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Exploration of (super-)heavy elements using the Skyreme-Hartfree-Fock model [Elektronische Ressource] / vorgelegt von Jochen Erler

105 pages
Exploration of (Super-)Heavy Elements using theSkyrme-Hartree-Fock modelZ=120Z=82N=184N=126Z=50N=82Z=28Z=20N=50N=28N=20Der naturwissenschaftlichen Fakult¨at der¨ ¨Friedrich-Alexander-Universitat Erlangen-NurnbergzurErlangung des Doktorgradesvorgelegt vonJochen Erleraus AnsbachiiAls Dissertation genehmigt von der Naturwissen-schaftlichen Fakult¨at der Friedrich-Alexander-Universit¨atErlangen-Nurn¨ bergTag der mundlic¨ hen Prufung:¨ 31. Januar 2011Vorsitzender der Promotionskommission: Prof. Dr. Rainer FinkErstberichterstatter: Prof. Dr. Paul-Gerhard ReinhardZweitberich Prof. Dr. Joachim A. MaruhnAbstractMotivatedbythesteadilyincreasingnumberofknownnucleiandnuclearproperties,theoriesof nuclear structure are presently a field of intense research. This work concentrates on theself-consistent description of nuclei in terms of the Skyrme-Hartree-Fock (SHF) approach.The extrapolation of nuclear shell structure to the region of super-heavy elements (SHE)using the SHF model, the dependence on different parameterization and the influence ofcollective correlation will be studied. The general scope of this work are large scale calcula-tion for a global survey of properties of SHE like binding energies, separation energies anddecay characteristics and lifetimes.
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Exploration of (Super-)Heavy Elements using the
Skyrme-Hartree-Fock model
Z=120
Z=82
N=184
N=126
Z=50
N=82
Z=28
Z=20
N=50
N=28
N=20
Der naturwissenschaftlichen Fakult¨at der
¨ ¨Friedrich-Alexander-Universitat Erlangen-Nurnberg
zur
Erlangung des Doktorgrades
vorgelegt von
Jochen Erler
aus Ansbachii
Als Dissertation genehmigt von der Naturwissen-
schaftlichen Fakult¨at der Friedrich-Alexander-Universit¨at
Erlangen-Nurn¨ berg
Tag der mundlic¨ hen Prufung:¨ 31. Januar 2011
Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink
Erstberichterstatter: Prof. Dr. Paul-Gerhard Reinhard
Zweitberich Prof. Dr. Joachim A. MaruhnAbstract
Motivatedbythesteadilyincreasingnumberofknownnucleiandnuclearproperties,theories
of nuclear structure are presently a field of intense research. This work concentrates on the
self-consistent description of nuclei in terms of the Skyrme-Hartree-Fock (SHF) approach.
The extrapolation of nuclear shell structure to the region of super-heavy elements (SHE)
using the SHF model, the dependence on different parameterization and the influence of
collective correlation will be studied. The general scope of this work are large scale calcula-
tion for a global survey of properties of SHE like binding energies, separation energies and
decay characteristics and lifetimes. These calculations were done in a collaboration with the
theory group of the GSI in Darmstadt and have the aim to develop a database of lifetimes
and reaction rates for α-, β-decay and spontaneous fission in a very wide range with proton
numbers 86≤ Z ≤ 120 and neutron numbers up to N ≈ 260 relevant for the astrophysical
r-process.
The results of this study for example predictions of a possible islands of very stable nuclei
and information of favored decay mode for each nuclei are also applicable in the recent ex-
perimental synthesis of exotic SHE.
For these calculation a framework to calculate β-decay half-lives within the SHF model has
been developed and the existing axial SHF code has been extended to compute β-transition
matrix elements and so to provide an estimation of half-lives. Furthermore the full self-
consistent description of fission lifetimes of [2] has been improved to a more stable recipe for
large scale calculations.
