Dissertation
submitted to the
Combined Faculty of Natural Sciences and Mathematics
of the Ruperto Carola University Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
Presented by
Kulikov Ivan
born in:
Maloyaroslavets, Russia
Oral examination: 05072021
Study of the A=100 deformation region through high precision mass
measurements of the neutron rich krypton isotopes
Referees:
Dr. Apl. Prof. Yuri Litvinov
Dr. Priv.Doz. Wolfgang Quint
Abstract
Nuclear structure effects such as shapetransition are observed as sudden changes in the mean
square charge radii, lowlying excitation energies and the binding energies evolution. The latter
is executable by the direct mass measurements. The deformation in the krypton isotopic chain
has been an open question for decades and this work presents the direct mass measurement of
96−98
Kr isotopes. The isotopes of interest have been produced at the ISOLDE facility at CERN.
The mass measurements were performed by use of the versatile ISOLTRAP mass spectrome
ter. The multireflection time of flight and the time of flight ion cyclotron resonance techniques
were applied for the mass determination. The mass of 98 Kr isotope was determined for the first
time. Results from the state of the art meanfield and beyond meanfield approaches are com
pared to the experimental binding energy trends.
Kernstruktureffekte wie der Formübergang können anhand von plötzlichen Änderungen des
mittleren quadratischen Ladungsradius, von niedrig liegenden Anregungsenergien und von der
Entwicklung der Bindungsenergien veranschaulicht werden. Mit Hilfe von hochpräzisen Massen
messungen kann man Letzere messen. Kernstrukturelle Verformungen in der Isotopenkette von
Krypton sind seit Jahrzehnten eine offen Frage, welche mit der in dieser Arbeit präsentierten
Präzisionsmassenmessung der Krypotonisotope 96−98 Kr ausführlich behandelt wird. Genannte
Isotope wurden an ISOLDE/CERN produziert und mit Hilfe des Massenspektrometers ISOLTRAP
gemessen, wobei die Mass von 98 Kr zum ersten Mal direkt bestimmt werden konnte. Für die
Massenbestimmung wurden die Multireflection Time of Flight und die Time of Flight Ion
Cyclotron Resonance techniken angewendet. Die Ergebnisse der Messungen werden mit mod
ernen Methoden der Molekularfeldtheorie verglichen.
vii
Contents
Abstract
vii
Contents
ix
List of Figures
xi
List of Tables
xvii
Acknowledgements
1
1
Introduction
3
1.1
Why do we measure the mass of the nucleus? . . . . . . . . . . . . . . . . . .
3
1.1.1
The motivation for astrophysics . . . . . . . . . . . . . . . . . . . . .
5
1.1.2
The motivation for fundamental interactions physics . . . . . . . . . .
6
1.1.3
The motivation for nuclear physics . . . . . . . . . . . . . . . . . . . .
7
1.2
The nuclear binding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3
Subject of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 The ISOLTRAP Mass Spectrometer
15
2.1
The ISOLDE facility and isotope production . . . . . . . . . . . . . . . . . . . 15
2.2
The buncher device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3
2.2.1
Pressure of the buffer gas . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2
Resonance frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
The multireflection time of flight device . . . . . . . . . . . . . . . . . . . . 24
2.3.1
The statistical uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2
The systematical uncertainty . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.3
The MRToF mass spectrometry data analysis . . . . . . . . . . . . . . 28
2.4
Alignment of the horizontal beam line . . . . . . . . . . . . . . . . . . . . . . 32
2.5
ISOLTRAP Penning trap tandem . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.1
Preparation penning trap . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.2
Precision penning trap . . . . . . . . . . . . . . . . . . . . . . . . . . 37
ix
Contents
x
2.6
3
2.5.3
The statistical and systematical errors . . . . . . . . . . . . . . . . . . 40
2.5.4
ToFICR data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
The shape coexistence in neutronrich krypton isotopic chain
49
3.1
A=100 region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2
Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3
3.4
3.2.1
ToFICR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2
MRToF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Theoretical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1
Liquid drop model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2
Shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.3
Selfconsistent mean field model . . . . . . . . . . . . . . . . . . . . . 62
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Conclusion
73
5
75
List of published works
Bibliography
77
List of Figures
1.1
The nuclear chart presenting all experimentally known isotopes. Each square
represents a nuclide and the colors represents main decay modes. . . . . . . . .
1.2
3
(a) the trend of the binding energy in calcium isotopic chain [H+ 17]. (b) one
neutron separation energy in calcium isotopic chain. (c) one neutron shellgap
energy in calcium isotopic chain. (d) Two neutron separation energy in calcium
isotopic chain. (e) two neutron shellgap energy in calcium isotopic chain. . . 10
1.3
(a) two neutron separation energy in zirconium isotopic chain [M+ 21]. (b)
charge radii in zirconium isotopic chain [A+ 13]. (c) the energy of the first 2+
excited state in zirconium isotopic chain [Lab]. (d) the ratio between 4+ and
2+ excited states in zirconium isotopic chain. . . . . . . . . . . . . . . . . . . 12
2.1
The schematic representation of the ISOLDE facility and the ISOLTRAP mass
spectrometer. The recent installation of the high ISOLDE is not included in the
figure [Jon17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2
The basic layout of the RFQ buncher. It consists of an eggcup electrode, rod
system, and ejection electrodes [And18]. . . . . . . . . . . . . . . . . . . . . . 17
2.3
Top: schematic representation of the DC potential for axial confinement and
bunch forming. Bottom: the triangle of stability for ion guiding in radial direc
tion [Wol90]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4
(a) dependency of efficiency on pressure. (b) evolution of the resolving power
with pressure. (c) evolution of the difference between mean values of Gaussian
and exponential gaussian hybrid probability functions, errors are smaller the the
sizes of points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5
Top: the electronic scheme of the resonance circuit used for Btrap. The same
schematic was used to simulate the frequency range in the LTspice software.
The main elements of the circuit are discussed in the text. Bottom: 1 variable
capacitor, 2 the coupler, 3 the stepper motor. . . . . . . . . . . . . . . . . . . 21
xi
List of Figures
xii
2.6
The results of the simulation in LTspice software. The capacitor varies between
30 and 80 pF. The VRF amplitudes for minimum and maximum of the achievable
capacitance are shown on the plot. The gray band represents the experimental
results with the high voltage amplifier. . . . . . . . . . . . . . . . . . . . . . . 22
2.7
Top the amplitude VRF of the resonance frequencies reached on the segments
of the buncher. Black dots correspond to the amplitude on the rods when the
count rate was maximized on the 133 Cs ions; red dots correspond to the ampli
tude when both 85 Rb and 133 Cs ions were observed in the same bunch. . . . . . 23
2.8
Top: the multi reflection time of flight device. Bottom: the in traplift technique
[W+ 13b] to store the bunch of ions between electrostatics mirrors. . . . . . . . 24
2.9
As an example, the time of flight spectrum of 49 Sc ions fitted by Gaussian and
Exponential Gaussian Hybrid fit functions. . . . . . . . . . . . . . . . . . . . . 27
2.10 The mean and sigma parameters of the Gaussian distribution in dependence of
the halfrange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.11 (a) the Gaussian fit of 49 Sc ions. (b)the difference between individual CT oFi
and weighted CT oF . The total uncertainty is shown with a grey band. (c)the
experimental mass excess values of
49
Sc are taken from [R+ 61, F+ 69, E+ 66,
W+ 66, G+ 68, V+ 68]. For a better visibility, the values from [O+ 56, M+ 56] are
excluded from the plot since they have uncertainties of 50 and 100 keV. . . . . 30
2.12 Same as figure 2.11 but for 50 Sc ions. The literature mass excess values of 50 Sc
are taken from [W+ 69, C+ 63, O+ 69, B+ 98]. . . . . . . . . . . . . . . . . . . . 31
2.13 Same as figure 2.11 but for 73 Br ions. The literature mass excess values of 73 Br
are taken from [T+ 01, R+ 74, H+ 87, M+ 70, H+ 01a, H+ 11, S+ 89, S+ 91b]. . . . . 32
2.14 The final radial position of the horizontal beam line parts. The switchyard and
the bender_out elements are taken as reference points. . . . . . . . . . . . . . . 33
2.15 Left: the trajectory of an ion. It consists of three motions: modified cyclotron,
magnetron and axial motion. Right: the electric field of a quadrupolar shape.
The pictures are taken from [Din16]. . . . . . . . . . . . . . . . . . . . . . . . 35
2.16 Top: the precision Penning trap [M+ 18]. Bottom: schematic representation of
the principle of the time of flight ion cyclotron resonance detection technique
[Din16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.17 (a) a typical ToFICR resonance of 70 As ions using a single rfexcitation pulse
of 1.2 seconds duration. (b) a ToFICR resonance of
97
Kr ions obtained by
Ramsey excitation with duration of 104010 milliseconds. The line represents
a fit of the theoretical lineshape to the data points [K+ 95b] . . . . . . . . . . . 39
List of Figures
xiii
2.18 The drift of the magnetic field over 75 hours. Top picture represents residual
fluctuations and bottom picture represents the long drift of the magnetic field.
Pictures are taken from [Din16]. . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.19 Same as Figure 2.11 but for
70
+
+
70
As ions. The literature mass excess values of
+
As are from [B 63, S 17, L 75, T+ 01]. . . . . . . . . . . . . . . . . . . . . 44
2.20 Same as Figure 2.11 but for
196
+
196
Hg ions. The literature mass excess values of
+
Hg are taken from [K 80, W 62, S+ 01]. . . . . . . . . . . . . . . . . . . . 45
2.21 Comparison of the mass excess values determined in this work (red symbols)
and the literature values taken from AME2016 [H+ 17]. The gray shaded back
ground represents the AME2016 uncertainties. . . . . . . . . . . . . . . . . . . 46
3.1
(a) the two neutron separation energy trend [H+ 17]. (b) the evolution of the
charge radii in A=100 region [A+ 13]. (c) the evolution of the energy of the first
2+ excited state [Lab]. (d) the trends of the ratio between 4+ and 2+ excited
states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2
(a) the single pulse ToFICR resonance of 97 Kr ions. (b) the spread of the Ricr
ratio of 97 Kr ions. (c) the dependence of counts on number of ions per ejection. 52
3.3
(a) the time of flight spectrum of 96 Kr ions. (b) the time of flight difference
between two peaks vs the number of ions per ejection. Every color represents
different set of files. (c) the difference between the individual Ctof ratios and
weighted CT oF ratio of
96
Kr ions. (d) the comparison between experimental
and literature values from AME16 [H+ 17] . . . . . . . . . . . . . . . . . . . . 53
3.4
(a) the time of flight spectrum of 97 Kr ions. (b) the time of flight difference
between two peaks vs the number of ions per ejection. Every color represents
different set of files. (c) the difference between the individual Ctof ratios and
weighted CT oF ratio of
97
Kr ions. (d) the comparison between experimental
and literature values from AME16 [H+ 17] . . . . . . . . . . . . . . . . . . . . 54
3.5
(a) the time of flight spectrum of 98 Kr ions. (b) the time of flight difference
between two peaks vs the number of ions per ejection. Every color represents
different set of files. (c) the histogram of the differences between ∆T oFi and
∆T oFmax,ions/shot for details see text. (d) the difference between of the indi
vidual Ctof ratios and weighted CT oF ratio of 98 Kr ions.
3.6
. . . . . . . . . . . . 56
The evolution of the two neutron separation energy in molybdenum, niobium,
zirconium, strontium, yttrium, rubidium and krypton isotopic chains. The liter
ature values taken from AME2016 [H+ 17]. The new experimental two neutron
separation energies obtained by ISOLTRAP mass spectrometer are colored in
red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
List of Figures
xiv
3.7
The dependence of the average binding energy per nucleon as a function of the
number of nucleons in the nucleus. . . . . . . . . . . . . . . . . . . . . . . . . 58
3.8
The evolution of the two neutron separation energy in neutronrich krypton,
strontium, zirconium and molybdenum isotopic chains. The black two neutron
separation energy trends are taken from AME2016 [H+ 17]. Red two neutron
separation energies calculated from the new 96−98 Kr mass measurements. The
blue values are taken from FRDM2012 [M+ 16b]. For the visibility error bars
are omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.9
The shell ordering with different potentials [MJ55] . . . . . . . . . . . . . . . 61
3.10 The evolution of the two neutron separation energy in neutronrich krypton,
strontium, zirconium and molybdenum isotopic chains. The blackcolored two
neutron separation energies are taken from AME2016 [H+ 17]. Red two neutron
separation energies calculated from the new 96−98 Kr mass measurements. Blue
color corresponds to the HFB31 calculations [G+ 13a]. For the visibility error
bars are omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.11 (a) The evolution of the two neutron separation energy in neutronrich kryp
ton, strontium, zirconium and molybdenum isotopic chains. The black colored
two neutron separation energies are taken from AME2020. Red two neutron
separation energies calculated from the new 96−98 Kr mass measurements. Blue
color corresponds to the SLy4 calculations [S+ 03]. (b) The evolution of the
two neutron separation energy in neutronrich krypton, strontium, zirconium
and molybdenum isotopic chains. The black colored two neutron separation
energies are taken from AME2016 [H+ 17]. Red two neutron separation en
ergies calculated from the new 96−98 Kr mass measurements. Blue/green color
corresponds to the UNEDF0/1 calculations [K+ 10]. For the visibility error bars
are omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.12 The evolution of the two neutron separation energy in neutronrich krypton,
strontium, zirconium and molybdenum isotopic chains. The blackcolored two
neutron separation energies are taken from AME2016 [H+ 17]. Red two neutron
separation energies calculated from the new 96−98 Kr mass measurements. Blue
color corresponds to the Gogny D1S calculations [DG80], green color corre
sponds to 5DCH parametrization [RG14]. For visibility error bars are omitted. . 68
List of Figures
xv
3.13 The evolution of the two neutron separation energy in neutronrich krypton,
strontium, zirconium and molybdenum isotopic chains. The blackcolored two
neutron separation energies are taken from AME2020. Red is two neutron sepa
ration energies calculated from the new 96−98 Kr mass measurements. Blue color
corresponds to the Gogny D1S 5DCH calculations [RG14]. Svmin colored in
greed [G+ 06]. The ANN calculations with Gogny D1S 5DCH parametrization
are colored in orange [L+ 20a]. For visibility error bars are omitted. . . . . . . . 69
3.14 (a) the first 2+ excited states in 38 Sr isotopes. (b) the first 2+ excited states in
36 Kr
isotopes. (c) the representation of the energy levels for 98 Kr isotope. The
HFB value is calculated by HFB approach with Gogny interaction, the AI value
calculated by the same approach but with artificial neural network, experimental
values are taken from [Lab]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
List of Tables
2.1
The achievable frequency ranges of new build matching boxes. The first col
umn represents the capacitance of the variable capacitor and does not include
the capacitance of segments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2
Summary of the Ricr and CT oF ratios as well as mass excess values of 49,50 Sc,
70
As,
73
Br and
196
Hg nuclides. Reference masses are taken from AME2016
[H+ 17] and T1/2 values are from NUBASE2016 [WA+ 17]. The last column
provides the significance of the ISOLTRAP results in the reevaluated mass
excess values if included into AME. . . . . . . . . . . . . . . . . . . . . . . . 47
3.1
Summary of the Ricr and CT oF ratios as well as mass excess values of 96−98 Kr
ions. Reference masses are taken from AME2016 [H+ 17] and T1/2 values are
from NUBASE2016 [WA+ 17]. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
xvii
Acknowledgements
I want to thank my primary supervisor Dr. Apl. Prof. Yuri Litvinov for all the help and
support he provided me during these four years. His knowledge about the physics of trapped
and stored ions and the positive attitude helped me evaluate the work. Yuri Litvinov was always
available for any help from physics until general management/administrative advising. He gave
me the best condition to study I have ever had in my life in terms of physics discussions and
funding. I could always rely on him. During the project, I knew if I make some mistakes Yuri
will watch my back. So it was all safe to do the research. I could always come to Yuri, dream
about the future and often unrealistic future :) and he would just smile or makes sarcastic jokes
which seems only Russians can understand. So thank you very much Yuri, it was joyful to work
with you during these years.
Second, I would like to thank my second supervisor Prof. Dr. Klaus Blaum. The person who
knows everything about traps and associated physics. His advices were always clear. During
the discussions, Klaus starts with simple ideas which are very easy to understand and follow.
He is always available for help, always smiling and at the same time serious researcher with
very deep knowledge in physics. He also provided me with the best conditions to work and
flourish. His funding sources seem to be infinite and I could always rely on it if I needed to buy
some materials for the lab. Thank you very much Klaus.
Third, I would like to thank my local teammates. With these guys, we went together through
all the stages, through thick and thin of the negotiations. I would like to thank Dr. Maxime
Mougeot for his advising as a local supervisor. Also, I would like to thank Lukas Nies, our
newest PhD student. I believe he will do a great job at ISOLTRAP. He already does. Last but
not least, I would like to thank Dr. Jonas Karthein. The person who was acting as a buddy from
the beginning of my project. It was very nice and easy to communicate with him and it was
always joyful to work together in the lab.
Fours, I would like to thank exISOLTRAPers. I would like to thank Dr. Frank Wienholtz
for his advicing on the equipment and how to choose a proper tool. I personally can build
a setup from wood and dirt; therefore, his help was essential for me. I would like to thank
Dr. Vladimir Manea. Although he has been far away, I could always send him a mail and
receive a manual/instruction answering all my questions and even more. A special thank goes
1
Acknowledgements
2
to Dr. Dinko Atanasov. This person was helping me at all stages of my project, especially with
socialization and understanding all the other colleagues. If I write about socialization, I also
must mention Dr. Dave Lunney. He was always making fun of all these ”Sheldons” around and
also surprisingly knows a lot about PhD culture. Thanks, thanks, thanks.
And fifth, I would like to thank all members of my big family. Brothers and sisters, mother
and father, grands, aunts and uncles. All your letters and postcards I reread in the darkest times
of my life. Nothing makes me warmer than this. Although there is a very big distance between
us, I feel your love and I will make it through my entire life. My wife Masha, these years have
been a real test for us. And I am very happy to see that together we did a really great job. Keep
it up.
I
Chapter 1
Introduction
Why do we measure the mass of the nucleus?
Proton Number Z
1.1
82
50
2016
28
20
stable
β+
8
p
n
α
SF
β−
8
20 28
50
82
126
Neutron Number N
Figure 1.1. The nuclear chart presenting all experimentally known isotopes. Each square rep
resents a nuclide and the colors represents main decay modes.
