7 0

8 0

M a s s N u m b e r A
Figure 3.2: Measured β-activity of 33 Si (T1/2 = 6 s) after the RFQ ion guides. The
chemistry for 32 Si was assumed to be identical but not measured due to its long half
life, T1/2 = 132 yr.

31

extracted activity was measured with a silicon detector after the RFQ ion guides [55]
as a function of the mass-to-charge state ratio, A/Q. As 32 Si possesses a long half
life, T1/2 = 132 years, the chemistry of the silicon isotopes was measured only with
33 Si+

ions (T1/2 = 6 s), which was found to come as bare, singly charged ions. For

32 Si+ ,

there was a strong beam of the isobaric contaminant O+ at A/Q = 32, enough
2

to reduce the signal strength in the Penning trap if not wholly hide it. Accordingly, a
small quantity of water vapor was introduced into the gas cell to create radiomolecular
ions. The remeasurement of the activity of 33 Si after the introduction of water is
shown in Figure 3.2; peaks were seen at A/Q = 33, 68, 86 consistent with 33 Si+ and
33 SiOH(H O)+ ,
2 n

n= 1, 2. For the best yield, 33 SiOH(H2 O)+ was selected. As 34 P was
2

measured after the silicon isotopes and encountered residual water vapor in the gas
cell, it was found to form a water adduct, too, 34 P(H2 O)+ .
2
As an accuracy check, two stable masses were measured as well: 32 S and 31 P. 32 S
is the Tz = 0 member of the A = 32, T = 2 quintet. Both masses had been previously
measured with Penning trap mass spectrometry at Florida State University (FSU)
with relative uncertainties below 10−9 [34, 88]. 32 S was found as an impurity in the
helium gas and ionized by an RF discharge in the gas cell. In the presence of water,
it formed HSO2 ; after the water was depleted, its molecular form was SO2 . The 31 P
ions were produced in the test ion source, operated in plasma-mode. A small reservoir
was loaded with a solution of 31 P2 O5 and heated. 31 P+ , 31 P+ , and H31 PO+ were
2
2
identified in the Penning trap. Since O+ was created as well and could be used as a
2
reference ion to calibrate the magnetic field (see Section 2.4.3), 31 P+ was chosen as
the chemical form with the closest A/Q to O+ .
2
After the ion guides, or the test ion source for the 31 P ions, the beam was directed
to the beam cooler and buncher [5]. During the beam time, the cooler was filled
with a neon buffer gas at about 1 × 10−2 mbar, which facilitated collision-induced
dissociation (CID) of the radioactive molecular ions. One water molecule was stripped
from the SiOH(H2 O)+ and 34 P(H2 O)+ , resulting in SiOH(H2 O)+ and 34 P(H2 O)+
2
2

32

which were accumulated and bunched in the buncher, which was operated as usual
with a helium buffer gas. The stable ions, on the other hand, were cooled with helium
buffer gas in the cooler, and therefore CID was not used for the stable molecular ions.
Next, the beam pulses were ejected from the buncher and sent to the Penning trap
system [57, 58].
The time-of-flight mass filter and the dynamic capture process in the Penning
trap ensured only the ion of interest and molecular isobars were trapped. All identified isobaric contaminant species were removed with a dipole excitation at their
reduced cyclotron frequency to prevent shifts in the measured cyclotron frequency.
Additionally, measurements were performed with a very small number of stored ions,
on average less than one detected ion per cycle with the exception of the 31 P+ and
O+ measurements. The average number of detected 31 P+ and O+ ions were five and
2
2
six ions per cycle, respectively.

3.2

Results

The cyclotron frequency 2πνc = q/m · B and the statistical uncertainty were determined by fitting the resonance curve with the theoretical line shape [49], as shown
for 32 SiOH(H2 O)+ in Figure 3.3. Other sources of systematic uncertainty will be discussed later. The first resonance for the silicon isotopes was taken with TRF = 500 ms,
and subsequent resonances were performed with TRF = 1 s. Measurements for 34 P(H2 O)+
and 31 P+ were obtained with TRF = 500 ms. Resonances of HSO+ and SO+ used a
2
2
combination of TRF = 500 ms and 1 s.
As discussed in Section 2.4.3, the primary experimental result is the ratio R = νc /νc,ref ,
where νc is the cyclotron frequency of the ion of interest and νc,ref that of the reference ion, linearly interpolated to the time of the measurement of νc . The reference
ions were contaminant ions formed in the gas cell and extracted with the ion of interest for the radioactive and 32 S ions. For 31 P, the calibration ion, O+ , was a byproduct
2

33

M e a n T i m e o f F l i g h t [ µs ]

6 6
6 4
6 2
6 0
5 8
5 6

3 2

5 4
-2

S iO H ( H
2

O )

+

-1

0

1

2

νR F - 2 1 5 1 0 4 3 . 6 0 7 1 [ H z ]

Figure 3.3: The mean time-of-flight as a function of the applied RF frequency in the
Penning trap, νRF . The excitation time was TRF = 1 s. The solid curve represents
the result from fitting the theoretical line shape [49] to the data.
of the molecular dissociation of 31 P2 O5 . The reference ions and the weighted average
¯
R of the individual frequencies are tabulated in Table 3.1.

3.2.1

Systematic Error Analysis

The experimental data was examined for evidence of systematic errors associated with
nonlinear magnetic field effects, trapping field imperfections, relativistic effects, and
frequency shifts arising from ion-ion interactions.
Magnetic field strength calibration: The magnetic field strength must be
known to great accuracy and precision; nonlinear changes in the magnetic field are
unaccounted for in the linear interpolation used to calculate the B at the time of the
measurement of the ion of interest. The decay of the magnetic field and the impact
of nonlinear effects on the cyclotron frequency have been thoroughly investigated
[61]; so long as the time between reference measurements is less than 1.5 hours, the
uncertainty in the measured cyclotron frequency was found to be ∆νc /νc = 2(1) ×
10−10 . The statistical uncertainties of all measurements were one to two orders of

34

Isotope

Molecular Ion

A

Reference Ion

Aref

32 Si

32 SiOH(H O)+
2
33 SiOH(H O)+
2
31 P+

67
68
31
52
65
65
64

C5 H+
7
13 C12 C H +
4 7
O2 +
C4 H+
5
CF+
3
H2 CO2 F+
CF+
3

67
68
32
53
69
65
69

13
7
3
5
8
7
3

¯
R

N

33 Si
31 P
34 P
32 S

34 P(H O)+
2
32 SO H+
2
32 SO H+
2
32 SO+
2

1.001
1.000
1.032
1.020
1.061
1.000
1.078

004
983
804
293
959
525
692

¯ ¯
δ R/R

962 8(48) 4.8×10−9
204(11)
1.1×10−8
708 2(32) 3.1×10−9
195(17)
1.7×10−8
895 0(55) 5.2×10−9
733 9(88) 8.8×10−9
991(10)
9.3×10−9

Table 3.1: For each mass measured, the isotope is listed with its mass number, its molecular ion, the molecular ion’s mass number,
¯
its reference ion and its mass number, the number of measurements N, the mean frequency ratio R, and the relative uncertainty.
The uncertainty includes both statistical and systematic uncertainties. The charge state of all ions was Q = 1, and so A/Q = A.

35

magnitude higher than this. In addition, the reference measurements were typically
spaced less than an hour apart. Therefore, non-linear changes in the magnetic field
were considered to be negligible.
Mass-dependent effects: Residual imperfections of the trapping field affect ion
motion and may shift the measured cyclotron frequency. If the ion of interest and
its reference are a mass doublet and undergo the same injection into the Penning
trap, then their motion in the trap will be identical, and the effects cancel in the
determination of the frequency ratio R. The reference ions should be chosen to have
A/Q as close to that of the ion of interest as possible even though earlier studies have
shown that mass-dependent effects at LEBIT affect the ratio less than 5(5) × 10−10 /u
[30]. Reference ions were chosen from the contaminants with A/Q as close as possible
to that of the ion of interest although mass doublets could only be achieved for the
silicon isotopes. The largest difference was between the 32 SO+ and CF+ , ∆A/Q = 5
2
3
u. For all ratios measured, the uncertainty due to field imperfections were more than
an order of magnitude less than the statistical uncertainty.
Relativistic effects: Relativistic mass shifts always occur and affect lighter
masses, like 31 P+ , more than heavier masses like 32 SiOH(H2 O)+ . The cyclotron frequency should written as 2πνc = q/γm · B, where γ = 1/ 1 − v 2 /c2 and m is the
rest mass. The velocity v is proportional to the initial magnetron radius ρ0 . If the ion
−
of interest and its reference are a mass doublet and have the same ρ0 , the Lorentz
−
factor γ cancels in the calculation of the frequency ratio. The initial magnetron radius
ρ0 is a parameter in the theoretical line shape fit to the resonance data. Using the
−
average value of ρ0 for each ion, the relativistic shift was calculated for each ratio.
−
The largest relativistic shift was found for the ratio of 31 P+ to O2 (∆A/Q = 1 u).
The average value of ρ0 obtained was 1.047(14) mm for 31 P+ and 1.089(14) mm for
−
O+ . The effect on the ratio was of order 2 × 10−10 , which was again insignificant
2
compared to the statistical uncertainty.
Ion-ion interactions: To account for unidentified impurities, a so-called counting

36

rate analysis was performed. The cyclotron frequency was measured as a function of
the number of stored ions; the observed cyclotron frequency was then linearly extrapolated to the value for one trapped ion. Significant frequency shifts were observed
for the stable isotopes and for 33 SiOH(H2 O)+ . The maximum ratio shift observed
was 2.2 × 10−9 for the frequency ratio of 32 SO+ to CF+ . These measurements also
2
3
suffered most from contamination, and for 32 S not all of which were identified during
the measurement.

3.2.2

Comparison of Measured Mass Excesses to Literature
Values

The ratios and uncertainties listed in Table 3.1 include the statistical and applicable
systematic uncertainties. The masses of 32,33 Si and 31,34 P were determined from these
ratios and the reference mass values taken from AME’03 [76]. The mass of 32 S was determined by another procedure since the measurement of HSO+ was performed with
2
two references ions simultaneously. That is, the measurement with CF+ as the refer3
ence ion was not independent of the measurement with H2 CO2 F+ as the reference ion.
Instead of the normal procedure, the mass of 32 S was extracted from the individual
ratios, and then the weighted mean was taken. Its mass excess, ME = Amu - M, and
the other mass excesses determined in this thesis are given in Table 3.2 and compared
to mass values compiled in AME’03.
For the stable masses 32 S and 31 P, there is a deviation between LEBIT and
AME’03 of order 1.0σ and 1.8σ respectively. However, since the publication of AME’03,
more recent data has been published by the Florida State University (FSU) Penning
trap mass spectrometry collaboration. The LEBIT mass values are in excellent agreement with the FSU values [34,88] (see Table 3.3), which were performed with a single
ion technique, in which the cyclotron frequency was determined by measuring the
signal induced by the ion motion, and achieved a precision of

37

1 eV. The authors of

Isotope

MELEBIT
keV

MEAM E 03
keV

∆ME
keV

32 Si

-24077.68(30)
-20514.30(70)
-24440.53(09)
-24548.71(81)
-26015.34(32)

-24080.86(77)a
-20493(16)
-24440.88(18)
-24558(5)
-26015.70(14)

3.18(83)
-21(16)
0.35(20)
9(5)
0.36(35)

33 Si
31 P
34 P
32 S
a

Based solely on [89], ME(32 Si) = -24080.81(05)
keV; authors later republished the value listed
above [90]

Table 3.2: Experimentally determined mass excesses and a comparison to literature
values. ∆ME = MELEBIT - MEAM E 03 is the difference between the LEBIT and
AME’03 values.

