Effective Interactions for
Nuclear Structure Calculations
By
Angelo Signoracci

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

ABSTRACT
EFFECTIVE INTERACTIONS FOR NUCLEAR STRUCTURE CALCULATIONS
By
Angelo Signoracci

Experimental interest in nuclei far from stability, especially due to proposed advancements in rare isotope facilities, has stimulated improvements in theoretical predictions of exotic isotopes. However, standard techniques developed for nuclear structure calculations, Configuration Interaction theory and Energy Density Functional
methods, lack either the generality or the accuracy necessary for reliable calculations
away from stability. Hybrid methods, which combine Configuration Interaction theory
and Energy Density Functional methods in order to exploit their beneficial properties,
are currently under investigation for improved theoretical capabilities.
A new technique to produce nuclear Hamiltonians has been developed, implementing renormalization group methods, many-body perturbative techniques, and Energy
Density Functional methods. Connection to the underlying physics is a primary focus,
limiting the number of free parameters necessary in the procedure. The main benefit
of this approach is the improvement in the quality of effective interactions outside of
standard model spaces.
In the Hybrid Renormalization Procedure developed in this dissertation, Skyrme
energy density functionals provide a realistic single particle basis that accounts for
the long tail of loosely bound orbits, especially significant for valence orbits of exotic
isotopes. A microscopic nucleon-nucleon potential is softened with renormalization
group techniques to eliminate the hard core of the nuclear interaction. Many-body
perturbative techniques, in the form of Rayleigh-Schrödinger theory, implement the
realistic basis to convert the low-momentum interaction into a model space of interest.

The basis is an important ingredient in the renormalization and greatly affects the
results obtained with the Hybrid Renormalization Procedure, specifically through
the single particle energies derived from Skyrme functionals. A comparison of the
standard harmonic oscillator basis and the realistic basis derived from energy density
functional methods illustrates the necessity of a realistic basis when a microscopic
nucleon-nucleon potential is renormalized into the nuclear medium. Because Skyrme
single particle energies are unreliable, other sources are desired for the determination
of this component of the effective interaction.
An sd shell interaction is produced for a proof of principle, and extensive results
are obtained in the island of inversion region and for 42 Si. One hundred nuclei are
calculated near the island of inversion region of the nuclear chart. Binding energies
and low-lying excitations agree well with available experimental data. The neutron
dripline is determined theoretically for isotopic chains near the island of inversion.
The boundaries of the island of inversion region are mapped out, suggesting an extension to lighter and more neutron-rich isotopes than measured experimentally thus
far. Reactions in the island of inversion region have also been studied and reproduce
experimental behavior, such that conclusions can be reached regarding the evolution
of shell structure and the properties of states of particular interest. The known states
of 42 Si are reproduced with effective interactions derived from the Hybrid Renormalization Procedure, but the detection of the 0+ and 4+ states are important to
2
1
distinguish between different theoretical approaches.
The viability of the Hybrid Renormalization Procedure is evident from the results
in the island of inversion region. Applications to other exotic regions of the nuclear
chart are desired for future research.

ACKNOWLEDGMENTS

This document represents the culmination of five years of research and development in nuclear structure physics at the National Superconducting Cyclotron Laboratory and the Department of Physics and Astronomy at Michigan State University. In
that time, I have been fortunate to receive instruction and guidance from a committee
of scientific researchers, comprised of Alex Brown, Carlo Piermarocchi, Jon Pumplin,
Michael Thoennessen, and Vladimir Zelevinsky. In particular, I would like to thank
Alex Brown as the advisor of my doctoral research for helpful discussions from the
formulation of the project to its completion.
As this project incorporated many aspects of nuclear physics, I am indebted to
experts in the subdisciplines relevant for my research: Alex Brown, for CI theory,
EDF methods, and general nuclear structure; Morten Hjorth-Jensen, for renormalization group methods and many-body perturbative techniques, and especially for
instruction on implementing his programs to produce effective interactions, which
were essential for this work; Thomas Duguet, for EDF methods and single particle
energies; Scott Bogner, for renormalization group methods; the UNEDF collaboration, for EDF methods and hybrid methods; Vladimir Zelevinsky, for general nuclear
structure; Filomena Nunes, Ian Thompson, and Kathryn Wimmer, for reaction calculations of 30 Mg(t,p); and Alexandra Gade, Paul Mantica, Wolfgang Mittig, and
Michael Thoennessen, for interpretations of experimental data.
The U.S. Department of Energy National Nuclear Security Administration Stewardship Science Graduate Fellowship program and the Krell Institute funded the majority of my graduate career and provided unique opportunities. I benefited greatly
from interaction with personnel from the DOE and national labs, and also with my
peers in the SSGF program. My practicum at Lawrence Livermore National Laboratory was a formative experience, and I gained new perspectives on nuclear structure