One of the fundamental properties of a nucleus is its mass, which reflects nuclear structure
through the binding energy [G+ 13a], answering questions about the element’s origin [Gor01]
and even testing the standard model [H+ 09]. There is an enormous amount of data collected
over more than a hundred years of experimental investigations, which can be summarized in
figure 1.1. This figure is the visual representation, details of the nuclear masses measured by
different methods have been collected in the Atomic Mass Evaluation [M+ 21]. The analysis of
the available data supports the modern understanding of the evolution of the mass surface by ap
3
4
1.1. Why do we measure the mass of the nucleus?
plying various mass filters along isotopic, isotonic or isobaric chains. Nuclearstructure effects
such as shell closures [W+ 13a] or onsets of deformation [M+ 13a] are revealed as irregularities
on the smooth mass surface. Therefore, new precision mass measurements are indispensable
for investigations of such effects. Furthermore, the mass values are an essential input for mod
eling the astrophysical processes [M+ 06]. For instance, explosive nucleosynthesis processes
involve exotic nuclides with presently unknown masses [Gor01]. The description thus relies
upon predictions by nuclearmass models [C+ 21], which in turn have to be constrained through
accurately measured masses [A+ 15]. Even in the investigations of the weak interaction, the
mass measurements may play a significant role. The Qvalue of the neutrinoless doublebeta
decay can answer the questions about fundamental properties of the neutrino particle [F+ 12].
Moreover, mass spectrometry contributes to tests of the unitary of the CabibboKobayashi
Maskawa quarkmixing matrix through constraining the |Vud | element [H+ 15].
Such diverse research directions require unique tools. J.J. Thomson and his student F.W.
Aston were the first to build mass spectrometers at the beginning of the last century [M+ 10].
Since then technologies related to mass spectrometry are under constant upgrade in terms of the
precision, focusing and effective manipulations of the ion motions using the electromagnetic
fields. For example, the first device had 10−3 level of relative precision and nowadays the
relative precision of mass spectrometers can reach 10−12 . Such remarkable improvement in the
field of mass spectrometry was achieved by introducing the Radio Frequency Quadrupole(RFQ)
spectrometers [PS53], Time of Flight systems [WP90], Penning traps [GK78] and Storage rings
[L+ 20b]. Moreover, not only the systems but the different mass measurement techniques are
under constant upgrade to reach a better resolving power and precision. A recent and one of
the most promising techniques is the phaseimaging ion cyclotron resonance technique [E+ 14]
applied in Penning traps. The central device of this work is the ISOLTRAP mass spectrometer.
It combines different devices and techniques the details of which are discussed in Section 2.2
2.5.
In the early years of mass spectrometry field researchers were studying stable species avail
able in nature. The discovery of radioactivity led to the development of the field such that
today 3437 masses of different nuclides are known [M+ 21]. The progress on the production
and separation methods of radioactive nuclei made it possible. Among the laboratories in the
world, the first five places in the number of discovered isotopes are occupied by the Lawrence
Berkeley National Laboratory (638) [S+ 09c], GSI Helmholtz Centre for Heavy Ion Research
(438) [H+ 12], the Joint Institute for Nuclear Research (221) [Oga10], the Cavendish Labora
tory (218) [Lon16] and CERN (115) [BB18]. These laboratories differ by the production and
separation methods of radioactive isotopes. The ISOLTRAP mass spectrometer was installed
at ISOLDE/CERN facility in yearly 1980s to measure masses of radioactive isotopes and since
then addressed more than 600 shortlived nuclides [Kow10]. The production and separation
1.1. Why do we measure the mass of the nucleus?
5
methods of the ISOLDE facility are discussed in Section 2.1.
1.1.1
The motivation for astrophysics
The cosmos is the unique and free laboratory existing in nature. All objects in the universe such
as galaxies, stars, interstellar environment, planets, meteorites have been sources of information
about the world we live in [WLM05]. Today researchers are particularly interested in how and
where the elements are produced [Z+ 16].
At first, the production of elements was carefully studied in the Sun. Through the ppI ther
monuclear cycle [SC05], four protons create a He nucleus with the release of a tremendous
amount of energy, electron neutrinos, photons, and positrons. The ppII and ppIII cycles are
responsible for producing the elements until boron, which can be achieved in Sunlike stars.
However, there are other stars of mass much more than the Sun, and through the CNO cy
cle [S+ 09b], triple αprocess [Sal52] and nuclear fusion reactions elements can be created until
the iron nucleus.
The significant invention was the proposed slow and rapid neutron capture processes [B+ 57,
A+ 07]. The slow neutron capture is the process that starts from stable iron and continues until
lead/bismuth. It creates mostly stable elements of the nuclear chart followed by a β − decay
transforming the neutron into a proton. The most likely place to observe such a process is the
Asymptotic Giant Branch (AGB) stars [Sie06]. The rapid neutron capture is the process when
the elements heavier than iron are produced through consequent (n, γ) reactions. The neutron
rich elements of the nuclear chart are produced through this process. The most probable place
where such extreme conditions can be achieved is the corecollapsing supernovae or neutron
star mergers environments [L+ 98]. ISOLTRAP mass spectrometer put many efforts to study r
process by performing precision mass measurements, as an example 129−131 Cd isotopes [A+ 15].
For the rapid neutron capture process the equilibrium between (n, γ) and (γ, n) reactions has to
be reached. The general and first approximation for the statistical balance can be expressed as:
log(
n(N + 1, Z)
3
5.04
) = log(nn ) − 34.07 − log(T9 ) +
Sn
n(N, Z)
2
T9
(1.1)
n(N, Z) : the number density of a nucleus with N,Z number of neutrons and protons
nn
: the number density of free neutrons
T9
: the temperature T/109 K
Sn
: one neutron separation energy
By precisely measuring the mass of the nucleus researcher can obtain the one neutron sep
aration energy (see the nuclear binding energy subsection 1.2) which is the contribution of the
1.1. Why do we measure the mass of the nucleus?
6
mass measurements in the nucleosynthesis calculations. The mass of 129−131 Cd isotopes were
used for the simulation of νdriven wind model of the corecollapse supernovae [W+ 11a] and
the decompression of the neutron star merger model [G+ 13b]. For the detailed description of
the reaction network calculations reader is refereed to [Lan99].
The rapid proton capture process also exists in nature [A+ 03], but protons have to overcome
the Coulomb barrier. In the rpprocess, protons are captured via a sequence of (p, γ) reactions
and β + decays [G+ 19]. The process runs on the neutrondeficient part of the nuclear chart. A
new element will be produced if the capture reaction is faster than the decay of the parent nucleus
or until the proton drip line has been reached. According to the current computer simulations,
this process can not produce all neutrondeficient nuclei. Therefore, scientists are looking for
other processes to explain the abundances, for example, νp process [S+ 09a]. The ISOLTRAP
mass spectrometer contributes and regularly proposes to measure masses of nuclei for these
processes, as an example [H+ 11]. These nuclei are of the high demand for the theoretical models
of type I xray burst and collapse scenario of supernova where these processes most probably
take place [P+ 13, SR06].
This is just a tiny overview aiming to show that mass measurements play a crucial role in
understanding how the elements are produced. Most of the isotopes are created in very extreme
environments, and a limited number of them can be reached at nuclear facilities because of
their short halflives and production yields. Therefore, the theoretical models try to describe the
production mechanisms, and precise mass measurements [N+ 20,A+ 16] are needed to constrain
the theoretical calculations.
1.1.2
The motivation for fundamental interactions physics
The electron is the first elementary particle, which was discovered in 1892 [Tho97]. In 1964
Murray GellMann and George Zweig assumed the composite nature of the proton and the neu
tron [GM64]. Now it is a scientific fact that they consist of u and d quarks. Attempts to describe
the observed phenomena led to the development of the standard model [Hig64]. The standard
model describes three of four fundamental interactions: weak, strong and electromagnetic.
One of the interactions which can be studied through precision mass measurements is weak
interaction. The interaction is accessible through βdecays of nuclei. This decay is nothing
else but the flavor change of quarks. The flavor change of quarks is taken into account in the
Cabibbo–Kobayashi–Maskawa(CKM) matrix [Cab63]. The mass spectrometrists contribute to
testing the unitary of this matrix through calculations of |Vud | element. Equation 1.2 links the
mass measurement contribution in the calculation of the F t value of a βtransition:
′
F t = f t(1 + δR )(1 + δN S − δC ) =
K
+ ∆VR )
2G2V (1
(1.2)
1.1. Why do we measure the mass of the nucleus?
ft
: the strength of the transition
δC
: the isospin symmetrybreaking correction
7
′
δR , δN S : transitiondependent part of the radiative correction
K
: constant
GV
: the vectorcoupling constant
∆VR
: transitionindependent part of the radiative correction
To obtain the Gv constant, superallowed Fermi decays are typically considered. The ex
perimental f t value is related to the vectorcoupling constant GV [H+ 15]. After obtaining the
vectorcoupling constant GV one can calculate |Vud | =
GV
GF
, where GF is a weakinteraction
constant. Until now, only fourteenth such decays participate in the worldaverage corrected F t
value. The strength of the transition f t is the product of the phasespace factor f , and the par
tial halflife of the transition t. The experimental halflife and branching ratio are included in
the partial halflife t determination, while phasespace factor f depends on the electron capture
energy QEC . The QEC energy has been in the close research circle of the ISOLTRAP mass
spectrometer [K+ 19]. From the past attempts to measure
70
Br isotope [A+ 17] until a recent
proposal to measure the 98 In isotope [N+ 20]. The latter is the heaviest nucleus which has not
been included in the worldwide average because of the lack of experimental information. The
′
radiative and nuclearstructuredependent corrections, such as δC , δR , δN S , ∆VR , are also indis
pensable inputs and have to be provided by theory.
The other topic where the mass measurements play a significant role is the neutrino mass
determination [And18]. This has been a hot topic since the discovery of this particle. The lower
limits on the neutrino mass can be set through a kinematic approach of the βdecay transitions.
When the decay occurs, the nucleus emits the positron/electron and the electron neutrino/anti
neutrino. The smaller the transition energy the more precise the measurement can be. And
again, this topic has been included in the scientific program of ISOLTRAP mass spectrometer
[E+ 08, B+ 10].
To sum up, mass measurements play a significant role even in weak interaction studies.
1.1.3
The motivation for nuclear physics
At the beginning of the 20th century, scientists knew only the proton, the electron, and the
photon [Rut20]. It took an enormous effort to develop an atomic model from a naive raisin
pie model [Tho04] to the model we know now as the planetary model [Rut11]. Furthermore,
it passed twenty years to discover neutron [Cha32] and start building the model to describe the
modern understanding of the nucleus. It began from a naive liquid drop model [Wei35] until
the discovery of magic numbers and shellmodel outbreak [May48].
1.1. Why do we measure the mass of the nucleus?
8
It is well known that the nucleus consists of protons and neutrons, and the simple sum of
them should give the total mass of a nucleus. However, the actual nuclear mass is written as:
mnuc (Z, N ) = Z × mp + N × mn +
Bnuc
c2
(1.3)
mnuc : the mass of a nucleus which has Z number of protons and N number of neutrons
mp
: the mass of a single proton
mn
: the mass of a single neutron
Bnuc
c2
: the binding energy of the nucleus
The last term in equation 1.3 is the nuclear binding energy, reflecting interactions inside
the nucleus. Since the mass of the proton and the mass of the neutron are well known, mass
spectrometrists obtain the binding energy and study the systematics of the deviations of the
nuclear binding energies. These studies can tell us about the inner kitchen of the nucleus and
how it is organized.
It has been observed that at some exceptional Z number of protons or N number of neutrons
or both of them simultaneously, the nucleus reaches an extra gain in its binding energy. Those
are the magic numbers, and they appear when protons or neutrons fill a shell [R+ 18]. The latest
example from ISOLTRAP is a recently measured 100 In isotope. With the Qvalue of 100 Sn decay,
the doubly magic origin of the tin isotope can be tested. However, this gain in the binding energy
also appears on the subshell level [X+ 19], as an example subshell closure in calcium isotopic
chain measured at ISOLTRAP [W+ 13a]. These shells and subshells are strongly affected by
the interactions between nucleons. That leads to a nuclear structure effect, namely breaking of
the shell order [M+ 18]. There are five regions on the nuclear chart where modifications of the
shell structure have been experimentally observed. They lie close to the neutron drip line at N=
8, 14, 20, 28, 40. This phenomenon has been in the focus of the ISOLTRAP research program
[A+ 19]. There is a further interesting effect that the mass measurements can study. There are
particular regions on the nuclear chart where extreme deviations can be observed. For example,
in light isotopes where the N /Z ratio is large, the neutrons can form a halo around the nucleus
[O+ 92,S+ 08]. The same effect has been observed with protons forming the proton halo [K+ 88,
KE+ 95]. An interesting effect has been seen when neutrons form a ”skin” on the nucleus surface
[HKE98]. Driven by nucleonnucleon interaction, the nuclei can form extraordinary shapes,
such as pearshaped nucleus [B+ 20]. In this thesis’s frame, the ISOLTRAP mass spectrometer’s
particular interest is the A=100 region known as the ”region of deformation” [M+ 13a, DR+ 17]
and discussed in Section 3.1.
1.2. The nuclear binding energy
1.2
9
The nuclear binding energy
The nuclear binding energy can be generally expressed as a sum of different contributions:
Bnuc
−1Z+1 + 1
−1N +1 + 1
=
B
(Z,
N
)
+
B
(Z,
N
)
+
δ
(Z,
N
)
+
δn (Z, N )
0
1
p
c2
2
2
(1.4)
B0 , B1 : the terms which link different theories
δp , δn : the terms which are responsible for oddeven staggering effect
For example, in the finiterange droplet model (FRDM), two terms B0 , B1 can represent
different contributions to the total binding energy. Such terms as the volume term, the surface
term and other terms (the details are provided in Chapter 3) can be included as B0 correction
and B1 will stay for deformed droplet shape. In the shell model, shellcorrections can also
be included in B0 , B1 terms. For example, B0 can be considered as the result of independent
particles orbiting in a spherical potential and B1 represents the rest interactions. The δp , δn
parts describe the oddeven staggering effect of protons and neutrons, respectively. This effect
can be seen in figure1.2(a) where at every even neutron number of calcium isotope there is a
gain in the binding energy. This gain is due to two neutrons forming a pair. However, nuclear
binding energy is a large quantity to study the underlying nuclear structure phenomena. As an
example, the 20, 28 sell closures and 32 subshell closure are highlighted by vertical dashlines
in figure1.2(a)) and they are not visible on the general trend of the nuclear binding energy. To
access other nuclear structure effects mass spectrometriests use socalled mass filters.
One such filter is one proton/neutron separation energy:
Sp (Z, N ) = Bnuc (Z − 1, N ) − Bnuc (Z, N )
Sn (Z, N ) = Bnuc (Z, N − 1) − Bnuc (Z, N )
(1.5)
δ1n (Z, N ) = Sn (Z, N ) − Sn (Z, N + 1)
Sp , Sn : one proton and one neutron separation energy correspondingly
δ1n
: the one neutron shell gap
This filter is equal to the energy being required to remove one proton/neutron from the
nucleus. Figure1.2(b) shows the Sn energies for calcium isotopic chain. First, one can observe
a more pronounced oddeven staggering effect of neutrons. Second, it shows significant drops
in the trend of the one neutron separation energy at 20, 28 and 32 neutron numbers compared
to other neutron numbers. In this case these drops reflect the shell and subsell closures. The
amplitude of this drop can be described by the socalled one neutron shell gap in equation
1.2. The nuclear binding energy
10
340
AME2016
ISOLTRAP
Bnuc (MeV)
360
380
400
420
440
460
20
22
24
26
28
30
32
Neutron Number N
34
(a)
16
AME2016
ISOLTRAP
3
(MeV)
12
10
8
2
1
1n
Sn (MeV)
14
0
6
1
4
20
22
24
26
28
30
Neutron Number N
32
2
34
(b)
20
22
24
26
28
30
Neutron Number N
32
34
(c)
7
AME2016
ISOLTRAP
30
AME2016
ISOLTRAP
6
20
(MeV)
15
2n
25
S2n (MeV)
AME2016
ISOLTRAP
4
5
4
3
2
10
1
20
22
24
26
28
30
Neutron Number N
32
34
(d)
20
22
24
26
28
Neutron Number N
30
32
(e)
Figure 1.2. (a) the trend of the binding energy in calcium isotopic chain [H+ 17]. (b) one neu
tron separation energy in calcium isotopic chain. (c) one neutron shellgap energy in calcium
isotopic chain. (d) Two neutron separation energy in calcium isotopic chain. (e) two neutron
shellgap energy in calcium isotopic chain.
1.2. The nuclear binding energy
11
1.5. One can observe the magnitude of the changes through one neutron shell gap shown in
figure1.2(c).
The second filter is the two proton/neutron separation energy. This filter is equal to the
energy being required to remove two protons/neutrons from the nucleus:
S2p (Z, N ) = Bnuc (Z − 2, N ) − Bnuc (Z, N )
S2n (Z, N ) = Bnuc (Z, N − 2) − Bnuc (Z, N )
(1.6)
δ2n (Z, N ) = S2n (Z, N ) − S2n (Z, N + 2)
S2p , S2n : two protons and two neutrons separation energy correspondingly
δ2n
: the two neutron shell gap
The effect of the S2n filter can be seen in figure1.2(d). First, this filter removes the odd
even staggering effect because in the equation 1.6 only odd or even nuclei are included. Second,
this filter together with two neutron shell gap makes the nuclear structure effects more visible
in the calcium isotopic chain in figure1.2(e).
However, these filters are also sensitive to other nuclear structure effects. For example, the
trend of S2n mass filter in chromium isotopic chain [M+ 18] shows approximately the same be
havior as for neutronrich calcium isotopes. There is a change in the two neutron separation en
ergy at N=36. This region is socalled ”island of inversion” where the inversion of shells occurs.
The trends in two neutron separation energy in A=100 region show that the nucleonic system
becomes less bound. These changes represent socalled ”island of deformation” [M+ 16a]. The
investigation of the region requires the usage of complementary information.
To overcome the lack of information through the direct use of the mass filters, comple
mentary properties are involved. As an example, the charge radii of the nuclei, the angular
momentum, the excitation spectrum actively participate in the understanding of nuclear phe
nomena. A=100 region is an excellent example of such synergy. One can draw complementary
nuclear observables, such as the charge radii and the energies of the first 2+ excited states. For
simplicity these properties are shown in figure 1.3 only for zirconium isotopes.
Let first have a look at two neutron separation energy in figure1.3(a). The decreasing
trend of the S2n energy up to the N=59 does not show any irregularities. However, the sudden
increase at N=60 indicates a structural change, especially, if taken into account the pronounced
increase in the charge radii at N=60 (figure1.3(b)). This increase of the charge radii suggests
that neutrons start occupying states of much larger radial extension and this results in the low
lying first 2+ excited state at N=60 (figure1.3(c)). The strong evidence of the onset of the
deformation at N=60 in zirconium chain is a ratio between first 4+ and first 2+ excited states
1.3. Subject of this work
12
which rises to almost 3.33 (figure1.3(d)). This indicates that zirconium isotopes act as a rigid
6.0
14.0
13.5
13.0
12.5
12.0
11.5
11.0
5.8
40Zr
<r2> (fm2)
S2n (MeV)
rotor at N>=60.