Isotope

MELEBIT
keV

31 P

-24440.53(09) -24440.541(1) 0.01(09)
-26015.34(32) -26015.534(1) 0.19(32)

32 S

MEF SU
keV

∆ME
keV

Table 3.3: Experimentally determined mass excesses and a comparison to mass values
obtained by Penning trap mass spectrometry at Florida State University (FSU) [34,
88]. ∆ME = MELEBIT - MEF SU is the difference between the LEBIT and FSU
values. N.B. MEF SU = MEAM E

38

AME’03, in preparation for the Atomic Mass Evaluation of 2013, released a preview
earlier this year (called AME’11 hereafter) [91], which includes the Penning trap mass
measurements.
Among the radioactive isotopes measured here, the AME’03 and the LEBIT values
agree on the masses of 33 Si and 34 P. Previously, the mass of 33 Si had been determined
by the transfer reactions 34 S(14 C,17 O) [92] and 36 S(11 B,14 N) [93] and by delayed β-γ
coincidence techniques [94]. Both techniques result in uncertainties in the tens of
keV, while the present uncertainty is more than 20 times smaller. In the case of 34 P,
a (d,α) reaction was used to determine the mass value with an uncertainty of a few
keV [76]; the LEBIT measurement reduces the uncertainty by a factor of six. The
measurements of this work essentially determine the value compiled in AME’11 [91]:
ME(33 Si) = -20514.326(699) keV and ME(34 P) = -24548.699(810) keV.
The only significant discrepancy between LEBIT and literature mass values is
for 32 Si. Originally the mass value of -24080.81(05) keV was based upon the AVOGADRO mass measurement, a series of (n,γ) reactions starting with 28 Si and ending
at 32 Si [89]. The authors themselves later published another value with a larger uncertainty -24080.86(77) [90] without clarification. The AVOGADRO neutron separation
energies, Sn , and mass values were recalculated using the FSU mass excess value for
28 Si

[88] and the AME’11 value for the neutron [91] (see Appendix A). The recalcu-

lated mass excess of 32 Si is -24080.87(77) keV, where the undocumented uncertainty
in [90] is kept; the 3 keV discrepancy persists. The AVOGADRO Sn values for 30 Si
and 31 Si agree with the AME’11 compilation; however, there is significant disagreement for the values of 29 Si and 32 Si. The latter results from the adoption of the
LEBIT mass value by the authors of AME’11. Given that the chain of masses is now
anchored by direct mass measurements at 28 Si [88] and 32 Si (this work), there must
be an erroneous value for a neutron separation energy in the chain from 29−32 Si. The
LEBIT measurement agrees with an older (t,p) study published only as a conference
abstract, ME = -24078.0(1.3) keV [95].

39

Nuc.

Tz

MEgs [keV]

Ex (T = 2) [keV]

MET =2 [keV]

32 Si

2

-24077.68(30)a
-24080.87(77)b

32 P

1

-24304.874(40)c

5072.44(06)d

-19232.43(07)

32 S

0

-26015.534(1)c

12047.98(08)e

-13967.55(08)

32 Cl

-1

-13334.6(6)f
-13335.115(939)c

5046.3(4)

32 Ar

-2

-2200.37(1.77)c

0

a This work
b ME of 28 Si from [91] and the
c [91]
d [87]
e Weighted average of [33, 96]
f Weighted average of [97, 98]
g [99]

-24077.68(30)
-24080.87(77)

0

g

-8288.3(7)
-8288.8(1.0)
-2200.37(1.77)

Sn of 29−32 Si from [90]

Table 3.4: The mass excesses used to fit IMME: the ground state mass excess, the
excited energy of the first T = 2 state, and the mass excess of the first T = 2 state.

3.3

Impact of the present

32

Si mass measurement

on IMME
To test the quadratic form of IMME, a parabola was fit to the experimentally determined mass excesses of the A = 32, T = 2 quintet given in Table 3.4. Penning trap
mass measurements supplied the ground state energy values for 32 S and 32 Ar. The
ground state mass of 32 P was determined by the weighted average of the Penning trap
mass measurements of 31 P from this work and [88] and the neutron separation energy
of 32 P in [100]. Values for the excitation energies of the first T = 2 state in 32 P, 32 S,
and 32 Cl were taken from the most recent and most precise values [33, 87, 99]. Two
values are listed for the mass of 32 Si: one from this work and the other from combining the AVOGADRO neutron separation energies [90] with the mass of 28 Si [91].
Two values are also provided for the ground state mass of 32 Cl: the weighted mean of
two recent and mutually consistent measurements [97, 98] and AME’11. As a result,

40

Data Set
a
ME(32 Si) [keV]
ME(32 Cl) [keV]

b

c

d

-24077.68(30)a -24077.68(30)a
-24080.87(77)b -24080.87(77)b
-13334.6(6)c
-13335.115(939)d -13334.6(6)c
-13335.115(939)d

a This work
b ME of 28 Si from [91] and the
c Weighted average of [97, 98]
d [91]

Sn of 29−32 Si from [90]

Table 3.5: The four data sets a, b, c, and d used to test IMME: two values were taken
each for the ground state energies of 32 Si and 32 Cl. Mass excesses are given in keV.
5

-5
-1 0
2

E x p

M E

2

(a ) χ/n = 5 8 .1

-1 5
5

- M E

IM M E

[k e V ]

0

(b ) χ/n = 5 6 .6

0
-5
-1 0
2

2

-1 5

(d ) χ/n = 7 .9

(c ) χ/n = 8 .1
S i

P

S

C l

A r

S i

P

S

C l

A r

Figure 3.4: Fit of the quadratic form of IMME to data with the four different data
sets indicated in Table 3.4. Rows result from different mass values of 32 Si, columns
from different mass values of 32 Cl.
four data sets, described in Table 3.5 were fit with IMME. Appendix A reviews the
experimental data in greater detail.
Residuals from the parabolic fit of IMME to the four data sets can be seen in Figure
3.4. Table 3.6 provides the coefficients and the χ2 /n for each fit. The 3 keV reduction
in binding energy in 32 Si, as measured at LEBIT, exacerbates the breakdown of the
quadratic form of IMME, χ2 /n > 56, where n is the degrees of freedom (data sets (a)
and (b) in Figures 3.4a,b); the shift in fact goes in the opposite direction needed for
41

[k e V ]

2

-2

2

2

(a ) χ/n = 0 .0 2

- M E

IM M E

0

(b ) χ/n =

0 .1 3

E x p

2

M E

0
-2

2

(c ) χ/n = 1 .1 0
S i

P

S

2

(d ) χ/n = 0 .1 3
C l

A r

S i

P

S

C l

A r

Figure 3.5: Fit of the cubic form of IMME to data with the four different data sets
indicated in Table 3.4. Rows result from different mass values of 32 Si, columns from
different mass values of 32 Cl.
a quadratic IMME. The quadratic fits to data sets with the AVOGADRO mass value
of 32 Si (data sets (c) and (d) in Figures 3.4c,d) are not as bad, having χ2 /n ≈ 8. The
different mass values for 32 Cl do not make an appreciable difference in the fitting of
a quadratic IMME to the data. None of the data sets can be fitted with a parabola,
indicating a departure from the quadratic IMME, and all require a significant cubic
coefficient. Figure 3.5 presents the residuals from a cubic fit to data; the coefficients
and χ2 /n are listed in Table 3.6. The χ2 /n is of order 1 or smaller, indicating an
excellent fit; indeed, the smallest χ2 /n is found for data set (a), χ2 /n = 0.02. Again,
there is no significant difference between either value for the ground state energy of
32 Cl.

On the other hand, the mass value of 32 Si has an appreciable effect on the size

of the cubic term. Data sets (a) and (b), which relied on the direct mass measurement
of this work, results in d = 0.89(1) and 0.90(3) keV respectively. Fits to data sets
(c) and (d) produces a cubic coefficient a factor of two smaller, 0.48(13) and 0.49(05)
keV respectively. An earlier discussion [98] had demonstrated that a 300 eV shift in

42

the binding energy of 32 P substantially impacts the quality of the fit and the size
of the cubic term. In contrast, the 500 eV difference of the two mass values for 32 Cl
considered here has a negligible effect.

3.3.1

Possible Reasons for a Non-zero d Coefficient

The constant term in IMME corresponds to the isoscalar Coulomb intraction, the
linear term to the isovector interaction, and the quadratic term to the isotensor interaction. I prefer more intuitive arguments in [82] based on an inert core with four
valence nucleons. The a coefficient corresponds to the energy associated with the core.
The b coefficient is related to the average interaction between a valence nucleon and
the core. The c coefficient reflects the interaction between the valence nucleons. The
isospin lowering operator converts a neutron to a proton. The newly charged nucleon
experiences a Coulomb interaction with the core; since the interaction between the
core and any valence proton is the same to first order, the energy associated with this
interaction is constant across the multiplet. As the number of protons increases, the
electric repulsion between the valence protons increases.
A non-zero cubic (or quartic) term may indicate higher-order Coulomb effects,
charge-dependent nuclear interactions, many-body interactions, or isospin mixing of
the isobaric analog state with neighboring states. Several authors have noted that the
higher-order perturbations from the Coulomb interaction should be mostly absorbed
by the a, b, and c coefficients [79–82]; however, here a and b coefficients do not change
substantially between the quadratic and cubic fits. For data sets (a), (b), and (c), the
c coefficient changes on the order of 1.5 keV between the quadratic and cubic fits. For
data set (d), the change in c is insignificant. Studies with a simple, non-perturbative
model [79] and a second-order perturbation calculation [80] indicate that a non-zero
cubic term arises from isospin mixing of the lowest T = 2 state with neighboring
T = 1 states for interior members of the quintet: 32 P, 32 S, and 32 Cl. Indeed, from the
estimates in [79], one expects that IMME should hold to d ≈ 1 keV, in accordance
43

Data Set
a
b
c
d
χ2 /n

set aquad

set acubic

-13967.80(58) -13967.55(1)
-5473.07(1.4) -5472.92(2)
208.68(99)
207.15(2)
0.89(1)
58.15
0.02

set bquad

set bcubic

set cquad

set ccubic

set dquad

set dcubic

-13967.78(58) -13967.55(3) -13967.59(22) -13967.54(8) -13967.60(22) -13967.55(3)
-5473.19(1.4) -5472.88(7) -5471.85(60) -5472.25(25) -5471.75(64) -5472.15(9)
208.76(1.0)
207.10(7)
207.06(56)
206.87(21)
206.98(60)
206.77(8)
0.90(3)
0.48(13)
0.49(5)
56.56
0.13
8.09
1.10
7.86
0.13

Table 3.6: The coefficients and the χ2 /n of the quadratic and cubic forms of IMME to the A = 32, T = 2 quintet for different
values of the ground state energies of 32 Si and 32 Cl (see Table 3.4).

44

with experiment. Unfortunately, the isospins of too few states in 32 P, 32 S, or 32 Cl
have been determined empirically for a fruitful discussion on which states contribute
and to what degree.
Recently, the first detailed calculation for isospin mixing in the A = 32, T = 2
quintet were performed using the USDB (universal sd-shell) interaction [101]; they
included charge-dependent and charge-symmetry-breaking interactions. According to
these calculations, a Jπ , T = 0+ , 1 state in 32 Cl just below the state of interest is
largely responsible for the deviation from the quadratic form of IMME. To a lesser
extent, two T = 1 states in 32 P mix with its T = 2 state and also contribute to the
non-zero cubic coefficient. The sum of these effects is d = 0.28 keV, far smaller than
observed in experiment. The authors also predict that isospin mixing of the T = 2
and T = 1 states in 32 Cl results in the isospin-forbidden proton decay to T = 1/2
states in 31 S and the isospin correction δc = 1.8% for the f t value of the superallowed
β-decay of 32 Ar.
The last possibility considered here is that of an erroneous mass value. The masses
of three members of the A = 32, T = 2 quintet have been measured with high
precision Penning trap mass spectrometry: 32 Si (this work), 32 S (this work and [34],
and 32 Ar [29]. These measurements are largely responsible for making the A = 32,
T = 2 quintet the most precisely known isobaric multiplet [28], and each deviated
from the previously accepted mass value. The mass of the ground state of the other
two members, 32 P and 32 Cl, have been measured indirectly. 32 P was determined
by adding the neutron separation energy to to the mass value of 31 P, which was
measured here and at FSU [88] by Penning trap mass spectrometry. 32 Cl was similarly
determined from the proton separation energy and the mass value of 31 S. Nonetheless,
as this work demonstrates, neutron separation energies may be wrong. An erroneous
mass value may explain why the χ2 /n values for the cubic fits of data sets (a), (b),
and (d) are 0.13 or smaller. It is imperative to measure the masses of 32 P and 32 Cl
directly to confirm the neutron and proton separation energies, the breakdown of the

45

quadratic form of IMME, and the size of the cubic term.