iiiivZusammenfassung
MotiviertvonderstetigzunehmendenAnzahlanbekanntenKernenandderenEigenschaften
ist die theoretische Kernstrukturphysik ein aktuelles und intensives Forschungsgebiet. Die
vorliegende Arbeit besch¨aftigt sich daher mit der selbst-konsistenten Beschreibung von Ker-
nenunterVerwendungdesSkyrme-Hartree-Fock(SHF)Ansatzes. DieExtrapolationsf¨ahigkeit
in das Gebiet super schwerer Elemente (SHE), die Abh¨angigkeit von verschiedenen Parame-
terisierungen und der Einfluss kollektiver Korrelationen werden untersucht. Im Mittelpunkt
des Interesses dieser Arbeit stehen großangelegte Berechnungen mit dem Ziel einer globalen
¨Ubersicht der Eigenschaften von super schweren Elementen, wie zum Beispiel Bindungsen-
ergien, Separationsenergien, Zerfallscharakteristiken and Zerfallslebenszeiten. Diese Berech-
nungen wurden in Zusammenarbeit mit der Theoriegruppe der Gesellschaft fur¨ Schweri-
onenforschung (GSI) in Darmstadt durchgefuhrt.¨ Hierbei bestand das Ziel in einer um-
fassenden Datenbank von Halbwertszeiten and Reaktionsraten der drei konkurrierenden
Zerfallsprozesse α-, β-Zerfall und spontaner Spaltung in einem fur¨ den astrophysikalischen
r-Prozess relevanten weiten Bereich mit der Protonzahl 86 ≤ Z ≤ 120 und bis zu einer
Neutronenanzahl von N ≈260.
Die Ergebnisse dieser Studie, wie zum Beispiel die Vorhersage von Bereichen h¨oherer Sta-
bilit¨at oder die des bevorzugten Zerfallskanals einzelner Nuklide, k¨onnen ebenfalls auf dem
Gebiet der experimentellen Synthese exotischer Kerne angewendet werden.
Fur¨ dieseBerechnungenmusstederRahmenzurBerechnungderβ-Halbwertszeiteninnerhalb
des SHF Modells geschaffen und in den bestehenden axialsymmetrischen SHF Code imple-
mentiert werden. Hierbei wurde die ebenfalls notwendige Erweiterung des Programmcodes
zur Berechnung ungerader Kerne im Rahmen einer parallel verlaufenden Diplomarbeit [1]
vollzogen. Des Weiteren wurde das bestehende Rezept [2] zur selbst-konsistenten Beschrei-
bung von Spalthalbwertszeiten verbessert, um eine stabile Berechnung fur¨ eine großfl¨achige
Anwendung zu gew¨ahrleisten.
vviContents
1 Introduction 1
2 The Skyrme-Hartree-Fock energy functional 3
2.1 A “force” and a functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The mean-field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Description of odd nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Theory of collective correlations 21
3.1 Generator coordinate method (GCM) . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Gaussian overlap approximation (GOA) . . . . . . . . . . . . . . . . . . . . 23
3.3 Collective particle-number correction . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Angular-momentum projection. . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Ground-state correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Theory of decay modes 27
4.1 Spontaneous fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 α-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 β-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Applications 33
viiviii Contents
5.1 Ground-state deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2 Binding and separation energies . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Spontaneous fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.4 α-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 β-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.6 Minimal lifetime plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6 Open problems 61
6.1 A variation of Skyrme forces . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2 A modified density dependence . . . . . . . . . . . . . . . . . . . . . . . . . 63
7 Conclusion 69
A 73
A.1 Relation of force parameters to functional paramaters . . . . . . . . . . . . . 73
B 75
B.1 Interface of SHF with collective Hamiltonian . . . . . . . . . . . . . . . . . . 75
C 77
C.1 Isospin-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
C.2 Derivation of β-decay matrix elements . . . . . . . . . . . . . . . . . . . . . 78
D 85
D.1 Some Standard Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
D.2 SV-min, SV-bas and its Relatives . . . . . . . . . . . . . . . . . . . . . . . . 86
References 87
Danksagung 951
Introduction
Thisyearelement112wasbaptized(onthename)copernicium(Cn)afterthefirstdetection
in 1996 [3] at the Heavy Ion Research Center (GSI) in Darmstadt synthesized by irradiation
206 70of Pb targets with Zn projectiles. The result was confirmed in 2000 [4] and in 2004 by
the RIKEN laboratory in Japan [5]. Copernicium is only one example for the considerable
progressin experimental synthesis ofnewelementsat theGSI,theDUBNA grouporothers.