5.6
5.4
5.2
5.0
40Zr
4.8
56
57
58
59
60
Neutron Number N
61
62
(a)
56
57
58
59
60
Neutron number N
61
62
(b)
1800
3.00
40Zr
1400
E(4 + )/E(2 + )
E(2 + ) (keV)
1600
1200
1000
800
600
400
200
2.75
2.50
2.25
40Zr
2.00
1.75
1.50
56
57
58
59
60
Neutron number N
61
(c)
62
56
57
58
59
60
Neutron number N
61
62
(d)
Figure 1.3. (a) two neutron separation energy in zirconium isotopic chain [M+ 21]. (b) charge
radii in zirconium isotopic chain [A+ 13]. (c) the energy of the first 2+ excited state in zirconium
isotopic chain [Lab]. (d) the ratio between 4+ and 2+ excited states in zirconium isotopic chain.
To conclude, the large amount of data on the nuclei complements and supports interpreta
tions of observed phenomena. Different points of view from different nuclear physics fields
form the basis of the modern understanding of nuclear structure.
1.3
Subject of this work
In the frame of this thesis precise mass measurements of exotic nuclei for the nuclear bind
ing energies determination will be discussed. The radioactive ions were produced at ISOLDE
radioactive ion beam facility. ISOLDE is the Isotope Separation OnLine (ISOL) type of fa
cility where exotic nuclei are produced through nuclear reactions of relativistic protons with
the thick target material. The isotopes from the lightest up to heavy uranium can be produced.
The ISOLTRAP mass spectrometer measures the masses of these isotopes by applying different
1.3. Subject of this work
13
techniques. In this work, the focus is set on the multireflection time of flight (MRToF) mass
spectrometry and the time of flight ion cyclotron resonance (ToFICR) technique.
Although the ISOLTRAP mass spectrometer has a diverse research program, krypton iso
topes for nuclear structure objectives are discussed in this work. Experimental results are com
pared to the theoretical calculations.
Theoretical models rely on the different approaches which can be grouped in macroscopic,
macroscopicmicroscopic and microscopic methods. In 1935 German physicist Carl Friedrich
von Weizsäcker proposed his famous formula [Wei35] based on liquid drop theory and empirical
measurements. This is an example, of a macroscopic model. This semiempirical mass model
went through modification to include microscopic parts which led to the development of the
macroscopicmicroscopic model namely the finiterange droplet model (FRDM) with the latest
upgrade FRDM2012 [M+ 16b]. The results of this global model will be compared particularly
with A=100 region.
The selfconsistent meanfield models with Skyrme type interactions are regarded as mi
croscopic approaches. The predictions of the global HFB31 model [G+ 13a] with Skyrme type
interaction are also compared to experimental values in A=100 region. The SLy4 parametriza
tion [S+ 03] of the Skyrme force has been specially tailored for the nucleonnucleon interaction
and the prediction of this model is also shown.
A new model based on the density functional theory(DFT) and Skyrme type force resulted
in the latest UNEDF0/UNEDF1 calculations [K+ 10] and is particularly interesting in predicting
nuclear properties in deformation regions. The UNEDF0/1 model was adjusted on spherical and
deformed nuclei which makes it particularly interesting for A=100 region.
The Gogny force is well presented in the literature. There is a subshellclosure at N=56
and predictions of the Gogny D1S model [B+ 91] can be tested on both subshellclosure and the
deformation.
Finally, the progress on the machine learning techniques made it possible to establish the
Svmin method [G+ 06]. The machine learning techniques and Gogny interaction make it possi
ble to provide new mathematical solutions and reduce the computational time [L+ 20a].
In this work, the experimental results from the ISOLTRAP mass spectrometer on the krypton
isotopes are compared to the aforementioned theoretical calculations.
The thesis is organized into three chapters:
• The first chapter describes different applications of precise mass measurements focusing
on nuclear physics particularly on nuclear binding energy. The different massfilters and
accessible nuclear structure effects are also discussed. The complimentary nuclear data
and their application for the investigation of the nuclear deformation are drawn.
• The second chapter introduces the ISOLDE facility and the ISOLTRAP mass spectrom
1.3. Subject of this work
14
eter. It describes the main components and techniques used for isotope production and
mass measurements. It also introduces the data analysis of the IS532 [B+ 14] and IS642
[A+ 17] experimental campaigns. The source of systematic uncertainty of MRToF mass
spectrometry is studied and discussed. The alignment of the horizontal beamline and
the developments of a new device for control of the resonance frequency of the buncher
device are also provided.
• The third chapter provides the detailed study of the A=100 deformation region through
high precision mass measurements of neutronrich krypton isotopes. It introduces the
data analysis of 96−98 Kr isotopes and investigations of sources of systematic uncertainty
of the MRToF mass measurement technique. It applies a general specification of differ
ent theoretical models and interpretation of the results of their calculations. Finally, the
experimental and theoretical results are compared.
• The last chapter provides a summary of the investigations of the work and an outlook.
Chapter 2
The ISOLTRAP Mass Spectrometer
Scientists have been interested in ”earthly” compounds since the Middle Ages. Centuries after
centuries, they discovered new stable elements that could be found in mines or obtained through
chemical reactions. Following the discovery of radioactivity, new elements have been found
in nature [Kas13]. Even today, identifying chemical properties is required to include the new
element in the periodic table [Oga10]. These elements form the Periodic table and nuclear
centers try to find new elements. To list some of these facilities: Lawrence Berkeley National
Laboratory [S+ 09c], Joint Institute for Nuclear Research [OK15], GSI Helmholtzzentrum fur
Schwerionenforschung [H+ 12].
The isotopes have the same chemical properties and the same number of protons as a stable
element but a different number of neutrons. The study of these isotopes led to discovering other
parts of the nuclear chart away from stability. The production of rare isotopes requires different
methods and techniques. One such technique is an inflight production method and separation
of radioactive ion beams. The synchrotron or a linear accelerator is used to accelerate a primary
beam to the energy of hundreds of MeV/u. The beam is focused on a relatively thin production
target where isotopes of interest are produced via projectile fragmentation or in the case of
238
U primary beam also via fission reactions. The positively charged products are separated
by socalled Bρ − ∆E − Bρ separation, where Bρ is the magnetic rigidity analysis, and ∆E
are energy losses in a specially shaped material. The degrader is needed for an additional Z
selectivity. The worldfamous facilities based on such method are GANIL, GSI, LEGNARO,
RIKEN, IMP [BNVD13].
2.1
The ISOLDE facility and isotope production
The isotopes measured by the ISOLTRAP mass spectrometer and presented in this work have
been produced at the ISOLDE facility at CERN [BB18]. The visual aid is shown in figure
2.1. The production technique applied at ISOLDE differs from the inflight method. Instead
15
2.1. The ISOLDE facility and isotope production
16
of hitting a thin target with an accelerated heavy primary beam, the isotopes of interest are
produced by bombarding the thick target with protons of 1.4 GeV delivered by the Proton Syn
chrotron Booster(PSB). The isotopes are produced via spallation, fragmentation, or fission nu
clear reactions. The structure, as well as the material of the target, varies depending on the
demands [Got16]. For example, UCx target was used to produce neutronrich krypton isotopes;
the tantalum foil target was used to produce scandium isotopes; bromine and arsenic isotopes
were produced from zirconium oxide powder target. To make the production of neutron rich
isotopes more efficient, tungsten rods can be installed parallel to the target body. They act as a
neutron converter. The target is heated up to 2000 ◦ C to make the diffusion and effusion pro
cesses more efficient. Via a transfer line, which is also heated, neutral atoms enter the ionization
part of the target. The isotopes’ production rate is limited by the chemical nature of the isotope
of interest on the one hand and by the halflife on the other. Some isotopes interact with the
target material so that they do not defuse out. While for the others, it takes so long that they
decay before being ionized.
Position-sensitive
Ion detector
targets
protons
1.4GeV
Precision
Penning Trap
B=5.9T
Preparation
Penning Trap
B=4.7T
mass separators
Ion time-of-flight
detector
Pulsed
drift tube
ISOLDE
30-50 keV ion beam
Alkali
ion source
radio-frequency
cooler buncher
3 keV
Pulsed
drift tube
< 100 eV
MR-ToF mass
spectrometer/separator
In-Trap lift
Quadrupole
Ion time-of-flight bender
detector
Figure 2.1. The schematic representation of the ISOLDE facility and the ISOLTRAP mass
spectrometer. The recent installation of the high ISOLDE is not included in the figure [Jon17].
A neutral atom effuses in the hot transfer line to the ionization section. At ISOLDE, three
different techniques are used to produce charged ions. One technique is the Resonance Ion
ization Laser Ion Source (RILIS) technique [F+ 17a, M+ 13b, MP+ 18]. The ions are created by
kicking out an electron. The level structure of an atom is unique for every isotope. The proper
laser frequency adjustment makes the ionization method highly selective and efficient. The
2.2. The buncher device
17
second technique is surface ionization, when atoms get ionized by contacting the heated wall
of the ion source cavity. The only parameter which can be varied is the temperature of the wall.
Therefore, this method relies on the chemical properties of the element. For example, the alkali
metals quickly lose their electron to the heated plate and get singly charged. The third method
of ionization is plasma ionization. This technique is exceptionally efficient for ionizing gasses.
The atoms enter the plasma ion source volume, consisting of an anode and a cathode. Electrons
get accelerated from the anode to the cathode. These electrons create an arc discharge, and neu
tral atoms get ionized. A particular type of plasma ion source (VADIS) is used at the ISOLDE
facility. It has a large working volume and less graphite, plus a watercooled transfer line to
reduce the contamination rate [P+ 10, DG+ 16].
The target and the ion source are floated with 3060 kV; therefore, singly/doubly charged
ions exiting the production section have the equivalent energy. The highresolution separa
tor(HRS) and generalpurpose separator(GPS) are the dipole based magnetic separators with
resolving power of 5000 and 1000, respectively [K+ 00]. The radioactive beam of isobars can
be selected by adequately adjusting the devices. The continuous isobaric beam is delivered to
the ISOLTRAP setup [M+ 08, K+ 13]. The ISOLTRAP is a multifunctional mass spectrometer
that consists of four different traps for cooling, bunching, isotope separation, and precise mass
measurements. The details are provided in the following sections.
2.2 The buncher device
Figure 2.2. The basic layout of the RFQ buncher. It consists of an eggcup electrode, rod
system, and ejection electrodes [And18].
The radiofrequency quadrupole (RFQ) buncher is a linear Paul trap system filled with He
buffer gas [H+ 01b]. The device’s primary purpose is to form a bunched beam and cool it down
by collisions with buffer gas [S+ 91a]. The entire device is placed inside a high voltage cage
powered to 3040 kV to adjust the setup to the energy of the incoming ISOLDE beam. Once ions
enter the buncher, the RFQ field radially confines the ions in the device [Daw93]. The particular
DC potential of a well configuration axially guides ions (see figure 2.3). The ions collide with
He buffer gas, lose the energy and slowly accumulate in the potential well. The accumulation
time varies between species of interest; however, it is in the order of tens of milliseconds. After
2.2. The buncher device
18
the bunch is formed, the DC potential of the last segment is switched off, and ions are ejected
from the buncher device.
The RFQ consists of four segmented rods to which a voltage Φ0 (t) = UDC (r, z)±VRF (r)sin(ωt)
is applied. Every rod is divided into twentysix segments. The segments form five groups: Bank
1, Bank 2, Segment 23, BTrap, and Segment 26 to create the potential well. Equation 2.1 de
scribes the evolution of the potential in the buncher device. The equation of ion motion 2.1(2)
can be transformed to the standard Mathieu equation 2.1(3). The stable solution of this equa
tion leads to stable ion trajectories in x and y directions and depends on au = ax = −ay and
qu = qx = −qy . One can notice that a stable solution mostly depends on the ion mass and
UDC , VRF amplitudes [BH08].
Φ0 (t) x2 − y 2
∗
2
r0 2
e
e
Φ0 x = 0 and ÿ −
Φ0 y = 0
ẍ +
2
mr0
mr0 2
Φ(⃗r, t) =
(2.1)
d2 u
+ (au − 2qu cos 2ξ)u = 0
dξ 2
au = ax = −ay =
qu = qx = −qy =
ξ=
ωt
2
4eUDC
mω 2 r0 2
2eVRF V
mω 2 r0 2
: constant which depends on amplitude of DC potential and ion mass
: constant which depends on amplitude of RF potential and ion mass
: constant which depends on the period of the RF signal
This is the principle of using the buncher as a mass filter because only a certain range of
masses can be stored in x and y direction for a given UDC , VRF parameters. The mass scan line
(figure 2.3) is described by a =
2UDC
q
VRF
and depends on the Paul trap parameters. The closer the
line to the hill, the better is the resolving power. Masses outside this stable region will be lost
in x, y, or both directions. Stored ions oscillate around the zaxis with finite amplitude. Since
ions are fixed in x and y directions, the amplitude decreases in the zdirection. The length z
depends on the interactions of ions with the buffer gas.
Therefore, such parameters as the pressure of the buffer gas and applied UDC , VRF play
a crucial role in cooling ions and bunch formation. These parameters are the subject of the
following subsections.
2.2.1
Pressure of the buffer gas
If UDC , VRF guides the ions through the RFQtrap and affects the bunch formation, the He buffer
gas is responsible for cooling down the beam [S+ 91a]. At first, the dependency of the count rate
on the pressure has been taken (figure 2.4(a)). There is an increase in the count rate when the
He pressure increases. One of the most crucial parameters of the ISOLTRAP mass spectrometer
2.2. The buncher device
19
Figure 2.3. Top: schematic representation of the DC potential for axial confinement and bunch
forming. Bottom: the triangle of stability for ion guiding in radial direction [Wol90].
is the achievable mass resolving power which is the quantity that allows separating the isobars
in the beam. For the evolution of the resolving power, the number of counts has to be fixed
because the resolving power depends on the number of counts. Figure 2.4(b) shows how the
resolving power depends on the pressure. It gets better with the pressure increase.
As second, the taken spectra were fitted with Gaussian and Exponential Gaussian Hybrid
probability density functions using maximum likelihood estimation (details are in Section 2.3).
The idea was to check how the peak shape asymmetry evolves with pressure. The criterion
was chosen to be the difference between the mean values of Gaussian and exponential gaussian
hybrid (EGH) probability density functions. As one can see from figure 2.4(c), the difference
is getting smaller the more He pressure is presented.
2.2.2
Resonance frequency
The second parameter to control in the buncher is VRF amplitude applied to the buncher rods.
In order to create the quadrupolar field, the RF frequency applied pairwise to rods such that
opposite electrodes have the same polarity. This is achieved by five matching boxes, where the
Φ0 (t) has been created by splitting the transformer into two parts and grounding the common
middle electrode. This configuration creates an RF signal of the same amplitude but the opposite
sign. The typical scheme is shown in figure 2.5 top. In parallel to the transformer, the variable
capacitor and buncher rods with their own capacitance are connected. According to equation
2.2. The buncher device
20
140000
Resolving power
Counts, ion/shot
4
3
2
1
0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Pressure *10 3, mbar
120000
100000
1.1
80000
60000
40000
0.4
0.6
1.0
1.1
0.7
0.8
0.9
Pressure *10 3, mbar
1.0
1.1
(b)
meangaus meanEGH, ms
(a)
0.5
10
8
6
4
2
0
0.4
0.5
0.6
0.7
0.8
0.9
Pressure *10 3, mbar
(c)
Figure 2.4. (a) dependency of efficiency on pressure. (b) evolution of the resolving power
with pressure. (c) evolution of the difference between mean values of Gaussian and exponential
gaussian hybrid probability functions, errors are smaller the the sizes of points.
2.2, the resonance frequency of the circuit can be controlled by changing the capacitance of
Cvar because Ctotal = Cvar + Cbuncher (C4 and C3 in figure 2.5 top). This gives access to the
mass scan line of the triangle of stability, where the slope of the line depends on
f0 =
1
√
2π Ltotal Ctotal
2UDC
VRF
ratio.
(2.2)
The variable capacitor has many metallic plates which form layers with an air gap in
between as a dielectric. By rotating the layers, one changes the distances between them, and
because Cvar = ε0 KS
(n − 1), the capacitance can be tuned. ε0 is the permittivity of free space
d
constant, K is the dielectric constant of the material in the gap (air), S is the area of the plates,
and d is the distance between the plates, n is the number of layers. Cvar can be varied between
30 and 80 pF. Cbuncher depends on the length of the segment of the buncher rods. For Bank 1,
Bank 2, it is 500 pF, and for Segment 23, BTrap, Segment 26 is 100pF. The inductance of the
2
transformer varies between segments as well and Ltotal = µ0 N K πD
, where µ0 is the vacuum
4d
permeability, N is the number of windings, D the diameter of the coil, d the diameter of the
2.2. The buncher device
21
Figure 2.5. Top: the electronic scheme of the resonance circuit used for Btrap. The same
schematic was used to simulate the frequency range in the LTspice software. The main elements
of the circuit are discussed in the text. Bottom: 1 variable capacitor, 2 the coupler, 3 the
stepper motor.
wire, and K is a factor which depends on the ratio between the diameter and the length.
The idea was to couple the variable capacitor and the stepper motor as shown in figure 2.5
bottom to tune via LabVIEW and Arduino software the resonance frequency. This would allow
using the matching boxes as steerer or lens to manipulate the count rate but based on a principle
of the mass separator (according to equations of motions 2.1 and triangle of stability in figure
2.3). On top of this, there was a proposal to use the capacitordiode, where the capacitance
depends on the voltage applied to the diode base. Instead of mechanically turning the capacitor,
2.2. The buncher device
22
one could change the capacitance by changing the voltage applied to the base. This electronic
component should survive a radio frequency signal of high voltage (more than 200 V), and no
such capacitordiode was found available on the market.
Amplitude, V
16
Bank 1
180 pF
230 pF
30 pF
80 pF
exper.
14
12
10
8
6
4
2
0
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Frequency, MHz
Figure 2.6. The results of the simulation in LTspice software. The capacitor varies between
30 and 80 pF. The VRF amplitudes for minimum and maximum of the achievable capacitance
are shown on the plot. The gray band represents the experimental results with the high voltage
amplifier.
To obtain the value of the resonance frequency with chosen circuit parameters and to check
the achievable frequency range, the simulation was performed in the LTspice software (figure
2.5 top). The variable capacitor is named C4, the buncher’s rods capacitance is named C3.
This schematic corresponds to either Segment 23, BTrap or Segment 26 because they have the
same length of the rods. L1, L2, L3 are the inductances of the solenoids while K1, K2, K3 are
the coupling parameters. R3, R4, Rgenerator , Rinductor are interior resistances of the elements.