46

Chapter 4
Motivation for and Foundation of
Stored Waveform Inverse Fourier
Transform
During the mass measurement of 32 S, several contaminant species were trapped simultaneously with the 32 S ions, and their presence shifted the measured cyclotron
frequency due to ion-ion interactions [70]. Nearly a day of running time cumulatively
was spent on identifying the contaminant species one by one. The identified ions were
each removed by dipolar excitation [46, 47] at their reduced cyclotron frequencies
(Equation 2.4); however, two other contaminants were detected but were present at
too low a level for timely identification. Thus significant time was lost to contaminant
“hunting” which would otherwise been spent on the mass measurements. Furthermore,
the chemistry and therefore the contaminants depend on the beam intensity and the
purity of the helium gas and in principle could change over time. A more optimal use
of beam time would be to specify mass bands around the radioactive ion to be ejected
from the Penning trap and, hence, eliminate the need to identify each contaminant.
The question becomes how best to generate a signal which can be applied with
a waveform generator for an arbitrary excitation in frequency space. The solution is

47

the Stored Waveform Inverse Fourier Transform (SWIFT) technique [51, 102–105].
SWIFT was designed for frequency-selective excitations used in Fourier transform
ion cyclotron resonance (FT-ICR) [106] and Fourier transform nuclear magnetic resonance (FT-NMR) spectrometry [51]. FT-NMR is the selection and manipulation of
the nuclear spin by its Larmor frequency. FT-ICR requires moving an ion from its initial radius and/or phase to a final radius and/or phase using its cyclotron frequency.
For dipole excitations, the final radius and phase are linearly related to the excitation strength, thus the inverse Fourier transform is the appropriate tool to transform
the excitation from the frequency domain into the time domain. (A brief overview
of the mathematical properties of Fourier transforms is presented in Appendix B, in
particular those important for SWIFT.)

4.1

Broadband excitation

At first glance, SWIFT appears simple to implement: define the desired excitation
spectrum and take the inverse Fourier transform. Let’s take for example a simultaneous uniform excitation of a frequency band, which by application of Equation 2.4
corresponds to a mass band. The discrete rectangular amplitude spectrum in frequency space Fk shown in Figure 4.1a was generated for N = 1028 points as follows

Fk =



0 if 0 ≤ k < N


8




N
7N
1 if 8 ≤ k < 8




0 if 7N ≤ k < N

8

(4.1)

The real part of the inverse Fourier transform of Equation 4.1 is shown in Figure 4.1b.

Two drawbacks arise for this time-domain signal. First, the large difference between the highest and lowest non-zero amplitudes, i.e. the dynamic range, may result
in distortion of the lower-amplitude section of the signal when digitized for use with a
48

Amplitude a.u.

Amplitude a.u.
Frequency Ν Hz
(a) In the frequency domain

Time t s
(b) In the time domain

Figure 4.1: (4.1a): A rectangular or uniform amplitude spectrum in frequency space
created with 1028 points. (4.1b) Its inverse Fourier transform, a sharply peaked high
amplitude sinc envelope.
programmable waveform generator. Second, the power is concentrated at the beginning and end of the excitation time and is near zero otherwise. Phase modulation is
the key to a power distribution which is well suited for the programmable waveform
generator.

4.2

Optimal phase modulation

The time-domain signal in Figure 4.1 results from the phase coherence of the frequency
signal F (ν). Cosine functions, the real part of the Fourier transform, with constant
phase add constructively near time t = 0 and destructively at larger t. The effect,
illustrated in Figure 4.2, is exacerbated as the frequency signal is broadened. Phase
modulation of the frequency signal F (k) destroys the phase coherence and achieves a
more optimal time-domain signal, in which the power is distributed evenly over the
excitation time.
While the simplest modulation, a linear phase modulation only serves to shift the
SWIFT signal in time without altering the signal shape (see the translation property
B.7); Figure 4.3, top, illustrates the shift. Any nonlinear phase modulation reduces
the dynamic range of the time-domain signal as was first noted in [102]. It was later

49

Figure 4.2: Red (dotted), blue (short dashed), and green (long dashed) curves are
cosine functions with similar frequencies (ν, 0.9ν, and 1.1ν respectively) and the
same phase. The sum of the colored curves is the black (solid) curve. Near t = 0, the
functions add constructively, leading to a large amplitude. At larger t, the functions
are no longer in phase and destructively interfere.
developed into a general algorithm for determining optimal phase modulation in [103].
The argument for the phase modulation scheme in [103, 107] is based on the
translation property of Fourier transforms. A time-domain waveform constructed from
a frequency signal whose phase depends linearly on frequency can be thought of as a
“wave packet”; its power is concentrated in a time interval whose center is determined
by the slope of the phase function. If the frequency spectrum is divided into two
segments, whose power spectra (square of the magnitude) is equal but whose phase
spectral slope differs, then the time-domain waveform is the superposition of two wave
packets separated in time. Should the slopes be sufficiently different, the wave packets
do not overlap, and the maximum time-domain amplitude is reduced by a factor of
two. Pursuing this argument, one may divide the frequency spectrum into many
segments of equal power spectral area and appropriately choose the phase spectral
slope for successive segments such that the successive wave packets just overlap. In
this manner, a uniform distribution of power across the excitation time is achieved.
For rectangular excitation profiles, such as SWIFT, the optimal phase modulation

50



Frequency-Domain
Magnitude Spectrum



Linear Phase
Modulation

Inverse F.T.

t

Random Phase
Modulation
Inverse F.T.


t



 Quadratic Phase
Modulation

Inverse F.T.

t


Time-Domain
Waveform

Figure 4.3: Comparison of three choices of the phase function φk (middle column). The
corresponding SWIFT time-domain waveforms (right column) were calculated from
the phase function and the rectangular frequency amplitude spectrum (left column).
A linear phase function (top row) temporally shifts power to the center of the SWIFT
waveform. A random phase function (center row) evenly distributes power over the
excitation period. A quadratic phase function (bottom row) centers and distributes
the time-domain power; the dynamic range is reduced; and, the time-domain amplitude is near zero at the beginning and end of the SWIFT waveform which reduces
distortion.
reduces to a quadratic phase function. As the quadratic term is increased, the power
spreads out.
The quadratic phase function may be written as

φk = A(k − ki ) +

B
(k − ki )2 for 0 ≤ k < N
2

(4.2)

where ki is the first frequency-domain point for which the excitation amplitude is
nonzero. In this work, the parameters A and B were taken from [102], A = π/2 and
B = -π/(kf -ki ), where kf is the last frequency-domain point for which the excitation
51

Figure 4.4: The details of the frequency-space signal are obtained by zero-filling the
time-domain excitation and performing a forward Fourier transform. Gibb’s oscillations and power leakage occur at discontinuities in the user-defined signal. Power
leakage can undesirably excite ions outside of the defined mass band.
amplitude is nonzero. The power was concentrated over half the excitation time T ,
centered at T /2, and reduced in magnitude compared to a constant or linear phase
modulation. The quadratic phase function is compared to a linear and a random phase
functions in Figure 4.3. A phase spectrum which alternates randomly between π/2 and
-π/2 was found to reduce the dynamic range by several orders of magnitude compared
to constant or linear phase modulation [108]; however, the phase discontinuities and
the truncation of the time-domain waveform can cause significant distortion.

4.3

Signal distortion

To examine the frequency spectrum of a particular excitation scheme, one “zero fills”
the time-domain signal before performing the Fourier transform [107]. The typical zero
fill doubles the number of points in a signal by adding N /2 zeroes to the beginning
and end of the signal. By doubling the frequency resolution, it permits examination
of the frequency signal not only at the originally defined points but also in between
them. The result is shown in Figure 4.4. Two types of distortion can be seen: “wiggles”
and magnitude “leakage” from high to zero amplitude segments. At discontinuities

52

in the original signal, the observed signal over- and undershoots, a phenomenon first
explained by J. Willard Gibbs (see pp. 73-74 in [109]). A sharp corner or jump cannot
be perfectly reproduced by a finite Fourier series. For a piecewise continuous function,
its Fourier series converges to the function at every point except at a discontinuity.
There, Dirichlet’s theorem [109] states that the limit converges to the average of the
values of the function on either side of the jump. Where the maximum overshoot
occurs depends on the number of terms in the Fourier series; as the number of terms
increases, the maximum overshoot moves closer to the discontinuity.
The power leakage leads to excitations outside of the desired mass band; however,
at LEBIT, the radioactive ion must not be disturbed by the cleaning process. Gibb’s
oscillations can be minimized by a window or apodization function, w(t) or wk for
continuous and discrete functions respectively. (Apodize is taken from the Greek
word apod, meaning “to remove the feet” in reference to the side lobes or power
leakage.) As the process is widely used not only in signal processing but also optics
and radioastronomy, many apodization functions have been studied [109]. They differ
by how quickly the function falls to zero and the size of the side lobes; and, they are
often tailored to the specific use. Guan describes another method [105, 107], which
smooths the original frequency (amplitude) spectrum before phase modulation and
the inverse Fourier transform. The suggested apodization in [102], referred to in this
work as the “quarter-wave sine” function, is given by

wk =



+ sin(2πk/N ) if 0 ≤ k ≤ N


8




if N ≤ k ≤ 7N
8
8
1




− sin(2πk/N ) if 7N ≤ k ≤ N

8

(4.3)

and was applied before the inverse Fourier transform. The Hann or Hanning window

53

(a) “Quarter-wave sine” window function

(b) Hanning window function

Figure 4.5: A comparison of frequency spectrum obtained by the Fourier transform
of a rectangular excitation in the time domain multiplied by different window or
apodization functions. Notice the side lobes are substantially reduced by the apodization functions, the “quarter-wave sine” window function [102] (left) and the Hanning
window (right).
function is also common to digital signal processing,
wk = sin2

πk
N

for 0 ≤ k < N.

(4.4)

It reaches zero so that the side lobes roll off at about 18 dB per octave. The effect
of both apodization functions can be seen in Figure 4.5. The Gibb’s oscillations are
suppressed although not eliminated. The drawback of apodization is reduced amplitude near the edges of the excitation profile, which may reduce the efficacy of cleaning
masses whose reduced cyclotron frequency ν+ = ω+ /2π lie at the edges of the cleaning band. However, suppressing power leakage outside the excitation band, to avoid
possible ejection of the radioactive ion, is preferred to uniformity inside the excitation
band.

4.4

SWIFT-based cleaning at LEBIT

To clean contaminant species from the Penning trap requires modifications to the
rectangular excitation scheme discussed above and indicated in Figure 4.1a. To clean

54

Figure 4.6: The desired excitation profile for beam purification at LEBIT. Two uniform rectangular excitations are separated by a gap of non-excitation. Ions whose ν+
lay within the excitation bands are cleaned from the Penning trap. The ν+ of the
rare isotope is within the gap to prevent driving its reduced cyclotron radius.
mass or frequency bands around the ion of interest, a signal consisting of two rectangular excitations separated by a gap of non-excitation is required as sketched in
Figure 4.6. Ions whose ν+ are within the excitation bands are excited and driven
from the Penning trap. The gap is centered on the ν+ of the ion of interest in order
to preserve its initial state. With SWIFT-based cleaning, only the reduced cyclotron
frequency of the rare isotope must be known, and the no-excitation gap should exceed
the uncertainty of ν+ .
Once the desired cleaning signal is defined, it is passed through the SWIFT signal
preparation process described above: phase modulation, apodization, and the inverse
Fourier transformation. Figure 4.7 shows how the quadratic phase modulation leads to
an optimal power distribution. To illustrate the power leakage into the no-excitation
gap, Figure 4.8 compares the cleaning excitation without apodization to excitations
which use either the quarter-wave sine or the Hanning window functions. As was
mentioned earlier, the prevention of undesired excitation of the ion of interest takes
priority over uniformity in the excitation bands. The Hanning window function was

55

Amplitude

t

Time t
Figure 4.7: The time-domain waveform of the frequency excitation shown in Figure
4.6 with quadratic phase modulation. The power is zero at the edges of the excitation
time and relatively evenly distributed through the middle. The dynamic range is
smaller than if the phase modulation had been constant or linear.

Figure 4.8: Comparison in frequency-space of apodized cleaning excitations. (a) The
entire excitation profile is shown. (b) Closer examination of signal strength in the gap.
The inset compares the size of the first side lobe. Apodization with the quarter-wave
sine gives less Gibb’s oscillations than the excitation without apodization. Use of the
Hanning window minimizes the Gibb’s oscillations and falls to zero amplitude fastest
in the gap. Since the gap is centered on the rare isotope’s ν+ , signal power in the gap
leads to the undesired excitation of the radioactive ion.

56

found to best suppresses the Gibb’s oscillations and featured the most well-defined
gap.