InthisyeartheDUBNAgroupevenreportsthesynthesisofelement117. Sothesegeneration
of new elements above the naturally existing ones, called super-heavy elements (SHE), has
attracted much attention in the past decades.
Beside the synthesis of new elements also precession measurements of binding energies of
SHE are performed using penning traps [6].
Nuclei are described by nuclear structure theory at all levels of refinement, from the more
phenomenological microscopic-macroscopic methods (see, e.g., [7]) through self-consistent
mean-field (SCMF) methods (see, e.g.,Ref. [8]) or shell-model calculations (see, e.g., Ref.
[9]) up to several ab initio techniques employing a given nucleon-nucleon interaction (see,
e.g., Refs. [10, 11, 12]). The description of SHE is a great challenge for nuclear structure
theory and is presently performed by microscopic-macroscopic and SCMF methods. These
models are employed, e.g. to confirm experimental data and to make predictions for binding
energies, possibledecaymodes, regionsofhigherstability(alsocalledislandsofstableSHE).
Another demanding field of activity is the description of the natural synthesis of the ele-
ments. While elements up to iron (Z=26) can be synthesized by fusion in heavy stars [13],
for heavier nuclei other processes have to be considered. There are three different mecha-
nisms which run in three different regions. They are naturally referred to as the p-, s- and r-
process. Protoncapture(p-)processandslowneutroncapture(s-)processoccurintheneu-
tron deficient regionand in thevalley of β-stabilityand playa less importantrole. However,
12 Introduction
the rapid neutron capture (r-) process is known to be responsible for the synthesis of about
half of the elements heavier than iron [14]. In case of the r-process the β-decay half-lives
τ are much longer than the neutron capture time τ (τ << τ ) which allows the processβ n n β
to reach the very neutron-rich regime by successive captures of neutrons. The necessary
20 −3high neutron number densities for this process (about n ≥10 cm ) can only be found inn
explosive environments like supernova explosions. After the decrease of the high density the
naturally occurring nuclei are synthesized by subsequent decays of these neutron-rich nuclei.
Due to the important role in the synthesis of elements heavier than iron the r-process is a
current field of activity of nuclear structure and astrophysics.
Here a very large amount of nuclear information like binding energies, separation energies,
decay characteristics and lifetimes for the possible decay modes spontaneous fission, β- and
α-decay is necessary in order to model the r-process [14].
This work concentrates on the self-consistent description of nuclei in terms of the Skyrme-
Hartree-Fock (SHF) approach to provide these necessary data. The thesis is outlined as
follows:
Chapter 2 gives a short overview on the Skyrme-Hartree-Fock (SHF) model, the used en-
ergy functional with its ingredients, pairing, and corrective terms with its various options.
Furthermore it addresses the phenomenological adjustment of the SHF functional and its
implications. Last the description of odd nuclei using the blocking approximation is dis-
cussed.
Chapter 3 provides a review of the theory of collective correlations. Here the used ansatz for
correlations, the generator coordinate method (GCM), and the Gaussian overlap approxi-
mation (GOA) and correction methods therein are discussed.
In chapter 4 the theory of the three competing decay channels is presented. Here fission
lifetimes are calculated on basis of the semi-classical WKB formula using microscopic input
from the SHF model and GCM+GOA. The β-decay half-lives are estimated by computation
of the β-transition matrix elements and for α-decay a semi-empirical formula, the Viola-
Seaborg relationship, is used to calculate the half-lives employing the energy release Q .α
Chapter 5 presents all results starting with nuclear ground state properties like binding and
separation energies and then going to decay properties of spontaneous fission like barriers,
lifetimes and the influence of asymmetric shapes. Furthermore benchmarks and systematics
of α- and β-decay are given. In the last section of the chapter the three decay modes are
compared by using minimal lifetime plots to see where the different decay modes dominate.
In the first part of chapter 6 the dependence of fission barriers and lifetimes on a variety of
different Skyrme parameterization related to different nuclear bulk properties is discussed.
The second part presents a new ansatz for the density dependence in the used energy func-
tional to explore a mismatch in binding energies of SHE as well a wrong trend in fission
barriers.

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