Finally, V1 is a frequency generator producing the sinusoidal signal of 1volt amplitude and
different ranges of frequencies. The simulation was also performed for C3=500 pF and the
result is shown in figure 2.6. With given circuit parameters, the Banks’ frequency could be
varied between 1.36 and 1.43 MHz, while for Segment 23, Btrap, and Segment 26 it was 1.05
1.22 MHz. Therefore, an additional capacitor of 150 pF was added in parallel in Bank 1 and
2.2. The buncher device
23
Bank 2 resonance circuits. That allowed an expansion of the frequency range to the lower
values of 1.251.3 MHz. The main results are listed in table 2.1. After the matching boxes
were constructed, their performance was checked with a frequency generator and a frequency
amplifier. The results are shown as a grey band in figure 2.6 and agree with the simulation.
Finally, the device was installed in place at the ISOLTRAP setup. It was possible to tune
the frequencies online. As expected, the count rate depends on the resonance frequency, which
depends on the configuration of the layers of the variable capacitors. Figure 2.7 shows the
current amplitude VRF applied to the buncher rods with the help of the new device. As one can
see, the Btrap region plays the most significant role in the bunch forming. It was possible to
tune capacitors so that the maximum number of 133 Cs ions have been counted, but at the same
time, 85 Rb ions were utterly lost in the buncher device. That was the expected result from the
triangle of stability.
Amplitude, V
100
200
300
400
500
only 133Cs
133Cs and 85Rb
Bank1 Bank2 Seg23 Btrap Seg26
Figure 2.7. Top the amplitude VRF of the resonance frequencies reached on the segments
of the buncher. Black dots correspond to the amplitude on the rods when the count rate was
maximized on the 133 Cs ions; red dots correspond to the amplitude when both 85 Rb and 133 Cs
ions were observed in the same bunch.
2.3. The multireflection time of flight device
24
Table 2.1. The achievable frequency ranges of new build matching boxes. The first column
represents the capacitance of the variable capacitor and does not include the capacitance of
segments.
Resonance Frequency, MHz
2.3
Capacitance, pF
Bank 1
Bank 2
Segment 23
Btrap
Segment 26
30
1.435
1.395
1.225
1.235
1.185
80
1.395
1.355
1.095
1.105
1.075
180
1.295
1.285
230
1.255
1.245
The multireflection time of flight device
After the bunch of ions is extracted from the buncher, it enters the pulsed drift tube where the
ions experience deceleration down to 3.2 keV by switching the drift tube’s potential. Down
stream there is a multireflection time of flight (MRToF) device [W+ 12] which has two work
ing modes. First, it can be used as an isobaric separator [W+ 11b]. Second, the time of flight
spectrum of ions can be recorded, which allows using the device as a mass spectrometer on its
own [W+ 13b].
Figure 2.8. Top: the multi reflection time of flight device. Bottom: the in traplift technique
[W+ 13b] to store the bunch of ions between electrostatics mirrors.
2.3. The multireflection time of flight device
25
The MRToF device schematically shown in figure 2.8. The bunch of isobars is trapped in
the middle of the device by applying in trap lift potential [W+ 17]. Once ions are stored, they
revolve between two sets of five by five electrostatic mirrors. The flypath of ions is increased to
kilometers, and eventually, the isobars with a slightly different mass can be separated. The main
parameter of the device is the mass resolving power
m
.
∆m
At ISOLTRAP, the MRToF device
reaches the resolving power of 130000. The trap lift potential can be set so that only the ions
of interest are transported to the next device, namely the precision Penning trap. This allows to
eliminate the contaminant species and increase the precision of the mass measurements in the
Penning trap.
The advantage of the MRToF device is that it can also be used as a mass spectrometer.
To achieve this goal, an electron multiplier detector is installed in combination with a multi
channel analyzer of the MCS6A model in the ISOLTRAP beamline. One of the most crucial
advantages of the MRToF mass spectrometry that it is fast. Typically only 20 ms are required
to split up the isobaric beam, which benefits when masses of fast decaying isotopes have to be
investigated. Another advantage is that the device can perform the mass spectrometry of beams
with low yields. The time of flight of an ion depends on its
mi
qi
ratio according to the equation
2.3
√
ti = α
ti
mi
+β
qi
(2.3)
: the time of flight of the ion
mi : the mass of the ion
qi
: the charge of the ion
α, β : calibration constants
To calibrate the device, at least two reference ions with wellknown masses are needed. For
this purpose, an offline alkali source is installed. It provides beams of 39 K, 85 Rb and 133 Cs ions.
The mass of the ion of interest can be determined through equation 2.4. However, to minimize
the uncertainties coming from different trajectories of the offline and online beams, usually, one
ToF spectrum from the offline ion source and one ToF spectrum from online measurement are
used. Since the uncertainties of the reference masses contribute to the uncertainty of the mass
of ion of interest, the measured mass is calculated as a CT oF ratio.
√
mi
1
= CT oF ∆Ref + ΣRef
qi
2
(2.4)
2.3. The multireflection time of flight device
26
2t−t1 −t2
: depends only on the ToF’s of the ion of interest and the reference
2(t1 −t2 ) √
√
∆Ref = mq11 − mq22 : depends only on the mass over charge ratio of reference ions
√
√
ΣRef = mq11 + mq22 : depends only on the mass over charge ratio of reference ions
CT oF =
2.3.1
The statistical uncertainty
The MCS6A card records counts and builds a histogram, the binning of which is equal to 0.8
ns based on the card’s resolution. The analysis based on binned maximum likelihood estima
tion was performed. The method assumes that the distribution Pi (θ) with θ as parameters is
Poisson distributed, and the probability to observe the event is given within likelihood func
tion. The maximization of this function will result in the most likely parameters of the distri
bution [Myu03].
The experimental data are represented by the number of counts in the chosen time of flight
window (figure 2.9). As a standard approach, all MRToF spectra are fitted with Gaussian
distribution by applying binned maximum likelihood estimation. The outcome of the fitting
procedure is a mean value of the distribution (most probable time of flight), the sigma value of
the distribution and its statistical uncertainty.
1 xi −µ 2
1
Pi (θ) = P (x, µ, σ) = √ e− 2 ( σ )
σ 2π
2.3.2
(2.5)
The systematical uncertainty
One of the sources of systematical uncertainty comes from the voltage applied to the mirrors of
the MRToF device [W+ 20]. The studies show that the most significant influence on the trajec
tories of the ions inside the device comes from the voltage applied to the last mirror [W+ 12].
The last mirror is a place where ions reflect, and the instability of the applied voltage results in
the time of flight distortions. Therefore, it is essential to take one reference peak from online
spectra (some stable isobars are always presented) and a reference peak from an offline source.
This shift is on the level of 1 ns. To include the error to the Ctof ratio, the partial derivatives are
taken and multiplied by the corresponding statistical uncertainties.
√
σCT oF ,stat =
∂CT oF
∂t
∂CT oF
∂t1
∂CT oF
∂t2
=
=
=
1
t1 −t2
t2 −t
(t1 −t2 )2
t−t1
(t1 −t2 )2
(
∂CT oF 2 2
∂CT oF 2 2
∂CT oF 2 2
) σt + (
) σt 1 + (
) σt 2
∂t
∂t1
∂t2
: the first coefficient
: the second coefficient
: the third coefficient
(2.6)
2.3. The multireflection time of flight device
Counts/0.8 ns
25
27
GAUS
EGH
data
25
20
20
15
15
10
10
5
5
16.856024
16.856124
Time of flight (ms)
Figure 2.9. As an example, the time of flight spectrum of
49
Sc ions fitted by Gaussian and
Exponential Gaussian Hybrid fit functions.
As it said earlier, the peak shape of the time of flight does not follow Gaussian distribu
tion [LJ01]. One of the reasons can be the buncher device, which introduces the asymmetry
through collisions with the buffer gas. The corresponding investigations are described in pre
vious chapters and are still ongoing. To include the systematic shift of the mean time of flight
value as the systematic uncertainty, another probability density function was used. A charac
teristic example is shown in figure 2.9. The exponential decay function well describes the right
tail. The full function consists of the normal distribution on the left side and the exponential tail
on the right side. This is the socalled exponential Gaussian hybrid (EGH) probability density
function. The difference between CT oF computed from Gaussian and EGH fit functions is taken
as systematic uncertainty and is added in quadrature to the statistical uncertainty.
σCT oF ,pdf = CT oF,Gauss − CT oF,EGH
(2.7)
The number of counts which are taken to build the histogram also affects the fitting proce
dure. The bigger the fitting window is, the more probable the situation when too much of the
data is included and the background dominates the total distribution. On the contrary, a small
fitting window may result that not all data is included in the fitting routine. For this purpose,
the study was conducted on the evolution of the output Gaussian parameters depending on the
2.3. The multireflection time of flight device
28
32
Sigma value / ns
Mean value / ns
+1.68564e7
50
49
30
28
26
24
22
48
20
60
80
100 120 140 160 180 200 220 240 260
60
80
Half-range / ns
(a)
100 120 140 160 180 200 220 240 260
Half-range / ns
(b)
Figure 2.10. The mean and sigma parameters of the Gaussian distribution in dependence of
the halfrange.
fit range. The fit range is the full width at half maximum. In figure 2.10 one can notice two
plateaus. One plateau between 120 and 200 ns, the second plateau between 220 and 260 ns. The
second plateau corresponds to the situation when too much of the data is included in the fitting
procedure and background starts to play a significant role. The middle plateau corresponds to
the situation when the most stable output fit parameters have been observed. There can be a
situation when the third plateau is observed at smaller ranges, but this also has a physics ex
planation as not enough data has been included. The middle and small plateau has not been
observed in every data set. Therefore, it can not be claimed as a universal criterion for the best
choice of the fitting window. Thereby, the fitting window was chosen on a casetocase ba
sis. One halfrange was chosen for the CT oF evaluation. Let us assume as an example 80 ns.
The difference in CT oF between smaller halfrange 60 ns and 80 ns and the difference between
bigger halfrange 100 ns and 80 ns were taken as systematical uncertainty coming from the
variation of the data included in the fitting procedure [K+ 20]. This is also part of the systematic
uncertainty which comes from the unknown peak shape and is expressed:
σCT oF ,window = 2CT oF,middle − CT oF,small − CT oF,big
(2.8)
The total uncertainty is the sum of all three contributions:
σCT oF ,total
2.3.3
√
= σC2 T oF ,stat + σC2 T oF ,pdf + σC2 T oF ,window
(2.9)
The MRToF mass spectrometry data analysis
1
1
This analysis has been published in [K+ 20] by the author of the thesis as a primary author. Therefore, some
pictures and text may be repeated
2.3. The multireflection time of flight device
49
Sc
Ions of
49
49
Sc were stored in the MRToF device for 500 revolutions. Approximately 14000
Sc ions were recorded in three data sets. The ToF spectra of 49 Sc+ were fitted by Gaussian,
and EGH probability density functions. The
39
29
49
Ti ions were used as an online reference, and
+
K was chosen as an offline reference. Online and offline references were likewise fitted by
Gaussian and EGH fit functions. The Gaussian fit function was used for computing the CT oF .
The weighted relative statistical uncertainty amounted to 2.5 × 10−7 . The difference between
scandium CT oF values obtained from Gaussian and EGH fits was 3.45 × 10−7 . This value was
added in quadrature to the total uncertainty as σCT oF ,pdf . The optimal fitting window was chosen
to be 80 ns. The obtained results for 100 ns and 60 ns fitting windows differed by 5.35 × 10−8
and 5.42 × 10−8 . Both values were added in quadrature to the total uncertainty as σCT oF ,window .
The weighted CT oF = 0.500203363 with the total uncertainty of 4.32 × 10−7 is listed in table
2.2
The first mass determination of
49
Sc was obtained from the Qvalues of
49
Ca(β − )49 Sc
[O+ 56,M+ 56] and 49 Sc(β − )49 Ti decays [R+ 61,F+ 69]. A few years later, this nucleus was used
in several nuclear reaction experiments, giving additional Qvalues [E+ 66,W+ 66,G+ 68,V+ 68].
The various number of β − decays and reaction experiments conducted with this nuclide con
strained the mass excess value to 46561.3(27) keV. The new mass excess value obtained by
our MRToF MS measurements agrees with AME2016 [H+ 17], deviating by 3 keV. Moreover,
the significance of the new direct mass excess value in the reevaluated AME is 29 %. The new
MRToF MS result is the first direct mass measurement of 49 Sc.
50
Sc
Ions of 50 Sc were stored in the MRToF MS for 500 revolutions. Altogether there were about
3600 ions of
50
Sc recorded in four data sets. Analogous analysis as for
49
Sc+ has been per
formed. The online reference was 50 Ti+ and the offline reference was 39 K+ . The Gaussian fit
function was used for computing CT oF . This led to the weighted CT oF =0.500633802 with the
statistical uncertainty 2.34 × 10−7 . The difference due to the choice of the probability density
function gave the systematic uncertainty of 4.01 × 10−7 . The systematic uncertainty from vary
ing the fitting halfrange amounted to 2.6 × 10−7 and 1.53 × 10−7 for the smaller and larger
windows, respectively. The weighted result is CT oF = 0.500633802 and the total uncertainty
is 8.4 × 10−7 .
The 50 Sc(β − )50 Ti decay [W+ 69, C+ 63] and the 48 Ca(He3 ,p)50 Sc reaction [O+ 69] were the
first experiments to provide the mass excess value of 50 Sc. The first direct mass measurement
was performed at the TOFI spectrometer in 1998 [B+ 98]. The uncertainty of the AME2016
[H+ 17] value was mainly determined by the result from the
48
Ca(He3 ,p)50 Sc reaction, which
2.3. The multireflection time of flight device
30
49
fit
data
Counts/0.8 ns
49
Ti
102
Sc +
0.0000015
+
101
101
100
16.8560096
Ctofi − < Ctof >
102
0.0000005
0.0000000
−0.0000005
100
16.8562596
0.0000010
16.8565096
1
Time of flight (ms)
2
3
Number of measurement
(b)
AME error
AME mean
experiments
ISOLTRAP
−46540
−46560
) 49
Ti
−
Sc
(β
−
50
49
Sc
(β
49
Ti
(t,
α) 49
Sc
48
Ca
( 3H
e, d
) 49
Sc
48
Ca
(d,
n) 49
Sc
48
Ca
(p,
γ) 49
Sc
MR
-T
oF
−46580
) 49
Ti
Mass excess (keV)
(a)
(c)
Figure 2.11. (a) the Gaussian fit of
49
Sc ions. (b)the difference between individual CT oFi
and weighted CT oF . The total uncertainty is shown with a grey band. (c)the experimental
mass excess values of
49
Sc are taken from [R+ 61, F+ 69, E+ 66, W+ 66, G+ 68, V+ 68]. For a
better visibility, the values from [O+ 56, M+ 56] are excluded from the plot since they have
uncertainties of 50 and 100 keV.
is 15 times more precise than the uncertainty obtained by TOFI. Altogether they defined the
mass excess value to 44547(15) keV. The new mass excess value measured by the MRToF
MS technique with uncertainty of 9.1 keV agrees perfectly with the AME2016 value and, if
included, contributes 73% to the reevaluated mass excess value.
73
Br
The 73 Br ions were stored in the MRToF MS for 250, 500, and 1000 revolutions. There were
three data sets recorded with a total number of about 4500 ions. The 61 Ni12 C molecule [SG89]
contaminated the spectra (see figure 2.13(a)). The mass resolving power R = m/∆m = t/2∆t
was 123000, while the minimum resolving power required to resolve these ion species is 118000
as calculated from AME2016. Thus, it was impossible to completely separate the
61
73
Br+ and
Ni12 C+ peaks. Therefore, a twoGaussian fit function was used for computing the CT oF .
2.3. The multireflection time of flight device
102
50
Ti +
50
Cr +
fit
data
31
102
0.000003
101
50
Ctofi − < Ctof >
Counts/0.8 ns
0.000002
101
Sc +
100
0.000001
0.000000
−0.000001
100
−0.000002
17.02672
17.02722
17.02772
17.02822
1
Time of flight (ms)
2
3
4
Number of measurement
(b)
−44400
−44600
−44800
-T
MR
FI
TO
48
Ca 3
(H
e, p
) 50
Ti
−
50
Sc
(β
−
Sc
(β
50
) 50
Sc
−45200
oF
AME error
AME mean
experiments
ISOLTRAP
−45000
) 50
Ti
Mass excess (keV)
(a)
(c)
Figure 2.12. Same as figure 2.11 but for 50 Sc ions. The literature mass excess values of 50 Sc
are taken from [W+ 69, C+ 63, O+ 69, B+ 98].
To employ the EGH and Gaussian fit functions, the fitting halfrange had to be reduced to 80
ns. Otherwise, the peaks for bromine ions and 61 Ni12 C molecules were recognized as a single
peak. The difference between the fit results with Gaussian and twoGaussian fit functions was
2.17 × 10−8 . The corresponding difference between EGH and twoGaussian functions was
2.63 × 10−7 . The systematic errors 1.23 × 10−6 and 1.0 × 10−6 were obtained as above by
varying the fitting window. The optimum window was found to be 80 ns. The latter small
window was the main source of systematic uncertainty. The final uncertainty of 1.63 × 10−6
was determined.
There are seven previous mass measurements of 73 Br: four beta decay studies [T+ 01,R+ 74,
H+ 87,M+ 70], one ESR storage ring experiment [H+ 01a], a ToFICR value measured by ISOLTRAP
[H+ 11], and two results obtained via deflectionvoltage ratios of 73 Br and 72 Br in a magnetic
dipole separator at the Chalk River TASCC facility [S+ 89, S+ 91b]. The evaluated mass excess
value obtained from those experiments is 63647(7) keV and is mainly determined by the ToF
ICR result from ISOLTRAP reported in 2011. The new mass excess value obtained by MRToF
MS agrees with the literature value within the error bars and contributes 14% to the reevaluated
mass excess.
2.4. Alignment of the horizontal beam line
32
73
Ge +
As +
Se +
fit
data
Counts/0.8 ns
73
73
73
Ga +
101
61
12
Ni C
0.0000015
Br +
Ctofi − < Ctof >
73
101
+
100
100
20.574149
20.574449
20.574749
20.575049
0.0000010
0.0000005
0.0000000
−0.0000005
−0.0000010
−0.0000015
20.575349
1
Time of flight (ms)
2
3
Number of measurement
(a)
Mass excess (keV)
(b)
−63000
AME error
AME mean
experiments
ISOLTRAP
−63200
−63400
−63600
−63800
) 73
Se
Br
− 72
Br
73
Br 7
- 2
Br
ES
R
To
F-I
CR
MR
-T
oF
+
73
) 73
Se
+
Br
(β
73
) 73
Se
+
Br
(β
73
Br
(β
73
73
Br
(β
+
) 73
Se
−64000
(c)
Figure 2.13. Same as figure 2.11 but for 73 Br ions. The literature mass excess values of 73 Br
are taken from [T+ 01, R+ 74, H+ 87, M+ 70, H+ 01a, H+ 11, S+ 89, S+ 91b].