57

Chapter 5
Implementation of SWIFT
Cleaning at LEBIT
The mass measurement of 32 S which was heavily contaminated by molecular isobars
provides a useful example for the motivation for the SWIFT technique. Other beam
purification methods at LEBIT removed contaminant ions with other A/Q values.
An attempt was made to identify each isobaric contaminant, and the identified ions
were removed from the Penning trap by dipole excitation at the appropriate reduced
cyclotron frequency ν+ , which drove each contaminant ion to a radius so large that
the ion struck a trap electrode. Sidebands in the frequency spectrum, resulting from
the finite cleaning time, were suppressed by amplitude modulation (AM) of the RF
signals with a Gaussian window function. Table 5.1 lists the contaminants identified
by dipole TOF resonance during the mass measurement of HSO+ and SO+ as well as
2
2
two other examples of contamination during found during beam times. The reduced
cyclotron frequency of molecular contaminant ions is typically a few kHz away from
the ion of interest, and the difference increases the more exotic the ion of interest
simply because the binding energy of radioactive nuclei decrease as one approaches
the driplines.
The stored waveform inverse Fourier transform (SWIFT) technique circumvents

58

HSO+
2
H2 CO2 F+
H3 C2 F+
2

SO+
2

∆ν+ [Hz]
-1167
-1726

CNF+
2
C3 N+
2
H2 C2 F+
2

∆ν+ [Hz]

64 Fe2+

-1338
O+
2
-1559 CH4 O+
-1781 C2 H+
8

26 Si+

∆ν+

-2709 CN+
-7825 C2 H+
2
-12931

-2290
-4969

∆ν+ [Hz]

Table 5.1: A list of contaminants identified during different mass measurements and
the separation of their ν+ from the ion of interest. Top row indicates the ion of
interest. During the measurement of HSO+ , two other contaminants were detected
2
but were too weak to identify.
the need for individual contaminant identification by using a broadband excitation
to drive out simultaneously a range of masses. The diagram in Figure 5.1 depicts the
excitation profile labeled with the four parameters needed to generate it. This chapter
will describe the implementation of SWIFT at LEBIT. To be considered successful,
the SWIFT scheme must:
• Provide high resolution cleaning at short cleaning times, Tclean ≈ 30 ms
• Excite a bandwidth of bL + bR = 30 kHz with a gap of non-excitation g = 1000 Hz
centered at ν+
• Suppress signal in the gap by a factor of 1000 compared to the amplitude of the
excitation bands

Figure 5.1: A cartoon of the desired SWIFT profile: two rectangular excitations of
bandwidth bL and bR , separated by a gap g. The gap should be centered on the
reduced cyclotron frequency ν+ of the ion of interest.
59

N
16 kS
64 kS
100 kS
1 MS
4 MS
16 MS
∆ν [Hz]

10 ms 20 ms
0.8
0.4
3.2
1.6
5
2.5
50
25
200
100
800
400
100
50

50 ms 100 ms
0.16
0.08
0.64
0.32
1
0.50
10
5
40
20
160
80
20
10

Table 5.2: The maximum cleaning bandwidth in MHz for an AFG with memory N
points for cleaning times Tclean = 10-100 ms. The reduced cyclotron frequency of
singly charged ions at LEBIT ranges 1-10 MHz, which can be covered by either large
N or amplitude modulation. The Agilent 33521 sample rate is 250 MS/s; should its
full 16 MS memory be required, the shortest cleaning time is 64 ms. The last row
indicates the frequency resolution in Hz.
As SWIFT requires relatively little equipment, a preliminary investigation was made
with available hardware.

5.1

Signal Synthesis

The SWIFT waveform was calculated in Labview, software version 8.2. A GUI was
developed that allowed the user to input the cleaning time Tclean , the bandwidth of
the left and right bands, the size of the gap g, and the reduced cyclotron frequency
ν+ . The cleaning time determines the frequency resolution, ∆ν = 1/Tclean , which
combined with the memory of N samples limited the maximum cleaning bandwidth.
Due to the Nyquist frequency, only N /2 points may be used. Table 5.2 tabulates the
total bandwidth available for different cleaning times and memory sizes N .
At LEBIT, the reduced cyclotron frequency ranges from 7.5 MHz for A = 19 to
1.6 MHz for A = 85. To cover this bandwidth, either an AFG with a large memory
(N ≥ 4 mega-samples for Tclean = 100 ms) or amplitude modulation (to frequency
shift the signal) must be used. If the latter is used, then a bandwidth of 30 kHz or
less would be sufficient.
Amplitude modulation was carried out by multiplying the SWIFT waveform with

60

Figure 5.2: Amplitude modulation of the SWIFT signal: (a) Half of the excitation
profile is defined at low frequencies (black, solid) and modulated at the carrier frequency νcarrier = ν+ . The band is shifted to the frequency range of ν+ (blue, solid)
and reflected νcarrier (blue, dashed). As there is typically some DC component from
the AFG (shifted to νcarrier ) and power leakage at νcarrier , the ion of interest may be
excited by the residual signal at νcarrier . (b) Defining both excitation bands at low
frequencies (black, solid) and then AM the signal avoids this potential noise at ν+ .
The carrier frequency is chosen to position ν+ in the center of the gap of the SWIFT
excitation (blue, solid). The reflected signal (blue, dashed) is wasted.
a sine function, sin(2πνcarrier t), where νcarrier is known as the carrier frequency. AM
shifts the waveform and creates two signals at ν+ ± νcarrier as illustrated in Figure
5.2. Thus, only one band of the excitation needs to be generated by the SWIFT AFG
and modulated at νcarrier = ν+ , and its mate will also be created. However, if there
is a DC component in the signal produced by the AFG, it, too, will be modulated to
νcarrier and could excite the ion of interest. Often, the carrier function bleeds through
the AM device and results in some signal strength at νcarrier . To avoid these peaks,
one generates a two excitation band waveform at low frequencies and then modulates
that signal. An additional SWIFT waveform results but does not interfere with the
desired excitation scheme.

61

The bandwidth (and cleaning time) should be chosen to ensure that the reduced
cyclotron frequency of all contaminant ions lay within the bands bL and bR . The gap
should be centered on the ν+ of the ion of interest and must be wide enough not to
excite it. The resolving power of SWIFT is approximately R ≈ ν+ /(g/2), and the
goal is to achieve a gap of 1000 Hz.
The signal must also be corrected for distortion due to digitization by the AFG.
When an AFG outputs an arbitrary waveform of N points in the time Tclean , every
output value is kept constant for a duration ∆t = Tclean /N . This corresponds to the
convolution of the signal with a rectangular function of duration ∆t; that is, in the
frequency domain, the excitation profile is multiplied by a sinc function. A sinc−1
function,
πi
,
sin(πi/N )

sinc−1 =
i

(5.1)

must be multiplied into the SWIFT amplitude profile before the inverse Fourier transformation to compensate for the effect of the digitization. This was later verified
experimentally, and it was found to lower the noise floor by 5 dB.
To summarize the steps for SWIFT signal synthesis:
1. The user defines an excitation profile in frequency space of two rectangular
excitation bands separated by a non-excitation gap.
2. The waveform is multiplied by a sinc−1 function to correct for the digitization
process.
3. The waveform is phase-modulated and apodized as described in Chapter 4
4. The digital waveform is loaded into the AFG.

5.2

Preliminary Studies

For the initial investigation, two arbitrary function (waveform) generators (AFG)
were used: Agilent 33250 and Agilent 33120. Both have 12 bit amplitude resolution
62

Figure 5.3: Unwanted signal power in the gap of a typical SWIFT signal due to
digitization with different amplitude resolution. The amplitude of excitation is 1 V,
and the desired noise suppression in the gap is a factor of 1000 or better. The Tektronix
3102 AFG, for the first implementation of SWIFT, has 14 bit resolution; the Agilent
33521, for the final and successful implementation of SWIFT, has 16 bit resolution.
for the output amplitude. The Agilent 33250 has a 64000-point memory and a 200
MS/s (mega-samples/second) sample rate while the Agilent 33120 has only 16000
points and 40 MS/s. Thus the former was used for the SWIFT signal synthesis and
the latter reserved for the amplitude modulation. The carrier sine wave was generate
with the Agilent 33120 sine wave generator which produces superior sine waveforms
compared to the arbitrary function generator feature. Amplitude modulation only
shifts the SWIFT signal to the desired frequency range (Equation B.8) and avoids
the need for large AFG memory. The modulation was carried out with an Analog
Devices AD835 “multiplier” chip. Various tests were performed using the LEBIT
system with stable 85,87 Rb isotopes from the test ion source.
The initial tests indicated that SWIFT held great promise at LEBIT: ions separated by two mass units could be individually cleaned. It was also noticed that
distortion from the digitization of the SWIFT signal could pose a problem, arising
from the round off errors in the integer representation in the AFG memory. In Figure
5.3, SWIFT signals are calculated with different amplitude resolution. To ensure that
the ion of interest was unaffected by the SWIFT excitation, the amplitude in the gap
should be at least 1000 times smaller than the excitation amplitude. An amplitude
63

Figure 5.4: SWIFT waveform output by Tektronix 3102 AFG and captured on an
Agilent EXA Signal Analyzer N9010A. (a) A SWIFT signal amplitude modulated (b
= 124 kHz, g = 2 kHz, νcarrier = 2.25 MHz) and captured on the Agilent spectrum
analyzer. (b) The gap of the same SWIFT waveform. The peak at 2 MHz is a “crossterm” from the amplitude modulation process.
resolution of 14 bits or higher can achieve this level of noise suppression.

5.3

First implementation of SWIFT

Following the initial studies, a dedicated AFG, Tektronix 3102, was purchased. It
features 14 bit amplitude resolution, N = 101 kS, a sample rate up to 250 MS/s for
N > 16 kS, and two output channels. It provides the minimum amplitude resolution
needed and offers more flexibility than the Agilent 33250 for longer cleaning times.
The SWIFT signal outputted by the Tektronix 3102 was first thoroughly investigated
off-line with an Agilent EXA Signal Analyzer N9010A (frequency range 9 kHz - 3.6
GHz). These studies demonstrated that amplitude-modulated SWIFT signals fail the
third criterion for success: a factor of 1000 signal suppression in the gap. An example
of a SWIFT signal can be seen in Figure 5.4. The gap can be seen, and the noise floor
is ≈60 dB (a factor 1000) below the amplitude of the excitation bands. The SWIFT
waveforms generated by the Tektronix AFG thus met the criteria outlined above and
can even produce frequency gaps smaller than 1000 Hz.
The next step was to shift the SWIFT signal into the frequency range of interest.

64

Figure 5.5: Amplitude-modulated SWIFT signals (b = 124 kHz, g = 2 kHz)
captured on the Agilent spectrum analyzer: (a) νcarrier = 2.277 MHz. (b)
νcarrier = 2.2501 MHz. The gap of the same SWIFT waveform. The Tektronix AFG
produces harmonics of the fundamental SWIFT and of the modulating sine signals,
all of which mix. The result is a SWIFT “family,” of which at least five members are
visible.
To amplitude modulate the signal, two devices were tested: the AD835 multiplier chip
mentioned earlier and a Mini-Circuit ZAD-3H+ mixer. At certain νcarrier , a peak was
seen in the gap with either AM device (see Figure 5.4b). When no source of noise
could be found, the peak was traced back to the amplitude modulation process itself.
The Tektronix AFG is linear to -50 dBc in the MHz regime, i.e. it suppresses higher
order harmonics of its signal by at least -50 dB. For example, there is the fundamental SWIFT signal at νSW IF T , and smaller SWIFT harmonics occur at 2νSW IF T ,
3νSW IF T , etc. and similarly for the modulating sine function. Thus, the fundamental SWIFT signal is modulated by sine waveforms at νcarrier , 2νcarrier , 3νcarrier , etc.
as is each of the SWIFT harmonics; thus, so-called “cross terms” are created and may
interfere with the desired SWIFT signal. For certain values of νcarrier , the harmonic
SWIFT signals constructively add with the fundamental signal, as in Figure 5.4. For
other values of νcarrier , the cross terms fill the gap of the fundamental SWIFT waveform as shown in Figure 5.5. Furthermore, AM devices are generally nonlinear and
may introduce noise themselves which may also hinder the creation of the desired
SWIFT profile.

65

Various attempts to make the AM process more linear failed. Most RF mixers are
designed for frequency ranges higher than the 1-10 MHz needed at LEBIT. To balance
the mixer, the typical amplitudes had to be so high they would overload the average
RF amplifier. The number of harmonic SWIFT and sine tones prevents centering an
amplitude-modulated SWIFT at any given frequency. Consequently, the decision was
made to abandon AM entirely in favor of simply using a very large memory AFG.