iv

research from the Nuclear Theory and Modeling Group. I would like to single out my
advisor for the practicum, Erich Ormand, for his guidance and support during and
after my time at LLNL. For additional funding, I thank the NSCL, the Department of
Physics and Astronomy, the College of Natural Science, and the Council of Graduate
Students.
In addition to those who directly influenced this research, I would like to thank
those people who contributed significantly to the successful completion of my degree
from Michigan State. The nuclear theory group at the NSCL offered a wonderful environment with a supportive staff that facilitated my development. Shari Conroy and
Lin Leslie were instrumental for their coordination of day-to-day affairs, enabling me
to focus solely on research. The collaborative nature of the NSCL and the consistent
visits by researchers from around the world expanded my knowledge and emphasized comparisons between my work and results obtained in experiments or in other
theoretical approaches.
From the Department of Physics and Astronomy, I must thank the Chair, Wolfgang Bauer, the Secretary for Graduate Student Affairs, Debbie Barratt, and the
Graduate Program Director, Scott Pratt. In addition, I am particularly indebted to
the former Graduate Program Director, Bhanu Mahanti, for his guidance and his role
in my teaching experience. I thank all the instructors of my graduate classes that
have given me the foundation for a career in physics. I have benefited greatly from
my fellow graduate students at Michigan State, not only from discussions regarding
research and homework but also from the relationships cultivated, especially those
which have spanned my entire graduate career.
Finally, I thank those who were not present in East Lansing during my stay, but
provided support and encouragement throughout the years. If not for my family, I
would have been unable to undertake this project and succeed. I thank you all, but
especially my parents, Jon and Judy.

v

TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures

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ix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Configuration Interaction Method . . . . . . . . . . . .
1.2 Energy Density Functional Methods . . . . . . . . . . .
1.3 Microscopic Nucleon-Nucleon Potentials . . . . . . . .
1.4 Renormalization Group Methods . . . . . . . . . . . .
1.5 Many-Body Perturbative Techniques . . . . . . . . . .
1.6 Hybrid Methods . . . . . . . . . . . . . . . . . . . . . .
1.7 Motivation . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Methodology and Layout . . . . . . . . . . . . . . . . .

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22
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31

2 Renormalization Procedure . . . . . . . . . . . . . . .
2.1 Bases for Renormalization . . . . . . . . . . . . . . . .
2.2 Application to sd Model Space . . . . . . . . . . . . . .
2.3 Dependence on Microscopic Interaction . . . . . . . . .
2.4 Dependence on RG method . . . . . . . . . . . . . . .
2.5 Dependence on Target Nucleus . . . . . . . . . . . . .
2.6 Dependence on Skyrme Interaction . . . . . . . . . . .
2.7 Dependence on Order of Perturbation Theory . . . . .
2.8 Number of Excitations . . . . . . . . . . . . . . . . . .
2.9 Issues . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Standard Implementation . . . . . . . . . . . . . . . .

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34
36
38
44
44
46
47
49
50
57
60

3 Comparison of Bases . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Application to sdpf model space . . . . . . . . . . . . . . . . . . . .
3.2 Calculations for 36 Si and 38 Si . . . . . . . . . . . . . . . . . . . . . .

62
63
76

4 Single Particle Energies . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Definition of Single Particle Energy . . . . . . . . . . . . . . . . . .
4.2 Comparison of Effective SPE and One-Nucleon Separation Energies
4.3 SPE Dependence of CI Calculations . . . . . . . . . . . . . . . . . .
4.4 Skyrme Single Particle Energies . . . . . . . . . . . . . . . . . . . .

82
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87
94
96

vi

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5 Applications . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Island of Inversion Region . . . . . . . . . . . . . . . .
5.1.1 Systematic Trends . . . . . . . . . . . . . . . .
5.1.2 Level Schemes of Representative Nuclei . . . . .
5.1.3 Ground State Occupations . . . . . . . . . . . .
5.1.4 β decay . . . . . . . . . . . . . . . . . . . . . .
5.1.5 26 Ne(d,p)27 Ne . . . . . . . . . . . . . . . . . . .
5.1.6 30 Mg(t,p)32 Mg . . . . . . . . . . . . . . . . . .
5.2 42 Si . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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101
101
107
113
118
121
128
130
133

6 Outlook and Conclusions .
6.1 Three-body Forces . . . .
6.2 Other Regions of Interest .
6.3 Single Particle Energies . .
6.4 Conclusion . . . . . . . . .