2.4
Alignment of the horizontal beam line
As one can see from figure 2.1, the ISOLTRAP mass spectrometer consists of two sections
one horizontal and one vertical section. All devices discussed so far are part of the horizontal
section.
The alignment of the vertical section was performed in 2015. It was imperative to inject
the beam into the lower trap as straight as possible. The injection to the upper trap where the
actual mass measurements are performed and the region between the upper trap and the top
multichannel plate detector was even more substantial [E+ 14]. The final vertical alignment
was checked with an electron beam.
The horizontal alignment procedure was done with the assistance of the trained CERN team.
All elements of the horizontal beamline were taken apart and covered with target holders. The
radial position was reconstructed based on the laser reflection on mirrors placed in the target
holders. Based on the reconstruction (fiducialization), the adjustment of elements was made to
minimize misalignments. Before the vacuum pumping, the positions of elements were checked
with the theodolite—the results of two different methods supported each other. At the final step,
2.5. ISOLTRAP Penning trap tandem
33
the beamline was pumped, and the final position of elements has been checked (red symbols
in figure 2.14). The vacuum pumping moved the beamline elements, and readjustment was
performed one more time. The overall misalignment was reduced to less than 1 millimeter in
0.0
−0.5
before pumping
after pumping
−1.0
t
Be
nd
er_
ou
ut
_o
OF
MR
-T
_in
MR
-T
e_
tub
rift
_
Pu
lse
d_
D
OF
ou
t
in
rift
d_
D
Pu
lse
Bu
nc
_tu
he
r_
be
_
ou
n
r_i
he
nc
Bu
hy
Sw
itc
t
−1.5
ard
Radial position, mm
the horizontal section.
Figure 2.14. The final radial position of the horizontal beam line parts. The switchyard and
the bender_out elements are taken as reference points.
2.5
ISOLTRAP Penning trap tandem
One of the most precise methods to measure the mass of an ion is based on cyclotron frequency
measurements. This can be achieved by applying special manipulations of the ion motions in
the electromagnetic field of a particular configuration inside the Penning trap [Bla06]. The
technique to confine ions was developed by W. Paul and H. Dehmelt [Deh90] based on the idea
of F.M. Penning [Pen36]. Nowadays, it is widely used for accumulation, bunching, cooling,
and charge breeding operations on the beams of ions. Moreover, the applications are broad and
include fields such as mass spectrometry, antimatter studies, spectroscopy, quantum computers.
At ISOLTRAP, two types of Penning traps are used. One of the Penning traps has a cylin
drical shape of electrodes, and it is used for cooling the ions and further purification from the
isobaric contaminants [S+ 91a]. The second trap has a hyperbolic shape of electrodes configu
2.5. ISOLTRAP Penning trap tandem
34
ration and is used for precision mass measurements. Both traps are placed in superconducting
magnets. A strong magnetic field is needed to confine ions radially. The electric field is used
to confine ions axially. The following paragraphs will describe the basics of the ion motions in
an ideal hyperbolic Penning trap. However, all equations and general physics can be applied to
cylindrical traps as well.
The quadrupolar electric potential can be written as equation 2.10 and created by the poten
tial V0 applied to a set of electrodes. The trap consists of three electrodes: one ring electrode,
and two end cups electrodes, the hyperbolic shape of which creates a quadrupolar potential. The
characteristic of the trap 2d2 = z02 + ρ20 /2 is the minimum distance between the trap center and
the electrodes. It depends on the distance between the endcup electrode and the ring electrode.
As it was said earlier, such a potential confines ions axially [B+ 90a].
V0 2
(ρ − 2z 2 )
(2.10)
2
4d
The magnetic field confines the ions in the radial direction and is aligned with the z axis
V (z, ρ) =
of the trap. The ion’s motion inside the Penning trap has been studied in great details by J.F.
Berger [B+ 91] and can be summarized by equations:
1
ẍ + ωc ẏ − ωz2 x = 0
2
(2.11)
1
ÿ + ωc ẋ − ωz2 y = 0
2
2
z̈ + ωz z = 0
√
ωz =
ωc =
qV0
md2
q
B
m
: axial oscillation frequency
: angular oscillation frequency
The axial motion among z axis (2.11(3)) is a simple harmonic motion and it is fully decou
pled from the radial motion. In turn the radial motion has a complex structure and consists of
two circular motions. The solutions of equation 2.11(12) gives angular eigenfrequencies:
√
1
ω± = (ωc ± ωc2 − 2ωz2 )
2
(2.12)
These two motions are called modified cyclotron motion ω+ and the magnetron motion ω− . The
magnetron motion is a drift motion in the electromagnetic field around the trap center.
The ωc is pure cyclotron frequency and from equation 2.12 one can obtain the invariance
theorem [BG86] which couples the cyclotron, magnetron, modified cyclotron and axial fre
quencies:
ωc = ω+ + ω−
(2.13)
2.5. ISOLTRAP Penning trap tandem
35
2
2
ωc2 = ω+
+ ω−
+ ωz2
Because the cyclotron frequency ωc is equal
q
B,
m
the precise measurement of the frequency
will result in a precise mass determination. A couple of techniques have been used to manipulate
these frequencies and trajectories of ions inside the Penning trap. First of all, to minimize the
amplitude of the axial motion, the voltage of the end cup electrode and the time of the injection
are optimized. Second of all, the ring electrode has been divided into four segments. This allows
the application of a bipolar or quadrupolar sinusoidal signal, which in turn allows addressing
the radial motions.
Electric field
20
15
10
z, mm
10
0
5
0
-10
-5
-20
-20
-10
0
r, mm
10
20
Figure 2.15. Left: the trajectory of an ion. It consists of three motions: modified cyclotron,
magnetron and axial motion. Right: the electric field of a quadrupolar shape. The pictures are
taken from [Din16].
In the case of the dipole field, the frequencies of the pulses are tuned to resonate with one of
the radial ion motion [W+ 05]. This allows increasing or decreasing one of the radial motions
depending on the phase and amplitude of the applied pulses. One of the applications of bipolar
excitation is bipolar cleaning. Because the modified cyclotron frequency is massdependent,
one can apply a bipolar field to this motion to remove contaminant species. The other appli
cation is to prepare all ions to have the same radius of the magnetron motion. The magnetron
motion is firstorder massindependent, and preparing the ions on the same magnetron radius is
crucial before applying quadrupolar excitation.
If the bipolar excitation gives the excess to manipulate the ion motions individually, the
quadrupolar excitation operates the sum of modified cyclotron and magnetron motion [K+ 95b].
Applying the quadrupole excitation results in coupling these two motions. This leads to a pe
riodic transformation of one motion into the other one. During the first part of the conversion,
the radius of the magnetron motion is decreasing while the radius of the cyclotron motion is
2.5. ISOLTRAP Penning trap tandem
36
increasing. This procedure continues until the whole conversion is achieved. At the end of the
cycle, the amplitude of the cyclotron motion is equal to the initial amplitude of the magnetron
motion. The period of the conversion is given by:
Tconv = 2πB
a
a2 ω+ − ω−
VRF
ωc
(2.14)
: the radius
VRF : the amplitude of the applied signal
2.5.1
Preparation penning trap
When entering the vertical section, ions are first decelerated to 100 eV by a second pulsed
drift tube. Then ions are trapped in the preparation Penning trap, which has a cylindrical shape
and placed in the superconducting magnet of 4.7 T strength [RH+ 97]. The primary purpose
of the device is to clean the beam from isobaric contaminants and to further cool it down by
interaction with He buffer gas [S+ 91a]. The trap consists of 20 cylindrical electrodes and filled
with He gas at 10−6 mbar. By collisions with the buffer gas, ions lose their kinetic energy to the
temperature of the buffer gas. The He gas is well known to have low chemical reactivity. This
makes it perfect for cooling because ions of interest will not chemically interact with the gas.
Moreover, the He atom has a fullfilled atomic structure, and the ion loss will be minimal due
to the charge exchange. The resulting damping force can be expressed by equation 2.15. The
damping coefficient δ depends on the buffer gas properties such as temperature and pressure.
−
→
−
→
F = −δm V
(2.15)
−
→
F : the dumping force
m : mass of an ion
δ : damping coefficient
−
→
V : the velocity vector of an ion
Without a buffer, gas ions have the motions, as discussed early. However, the presence
of the gas damps the axial motion by buffer gas collisions. Then the magnetron motion is ex
cited by bipolar excitation to the radius bigger than the aperture of the ejection electrode. The
quadrupolar excitation is then applied at the cyclotron frequency of the ion of interest. This
leads to the conversion of the magnetron motion to the cyclotron motion. The radius of mag
netron motion decreases during the conversion, while the radius of modified cyclotron motion
2.5. ISOLTRAP Penning trap tandem
37
is reduced by the collisions with buffer gas. Consequently, the ions of interest are centered in
the Penning trap at a smaller radius while the contaminant ions stay on the initial magnetron
radius bigger than the aperture of the end cap electrode. During the ejection, the ions of interest
are transported to the precision Penning trap, but contaminant ions hit the end cap electrode.
This is the socalled massselective resonant buffergas cooling technique [S+ 91a].
In conclusion, the preparation Penning trap can be used for purification and cooling of the
beam. However, the main disadvantage of the technique is the preparation time. It takes ap
proximately 150 ms to purify the beam, which is a significant limitation for the shortlived ions
of interest. On the other hand, the MRToF device with the same resolving power of 105 per
forms the purification in 20 ms. Therefore, at the current stage, the MRToF device is preferable
for the purification of the beam, and the preparation Penning trap is only used for cooling the
beam.
2.5.2
Precision penning trap
After the ions are well cooled and all contaminants have been removed in the previous sections
of the setup, they are injected into the precision Penning trap. At first, the ions are prepared by
bipolar excitation on the smaller magnetron radius. In this way, ions experience fewer imper
fections of the electric field. Second, the quadrupolar excitation is applied at νrf = ν+ + ν−
with amplitude Vrf . If the period is correct, the magnetron motion will convert to pure cyclotron
→
→
motion. The radial motion bears a certain magnetic moment −
µ = Er −
e . When the ions are
B
z
ejected from the precision Penning trap, they experience a strong gradient in the magnetic field
(from 6 T to some mT). This leads to a force accelerating the ions.
−
→
−
→−
→
→
F = −−
µ (∇ B )
(2.16)
The radial kinetic energy Er strongly depends on the modified cyclotron frequency. If the
conversion from magnetron motion to modified cyclotron motion reaches maximum the ions
will be accelerated most by force according to equation 2.16. In other words, resonant ions will
reach the multi channel plate detector faster. The figure 2.16 is the schematic representation.
This is a socalled time of flight ion cyclotron resonance (ToFICR) mass measurement tech
nique [B+ 90a]. During the experiment, the frequency of the quadrupolar excitation is scanned
around the approximate cyclotron frequency of the ion of interest νc . Every new ion bunch
ejected out of the trap corresponds to a new frequency step. The typical ToFICR resonance
can be recorded. The deep minimum in figure 2.17(a) corresponds to a pure cyclotron fre
quency. Every point represents the mean value and the standard deviation of multiple ejected
ions at the scanned excitation frequency. The shape of the resonance can be explained by the
Fourier transform of the square shape of the time profile of the applied excitation. The resolving
2.5. ISOLTRAP Penning trap tandem
38
B
z0
Trap centre
zmax
Drift electrodes
Detector
Figure 2.16. Top: the precision Penning trap [M+ 18]. Bottom: schematic representation of
the principle of the time of flight ion cyclotron resonance detection technique [Din16].
power and the precision of the technique essentially depend on the length of the quadrupolar
excitation TRF . The deeper and narrow the minimum, the higher is the precision. The longer the
pulse the higher the resolving power. At ISOLTRAP, the length of the quadrupolar excitation
TRF varies between 0.2 and 1.2 seconds. This depends on the halflife of the studied ions.
√
νc
N
R=
= νc TRF
δνc
c
(2.17)
2.5. ISOLTRAP Penning trap tandem
39
N : number of events
c : parameter ≈ 0.9 which slightly depends on the quality of the data
One single pulse of quadrupolar excitation couples two radial motions [GKT80]. Two pulses
with some waiting time in between can be applied during the conversion. This is the socalled
Ramsey type excitation. The typical resonance is shown in figure 2.17(b). The resonance is
symmetric around the cyclotron frequency. The Ramsey type [Ram90] excitation results in
125
125
120
120
115
115
110
110
105
105
100
70
95
100
As+
95
90
85
80
90
T
rf. =
1.2 s
fit
data
85
Mean time of flight (μs)
Mean time of flight (μs)
three times nigher precision for the same excitation time as a single pulse ToFICR.
450
Kr
400
400
350
350
300
fit
data
80
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0
−40
Excitation frequency-1302184.923 (Hz)
(a)
T rf. = 10-40-10 ms
97
+
450
−20
0
20
300
40
Excitation frequency-939344.832 (Hz)
(b)
Figure 2.17. (a) a typical ToFICR resonance of 70 As ions using a single rfexcitation pulse
of 1.2 seconds duration. (b) a ToFICR resonance of 97 Kr ions obtained by Ramsey excitation
with duration of 104010 milliseconds. The line represents a fit of the theoretical lineshape to
the data points [K+ 95b]
A new technique has been developed recently. It is based on the projection of the ion motion
on the positionsensitive detector. It is socalled phase image ion cyclotron resonance technique
(PIICR) [E+ 14]. The new approach is 25 times faster than the regular ToFICR and allows for
40 times higher precision. Unlike ToFICR, it is a noscanning technique, and every single ion
contributes to the cyclotron frequency measurement.
Applying ToFICR and PIICR techniques results in the loss of ions on the detector. These
are destructive techniques, and after the mass measurement is performed, the ions of interest
are entirely lost in the detector. However, there is one more technique when the beam has
not been absorbed by the detector. It is a Fouriertransform ion cyclotron resonance technique
(FTICR) [C+ 74]. The trapped ions induce electrical signals on the electrodes of the Penning
trap which follow the oscillations of the ions inside of the trap [M+ 02]. After applying the
Fourier transformation, one can reconstruct the frequencies while ions are still trapped. The
only disadvantage of the technique is the noise. Therefore, such a technique requires electrodes
to be cryogenically cooled and special low noise electronic devices and cables.
2.5. ISOLTRAP Penning trap tandem
40
2.5.3
The statistical and systematical errors
In reality, the Penning trap is not an ideal system. Some imperfections lead to shifts in frequen
cies affecting the mass determination. Being aware of these effects, one can include them as a
systematic error.
The magnetic field B has to be stable and known with high precision. Calibration of the
magnetic field is performed using wellknown reference masses provided from the offline ion
source. The cyclotron frequency νc,ref of one of the 39 K+ , 85 Rb+ or 133 Cs+ reference isotopes
is measured before and after measuring νc of the ions of interest and its value at the time when
the ion of interest was measured is linearly interpolated. The atomic mass is given:
m=
νc,ref q
(mref − qref · me ) + q · me ,
νc qref
(2.18)
where me is the electron mass, mref is the mass of the reference ion and q, qref are the charge
states of ion of interest and reference ion. Equation 2.18 assumes that the electron binding
energy of the removed electrons is negligible. To account for a possible readjustment of the
employed reference masses in future measurements, the mass of the isotope of interest is re
ported as a measured primary frequency ratio ricr = νc,ref /νc .
Three factors limit the statistical uncertainty of the ToFICR method. All three of them rise
from the beam production, contamination conditions, and decay properties of the studied cases.
The first factor is the number of detected ions. A higher number of detected ions leads to a
smaller statistical uncertainty. The production rate of the ion of interest, the transport efficiency
of the setup, and the halflive of measured species are often the limitations of the very precise
mass measurements. The second factor is the quality of the recorded spectra. The ripple of
voltages, the ionization of the gas by the decay products of the studied ions, the optimization of
trap excitation pulses results in a deviation of the spectrum from the theoretical line shape. As
we saw, contamination leads to the shift of the cyclotron frequency and is usually included as a
systematic uncertainty. The last factor is the duration of the excitation. The longer the excitation
is, the more precise the measurement is. The halflive of the studied case is the limitation of
the excitation time. All of these contributions can be summarized in the empirical equation
2.19. However, these factors do not strongly affect the reference measurements provided by the
offline ion source. That means the statistical uncertainty of the cyclotron frequency of the ion
of interest mostly dominates the final statistical uncertainty of the mass value.
(
δνc
1
c
)stat = √
νc
νc N TRF
(2.19)
The first imperfection which contributes to the systematical uncertainty is the electric field
[B+ 90a]. Due to the finite size of electrodes, manufacturing tolerances, and the injection/ejection
holes, the final configuration of the electric field deviates from the ideal quadrupolar shape.
2.5. ISOLTRAP Penning trap tandem
41
Such effect is treated by storing ions with a small amplitude of the axial motion. This is achieved
by properly cooling the ions. Moreover, the correction electrodes are introduced around the in
jection/ejection holes to compensate for the field imperfections. The imperfection of the electric
field leads to socalled massdependent shift. The deviation from a cyclotron frequency ratio
is induced through the mass difference between the reference ions and the ions of interest. The
effect can be summarized by:
(
δr
)m = 1.6 ∗ 10−10 (m − mref )/u
r
(2.20)
Figure 2.18. The drift of the magnetic field over 75 hours. Top picture represents residual
fluctuations and bottom picture represents the long drift of the magnetic field. Pictures are
taken from [Din16].
As one can notice, the measurement of the cyclotron frequency requires a very stable and
homogeneous magnetic field. Currently, at ISOLTRAP, two superconducting magnets are in
use, the homogeneity of which is in the order of ∆B/B < 10−7 . The temporal magnetic
stability (∆B/B)/∆T ≈ 28ppb/h. In figure 2.18 one can see the magnetic field drift over three
days. To improve the magnetic field properties, usually, oxygenfree highconductivity copper
and ceramics are used for the trap components. For example, the trap’s electrodes themselves
are made out of copper. The linear drift of the magnetic field presented in figure 2.18 taken
into account by linear interpolation of the magnetic field between two reference measurements.
There is a nonlinear part in the drift coming from the temperature and pressure fluctuations of
2.5. ISOLTRAP Penning trap tandem
42
the helium inside the cryostat. The nonlinear contribution has been quantified in and expressed
by:
δνc
)B = 6.35 ∗ 10−11 ∆T /min
(2.21)
νc
The alignment of the magnetic field axis in the ideal case shall be parallel to the trap’s axial
(
plane. However, there is always a presence of a tilt, which leads to the shift in frequencies. This
tilt was minimized during the first long shut down at CERN in 2013. The misalignment was as
small as 0.08 mrad. Usually, such effects are taken as residual systematic uncertainty and show
the limit of the spectrometer performance. The limit of the ISOLTRAP mass spectrometer’s
accuracy has been estimated in the same study. The scattering of the cyclotron ratios was still
greater than one (χ2 >1). The carbon cluster source was used to estimate the residual systematic
uncertainty of the ISOLTRAP setup which equals:
(
δr
)res = 8 ∗ 10−9
r
(2.22)
The last known systematic effect comes from the ionion interaction [K+ 03]. When many
ions are stored together in a small volume, the Coulomb force modifies the ion motions. If
the difference in mass between the ions of interest and the contamination ions is ample, the
contaminating species introduce a shift in the resonance frequency. If the resolving power is
sufficient to resolve both species, a double resonance is observed. However, each frequency
will be shifted towards a lower value. This shift is proportional to the number of the contami
nating ions. Such effect socalled zclass analysis is treated by recording less than five ions per
injection and using the MRToF and preparation Penning trap for the purification.