5.4

Second implementation of SWIFT

The recently issued Agilent 33521-02 offers the most memory of bench-top AFGs, 16
MS. Even for the longest cleaning time anticipated at LEBIT, Tclean = 100 ms, the
Agilent 33521-02 can cover a frequency range up to 80 MHz. Its specifications for
harmonic distortion and spurious noise are on par with other top-rated AFGs on the
market, ≈-50 dBc in the MHz regime. Moreover, the amplitude resolution is 16 bits,
which further improves the signal suppression in the gap of the SWIFT waveform.

5.4.1

Characterizing the SWIFT waveform

The capabilities of the Agilent 33521 AFG were explored with the same Agilent
spectrum analyzer as used with the Tektronix AFG. Particular attention was paid
to the two built-in filters to smooth the transition between points of an arbitrary
waveform. The effects of filtering in time and frequency space can readily be seen for
the test function sin(k/2) at a sample rate of 500 kS/s (see Figure 5.6). Without a
filter, the transition between points is abrupt and takes ≈10 ns; in frequency space,
the noise floor is higher than with the filters. The step filter response is -3 dB at
13% of the sample rate and intended to minimize ringing and overshooting on (timedomain) step functions. It provides smooth transitions between points and yields the
narrowest peak in frequency space; however, the filter reduces the amplitude. The
normal filter response is -3 dB at 27% of the sample rate; with this filter option, the

66

Figure 5.6: Sine waveform captured with the Agilent spectrum analyzer. The effects
of the Agilent 33521 AFG built-in filters on sin(i/2) at a sample rate of 500 kS/s
on the fundamental tone are compared. The spectrum analyzer time-averaged the
waveforms. The inset depicts the time-domain signal.
Parameter

Symbol

Typical value

Cleaning time [ms]
Bandwidth of left and right band [Hz]
Gap bandwidth [Hz]
Cleaning amplitude [Vpp ]
Apodization function
Ion species

Tclean
b
g

20
1000
1000
180
Hanning
23 Na+

output is smooth in the time domain and and broad and flat in frequency space.
The effect of the two filters was also examined with a sample SWIFT signal, as
shown in Figure 5.7. The typical parameters used for the SWIFT waveforms studied
in this chapter are listed in Table 5.4.1. The gap was intentionally chosen to be very
small (g = 5/Tclean Hz = 50 Hz) to investigate if the filters offered an advantage in
gap size. The step filter waveform had a very shallow slope although its measured
gap was equal to the gap of the normal filter waveform. The normal and unfiltered
waveforms appeared nearly identical when time-averaged, but the unfiltered waveform
contained more noise. The normal filter was chosen to minimize noise and to have the
most well-defined gap.

67

Figure 5.7: Effect of Agilent 33521 AFG built-in filters on the SWIFT signal as captured on the Agilent spectrum analyzer. The waveform was created using parameters
as listed in Table 5.4.1 except Tclean = 100 ms and g = 50 Hz. The amplitude was
1 Vpp not to overload the spectrum analyzer. (a) The entire SWIFT profile. (b) The
gap of the same SWIFT waveforms and time-averaged. Unfiltered signals suffer more
noise than filtered signals.
By itself, the Agilent AFG output was superior to the SWIFT waveform output by
the Tektronix AFG and amplitude modulated to MHz frequency range. No harmonics
were found to interfere with the fundamental tone. There are no AM devices needed
to raise the noise floor. But how well would SWIFT clean ions?

5.4.2

Determining the cleaning amplitude

The output of the Agilent AFG 33521 was sent to an RF amplifier (T&C Power
Conversion AG1021 at 100% voltage gain), and then applied to one of the ring segment
electrodes of the Penning trap. Tests of the SWIFT excitation scheme were first
performed with 23 Na+ and 39 K+ produced by heating the tungsten filament of the
test ion source.
To find the required cleaning amplitude for a given cleaning time and bandwidth,
one excitation band was generated and centered on the reduced cyclotron frequency
ν+ of the 23 Na (determined earlier by a dipole time-of-flight measurement) and the
amplitude scanned. Figure 5.8 shows the relative number of ions detected after ejection from the Penning trap as a function of the cleaning amplitude. The cleaning

68

1 .2

C o u n ts [a .u .]

1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
0

2 0

4 0

6 0

C le a n in g A m p litu d e [V

8 0
p p

1 0 0

]

Figure 5.8: Scan of the SWIFT amplitude determined with 23 Na+ . The red line is a
linear fit to determine the amplitude needed to clean ions. The cleaning amplitude
scales as 1/Tclean and increases with the bandwidth. The fluctuations are due to the
ion source. (SWIFT waveform parameters are listed in Table 5.4.1 except here a single
1 kHz-wide band was used.)

69

1 .0

Io n s C le a n e d [a .u .]

0 .8
0 .6
0 .4
0 .2

N A
H a n n in g
1 /4 - S in e

0 .0
-7 0 0

-6 0 0

νS

-5 0 0
W IF T

-4 0 0

-3 0 0

− ν+ [ H z ]

Figure 5.9: Measured side lobes of different apodization windows. The edge of a rectangular excitation band was scanned over ν+ of 23 Na+ . The Hanning wave function
was chosen to minimize power leakage into the gap. (Parameters are listed in Table
5.4.1 except here only one 1 kHz band was used.)
amplitude was determined by a linear fit to the slope in the counting rate distribution. It increases with the bandwidth, b, and with the inverse of the cleaning time,
1/Tclean .

5.4.3

Measurement of the SWIFT waveform with ions

The apodization window was checked by scanning the position of the edge of an
excitation band over the reduced cyclotron frequency of 23 Na+ . Three cases were
considered: the Hanning (Equation 4.4), the “quarter-wave sine” (Equation 4.3), and
no apodization functions. Figure 5.9 displays the results of the scans. All cases still
contained Gibb’s oscillations. The Hanning window features the smallest side lobe
although it reduces the amplitude at the edge of the excitation band. Since the biggest
concern is power leakage into the gap and less so the uniformity of the excitation
bands, the Hanning window was chosen for apodization just as it was in Section 4.3.

70

Sub-figure
a,b
c,d
e,f

Ion
39 K+
39 K+
23 Na+

Tclean [ms]

b [Hz]

g [Hz]

Amp. [Vpp ]

Apodization

20
20
20

500
500
1000

100
300
500

95
95
180

Hanning
Hanning
Hanning

Table 5.3: Parameter values used to generate the excitation profiles depicted in Figure
5.10
To determine the effect of the full SWIFT excitation scheme on the ions, the
excitation bands were shifted across ν+ of 23 Na+ and 39 K+ ; this was accomplished
by increasing the center of the gap in 25 Hz increments. Ions were injected into the
Penning trap on axis, and the number detected at BOB6 was recorded as a function
of the center or SWIFT frequency. Figure 5.10 compares the expected response (as
calculated by a forward Fourier transform) to the counting rate distribution. Table
5.3 summarizes the parameter values used to generate the SWIFT excitations. The
distribution of the number of ions cleaned from the Penning trap as a function of
the SWIFT frequency closely resembles the excitation profile. The distributions of
cleaned ions are flatter than the calculated waveforms because the amplitude used
for the SWIFT excitation exceeded that needed to clean ions (indicated by the red
dashed line) and to a lesser extent from fluctuations in the ion source. Side lobes can
be seen at the edges of the excitation band. With the cleaning time held constant,
Tclean = 20 ms (∆ν = 50 Hz), the gap size was varied for the three excitation schemes
shown in Figure 5.10. The gap is clearly visible. In examining the, one notices that
power leakage reduces the gap by about 50 to 100 Hz on either side. Figure 5.11
shows an expanded view of the the gap shown in Figure 5.10d. Small fluctuations in
counts within the gap indicate the variation in the signal power. The gap set to 500
Hz measures ≈375 Hz; the gap set to 300 Hz measures ≈200 Hz; and, the gap set to
100 Hz is filled in with its amplitude above the noise floor. From Figures 5.10 and
5.11, one can quickly ascertain that when Tclean = 20 ms the smallest gap that can
be set is 300 Hz and hence obtained is ≈200 Hz.

71

0 .6

(a )

1 .0

(b )

0 .4
0 .5

0 .0
0 .6

(c )

Io n s C le a n e d [a .u .]

C le a n in g A m p litu d e [V ]

0 .2

0 .4

0 .2

0 .0
0 .6

0 .0
1 .0

0 .5

0 .0
1 .0

(e )

(d )

(f)

0 .4
0 .5
0 .2
0 .0

0 .0
-1 0 0 0

νS

0
W IF T

1 0 0 0

-1 0 0 0

- ν+ [ H z ]

νS

0
W IF T

1 0 0 0

- ν+ [ H z ]

Figure 5.10: Comparison of expected SWIFT excitation profiles and observed ion
profiles. The red dashed line indicates the amplitude required to remove ions. Left:
The forward Fourier transform of the SWIFT signal. Right: Ions cleaned defined as
(1 - normalized counts detected at BOB6) as the SWIFT signal was shifted over ν+
in frequency space. The gap size was set as follows (a,b) g = 100 Hz, (c,d) g = 300 Hz,
and (e,f) g = 500 Hz. (b,d) were experimentally determined with 39 K+ and (f) was
measured using 23 Na+ . (The parameter values to generate the SWIFT waveforms
can be found in Table 5.3.)
72

Io n s C le a n e d [a .u .]

1 .0

0 .5

0 .0
-2 0 0

-1 0 0

νS

0

W IF T

- ν+ [ H z ]

1 0 0

2 0 0

Figure 5.11: A closer look at the gap of the SWIFT excitation profile experimentally
determined with 39 K+ shown in Fig 5.10c. Ions cleaned is the difference of one and
the normalized, detected count rate. The gap was set to 300 Hz and measures ≈200
Hz. At Tclean = 20 ms, gaps are 100-200 Hz smaller than their set value. Generalized
to any cleaning time, the gap is 2/Tclean - 4/Tclean smaller than its set value. (All
other parameter values can be found in Table 5.4.1.)

73

2 8

M e a n T O F [ µs ]

C o u n ts [a .u .]

1 .0

0 .5

0 .0

2 6

2 4

2 2

0

2

4

6

G a p x T

8

1 0

0

2

4

6

G a p x T

c le a n

8

1 0

c le a n

Figure 5.12: (a) Detected 23 Na+ ions as the gap size was varied from 0 to 500 Hz. (b)
Mean TOF of 23 Na+ ions as a function of gap width. Although these measurements
were done with Tclean = 20 ms, the results are identical with other cleaning times.
(The parameter values used to generate the SWIFT waveform can be found in Table
5.4.1.)

5.4.4

Isobaric separation

Two quantities were measured to determine the narrowest gap achievable: the mean
time of flight of the ions from the trap to the ion detector and the number of detected ions. Ions were injected on axis; the SWIFT excitation was applied, and then
a quadrupole excitation followed. If the gap is too small, the ion will be excited by
the SWIFT signal. If the amplitude is high enough, the ion will be cleaned. If the
amplitude is small, then the ion is only driven off axis, and then during the quadrupole excitation at νc , its magnetron motion will be converted to cyclotron motion as
described in Section 2.4.2 resulting in a shorter time of flight. Figure 5.12 shows the
distribution of counts and mean time of flight as the gap was varied with Tclean =
20 ms. The scan was repeated for Tclean = 10 ms and 50 ms with identical results.
Conservatively, the gap must be at least 6/Tclean wide, which corresponds to 120
Hz, 300 Hz, and 600 Hz for Tclean = 50 ms, 20 ms, and 10 ms respectively, to avoid
disturbing the ion of interest. The findings shown in Figure 5.12 match those found

74

in the previous section and illustrated in Figure 5.11.
The measurements just described demonstrate the cleaning capabilities of the
SWIFT technique; however, as easily seen in Figure 5.11, residual power leaks into
the gap in spite of the apodization and the 16 bit amplitude resolution of the Agilent
AFG. There will be some dipolar excitation of the ion of interest; so one is left with
the question, how much will the residual dipolar excitation effect the accuracy of mass
measurements?

5.4.5

Mass measurement with SWIFT

A mass measurement was performed with 39 K+ following the method described in
Section 2.4.2. “Measurements” with 39 K+ performed with SWIFT cleaning were interleaved with calibrations with 39 K+ performed without cleaning. Ions were injected
into the Penning trap with a non-zero magnetron radius whereas all of the measurements described above were performed with the ions injected on axis. The parameters
used to generate the SWIFT waveform can be found in Table 5.4.1. The measured
frequency ratios are shown in Figure 5.13. As expected, the mean ratio was found
to be 1.000 000 002 7(48), or a relative deviation from unity by 2.7(48) × 10−9 ; the
uncertainty corresponds to 176 eV. The Birge ratio [110] is 1.02(13), which indicates
extremely good agreement. The use of SWIFT does not affect the accuracy of the
high precision mass measurement.

5.4.6

Measurements with CO and N2

The ultimate test of SWIFT was to trap simultaneously two isobaric species, to clean
one, and to obtain a resonance of the other. For this purpose, a mixture of carbon
monoxide (CO) and nitrogen (N2 ) was ionized in the test ion source operated in
plasma-mode and delivered to the Penning trap, where the SWIFT cleaning was
applied. The ratio of N+ to CO+ in the Penning trap was roughly 1 to 2. The CO-N2
2

75

R a tio - R a tio A v g [x 1 0

-8

]

6
4
2
0
-2
-4
-6
0

5

1 0

1 5

M e a s u re m e n t
Figure 5.13: Deviations from the mean value of the mass measurement of 39 K+ with
SWIFT-based cleaning. Calibrations were performed without cleaning. Measurements
were performed with SWIFT cleaning. (SWIFT waveform parameters are listed in
Table 5.4.1.) The mean ratio is 1.000 000 002 7(48); and, the Birge ratio [110] is
1.02(13).

76

pair, whose reduced cyclotron frequencies are separated by only 2061 Hz, simulates a
typical rare isotope measurement at LEBIT; isobaric contaminants are usually a few
kHz away from the radioactive ion. First, a resonance was taken with both species in
the trap; with an excitation time of 7 ms and a 2700 Hz range; a resonance for each
species was observed, see Figure 5.14a. Next, a SWIFT signal was generated with a 1
kHz gap and 5 kHz bands; see Table 5.4.1 for the other parameters. First the gap was
centered on the reduced cyclotron frequency ν+ of N+ ; that is, the SWIFT excitation
2
cleaned CO+ from the Penning trap. As evidenced in Figure 5.14b, only the resonance
for N+ ions can be seen. Similarly, when the gap was centered on ν+ (CO+ ) and N+
2
2
was cleaned, only a resonance from CO+ ions developed (Figure 5.14c) of CO+ . The
measurements were performed over 3.5 hours, during which time the decay of the
magnetic field caused a shift of 7 Hz in the measured νc . The simultaneous presence
of contaminant ions reduced the signal strength; in Figures 5.14, the resonances in
(b) and (c) clearly look better than those in (a). The resonance dip is deeper, and
the sidebands are more pronounced.

5.5

Final Remarks

The measurements described in this chapter demonstrate the efficacy of SWIFT cleaning. Its successful implementation at LEBIT relied on high amplitude resolution to
lower the noise floor, a large AFG memory to avoid amplitude modulation, and sufficient amplification to drive the ions. Not only has the SWIFT cleaning scheme met
the criteria outlined above, but also the minimum gap size is a mere 6/Tclean (600 Hz
at Tclean = 10 ms, the shortest time envisioned at LEBIT). The resolving power R ≈
ν+ /(g/2) = ν+ · Tclean /3. For the case of 23 Na+ explored here, the resolving power
ranges from R = 4.2 × 104 for the typical LEBIT cleaning time Tclean = 20 ms to R
= 2.1 × 105 for Tclean = 100 ms. The standard or benchmark of mass spectrometry
is the separation of the CO-N2 mass doublet, which was demonstrated in Figure 5.14.

77

N

2 5

+

+

C O

2

M e a n T O F [ µs ]

2 4
2 3

(a )

2 4
2 1
1 8

N

+

(b )

2

2 4
2 1
1 8

C O

(c )
-3 0 0

0

3 0 0

νR
F

- ν+ ( N

1 8 0 0
2

+

2 1 0 0

+
2 4 0 0

) [H z ]

Figure 5.14: Quadrupole resonances (TRF = 7 ms) with CO+ and N+ simultaneously
2
loaded in the Penning trap. (a) SWIFT off. A small resonance is visible for each
species. (b) SWIFT cleaning CO+ . Only the N+ resonance is observed. (c) SWIFT
2
+
cleaning N2 . Only the CO+ resonance is seen. The red curves are theoretical line
shapes [49] to data. The resonances were measured over 3.5 hours, during which time
the decay in the magnetic field caused a shift of 7 Hz in the measured νc of CO+ . Note
that the scale of the y-axis in (a) differs from that in (b) and (c). (SWIFT waveform
parameters are as listed in Table 5.3 with b = 5 kHz here.)
78

Unfortunately, SWIFT could not be tested with rare isotope beams within the time
frame of this work. The upgrades to the gas stopping station, which began two years
ago, are still underway. Nonetheless, one can be confident of its success during future
experimental work.
In comparison to other methods of isobaric purification, SWIFT has many advantages. The traditional dipole cleaning used at LEBIT requires the exact identification
of each contaminant species; SWIFT cleaning eliminates this time-consuming process
and allows more beam time to be devoted to the measurement of rare isotopes. Massselective buffer gas cooling developed at ISOLTRAP [48–50] necessitates much more
hardware, occupies a significant amount of space, and most importantly is slow. At
JYFLTRAP [111], for example, the best resolution R = 1.5 × 105 was achieved with a
total cleaning time of 100 ms; to reduce the total cleaning time to 54 ms dropped the
resolution to R = 2.0 × 104 . SWIFT cleaning is fast (Tclean = 20 ms) and efficient
(no contaminant identification), and it provides excellent isobaric purification.

79

Chapter 6
Summary and Outlook
In this thesis, the high precision mass measurement of 32 Si and its contribution to
the test of the isobaric multiplet mass equation (IMME) are described. The 3 keV
reduction in binding energy amplified the breakdown of the quadratic form of IMME
in the A = 32, T = 2 quintet. Recent mass measurements of 32 Cl [97, 98] have
confirmed the findings published first in [112] and updated in this work. The wealth
of recent data on the A = 32, T = 2 quintet requires a large cubic coefficient, of order
1 keV, for a good fit. The newly achieved precision supports theoretical predictions for
IMME from models dating from the 1970’s [79, 80] to the first detailed calculations
in the sd-shell [101] of today. The possibility of an erroneous mass has not been
eliminated, and only three of the five ground state energies, 32 Si, 32 S (this work
and [34]), 32 Ar [29] have been measured with directly. Given that this quintet is
the most stringent test of IMME, it behooves the Penning trap mass spectrometry
community to measure the ground state masses of 32 P and 32 Cl as well.
The history of the A = 32, T = 2 quintet offers another lesson, namely the importance of direct mass measurements. The measurements of 32 Si presented here shifted
the mass value by 4σ. The LEBIT mass values for 32 S and 31 P agree with other Penning trap mass measurements [34, 88] and confirm a more than 1σ shift from indirect
mass measurements [76]. The latter mass value in turn shifted the mass determination

80

of radioactive 32 P, a member of the A = 32, T = 2 quintet.
LEBIT would be an excellent facility to measure 32 P and 32 Cl, which can be more
difficult to produce at ISOL facilities. The LEBIT mass measurement campaign will
resume shortly following upgrades to the gas stopping station and the installation of
the NSCL 3 MeV/u re-accelerator, ReA3. The upgrades include linear gas stopping
with improved efficiency and cleaner beams and, eventually, a cyclotron stopper [113]
with shorter extraction times and improved stopping of light ions.
In parallel to its mass campaign and then the NSCL developments, the LEBIT
collaboration has pursued upgrades of its own, including improved beam purification. SWIFT cleaning reduces the systematic uncertainty in mass measurements.
Compared to dipole cleaning, it eliminates the need for individual identification of
contaminant ions. Compared to mass=selective buffer gas cooling, SWIFT provides
high resolution cleaning at short cleaning times, occupies less space, and requires less
hardware. The criteria for successful implementation were met. Noise suppression in
the gap is a factor of 1000 compared to the excitation band amplitude. Gaps narrower than 1 kHz were demonstrated. When CO+ and N+ ions were simultaneously
2
trapped, SWIFT removed one species without disturbing the other. A resolving power
in better than 107 was achieved. SWIFT cleaning is certain to become a heavily used
tool in future LEBIT mass measurements.

81

APPENDICES

82

Appendix A
Experimental Data of the A = 32,
T = 2 Quintet
This appendix supplements the data and text in Chapter 3. The most recent and
most precise measurements available for each nuclei are reviewed and compared to
the accepted literature values in the Atomic Mass Evaluation 2003 (AME’03) [76] and
the preview of the Atomic Mass Evaluation 2013, released earlier this year (referred to
as AME’11) [91]. (As AME’11 is in a preliminary form, the evaluation and references
are undocumented.) This appendix is not a comprehensive review of all experimental
data relevant to the quintet. The ground states of the interior members 32 P, 32 S, and
32 Cl

have isospin T = 1, 0, 1 respectively; since the first T = 2 state is of interest,

measurements of its excitation energy are also described.

A.1

Mass Value of the Ground State of

32

Si

The AVOGADRO group measured the masses of 29−32 Si via thermal neutron capture on 28 Si [89]. This measurement is the basis for the AME’03 compiled value
-24080.81(05) keV [76] although the authors themselves republished the data in 2001,
with ME = -24080.86(77) [90] without explanation for the larger uncertainty. The

83

A

AVOGADRO

29
30
31
32

8473.551(05)
10609.18(1)
6587.40(3)
9203.22(6)

IAEA

AME’11

8473.537(23) 8473.6(0)
10609.23(03) 10609.20(2)
6587.39(03)
6587.39(4)
9199.97(30)

∆
-0.048(5)
-0.02(2)
0.01(5)
3.25(31)

Table A.1: Comparison of neutron separation energies Sn (in keV) for several Si
isotopes from the AVOGADRO project [89, 90], the IAEA database [100], and the
preview of the next Atomic Mass Evaluation (AME’11) [91]. Since the IAEA database
and AME’11 agree, only the difference of the AVOGADRO project and the AME’11
(∆ = MEAV OGADRO - MEAM E 11 ) is provided.
values of γ-ray energies were unchanged although their uncertainties were an order
of magnitude higher than originally documented. Propagation of the new error does
not result in σ = 0.77 keV.
Since then, the mass of 28 Si was measured with a precision of 0.6 eV at the
Florida State University (FSU) Penning trap mass spectrometer [88]. Although a
small discrepancy was found with AME’03 (2.3 ± 1.9 eV), it is negligible for this discussion. The newly adopted mass value in AME’11 is used to recalculate the neutron
separation energies Sn from the AVOGADRO-measured γ-ray energies. Table A.1
compares the neutron separation energies from AME’11, IAEA, and the recalculated
AVOGADRO values. AME’03 (and subsequently AME’11) also compiled measurements performed at McMaster University [114,115] and Los Alamos National Laboratory [116], which deliver precise (σ ≤ 140 eV) and mutually consistent measurements.
The IAEA [100] tracks thermal neutron capture data, including neutron separation
energies, from the Lone database, the Reedy and Frankle database, ENSDF, and
the Budapest reactor; unfortunately, it does not include the Sn of 32 Si. Nonetheless,
its Sn values for 29 Si, 30 Si, and 31 Si are in excellent agreement with AME’03 and
AME’11. The AVOGADRO Sn data agrees with the IAEA compilation for 29 Si and
31 Si

but not for 30 Si, 51(31) eV. With respect to the compilation in AME’11, agree-

ment is found for 30 Si and 30 Si, while for 29 Si deviation is significant, 48(5) eV, and
for 32 Si 3.25(31) keV. The latter is essentially the discrepancy in the mass of 32 Si

84

Ref.

Method

ME [keV]

[95]a
[90]
this work
AME’11 [91]
[88, 90]b

30 Si(t,p)

-24078.(1.3)
-24080.86(77)
-24077.68(30)
-24077.687(297)
-24080.87(77)

28 Si(n,γ)

PTMS

a Published only in
b ME value of 28 Si
of 29−32 Si [91]

a conference abstract
[88] and the Sn values

Table A.2: Recent ground state mass measurements and values of 32 Si: the reference,
the technique, and the mass excess value are given in chronological order.
given that the AME’11 adopted value is the LEBIT mass value. The mass values are
compared in Table A.2. The LEBIT mass value agrees with an undocumented (t,p)
measurement [95].

Ref.

Ground State Mass Excess
Method
ME [keV]

AME’03 [76]
AME’11 [91]
[91, 100]a

-24305.218(187)
-24304.874(040)
-24304.82(23)

First T = 2 Excitation Energy
Ref.
Method
Ex [keV]
[117]
[118]
[119]
[120]
NNDC [87]
a

30 Si(3 He,pγ)
30 Si(3 He,pγ)
31 P(p,γ)
31 P(p,γ)

5072.9(1.5)
5073.4(1.5)
5072.42(06)
5072.535(123)
5072.44(06)

ME value of 31 P [91] and proton separation
energy Sp of 32 P [100]

Table A.3: Recent ground state and excited state energy values of 32 P: the reference,
the technique, and the mass excess value are given in chronological order.