For the data analysis, the EVA software is wildly used among the trap experiments and the
uncertainty of a single measurement can be expressed as:
√
δri = (δri )2stat + (δri )2B
(2.23)
The measurement consists of a set of measurements which yield the weighted cyclotron
ratio:
∑
ri /(δri )2
r= ∑
1/(δri )2
(2.24)
with an associated error:
√
δr =
∑
1
1/(δri )2
(2.25)
The total uncertainty can be expressed:
δrtotal =
√
(δr)2 + (δrm )2 + (δrrest )2
(2.26)
2.5. ISOLTRAP Penning trap tandem
2.5.4
43
ToFICR data analysis
2
70
As
The mass measurement of
70
As+ was done with the Penning trap by applying the ToFICR
technique. A singleexcitationpulse ToFICR technique was used. Figure 2.16 (top) shows the
measured ToF spectrum of 70 As+ . Two ToFICR spectra were taken for two excitation times
of 0.1 and 1.2 seconds. The reference spectrum of 85 Rb ions was taken every time before and
after the mass measurement to account for changes in the magnetic field. The total statistical
uncertainty was 1.8 × 10−8 . The correction for the difference in the mass between the ion of
interest and the reference ion gave an error of 4.3 × 10−9 . To avoid ionion interaction shifts in
the trap, the number of ions detected per ejection was fixed to be ≤ 5. Furthermore, the residual
systematic uncertainty 8 × 10−9 [K+ 03] of the ISOLTRAP mass spectrometer was added in
quadrature. The frequency ratio is ricr =0.8235704146 with the total uncertainty 1.8 × 10−8 .
The mass of 70 As was known from Qvalues of 70 As(β + )70 Ge [B+ 63, S+ 17] and 70 Se(β + )70 As
decays [L+ 75, T+ 01]. Since 70 Ge is a stable nucleus with a wellknown mass, the Qvalue of
the former decay is more precisely known. It constrains the mass excess value of 70 As yielding
64340(50) keV. The new mass excess value measured by ToFICR technique is 36 times more
precise. Therefore, the reevaluated mass excess value is now entirely determined through this
ISOLTRAP measurement.
196
Hg
The mass measurement of 196 Hg2+ was done with the Penning trap by applying the ToFICR
technique with a singleexcitationpulse. Four ToFICR spectra were taken. One spectrum with
0.4 seconds and the other tree with 0.6 seconds of excitation time. The reference spectrum of
39
K ions was taken every time before and after the mass measurement. The total uncertainty
was 1.6 × 10−7 . The dominant part in this uncertainty was the correction for the mass differ
ence between 196 Hg2+ and the reference 39 K+ . This resulted in the massdependent systematic
uncertainty of 1.4 × 10−7 .
196
Hg2+ being doubly charged, the ToFICR resonance was con
taminated by chargeexchange products. As a result, the analysis was found to be dependent
on the halfrange chosen for the initial timeofflight cut, leading to an additional systematic
uncertainty of 2.1 × 10−7 . The residual systematic uncertainty of the ISOLTRAP mass spec
trometer of 8 × 10−9 [K+ 03] was added in quadrature as well. The number of ions detected
per ejection from the trap was fixed to ≤ 5. Finally, the frequency ratio is ricr =2.5147440758
2
This analysis has been published in [K+ 20] by the author of the thesis as a primary author. Therefore, some
pictures and text may be repeated
2.6. Summary
125
125
120
120
115
115
110
110
105
105
100
70
95
100
As+
90
T
rf. =
fit
data
1.2 s
0.82357038
85
80
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0
1
Excitation frequency-1302184.923 (Hz)
2
Number of measurement
(b)
−64400
−64600
CR
F-I
) 70
As
70
Se
(β
+
) 70
As
+
Se
(β
+
70
As
(β
+
As
(β
70
70
−64800
) 70
Ge
AME error
AME mean
experiments
ISOLTRAP
) 70
Ge
Mass excess (keV)
(a)
To
80
0.82357040
0.82357039
95
90
85
0.82357041
Ricr
Mean time of flight (μs)
44
(c)
Figure 2.19. Same as Figure 2.11 but for 70 As ions. The literature mass excess values of 70 As
are from [B+ 63, S+ 17, L+ 75, T+ 01].
with the total uncertainty 2.1 × 10−7 . The mass of stable 196 Hg was mostly determined from
deflectionvoltage ratios of 196 Hg and 198 Hg35 Cl molecule [K+ 80]. Combined with the Qvalue
of the 196 Au(β − )196 Hg decay [W+ 62] the evaluated mass excess value is 31825.9(29) keV. The
mass of 196 Hg was as well determined by ISOLTRAP in the past [S+ 01]. The new mass value
measured by ToFICR technique is in excellent agreement with all literature values and con
tributes 4% to the new reevaluated value.
2.6
Summary
The technical developments of the ISOLTRAP mass spectrometer have been presented. The
realignment of the horizontal beamline has been performed, overall misalignment of below 1
mm has been achieved. The performance of the buncher device was examined. The count rate
and the peak shape of the bunched beam do depend on the pressure of the He buffer gas. The
difference between mean values of Gaussian and EGH fit functions was checked independence
on the buffer gas pressure. The new device has been built to control the bunch forming. The
groundwork on the device has been made. The simulations in LTspice software and the experi
475
T
rf. =
196
450
45
600 ms
475
Hg2 +
2.5147442
450
425
425
400
400
375
2.5147440
Ricr
2.5147438
375
2.5147436
fit
data
350
−4
−2
0
2
weighted Ricr
350
weighted error
4
1
Excitation frequency-929432.161 (Hz)
2
3
4
Number of spectra
(b)
−31820
19
6
CR
F-I
CR
F-I
To
) 196
Hg
−
Au
(β
−31860
AME error
AME mean
experiments
ISOLTRAP
19
6
−31840
Hg
- 198
Hg 3
5
Cl
Mass excess (keV)
(a)
To
Mean time of flight (μs)
2.6. Summary
(c)
Figure 2.20. Same as Figure 2.11 but for 196 Hg ions. The literature mass excess values of 196 Hg
are taken from [K+ 80, W+ 62, S+ 01].
ment results with a frequency generator plus an amplifier support each other. The performance
of the device as part of the ISOLTRAP mass spectrometer was checked. With this device, the
online tuning of the resonance frequency has been enabled.
The masses of the shortlived 49,50 Sc, 70 As, and 73 Br as well as stable 196 Hg nuclides have
been investigated by the versatile mass spectrometer ISOLTRAP. The results are summarized in
figure 2.21 and table 2.2. Overall, the new experimental values agree with the literature values
within the reported error bars. The uncertainties were reduced by factors 1.6 and 35 for 50 Sc and
70
As nuclei, respectively. These two isotopes have the biggest impact on the reevaluated values
in the updated AME. The agreement of the new data obtained with the MRToF MS confirms
the reliability of this complementary mass measurement technique. Considering the achieved
small uncertainties, all isotopes addressed in this work can be used as online references in future
experiments at ISOLTRAP or elsewhere.
2.6. Summary
MISOLTRAP - MAME (keV)
46
50
50
40
40
30
30
20
20
10
10
0
0
−10
−10
−20
−20
−30
−30
AME16 weighted error
base line
ISOLTRAP
−40
−50
−40
−50
−60
−60
49
Sc
50
Sc
70
As
73
Br
196
Hg
Figure 2.21. Comparison of the mass excess values determined in this work (red symbols) and
the literature values taken from AME2016 [H+ 17]. The gray shaded background represents the
AME2016 uncertainties.
2.6. Summary
47
Table 2.2. Summary of the Ricr and CT oF ratios as well as mass excess values of 49,50 Sc, 70 As,
73
Br and 196 Hg nuclides. Reference masses are taken from AME2016 [H+ 17] and T1/2 values
are from NUBASE2016 [WA+ 17]. The last column provides the significance of the ISOLTRAP
results in the reevaluated mass excess values if included into AME.
Mass excess (keV)
Ricr , CT oF
This work
AME2016
reevaluated
Sign.
49
Sc
57.2 min
49
Ti, 39 K
CT oF =0.500203363(432)
46564.7(42)
46561.3(27)
46562.4(23)
29 %
50
Sc
102.5 sec
50
Ti, 39 K
As
52.6 min
Nuclide
70
73
Br
196
Hg
T1/2
3.4 min
stable
References
CT oF =0.500633802(840)
44546.5(91)
44547(15)
44546.8(78)
73 %
Rb
ricr =0.823570415(18)
64334.0(14)
64340(50)
64334.0 (14)
100 %
Se, 85 Rb
CT oF =0.49957242(163)
63635(19)
63647(7)
63645.8(67)
14 %
ricr =2.514744076(209)
31822(15)
31825.9(29)
31825.8(29)
4%
85
73
39
K
Chapter 3
The shape coexistence in neutronrich
krypton isotopic chain
At first, the nuclear deformation was observed in the rareearth region. The γray emission
spectra were showing the rotationallike behavior. During the γspectroscopy study of fission
fragments of 252 Cf, similar behavior has been noticed at A=100. Furthermore, the energies of
the first 2+ excited state indicated the nuclear deformation. The A=100 region was identified
as a new region of deformation [Joh65]. Since then, many experimental and theoretical efforts
were addressed to study this region. It has been studied in great detail for the molybdenum,
niobium, zirconium, strontium, yttrium, and rubidium isotopic chains. All the experimental
observables indicate the shape deformation, especially in zirconium and yttrium cases. The
krypton isotopic chain attracted much scientific attention over the last ten years because of the
unobservable shape transition at N=60.
3.1
A=100 region
Many mass spectrometers extensively studied the A=100 region. The direct observation of ir
regularities of two neutron separation energies was noticed by the JYFLTRAP Penning trap
system at the IGISOL facility. There was evidence of a strong correlation between the nuclear
deformation and the two neutron separation energies in neutronrich strontium, zirconium, and
niobium isotopic chains [H+ 06]. A year later, this experimental group reported even more
noticeable changes in the yttrium isotopic chain and smooth irregularities in the molybdenum
isotopes [H+ 07] [R+ 07]. The TITAN Penning trap experiment contributed to measurements
of masses of rubidium and strontium isotopes and also observed irregularities in two neutron
separation energies [K+ 16] [S+ 12]. The endpoint of the mass measurements in rubidium and
strontium isotopic chains was done by the ISOLTRAP mass spectrometer, where confirmation
of the deformation was shown until the neutron number N=65 [DR+ 17]. However, one ques
49
3.1. A=100 region
50
tion remained open. Where is the lower Zbound of the deformation region? Is it the krypton
isotopic chain? And if so, at which neutron number? In 2010 the mass measurement from
the ISOLTRAP mass spectrometer got the mass of
97
Kr, and no irregularities in two neutron
+
separation energy were found [N 10]. It was established as a critical point boundary [A+ 12].
AME 2016
ISOLTRAP
JYFLTRAP
TITAN
42Mo
41Nb
Zr
39Y 40
38Sr
37Rb
36Kr
58
60
62
Neutron Number N
64
<r2> (fm2)
14
13
12
11
10
9
8
7
42Mo
7
5
41Nb
40Zr
39Y
4
38Sr
3
37Rb
6
36Kr
1
66
(a)
8
2
57
58
59
60
61
Neutron number N
62
63
(b)
1800
42Mo
40Zr
38Sr
E(2 + ) (keV)
1600
1400
1200
36Kr
1000
800
600
400
3.25
3.00
E(4 + )/E(2 + )
S2n (MeV)
Authors conclude in [M+ 13a] that the nuclear deformation may start at 98 Kr.
200
2.75
2.50
Neutron number
(c)
40Zr
2.00
1.75
1.50
52 53 54 55 56 57 58 59 60 61 62 63 64
42Mo
2.25
38Sr
36Kr
52 53 54 55 56 57 58 59 60 61 62 63
Neutron number
(d)
Figure 3.1. (a) the two neutron separation energy trend [H+ 17]. (b) the evolution of the
charge radii in A=100 region [A+ 13]. (c) the evolution of the energy of the first 2+ excited
state [Lab]. (d) the trends of the ratio between 4+ and 2+ excited states.
The change in the shape of a nucleus is experimentally observed through the evolution of the
meansquare charge radii [A+ 13], extracted from the isotopeshift measurements and hyperfine
structure measurements [K+ 95a, T+ 81, B+ 90b, L+ 91, C+ 07, C+ 02]. The meansquare charge
radii show a rising trend at N=60 for zirconium, yttrium, strontium, and rubidium isotopes,
which is a clear indication of the deformation [C+ 97, F+ 03, C+ 16]. Unlikely, the charge radii
for 97 Kr and 98 Kr are not yet available.
Another experimental evidence is the reduced energies of the first 2+ excited states, which
have been experimentally observed for strontium, zirconium, and molybdenum isotopes at
N=60. There were contradictions between mass measurements [M+ 13a] and γray spectroscopy
3.2. Data analysis
51
measurements [M+ 09] on the 96 Kr isotope. Authors of [M+ 09] measured the energy of 2+ state
and concluded the deformation of this isotope. However, newer results from γray spectroscopy
and Coulomb excitation experiments do not observe the energy drop of 2+ excited state at 96 Kr
isotopes [D+ 17, A+ 12]. The γray spectroscopy research [F+ 17b] up to
100
Kr isotope spot
lights the drop at 98 Kr. Moreover, the ratio between 4+ and 2+ excited states indicates that the
deformation starts at N=62 in the krypton isotopic chain.
3.2
Data analysis
Two different campaigns have taken place at CERN in 2015 and 2017. Radioactive ions were
produced and delivered by the ISOLDE facility. In both cases, krypton isotopes were produced
by bombarding the UCx target with 1.4 GeV proton pulses and an average intensity of 1.3 µA.
Nuclides diffused and effused out of the target and were ionized by a cold plasma ion source
in VADIS mode. The ToFICR mass spectrometry was performed on the 97 Kr isotope in 2015.
The MRToF mass spectrometry was performed on 96−98 Kr isotopes in 2017.
3.2.1
97
ToFICR
Kr
Three spectra of
97
Kr isotope were taken in 2015. Two spectra with 50 ms excitation time
and one TOFICR spectrum of Ramsey scheme with excitation of 104010 ms. The Ramsey
resonance is shown in figure 2.17(b) and a single pulse ToFICR resonance is shown in figure
3.2(a). The 39 K ions were chosen to make a linear extrapolation of the drift of the magnetic
field. The total statistical uncertainty is equal to 7.2 × 10−7 . The correction for the difference
in mass between 39 K and 97 Kr ions is equal to 5 × 10−8 . The last systematic uncertainty is the
residual uncertainty of the ISOLTRAP mass spectrometer which is equal to 8 × 10−9 . As one
can see from figure 3.2(b) the Ricr ratio obtained by Ramsey type of excitation (third point on
the figure) agrees well with the single pulse ToFICR scheme but is two times more precise.
As a check of Coulomb interactions inside the Penning trap, the histogram in figure 3.2(c) was
plotted. As one can see, most ejections have 0 number of ions, and only a few have more than
three ions per ejection. That means that ”z”class analysis is unneeded because mostly it was
one ion per ejection. The new mass excess value (figure 3.4(d)) agrees well with previous
mass measurement which was done in 2010 in the preparation Penning trap applying different
technique [N+ 10]. Moreover, it agrees well with MRToF mass spectrometry value provided
later in work. Final mass excess value is listed in table 3.1.
3.2. Data analysis
450
T
rf. =
weighted Ricr
50 ms
2.488211
450
weighted error
2.488210
400
400
350
350
300
300
250
97
fit
data
Kr+
−20
0
Ricr
Mean time of flight (μs)
52
2.488209
2.488208
2.488207
2.488206
250
20
1
2
Excitation frequency-939343.347 (Hz)
3
Number of spectra
(b)
(a)
1250
1000
1000
750
750
500
500
250
250
Counts
1250
0
0
1
2
3
4
0
Number of ions/ejection
(c)
Figure 3.2. (a) the single pulse ToFICR resonance of 97 Kr ions. (b) the spread of the Ricr
ratio of 97 Kr ions. (c) the dependence of counts on number of ions per ejection.
3.2.2
96
MRToF
Kr
The MRToF data was recorded in 2017. For the mass determination of 96 Kr ions, 150 files were
recorded with 201 ions of interest in total. Files were summed and grouped into 12 data sets.
The number of revolutions was the same in all spectra and equaled 1000 revolutions. The 96 Kr
ions were measured as doublecharged ions. As an online reference, 48 SO molecule was chosen,
and 85 Rb as an offline reference. It was fitted by Gaussian and EGH fit functions based on the
3.2. Data analysis
53
102
96
Kr2 +
101
101
100
ToF, ns
102
Counts/0.8 ns
775
fit
data
S O
32 16
16.687502
785
790
795
100
16.686502
780
5
16.688502
Time of flight (ms)
(a)
15
Ions/shot
20
25
(b)
−53040
Mass excess (keV)
0.0000015
Ctofi − < Ctof >
10
0.0000010
0.0000005
0.0000000
−0.0000005
1
2
3
4
5
6
7
8
9
10
11
AME error
AME mean
−53060
−53080
−53100
−53120
96
−53140
ISOLTRAP
ToF-ICR
2010
12
Number of measurement
Kr
Titan
MR-ToF
2020
ISOLTRAP
MR-ToF
2020
(d)
(c)
Figure 3.3. (a) the time of flight spectrum of
96
Kr ions. (b) the time of flight difference
between two peaks vs the number of ions per ejection. Every color represents different set of
files. (c) the difference between the individual Ctof ratios and weighted CT oF ratio of
96
Kr
ions. (d) the comparison between experimental and literature values from AME16 [H+ 17]
maximum likelihood estimation and yielded the systematic uncertainty σCT oF ,pdf =2.8 × 10−7
coming from an unknown peak shape. The Gaussian fit function was used for computing the
CT oF =0.499858112. The weighted statistical uncertainty amounted to 6.7 × 10−8 . The range
study was performed. The optimum range was 70 ns for
96
Kr ions and reference ions. The
σCT oF ,window systematic error was not considered in this analysis. The other study on the MR
ToF systematics was performed instead. The idea was to see how the time of flight difference
between two peaks depends on the number of ions present in the device. Figure 3.3 (b) repre
sents this dependency. As one can see the difference between the two peaks is getting smaller
with more ions in the shot. The systematic error coming from the number of ions present in the
device can be expressed:
σCT oF ,z−class = CT oF (min + next, ions/shot) − CT oF (max, ions/shot)
(3.1)
CT oF (min + next, ions/shot) : calculated for the preminimum number of ions/shot.