85

A.2

Mass Value of Isobaric Analog State of

32

P

The ground state mass of 32 P has been measured at several institutions through βdecays of 32 P [121] and of 32 Si [122] and via 31 P(n,γ) [100,119,120]. The latter requires
accurate knowledge of the mass of 31 P. The Penning trap mass measurement of 31 P
in this thesis and at FSU [88] agree extremely well and achieve a higher precision than
the compiled value in AME’03 [76] and shift the mass value by about 300 eV. The
AME’03 value of 31 P was based on several 31 P(p,γ) reactions, but AME’11 appears
to adopt the weighted average of the Penning trap mass measurements for 31 P. The
average 31 P value can be combined with the 32 P neutron separation energy for a high
precision mass excess value for 32 P.
The excitation energy of 32 P was first explored via charge exchange [117,118] and
later by neutron capture [119, 120]. Measurements are mutually consistent. In this
thesis, the accepted literature value [86] of Ex is adopted.
Ground State Mass Excess
Method
ME [keV]

Ref.
AME’03 [76]
[34]
this work
AME’11 [91]
Ref.
[96]
[95]a
[33]
a

PTMS
PTMS

-26015.70(14)
-26015.5344(15)
-26015.34(32)
-26015.53374(133)

First T = 2 Excitation Energy
Method
Ex [keV]
31 P(p,p), 31 P(p,α)
31 P(p,γ)
31 P(p,γ)

12049(2)
12045..0(4)
12047.96(28)

Published only in a conference abstract

Table A.4: Recent ground state and excited state energy values of 32 S: the reference,
the technique, and the mass excess value are given in chronological order.

86

A.3

Mass Value of the Isobaric Analog State of 32S

A stable isotope, 32 S is highly accessible. Despite this, the uncertainty of the AME’03
value for the ground state energy is 140 eV. The precision in the AME’11 value is
improved by the Penning trap mass measurements of this work and in [34]. The latter
finds a small but acceptable deviation of 166(140) eV from the AME’03 value. The
LEBIT mass value exhibits a similar, acceptable deviation from AME of 360(349) eV.
AME’11 appears to incorporate the two Penning trap measurements, for an uncertainty of ≈1 eV.
Surprisingly, few measurements have been made of the first T = 2 excitation
energy [33, 95, 96]. Unfortunately, the authors of [95] did not document their work
further than a conference abstract, and their measured Ex disagrees with the more
recent measurements. The (p,p) and (p,α) measurement [96] suffers from a large
uncertainty, 2 keV. The (p,γ) measurement [33] offers the smallest uncertainty, 0.28
keV. In this work, the ground state mass excess is taken from AME’11 and the excited
state energy is the weighted average of [33, 96].

A.4

Mass Value of the Isobaric Analog State of
32

Cl

The AME’03 mass value for the ground state of 32 Cl was based on charge exchange
measurements made more than 25 years earlier [123–125] and suffered a large ≈7 keV
uncertainty. Within the past year, two high precision measurements have been made
[97, 98]. As there is a small difference in the values of the weighted average of [97, 98]
and the AME’11 value, both are used to fit IMME although the difference was found
to be insignificant. The NNDC [87] gives the first T = 2 excited state energy Ex =
5024(8) keV but has not been updated to include the most recent measurement [99],
which found Ex = 5046.3(4) keV. The difference of 22 keV most likely arises from

87

Ground State Mass Excess
Ref.
Method
ME [keV]
AME’03 [76]
[97]
[98, 99][
a]
AME’11 [91]

32 S(3 He,t)

-13329.771(6.593)
-13333.8(1.2)
-13334.88(65)
-13335.115(939)

First T = 2 Excitation Energy
Ref.
Method
Ex [keV]
[126]
[125]
[127]
NNDC [87]
[99]
a

32 Ar(β)
32 S(3 He,t)
32 Ar(β)
32 Ar(β)

5057(12)a
5018(10)
5052(9)a
5045(8)a
5046.3(4)

21 keV has been added for the newer mass
measurement of 32 Ar [29]

Table A.5: Recent ground state and excited state energy values of 32 Cl: the reference,
the technique, and the mass excess value are given in chronological order.
the 2003 direct mass measurement of 32 Ar, whose value changed by 21 keV. The
measurement from [99] is used exclusively in this analysis.

A.5

Mass Value of the Ground State of

32

Ar

The only direct mass measurement of 32 Ar and the sole basis for the AME’03 [76]
and AME’11 [91] values is a Penning trap mass measurement from the ISOLTRAP
collaboration [29], ME = -2200.2(1.8) keV, a 21(50) keV shift from the previously
accepted value. AME’11 updates the reference mass used to calibrate the magnetic
field, yielding ME = -2200.369(1.770) keV, the value adopted in this analysis.

88

Appendix B
Fourier Transforms and Their
Properties
Fourier analysis developed from the study of Fourier series, introduced by Joseph
Fourier. The French physicist and mathematician was solving the problem of heat
transfer through a metal plate; he proposed modeling heat sources as a linear sum of
trigonometric functions. This superposition, known as the Fourier series, allows any
complicated periodic function to be decomposed into sine and cosine functions.
This decomposition process led to the Fourier transform. The value of the transform is the dual perspective of a function in both time and frequency space. The
forward Fourier transform (FT) permits the analysis of the amplitude and phase
of each frequency component, while the inverse Fourier transform (IFT) allows the
synthesis of a time-domain signal from any arbitrary frequency signal.
Below is a brief introduction to Fourier transforms and the properties relevant
to the stored waveform inverse Fourier transform (SWIFT) technique in Chapters
4 and 5. More comprehensive discussion of Fourier transforms can be found in, for
example, [109, 128].

89

B.1

Definition

There are several common conventions to define the Fourier transform. In this work,
the unitary transformation is taken here:
+∞

f (t) e−2πiνt dν

F (ν) =

+∞

F (ν) e+2πiνt dν

⇐⇒ f (t) =

−∞

(B.1)

−∞

The latter is the inverse Fourier transform. Here the independent variable t represents
time and the transform variable ν frequency. Clearly the time-domain signal f (t) and
its Fourier transform F (ν) are complex. The Fourier transform can be written as a
plane wave of amplitude |F (ν)| and with phase φ(ν): F (ν) = |F (ν)| eiφ(ν) . Or, using
Euler’s formula, eiθ = cos θ + i sin θ, it can be decomposed into its real part R(ν) and
imaginary part I(ν), which relate to the amplitude and phase by
|F (ν)|2 = R(ν)2 + I(ν)2

(B.2a)

tan(φ(ν)) = I(ν)/R(ν).

(B.2b)

Perhaps the most intuitive example is a sine wave in the time domain at a frequency
ν0 and its transform a delta function in frequency spaced centered a ν0 .

f (t) = sin(2πν0 t) ⇐⇒ F (ν) = δ(ν − ν0 ) + δ(ν + ν0 )

(B.3)

Another common pair, including in the SWIFT waveforms, is the rectangle (or “top
hat”) function and the sinc function.

f (t) =



1, if |t| ≤ T

2

⇐⇒ F (ν) = sincT (ν) =


0, if |t| > T

2

90

sin(πνT )
.
πν

(B.4)

B.1.1

Basic Properties

To describe the properties of Fourier transforms relevant to SWIFT, let F (ν), G(ν),
and H(ν) be the Fourier transforms of the integrable functions f (t), g(t), and h(t)
respectively.
Linearity: For any complex number a and b,

h(t) = a f (t) + b g(t) ⇐⇒ H(ν) = a F (ν) + b G(ν)

(B.5)

Scaling: For any non-zero, real number a,

g(t) = f (at) ⇐⇒ G(ν) =

1
ν
F
|a|
a

(B.6)

Translation: A linear translation in time t0 shifts the phase in the frequency domain.
g(t) = f (t − t0 ) ⇐⇒ G(ν) = e−2πit0 ν F (ν)

(B.7)

Modulation: Similarly, for any real number ν0 ,

g(t) = e2πitν0 f (t) ⇐⇒ G(ν) = F (ν − ν0 )

(B.8)

Complex conjugation:

g(t) = f ∗ (t) ⇐⇒ G(ν) = F ∗ (−ν)

(B.9)

Convolution:
+∞

f (τ )g(t − τ ) dτ f (t) ∗ g(t) ⇐⇒ H(ν) = F (ν) · G(ν)

h(t) =
−∞

91

(B.10)

Similarly, h(t) = f (t) · g(t) ⇐⇒ H(ν) = F (ν) ∗ G(ν) .
Parseval’s theorem: If f (t) and g(t) are square integrable, then
+∞

f (t)g ∗ (t) dt =

+∞

F (ν)G∗ (ν) dν

(B.11)

−∞

−∞

Perhaps the most important physical consequence of Parseval’s theorem is the
conservation of energy. That is, the total energy stored in the time-domain
spectral density |f (t)|2 equals that in the frequency-domain density |F (ν)|2 .

B.2

The Discrete Fourier transform

Often in experiment, N data points are collected at times separated by ∆t into the
data set fn where n = 0, 1, . . ., N. The total measurement time T is their product N ∆t
and results in a frequency resolution of 1/T . Since frequency and time are reciprocal, if
the duration T is decreased while N stays constant, the frequency bandwidth increases
while the frequency resolution drops. The discrete Fourier transform (DFT) Fm and
its inverse fn replace their continuous counterparts:
1
Fm = N

N −1

fn e2πinm/N

N −1

⇐⇒ fn =

n=0

Fm e−2πimn/N

(B.12)

m=0

The definition implies the functions in the time and frequency are periodic over the
N/2

intervals [0, T ] and − T ,

N/2
T

“wrap around” to the interval

respectively. In this sense, negative frequencies
N/2
T ,

N . Another way to look at the DFT is to

consider the data set [fn ] as a vector multiplied by an n × n square matrix of the
exponentials to obtain another vector [Fm ] of length n. Properties of the continuous
transform can also be applied to the discrete transform.

92

B.3

Fast Fourier transform

Calculation of the discrete Fourier transform requires N 2 operations. The invention of the Fast Fourier Transform (FFT) [129] reduces the number of operations
to 2N log2 N , a substantial savings in time. If N is divisible by 2, [an ] could be
broken into two vectors, each of length N /2 and subjected to a Fourier transform
of only half the length to give two vectors, which when multiplied by the so-called
exponential “twiddle parameters” and re-combined to yield [Fm ]. This process can
be repeated until only trivial 2 × 2 matrices are left. Furthermore, N can be any
non-prime number; the FFT relies on factorization of the Fourier transform matrix.
More information on FFT algorithms can be found in [129, 130]

B.4

Sampling theorem

In digital signal processing, how frequently must a signal be measured to determine
accurately its frequency spectrum? The sampling theorem supplies the answer: the
Nyquist or folding frequency νN , which is twice the highest frequency component. The
N/2

N/2

bandwidth of the spectrum is - T to + T . The theorem is more easily understood
schematically as illustrated in Figure B.1. If the signal is non-zero above the Nyquist
frequency, the power will be “folded” back into the spectrum and appears at a lower
frequency. A frequency at νN + ∆ν is mapped to νN - ∆ν. Aliasing, as it is called,
is the result of beating between the frequency and the sampling rate.

93

(a) In the time domain

(b) In the frequency domain

Figure B.1: Illustration of the sampling theorem. Left: A 7 Hz sine wave (blue) is
sampled 10 times over 1 second (black dots). Since the Nyquist frequency νN = 5 Hz,
the measured function (black curve) is at 3 Hz. Right: In the frequency domain, the
true frequency (blue) is mapped into the bandwidth [-5 Hz, 5 Hz] at 3 Hz (black).

94

Appendix C
Simulations of RFQ Mass Filter:
Calculations of Transmission and
Emittance
The RFQ ion guides were designed and constructed as part of Pete Schury’s thesis
work [45]; however, the emittance of the ion guides and in particular of the mass filter
was not studied at the time. To my knowledge, there have been no published studies
of the emittance as a function of mass filtering strength. This appendix presents the
results of simulations performed in SIMION 8.0 of the emittance and transmission
through the LEBIT RFQ mass filter; and, they complement experimental studies
performed by and under analysis by Rafael Ferrer. The ion guides were described in
Section 2.2, and their design was shown below in Figure C.1. The ion motion in an
RFQ trap are well known [59, 60, 131]; hence, the equations of motion will be stated
here without derivation.

95

Figure C.1: Mechanical design of the RFQ ion guides. The three sections are followed
by a 5 keV acceleration region and end at BOB1.

C.1

Ion Motion in an RFQ Ion Guide

In an RFQ, there are four rods, each phase-shifted 180◦ from its nearest neighbor. If
the potential difference between the rods is VDC −URF cos(ωt+φ), then the equations
of motion are
e
(V
− URF cos(ωt))x = 0
2 DC
mr0
e
y−
¨
(V
− URF cos(ωt))y = 0
2 DC
mr0

x+
¨

z=0
¨

(C.1a)
(C.1b)
(C.1c)
(C.1d)

where the z-direction is taken along the beam axis. They can be rewritten using
the dimensionless parameter ξ = ωt/2 for the phase of the driving RF field and the
Mathieu parameters a and q:
4eVDC
2
mω 2 r0
2eURF
qx = −qy =
2
mω 2 r0

ax = −ay =

(C.2a)
(C.2b)
(C.2c)

96

Then the transverse equations of motion may be written as
d2 u
+ [au − 2qu cos 2ξ]u = 0
dξ 2

(C.3)

where u represents x or y. Setting aside the trivial solution, the ion may follow either
a stable oscillatory trajectory or an unstable exponentially increasing trajectory. This
defines regions of stability in in the Mathieu parameter space (a, q) for the x- and
y-directions, and where they overlap the ion motion is stable in the transverse plane.
The first region of stable x- and y-trajectories is commonly referred to as the stability
diagram and by convention is shown only for a > 0 since it is symmetric about a
(see Figure C.2. The left edge of the stability diagram represent a heavy-mass cutoff
and the right edge a light-mass cutoff. At the tip of the stability diagram, (a,q) =
(0.237,0.706), only ions of a particular mass have stable trajectories. For a = 0, the
RFQ ion guide becomes a bandpass mass filter. The mass filter strength at LEBIT is
measured by the ratio α = VDC /URF . As ions enter the mass filter (a = 0), they pass
through a region of stability as indicated in Figure C.2. If VD C is slowly increased,
the ion always stays within the stable region; a Brubaker or delayed DC-field [71]
accomplishes this and was implemented at LEBIT [45].
Two sets of ions were considered, an “ideal” case and a damped version for the
masses A = 28, 45, and 85. In the ideal case, the ions were assumed to have been
damped in the segmented RFQ (due to buffer gas leaking from the gas cell) and to
have reached thermal equilibrium in the µRFQ; i.e. their initial conditions were ideal.
Following the discussion in [131], the equilibrium density distribution of the ions was
a Gaussian distribution in the x- and y-directions with standard deviations σu and
σv in phase space.
σu =

1
ωu

kB T
, σv =
m

kB T
m

(C.4)

where kB is the Boltzmann constant, T is the temperature at 20◦ C, and ωu is the
eigenfrequency of the macromotion. For sufficiently small Mathieu parameter q, ωU
97

Figure C.2: Stability diagram in the Mathieu parameter space (a,q). The red line
corresponds to a constant α. As the ion enters the mass filter, it passes through a
region of instability. By slowly increasing the DC-quadrupole or mass filter strength
α (black line), the ion always stays within the stable region.
√
may be approximated by qu ωRF / 8. The RF phase was randomly sampled; the RF
drive is a periodic transformation of the initial phase-space distribution into another
of the same area.






 



γu
0   u 
 u  
 =
· 
ω
v
γu RF tan θr γ1
v
2
u
macro

(C.5)

where the stretch and rotation parameters γ and θr are complicated functions of the
RF phase and the operating parameters. The following parameterizations of γ and
θr [132], valid for |q|

0.7, were used:

γu =

1 + p0 ep1 |q| − p2 q cos (2ξ) (1 + p3 |q|p4 )

(C.6a)

tan θr =

q (1 + r0 |q|r1 ) · sin( 2ξ + r2 q sin (2ξ) ).

(C.6b)

The parameters pi are listed in Table C.1. For any given RF phase, the ion trajectory

98

i

pi

ri

0
1
2
3
4

0.00371
5.53052
0.51347
1.44023
3.27271

1.02035
2.74320
0.45510

Table C.1: Fit parameters for Equations C.6.
P [mbar]
1.610−1
1.010−5
4.010−7

Cross A
Cross B
Cross C

Table C.2: Typical pressures P measured along the RFQ ion guides. Cross A housed
the segmented RFQ. Cross B was just downstream of the µRFQ. The mass filter lay
between Cross B and Cross C.
touches the same ellipse in phase space; to achieve this in SIMION simulations, a
trajectory quality (“TQual”) of at least 120 is required. For each mass 3000 ions were
generated.
In the damped case, the damping in the segmented RFQ and the µRFQ was
modeled with hard-sphere collisions [131]. Typical pressures in the three sections of
the ion guides are listed in Table C.2. I received ion distributions for 5000 ions which
had begun at rest in the segmented RFQ and flown through the µRFQ from Stefan
Schwarz [132]. The transmission from the segmented RFQ to the beginning of the
mass filter is provided in Table C.3. The transmissions I calculated through the mass
filter are normalized to Table C.3.
A

T [%]

28
45
85

59.7
80.7
83.5

Table C.3: Transmission from the segmented RFQ to the mass filter using hard-sphere
collisions [132].

99

r0 [mm] URF (28) [V0p ] URF (45) [V0p ] URF (69) [V0p ]
Seg. RFQ
µRFQ
Mass Filter

7.8
3.0
6.2

79.5
18.0
77.0

114.5
29.0
125.0

196.5
55.0
235.0

Table C.4: The inter-rod distance r0 and the RF amplitudes for A = 28, 45, 85 of
the segmented RFQ, the µRFQ, and the mass filter. The electrostatic potentials are
listed in Table C.5.
Seg. RFQ
Potential [V]
µRFQ
Potential [V]

DC1
60

DC2
43

DC3
38.5

µplate1 µDC1 µDC2 µplate2
0
22.5
22.5
0

Mass Filter
Potential [V]

DC4
20

DC5
10

DC6
10

DC7
0

Acceleration
Potential [kV]

Acc1
-0.35

Acc2
-3

Acc3
-1.25

DC8
-350

Acc4
-5

Table C.5: The electrostatic potentials for the electrodes in the ion guides. The RF
amplitudes are listed in Table C.4.

C.2

Simulation parameters

The inter-rod spacing or inscribed radius and the RF amplitudes are given in Table
C.4. The RF amplitudes for the µRFQ and the mass filter correspond to q = 0.706 for
the masses used in this study; for the segmented RFQ, values used in experiment were
taken. The voltages for the electrostatic potentials based on voltages typically used for
the RFQ ion guides and the acceleration region directly downstream from the RFQ
ion guides are summarized in Table C.5. Although the transmission and emittance
were recorded immediately downstream of the DC8 electrode, the accelerating field
penetrated DC8.

C.3

Transmission through the Mass Filter

The transmission T is plotted as a function of mass filter strength α in Figure C.3
for the three masses under both conditions. For ions with ideal initial conditions,

100

T r a n s m is s io n ( % )

1 0 0
8 0
6 0
A = 2
A = 2
A = 4
A = 4
A = 8
A = 8

4 0
2 0
0
1 5 .2

8 d a
8 id e
5 d a
5 id e
5 d a
5 id e
1 5 .6

m p e d
a l
m p e d
a l
m p e d
a l
1 6 .0

1 6 .4

M a s s F ilte r S tr e n g th α( % )
Figure C.3: Simulated transmission as a function of mass filter strength α =
VDC /URF . Solid curves correspond to ions which underwent damping in a hardsphere collision model. The dashed lines correspond to ions whose initial conditions
were defined with the Mathieu equations.
the transmission is almost 100% until high mass resolving power is achieved; at α ≈
αmax , the transmission drops to ≈ 85% for A = 24, 45 and 95% for A = 85. In general
lighter masses are more difficult to transport than heavier masses as a consequence
of the Mathieu equations C.3. Similarly for the damped case, more A = 85 ions are
transported than A = 28 or 45 ions. However, the difference in transmission between
heavy and light masses is exacerbated relative to the ideal case. Transmission for A
= 28 drops from 73% at α = 15.1% to 39% at α = 16.5% (normalized to the value
in Table C.3). Damping leads to extraordinary losses especially for light ions. Most
losses for ions with ideal or damped initial conditions occurred at the exit of the mass
filter; the ions with damped initial conditions also frequently hit an electrode at the
beginning of the mass filter. In general, the ion trajectories of the ions with damped
initial conditions had larger motional amplitude as can be seen in Figure C.4

101

Figure C.4: Ion trajectories of A = 28 ions in the mass filter. Above “ideal” initial
conditions; below damped initial conditions.

102

60
40
20
0

15.2

100

x-emittance [ mm mrad]

A = 28

15.6

16.0



ideal
damped

80

A = 45

40
20
15.2

15.6

16.0

80

ideal
damped

60

20
0

15.2

ideal
damped

80

20

0

15.2

15.6

16.0



16.4

A = 45

40
20
0

15.2

15.6

16.0

16.4



y-emittance [ mm mrad]

x-emittance [ mm mrad]

40

16.0

60

16.4

A = 85

15.6



100


ideal
damped

A = 28

40

16.4

60

0

y-emittance [ mm mrad]

ideal
damped

y-emittance [ mm mrad]

x-emittance [ mm mrad]

80

16.4

ideal
damped

40

A = 85

20

0

15.2

15.6

16.0

16.4



Figure C.5: Simulated x- and y-RMS emittances as a function of mass filter strength
α. Emittance is normalized to 5 keV.

103

C.4

Emittance and Beam Profile of Mass-filtered
Beams

One would expect that as α increases, the emittance grows. Figure C.5 plots the
RMS emittance for the three masses as a function of α. The emittance of the damped
ions steadily increases with α, except for the y-emittance of A = 28. The emittance
of ions with ideal initial conditions does not follow the expected trend; moreover no
general trend can be observed for the x- or y-emittance or for a particular mass in
the ideal case. To some extent, this may be explained by the falling transmission.
As α increases, the ion trajectories resemble more unstable trajectories. Ions lost
were those with the largest motional amplitude and highest velocity. Given that the
y-emittance is larger than the x-emittance for all cases indicates that the pseudopotential (confining the ions in the radial direction) is elongated in the y-direction;
that is, the restoring force is weaker in the y-direction. Examination of the beam area
in phase space supports this hypothesis.
Figures C.6-C.11 show the x- and y-emittance for A = 28, 45, 85. The outline
of the beam area in phase space is roughly the same between the ideal and damped
cases although the latter has more diffuse edges. To first order, the orientation of the
areas (the direction along which the area is longest) is 90◦ out of phase between the
x and y phase-space plots as expected from the Mathieu equations. Ions are clustered
in a small phase space area with fewer ions far from the origin; however, there is
sufficient “scatter” to enhance the emittance. The size of the high density cluster and
of the overall scatter is generally larger for the phase space in the y-direction. The
enhancement in the y-direction is also visible in the beam profiles shown alongside the
emittances in Figures C.6-C.11. The profile is always star-shaped, as was observed in
experiment.
These simulations demonstrate that transmission through the mass filter falls off
drastically for lighter masses as the mass filter strength α increases. The emittance

104

may increase because the ion trajectories more resemble unstable trajectories or fall if
the ions farthest from the origin in phase space are lost. If the beam encounters heavy
damping and is mass filtered at single mass resolving power, its emittance is larger
than the acceptance of the beam cooler and buncher (41.5 π mm mrad at 5 keV [132];
even for ions with ideal initial conditions, high α causes the emittance for light ions
to be larger than the acceptance of the cooler and buncher. Further simulations are
needed to confirm the elongated pseudo-potential hypothesis.

105

Figure C.6: Simulated x- and y-emittance and beam profile for A = 28 ions with damped initial conditions.
106

Figure C.7: Simulated x- and y-emittance and beam profile for A = 28 ions with “ideal” initial conditions.
107

Figure C.8: Simulated x- and y-emittance and beam profile for A = 45 ions with damped initial conditions.
108

Figure C.9: Simulated x- and y-emittance and beam profile for A = 45 ions with “ideal” initial conditions.
109

Figure C.10: Simulated x- and y-emittance and beam profile for A = 85 ions with damped initial conditions.
110

Figure C.11: Simulated x- and y-emittance and beam profile for A = 85 ions with “ideal” initial conditions.
111

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