3.2. Data analysis
54
It yielded 6.2 × 10−8 of the systematic uncertainty. The final CT oF values which were used
to compute CT oF were taken at maximum ions per shot. The total uncertainty 3.2 × 10−7 can
be expressed as square root of the sum of contributions:
√
σCT oF ,total = σC2 T oF ,stat + σC2 T oF ,pdf + σC2 T oF ,z−class
(3.2)
The new mass value agrees well (see figure 3.3 (d)) with ToFICR measurement from
ISOLTRAP and with MRToF mass measurement from TITAN experiment based at the TRI
UMF facility [M+ 20]. The final results are listed in table 3.1.
Kr
Counts/0.8 ns
97
194
fit
data
Mo+
1845
Hg2 +
101
101
97
Kr+
ToF, ns
97
1850
1855
1860
100
100
1865
8.325002
8.326002
8.327002
10
8.328002
Time of flight (ms)
(a)
12
14
16
18
Ions/shot
20
22
(b)
Mass excess (keV)
−47300
0.000010
Ctofi − < Ctof >
24
0.000005
0.000000
−0.000005
AME error
AME mean
−47350
97
Kr
−47400
−47450
−47500
−47550
1
2
ISOLTRAP
Prep. Trap
2010
3
Number of measurement
(c)
(d)
Figure 3.4. (a) the time of flight spectrum of
97
ISOLTRAP
MR-ToF
2020
ISOLTRAP
ToF-ICR
2020
Kr ions. (b) the time of flight difference
between two peaks vs the number of ions per ejection. Every color represents different set of
files. (c) the difference between the individual Ctof ratios and weighted CT oF ratio of
97
Kr
ions. (d) the comparison between experimental and literature values from AME16 [H+ 17]
A limited statistic is available for 97 Kr isotope. Only four files were recorded with 19 ions
in total at 350 revolutions. The spectra were fitted by Gaussian and EGH fit functions.
97
Mo
was taken as an online reference and 85 Rb as an offline. It was fitted by Gaussian, and EGH fit
3.2. Data analysis
55
functions based on the maximum likelihood estimation and yielded the systematic uncertainty
σCT oF ,pdf =1.8 × 10−6 coming from an unknown peak shape. The Gaussian fit function was
used for computing the CT oF =0.503474279. The weighted statistical uncertainty amounted to
3.6 × 10−6 . The range study was performed. The optimum is 70 ns for references and 150 ns
for the ion of interest. However, it was not included in the systematic uncertainty. The same
analysis was performed as for the 96 Kr isotope. The trend of the differences shows the same
behavior as for 96 Kr ions. It resulted in the 3.9 × 10−6 of systematical uncertainty coming from
the ”zclass” analysis. The total uncertainty is equal to 5.6 × 10−6 . The comparison of results
between different techniques is shown in figure 3.4(d). As one can see the result is in good
agreement with previous mass measurement from 2010 and also with a new mass excess value
obtained by ToFICR technique. The final results are listed in table 3.1.
98
Kr
The mass of
98
Kr isotopes was not previously measured. It is an exotic isotope with a tiny
production. There are 182 files with 37 ions of interest in total. To make sure that observed
ions are krypton ions one spectrum was taken without any protons on the target. That led to the
disappearance of krypton ions in the ToF spectrum. The spectra were grouped in 9 files and were
fitted by Gaussian and EGH fit functions with maximum likelihood estimation. The systematic
uncertainty is σCT oF ,pdf =3.5 × 10−6 and comes from unknown peak shape. Since the count rate
of
98
Kr was low, it had no sense to fit the spectra at small ranges and 150 ns was chosen for
all files.
98
Mo was chosen as an online reference. The optimal range for Gaussian fit of 98 Mo
was chosen 60 ns for all spectra.
85
Rb was chosen as an offline reference with 60 ns range. The
Gaussian fit function was used for computing the CT oF =0.503501959. The weighted statistical
uncertainty amounted to 3.9 × 10−6 . The actual MRToF spectrum is shown in figure 3.5(a).
To see how much of the data is included by equation 3.1 the histogram 3.5(c) was built. It
shows the difference between individual differences and the difference at maximum ions per
injection. The mean value of the distribution is equal to 0.851 ns. The exponential fit is plotted
to show the trend and red counts are those which have been excluded. As one can see, most
of the deviations are included by the applied equation 3.1. Overall this analysis resulted in the
6.0 × 10−6 of systematical uncertainty. The total uncertainty is equal to 7.98 × 10−6 . The final
results are listed in table 3.1.
The results of the data analysis can be summarised in figure 3.6. The observed trend of two
neutron separation energy does not indicate a clear shape transition in the krypton chain up to
N=62.
3.3. Theoretical approaches
56
196
98
5600
5575
Hg2 +
Mo+
102
102
101
98
ToF, ns
Counts/0.8 ns
fit
data
101
Kr+
4730
4720
415
433
213
232
254
271
285
366
382
392
4060
100
4040
100
7.177042
7.178042
7.179042
5
4730
4720
4060
4040
10
7.180042
20
30
40
Ions/shot
Time of flight (ms)
(a)
5600
5575
(b)
0.00004
Ctofi − < Ctof >
12
Counts
0.00005
Exponent
mean=0.851 ns
Data
14
10
8
6
4
2
0
0.00003
0.00002
0.00001
0.00000
−0.00001
−0.00002
−0.00003
0.02 3.02 6.02 9.02 12.02 15.02 18.02 21.02 24.02
Offset, ns
1
2
3
4
5
6
7
8
9
Number of measurement
(d)
(c)
Figure 3.5. (a) the time of flight spectrum of
98
Kr ions. (b) the time of flight difference
between two peaks vs the number of ions per ejection. Every color represents different set of
files. (c) the histogram of the differences between ∆T oFi and ∆T oFmax,ions/shot for details
see text. (d) the difference between of the individual Ctof ratios and weighted CT oF ratio of
98
Kr ions.
3.3
Theoretical approaches
The main aim of theoretical nuclear models is to describe the experimentally observed properties
of nuclei. The nucleus consists of protons and neutrons, which also have compound nature (they
consist of quarks). All together, they form a complex system, and understanding all interactions
within the system is a challenging task. Modern theoretical models can describe some nuclear
properties. However, there is no such model that can describe everything at once.
If one wants to find the mass of the nuclei, the naive approach would be to sum up the mass
of all protons and neutrons constituting the nuclei. With developments of mass spectrometers
[BL13], scientists quickly realized that this is not the case and the mass of a nucleus is less than
the sum of the mass of its constituents. This statement has been expressed by equation 1.3. This
mass difference was identified as the energy that bounded the nucleus and was named binding
energy.
3.3. Theoretical approaches
57
18
S2n (MeV)
16
14
42Mo
41Nb
40Zr
39Y
38Sr
12
10
8
52
54
56
58
60
Neutron Number N
62
36Kr
37Rb
64
Figure 3.6. The evolution of the two neutron separation energy in molybdenum, niobium,
zirconium, strontium, yttrium, rubidium and krypton isotopic chains. The literature values taken
from AME2016 [H+ 17]. The new experimental two neutron separation energies obtained by
ISOLTRAP mass spectrometer are colored in red.
Figure 3.7 summarizes the average binding energy per nucleon as a function of nucleon
number. First, the rapid increase can be spotlighted in the light nuclei region until sodium.
Each additional nucleon is attracted by other nearby nucleons, and thus more tightly bound to
the whole system by the shorthanded nuclear force. Second, the steady decrease of the binding
energy after iron can be explained as nuclei have become big enough that the nuclear force is
almost equal to the electromagnetic forces between protons.
3.3.1
Liquid drop model
The first attempts to describe the nucleus and observed trends of the binding energy were done
by Gamow and further developed by Bohr and Wheeler [GR30]. They assumed that the nucleus
is a drop of incompressible fluid of very high density, held together by the nuclear force. In 1935
German physicist Carl Friedrich von Weizsäcker proposed his famous formula [Wei35] based
on liquid drop theory and empirical measurements:
Bnuc = avol A − asurf A2/3 − ac
Z(Z − 1)
(N − Z)2
+ δ(N, Z)
−
a
I
A1/3
A
(3.3)
The first term is the volume term, and it is directly proportional to the sum of nucleon masses.
This term represents shorthanded nuclear interaction meaning that nucleon can interact only
with its neighbors. The second term is the surface term. The term can be associated with the
surface tension of the liquid. This is a correction for the volume term because nucleons on the
surface have fewer neighbors. The third term represents the Coulomb force. The fourth term
3.3. Theoretical approaches
58
Figure 3.7. The dependence of the average binding energy per nucleon as a function of the
number of nucleons in the nucleus.
considers the asymmetry energy coming from the Pauli principle. The last term considers the
oddeven staggering effect through the function δ which depends on the number of protons Z
and the number of neutrons N and their tendency to form pairs.
The liquid drop model is a macroscopic approach to describe the nucleus. The advantage
of the liquid drop model is that it describes the binding energy trend shown in figure 3.7. Also,
it provides a good description of the fission mechanism and nuclear reactions at low energies.
However, it falls short of describing peaks at He4 , C12 , O16 б socalled magic numbers where
the gain of the binding energy appears and the nuclear deformation [M+ 95].
In order to obtain the avol , asurf , ac and aI constants one have to fit the model to the entire
nuclear chart. The latest FRDM2012 model [M+ 16b] which also includes some microscopic
contributions shows rather well agreement with the experimental data collected in AME2003.
The rootmeansquare(RMS) deviation is the quantity that shows the ability of the mass model
to describe the experimental values. For the FRDM2012 the RMS is equal to 560 keV. This
calculation is based on 2169 nuclei of the nuclear database. Figure 3.8 shows the trends of two
neutron separation energy in A=100 region based on the experimental values and the values
calculated by this model. The model indicates some nuclear structure effects at N=56 and N=60.
The experimental trends (black dots) represent the N=56 subshell closure and the deformation
at N=60. The model, indeed, is not sensitive to separate these two effects well. Especially
3.3. Theoretical approaches
59
in strontium and zirconium chains model mixes these two effects which lead to the average
increase of the two neutron separation energy between N=56 and N=60.
18
FRDM2012
AME2016
ISOLTRAP
S2n (MeV)
16
14
12
42Mo
10
40Zr
38Sr
8
36Kr
52
54
56
58
60
Neutron Number N
62
64
Figure 3.8. The evolution of the two neutron separation energy in neutronrich krypton, stron
tium, zirconium and molybdenum isotopic chains. The black two neutron separation energy
trends are taken from AME2016 [H+ 17]. Red two neutron separation energies calculated from
the new
96−98
Kr mass measurements. The blue values are taken from FRDM2012 [M+ 16b].
For the visibility error bars are omitted.
Overall the predictive power of the modern FRDM2012 model is not sufficient to conclude
the effects discussed in this work.
3.3.2
Shell model
The observed periodicity in the nuclear binding energy was quickly rethought with the shell
structure of an atom. Mayer and Haxel proposed the shell structure of a nucleus in the fifties. In
this model, the neutrons and protons fulfill independent shells, and the shell closure corresponds
to the fulfilled states [May48].
In nuclear structure calculations, one has to solve the Schrodinger equation:
Ĥ|ψ >= E|ψ >
with Hamiltonian defined as:
(3.4)
3.3. Theoretical approaches
60
A
A
∑
∑
p2i
Ĥ = T̂ + V̂ =
+
Vij
2mi
i=1
i<j
(3.5)
The first term is kinetic energy, and the second term is potential energy considering only
twobody interactions. For simplicity, the higher orders of interaction have been omitted. It
is challenging to find the exact solution when calculating the interactions in the Hamiltonian
by summing over all states. It is the famous manybody problem [Rin00]. It has been solved
strictly only for the lightest nuclei [P+ 01]. In most models only twobody interactions are
considered. In the shell model the system of twobody interacting bodies is replaced by a system
of independent particles interacting in a wellchosen average potential. Therefore, the potential
term in the Hamiltonian can be written as:
V̂ = V̂0 + V̂rest
(3.6)
In the first independent particle model the rest interaction was neglected. The shell model
treated the nucleus as a system of independent particles orbiting in the spherical potential. The
model was built on a simple harmonic oscillator (S.H.O) [May48] for neutrons and protons
separately:
1
V0 (⃗r) = h̄ω 2⃗r2
(3.7)
2
It reproduces well the energy levels up to magic number 20. The S.H.O. potential has two
weak points: the nucleons are infinitely bound at a large radius, and each level is actually a
degenerate multiplet, unresolved using the oscillator’s quantum number. The second attempt
was a slight modification of the S.H.O potential to a WoodsSaxon [C+ 87] shape:
V0 (⃗r) =
V0
1 + exp ⃗r−R
a
V0
: the depth of the potential
⃗r
: the distance from the center of the potential
(3.8)
R = 1.2A1/3 : the radius
a
: is a measure of how rapidly the potential falls to zero
Even this modification could not replicate all the magic numbers. So the last modifica
tion takes into account the spinorbit interaction. After applying the spinorbit term into the
singleparticle Hamiltonian the success of the shell model was confirmed. The potential can
be described by equation 3.9. For the visualization all the potentials and corresponding shell
structure are summarized in figure 3.9.
3.3. Theoretical approaches
61
dV0 (⃗r)⃗
l⃗s
V0 (⃗l, ⃗s) = −Vls
d⃗r
(3.9)
Vls : the strength constant
⃗l : the orbital angular momentum
⃗s : the intrinsic spin of the nucleon
Figure 3.9. The shell ordering with different potentials [MJ55]
.
For simplifying the calculations the restriction on the number of active nucleons outside of a
certain core is often applied in shell model calculations. In most cases the inert core corresponds
to a doubly magic nucleus and the orbits are forced to be full. At the same time, the valence
nucleons are those which occupy valence orbits. The deformation at A=100 firstly has been
studied via shell model calculations on zirconium isotopes. An inert core of 88 Sr was assumed
for the shell model calculation of even 96−100 Zr isotopes. Authors determined the deformation
as the results of the protonneutron correlation between 1g9/2 orbital for protons and 1g7/2 , 3s1/2
and 2d3/2 orbitals for neutron [FP79]. The other method to keep the sizes of the Hamiltonian
matrices manageable is to use Monte Carlo mathematical method. The largescale Monte Carlo
3.3. Theoretical approaches
62
Shell Model (MCSM) calculations on eveneven Zr isotopes have been studied [T+ 16]. Both
methods reproduced the experimentally observed trend of the first 0+ and 2+ excited states of
even zirconium isotopes. Finally, there are other methods studying the region the projected
shell model, discussed in detail in [V+ 08].
The experimentally measured masses of neutron rich
96−98
Kr isotopes are a clear call to
perform shell model calculations. So far, only calculations for zirconium and strontium isotopes
are well presented in the literature. A quantum phase transition has been shown to occur in
zirconium and strontium neutronrich isotopes.
3.3.3
Selfconsistent mean field model
The nuclear shell model assumes that the residual interactions are small compared to the mean
field. And this is true for the reproduction of the magic periodicity of the binding energy.
However, the energy spacing between subshells can have values close to the residual interaction
energy scale. Although residual interactions make a relatively small contribution to the total
binding energy of the nucleus, they largely determine the bond of valence nucleons and the
spin, magnetic, and quadrupole moments of nuclei. However, how to divide the singleparticle
information and twobody interactions in the total sum?
A special method was developed by Hartree and Fock (HF) [Rin00] to replace the exact
Hamiltonian 3.5 by its twobody approximation. In this method the potential for nucleons is
computed from the nucleonic wave functions. In the following only the main results of the
formalism will be highlighted [Rin00].
The methods starts from the general variational principle. It has been shown that exact
Schrodinger equation 3.4 is equivalent to the variational equation:
δE[ψ] = 0
E[ψ] =
(3.10)
< ψ|Ĥ|ψ >
< ψ|ψ >
The wave functions |ψ > are restricted to be trial and simple. From this variational method
we do not know the exact wave function ψ, we can find only an approximation |ϕ >. This
approximation corresponds to the variationaly optimized singleparticle operator with Hamil
tonian of the spherical harmonic oscillator:
Ĥ =
∑
i,j
tij a†i aj +
1 ∑
ν ij,kl a†i a†j al ak
4 i,j,k,l
ti,j =< i|t̂|j >
(3.11)
3.3. Theoretical approaches
63
ν ij,kl =< ij|ν|kl > − < ij|ν|lk >
The first and the second term are the matrix elements of the kineticenergy and twobody
potential between states. a†i and ai represents the creation and annihilation operators for the
state i. The latter includes the Pauli principle through the anticommutation relations. The Slater
determinant |ϕ > can be expressed as product of creation operators of A particles corresponding
to the arbitrary but orthogonal single particle wave functions ϕ:
|ϕ >=
A
∏
a†i |− >
(3.12)
i=1
The expectation value of the energy of the unknown state |ϕ > can be expressed as a func
tional of the singleparticle density matrix ρij =< ϕ|a†j ai |ϕ >. The minimization of the energy
with respect to the Slater determinant is equal to the minimization with respect to ρ when ρ2 = ρ:
δ
< ϕ|Ĥ|ϕ >= 0
δϕ
(3.13)
∑
δ
1 ∑
E[ρ] =
tij ρij +
ρki ν ij,kl ρlj = 0
δρ
2
i,j
i,j,k,l
which leads to the HartreeFock equations:
∑
hij Djk = ϵk Dik
(3.14)
j
∑
hij = tij + Γij = tij +
l, kν ik,jl ρlk
∑
b†j =
Dij a†i
i
where hij is a single particle Hamiltonian and Dij is the transformation form for the initial
singleparticle to the eigenbasis of ĥ. The singleparticle Hamiltonian contains the kinetic en
ergy term and selfconsistent field Γ. It is a onebody field and the average over all twobody
interaction. By plugging the transformation D to the singleparticle density matrix one can get
a selfconsistent equations for the transformation from the initial to the HF basis:
∑
j
(tij +
A
∑∑
∗
ν ik,jl Dlm Dkm
)Djk = ϵk Dik
(3.15)
l,k m=1
By solving these equations iteratively one can get the resulting HartreeFock field Γ which
is a onebody approximation of the given twobody interaction νij . Usually, parameters of the
transformation Dij are found from the fit to the experimental data.
The method can reproduce the ground state properties of spherical nuclei. However, for
middlemass nuclei, the pairing interaction between nucleons plays a significant role. The HF
3.3. Theoretical approaches
64
approach takes into account only the longrange part of the nucleonnucleon interaction. For the
shortrange interaction, the Bardeen–Cooper–Schrieffer correction has been applied [BZ13].
HartreeFockBogoliubov(HFB) method [B+ 03] also is widely used. However, the detailed
description of the HFB approach is outside of the scope of this work.
Skyrme Interaction
The attempts were made to derive νij effective interactions directly from the bare nucleon
nucleon interaction. The breakthrough came when the connection to the bare nucleonnucleon
force was abandoned, and effective interactions tailored to meanfield calculations were directly
adjusted to the observables of the finite nuclei.
One of such interaction is zerorange Skyrme interaction written for twobody part in the
coordinate representation as:
νij = t0 (1 + x0 P σ )δ(⃗rij )+
(3.16)
1
+ t1 [δ(⃗rij )⃗k 2 + ⃗k 2 δ(⃗rij )] + +t2⃗kδ(⃗rij )⃗k+
2
+iW0 (⃗σ 1 + ⃗σ 2 )⃗k × δ(⃗rij )⃗k
The first term is a pure zerorange delta force with a spin exchange. The second and third
term simulates an effective range and the last term represents the spinorbit term [Rin00]. The
five constant t0 , t1 , t2 , t3 , x0 and W0 are adjusted to the experimental binding energies.
Similar to the FRDM2012 model, the HFB method with Skyrme interaction was applied
to fit the global nuclear chart and led to the development of HFB31 model [G+ 13a]. The
total binding energy can be given, as an example, as the sum of the kinetic energy, the Skyrme
energy functional (the effective interaction between particles), the Coulomb energy, the pairing
energy and the correction for the spurious motion (as in formula 1.4). The HFB31 model has
an RMS deviation of 576 keV based on 2353 nuclei of the global nuclear chart. The figure
3.10 shows a comparison between experimental data and the global calculations. First of all
the model on average reproduce the experimental trend. However, it is absolutely inconsistent
with the observed experimental trends (black dots) representing the N=56 subshell closure and
the deformation at N=60 in the region. The model instead averages the general trend and is not
sensitive to the deviations.
In the frame of this work the considered parametrizations of the Skyrme force is SLy4
+
[S 03]. This parametrization depends on three parameters, the strength of the pairing inter
action and how the pairing strength is distributed as a function of the nuclear density. Conse
quently, different types of pairing can be obtained depends on the distribution of the pairing
3.3. Theoretical approaches
65
18
HFB31
AME2016
ISOLTRAP
S2n (MeV)
16
14
12
42Mo
10
40Zr
38Sr
8
6
36Kr
52
54
56
58
60
Neutron Number N
62
64
Figure 3.10. The evolution of the two neutron separation energy in neutronrich krypton, stron
tium, zirconium and molybdenum isotopic chains. The blackcolored two neutron separation
energies are taken from AME2016 [H+ 17]. Red two neutron separation energies calculated
from the new
96−98
Kr mass measurements. Blue color corresponds to the HFB31 calcula
tions [G+ 13a]. For the visibility error bars are omitted.
strength, as an example, two extreme cases are the volumetype pairing and the surface type
pairing can be parametrized. The SLy4 model was adjusted only to five doubly magic nuclei.
As we noticed from equation 3.13(2) the binding energy is a function of the nuclear density.
This concept can be linked to density functional theory(DFT) [Rin00]. The second parametriza
tion is UNDEF0 which is also an optimization of a nuclear energy density of Skyrme type in
the frame of extension of the DFT theory to a nuclear manybody problem. The particle inter
actions are optimized simultaneously. This model includes a description of both spherical and
deformed nuclei and a new modelbased, derivativefree optimization algorithm [K+ 10].
The model which describes the region of interest the best is UNEDF0. One can see in figure
3.11(b) an excellent agreement between the experimental data and the model in the strontium
isotopic chain. First of all, the model predicts the nuclear subshell closure at N=56 in the
strontium isotopic chain. Second, it also does predict the deformation at N=60 for strontium
isotopes. This model predicts a weaker shape transition in the krypton isotopic chain which
agrees with observed experimental trends. The SLy4 model does not predict trends of the entire
3.3. Theoretical approaches
66
18
SLy4
AME2016
ISOLTRAP
S2n (MeV)
16
14
12
42Mo
10
40Zr
38Sr
8
52
54
56
58
60
Neutron Number N
62
64
36Kr
(a)
18
UNEDF0
UNEDF1
AME2016
ISOLTRAP
S2n (MeV)
16
14
12
42Mo
10
40Zr
38Sr
8
6
36Kr
52
54
56
58
60
Neutron Number N
62
64
(b)
Figure 3.11. (a) The evolution of the two neutron separation energy in neutronrich krypton,
strontium, zirconium and molybdenum isotopic chains. The black colored two neutron separa
tion energies are taken from AME2020. Red two neutron separation energies calculated from
the new 96−98 Kr mass measurements. Blue color corresponds to the SLy4 calculations [S+ 03].
(b) The evolution of the two neutron separation energy in neutronrich krypton, strontium,
zirconium and molybdenum isotopic chains. The black colored two neutron separation ener
gies are taken from AME2016 [H+ 17]. Red two neutron separation energies calculated from
the new 96−98 Kr mass measurements. Blue/green color corresponds to the UNEDF0/1 calcula
tions [K+ 10]. For the visibility error bars are omitted.
3.3. Theoretical approaches
67
region. For visibility the theoretical calculations have a 1 MeV offset. However, the model
seems to globally consider nucleonnucleon interaction because it indicates nuclear structure
but with a shift.
Despite the success of interest of Skyrme interaction the models are not able to adequately
describe the interaction in the nuclei. As a result in figures 3.10 and 3.11(a) the HFB31 and
SLy4 models do not efficiently separate the subshell closure at N=56 and the deformation at
N=60. The UNEDF0 model is adjusted to spherical and deformed nuclei and this can be a
reason for a good agreement between this model and the experiment.
Gogny Interaction
In 1970 Gogny suggested adding a densitydependent interaction as well as a spinorbit term.
He subsequently proposed a finiterange interaction which is now called the Gogny force.
νij =
2
∑
e−⃗rij /µi (Wi + Bi P σ − Hi P τ − Mi P σ P τ )+
2
2
(3.17)
i=1
+iW0 (⃗σ 1 + ⃗σ 2 )⃗k × δ(⃗rij )⃗k+
1
+t3 (1 + P σ )δ(⃗rij ))ρ1/3 ( ⃗rij )
2
The first term represents the central finite range part of the force, the second term is the spin
orbit term having a zero range, the last part represents the densitydependent term [Rin00].
The Gogny D1S model [DG80] [B+ 91] was adjusted to the entire nuclear chart in order
to find constants in equation 3.17. Also, there is a parametrization [B+ 07] called 5DCH to
calculate better correlation energies and shapes [RG14]. Figure 3.12 provides the experimental
trends of the two neutron separation energies of both models. On the one hand, the Gogny
D1S model(blue) seems to overestimate the N=56 subshell closure. On the other hand, it does
not reproduce the trend of two neutron separation energies at N=60 for both strontium and
krypton isotopes. The 5DCH parametrization treats the overestimation of the subshell. Still, it
shows only a slight decrease in two neutron separation energy at N>62, which results from the
nature of this interaction. This interaction is particularly well suited for representing the pairing
correlation which is experimentally visible at shell and subshell levels.
Artificial neural network
An artificial neural network(ANN) is a mathematical model built on the principle of the organi
zation and functioning of biological neural networks as in the brain of the biological organism.
Nowadays, such models are gaining more popularity and have been applied in nuclear physics.
ANN is a system of simple processors (artificial neurons). Such processors are usually quite
simple. Each processor of such a network deals only with signals that it periodically receives
3.3. Theoretical approaches
68
18
Gogny_D1S
5DCH
AME2016
ISOLTRAP
S2n (MeV)
16
14
12
42Mo
10
40Zr
38Sr
8
36Kr
6
52
54
56
58
60
Neutron Number N
62
64
Figure 3.12. The evolution of the two neutron separation energy in neutronrich krypton, stron
tium, zirconium and molybdenum isotopic chains. The blackcolored two neutron separation
energies are taken from AME2016 [H+ 17]. Red two neutron separation energies calculated
from the new 96−98 Kr mass measurements. Blue color corresponds to the Gogny D1S calcula
tions [DG80], green color corresponds to 5DCH parametrization [RG14]. For visibility error
bars are omitted.
and sends to other processors. And, nevertheless, when connected in an extensive enough net
work with controlled interaction, such individually simple processors together are capable of
performing rather complex tasks. The processor receives an actual number, and the output of
each processor is computed by some nonlinear function of the sum of its inputs. Output de
pends on the weights and the whole network adjusts with learning. The weight decreases or
increases the strength of the signal. Technically, training consists of finding the coefficients of
connections between neurons.
The ANN approach [L+ 20a] was used to predict the groundstate properties and exciting
energies for only strontium and krypton isotopes with Gogny 5DCH effective interaction to
compile the properties. Figure 3.13 provides the results of the calculations. Because of the
chosen interaction the new approach does almost the same as the standard Gogny D1S 5DCH
approach but in a much shorter time. One of the reasons for this might be that not enough
information about deformation is taken into the model. Another model based on similar math
3.3. Theoretical approaches
69
ematical solutions is Svmin. There is no clear description in the available literature on the
underlying interaction used for the model [G+ 06].
18
Gogny_5DCH
Svmin
Gogny_5DCH_AI
AME16
ISOLTRAP
S2n (MeV)
16
14
12
42Mo
10
40Zr
38Sr
8
36Kr
6
52
54
56
58
60
Neutron Number N
62
64
Figure 3.13. The evolution of the two neutron separation energy in neutronrich krypton, stron
tium, zirconium and molybdenum isotopic chains. The blackcolored two neutron separation
energies are taken from AME2020. Red is two neutron separation energies calculated from the
new
96−98
Kr mass measurements. Blue color corresponds to the Gogny D1S 5DCH calcula
tions [RG14]. Svmin colored in greed [G+ 06]. The ANN calculations with Gogny D1S 5DCH
parametrization are colored in orange [L+ 20a]. For visibility error bars are omitted.
The comparison of the first 2+ excited state predicted by Gogny D1S 5DCH interaction
and the new approach is shown in figure 3.14(ab). The black dashed line corresponds to
the experimental values. The trend of the 2+ excited states is reproducible by both models.
The spotlight of the deformation is a sudden decrease of the excited state energy as shown in
figure 3.1(c) and both models predict this behavior. However, calculations show a more steady
decrease of energy compares to the experimental values. The new calculations based on neural
networks reproduce the general trends of the energy. However, the major disadvantage is that
it does not reproduce the absolute value of the energy as in figure 3.14(c).
3.4. Summary
70
3000
38Sr
D1S+5DCH
D1S+5DCH+AI
NNDC
2000
1500
1000
500
0
36Kr
2500
E2 + (keV)
E2 + (keV)
2500
D1S+5DCH
D1S+5DCH+AI
NNDC
2000
1500
1000
500
54
55
56
57
58
59
60
Neutron number, N
61
62
54
(a)
56
58
60
62
Neutron number, N
64
(b)
98
4
+
Kr
Energy (MeV)
1.2
1.0
4+
2+
0+, 4+
0.8
?
0+, 2+
0.6
0.4
2+
2+
0.2
0.0
0+
0+
HFB
0+
AI
Exp.
(c)
Figure 3.14. (a) the first 2+ excited states in 38 Sr isotopes. (b) the first 2+ excited states in
36 Kr
isotopes. (c) the representation of the energy levels for
98
Kr isotope. The HFB value
is calculated by HFB approach with Gogny interaction, the AI value calculated by the same
approach but with artificial neural network, experimental values are taken from [Lab].
3.4
Summary
The new mass measurements of
96−98
Kr isotopes using the versatile ISOLTRAP mass spec
trometer have been measured and presented.
96
Kr ions were measured by MRToF mass mea
surement technique. The new mass value agrees well with previous experimental values. One
measurement was done applying ToFICR technique at ISOLTRAP and another applying MR
3.4. Summary
71
ToF mass measurement technique at the TITAN experiment. The mass value does not affect
the smooth trend of the two neutron separation energies.
97
Kr ions were measured with two
different techniques (ToFICR and MRToF) and in different years (2015 and 2017). Both
measurements agree well with the previous mass measurement which was also performed at
ISOLTRAP but in the preparation Penning trap. New mass measurement confirms a previously
observed trend of the two neutron separation energies but with a much smaller uncertainty.
The mass of 98 Kr isotope is determined for the first time by applying MRToF mass spectrom
etry. The observed trend of the two neutron separation energies possibly indicates a smooth
shape coexistence starting at 98 Kr isotope. This result agrees with another experimental prop
erty namely the energy of the first 2+ excited state. A comparison with theoretical calculations
available in the literature has been drawn. The macromicro models FRDM2012 and HFB31 do
not predict the region well. The UNEDF0 parametrization shows the best agreement with the
experimental values, possibly because of its adjustment of the spherical and deformed nuclei.
Gogny D1S and its 5DCH parametrization; Skyrme and its SLy4 parametrization do not predict
the region well. The new 5DCH calculations base on the deep neural network methods were
made and compared to the wellestablished methods. The new approach predicts the ground
state properties but does not predict the excited state energies.
Table 3.1. Summary of the Ricr and CT oF ratios as well as mass excess values of 96−98 Kr ions.
Reference masses are taken from AME2016 [H+ 17] and T1/2 values are from NUBASE2016
[WA+ 17].
Mass excess (keV)
Isotope
T1/2
References
96 Kr
80 ms
48 SO,85 Rb
97 Kr
62.2 ms
39 K
97 Kr
62.2 ms
97 Mo,85 Rb
98 Kr
42.8 ms
98 Mo,85 Rb
Ricr , CT oF
This work
AME16
53124(13)
53080(20)
Ricr =2.4882083741(720) 47502(26)
47420(130)
Ctof =0.503474279(556)
47442(64)
47420(130)
Ctof =0.503501959(798) 44217(100)
44310(300)
Ctof =0.499858112(320)
Chapter 4
Conclusion
The work presented in this dissertation can be divided into two sections. The first section de
scribes the measurement steps from the production of radioactive ions at the ISOLDE facility
until these ions are detected by ISOLTRAP mass spectrometer. For the mass measurement,
different techniques are used, namely MultiReflection Time of Flight mass spectrometry and
Time of Flight Ion Cyclotron Resonance technique. The techniques, as well as the ISOLTRAP
mass spectrometer, have been described. The IS532 [B+ 14] and IS642 [A+ 17] experimen
tal campaigns were claiming to measure the nuclear subshell closure at N=32 in neutron rich
50−55
Sc isotopes and contribute to the determination of F tvalue of 70 Br isotope. However, the
desired ions were not produced in enough quantity and/or were highly contaminated by stable
isotopes. However, the data analysis was performed on 49 Sc,
50
Sc,
70
As,
73
Br ions. The idea
was to establish a protocol for the MRToF mass spectrometry data analysis. For this, the stabil
ity of the output fit parameters of Gaussian and EGH fit functions was investigated. The results
of the analysis have been published in Nuclear Physics A. These two experimental campaigns
and the followed data analysis raised again the question about the transport efficiency of the
ISOLTRAP mass spectrometer. It was estimated to be only 0.25 %. The significant losses hap
pen during cooling and bunching in the buncher device. For this, a new set of radiofrequency
coils was built. The new device can increase the count rate by remotely tuning the resonance
frequency. Also, the effect of the pressure inside the buncher has been studied. The idea was to
see how the pressure of the gas inside the buncher affects the peak shape of the beam. Another
treatment of the transport efficiency problem was the realignment of the horizontal beamline.
Less than 1 mm has been achieved.
The second section of the work is devoted to the deformations at A=100 region. This has
been a hot topic for many years. The new mass measurements of the 96−98 Kr isotope demon
strate the evolution of collectively beyond N=60. Masses of 96,97 Kr isotopes demonstrate good
agreement with previous mass measurements in the region. The mass of 98 Kr isotope has been
determined for the first time. The uncertainties have been reduced for all cases. A compari
73
74
son with micromacro and meanfield calculations available in the literature has been drawn.
The Skyrme interaction was used in the calculations with the UNEDF0 parametrization. Be
cause this interaction was adjusted on the spherical and deform nuclei the calculations give
the best description of the region. It agrees well with experimentally observed trends of the
two neutron separation energies. Also, it predicts the smooth shape transition in the krypton
isotopic chain. The second nucleonnucleon interaction which was considered is Gogny with
the D1S parametrization. The model does not predict the trends in the region as well as five
dimensional collective Hamiltonian (5DCH) extension. The new 5DCH calculations based on
the deep neural network methods were considered. The new calculations require less compu
tational time. However, it predicts the region similar to the previously mentioned theoretical
calculations. Moreover, the new approach misses predicting excited state energies. It predicts
only the ground state properties. The experimental evidence from both γray spectroscopy and
direct mass measurements lead to the same conclusion the smooth shape transition occurs at
N=62 in the krypton isotopic chain.
An outlook of this work is to perform potential energy calculations with HFBTHO and
UNEDF0. The evolution of oblate/prolate/global two neutron separation energies would show
the energy depth of the two shapes in 96−98 Kr isotopes compared to strontium and zirconium
and answer the question of why the kink in two neutron separation energies occurs in strontium
isotopic chain but only a flattening in krypton. Also, BMF calculations using the SLy4 Skyrme
interaction, within a symmetryrestored generator coordinate method (GCM) are in process.
The results are planned to be submitted to the peerreviewed journal.
Chapter 5
List of published works
• I. Kulikov, A. Algora, D. Atanasov, P. Ascher, K. Blaum, R.B. Cakirli, A. Herlert, W.J.
Huang, J. Karthein, Yu.A. Litvinov, D. Lunney, V. Manea, M. Mougeot, L. Schweikhard,
A. Welker, F. Wienholtz, Masses of shortlived 49Sc, 50Sc, 70As, 73Br and stable 196Hg
nuclides, Nuclear Physics A, Volume 1002, 2020, 121990
• M. Mougeot, P. Ascher, D. Atanasov, K. Blaum, K. Chrysalidis, G. Hagen, J.D. Holt, W.
Huang, G.R. Jansen, J. Karthein, I. Kulikov, Y. Litvinov, D. Lunney, V. Manea, T. Miyagi,
T. Papenbrock, L. Schweikhard, A. Schwenk, T. Steinsberger, Z. H. Sun, A. Welker, F.
Wienholtz, S.G. Wilkins, and K. Zuber, Challenging abinitio nuclear theory near the
iconic 100Sn through mass measurements of 99101In, Submitted to Nature Physics
• K. Blaum, J. Karthein, I. Kulikov, Y. Litvinov, D. Lunney, V. Manea, M. Mougeot, L.
Nies, W.J. Ong, H. Schatz, L. Schweikhard, A. Schwenk, F. Wienholtz, K.Zuber, Mass
measurement of the selfconjugate 98In for nuclear and astrophysical studies, Proposal
to the ISOLDE and Neutron TimeofFlight Committee
75
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