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142

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A Abbreviations and Definitions of Selected Terms . . . . . . . . . . 147
Bibliography

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vii

LIST OF TABLES

1.1

Skyrme parameterizations . . . . . . . . . . . . . . . . . . . . . . . .

12

3.1

Single particle energies in the SHF basis . . . . . . . . . . . . . . . .

66

3.2

Single particle energies in the HO basis . . . . . . . . . . . . . . . . .

66

3.3

TBME components in the sdpf model space . . . . . . . . . . . . . .

74

4.1

Spectroscopic factors for proton capture on 22 O . . . . . . . . . . . .

88

4.2

Comparison of SPE for 23 F . . . . . . . . . . . . . . . . . . . . . . .

90

4.3

Neutron SPE for the sd orbits . . . . . . . . . . . . . . . . . . . . . .

95

4.4

16 O

neutron SPE for different Skyrme interactions . . . . . . . . . . .

96

4.5

22 O

neutron SPE for different Skyrme interactions . . . . . . . . . . .

96

5.1

Parameters in the IOI interaction . . . . . . . . . . . . . . . . . . . . 105

5.2

Spectroscopic factors for states in 27 Ne . . . . . . . . . . . . . . . . . 129

viii

LIST OF FIGURES

1.1

Mean field for protons and neutrons . . . . . . . . . . . . . . . . . . .

5

1.2

Single particle states in 132 Sn . . . . . . . . . . . . . . . . . . . . . .

6

1.3

Diagrammatic description of the three-body Hamiltonian . . . . . . .

8

1.4

One-body Goldstone diagrams . . . . . . . . . . . . . . . . . . . . . .

19

1.5

Two-body Goldstone diagrams . . . . . . . . . . . . . . . . . . . . . .

20

1.6

Chart of nuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

1.7

Energy differences in sd shell nuclei with the USDB interaction . . . .

27

1.8

Standard model spaces . . . . . . . . . . . . . . . . . . . . . . . . . .

28

1.9

Comparison of silicon and calcium isotones . . . . . . . . . . . . . . .

29

2.1

18 O

level scheme with a second order interaction . . . . . . . . . . . .

39

2.2

18 O

level scheme with a third order interaction . . . . . . . . . . . . .

40

2.3

Comparison of empirical and microscopic TBME . . . . . . . . . . . .

41

2.4

TBME for different N N interactions . . . . . . . . . . . . . . . . . .

43

2.5

TBME for different RG methods

. . . . . . . . . . . . . . . . . . . .

45

2.6

TBME for different target nuclei . . . . . . . . . . . . . . . . . . . . .

46

2.7

TBME for different Skyrme interactions . . . . . . . . . . . . . . . .

48

2.8

TBME for different orders of perturbation theory . . . . . . . . . . .

49

2.9

Single particle orbits in 16 O . . . . . . . . . . . . . . . . . . . . . . .

51

2.10 2 ω excitation of 16 O from promotion of a pair of protons . . . . . .

52

2.11 2 ω excitation of 16 O from promotion of one neutron . . . . . . . . .

53

ix

2.12 TBME for 4 ω and 6 ω excitations . . . . . . . . . . . . . . . . . . .

55

2.13 TBME for 6 ω and 8 ω excitations . . . . . . . . . . . . . . . . . . .

56

2.14 TBME for different vlowk cutoffs

. . . . . . . . . . . . . . . . . . . .

59

3.1

Single particle orbits in 40 Ca . . . . . . . . . . . . . . . . . . . . . . .

64

3.2

Single particle orbits in 34 Si . . . . . . . . . . . . . . . . . . . . . . .

65

3.3

Valence neutron orbits for 40 Ca and 34 Si . . . . . . . . . . . . . . . .

67

3.4

Comparison of radial wavefunctions in the HO and SHF bases . . . .

67

3.5

Comparison of TBME in HO basis . . . . . . . . . . . . . . . . . . .

69

3.6

Comparison of TBME for 40 Ca . . . . . . . . . . . . . . . . . . . . .

70

3.7

Comparison of TBME in SHF basis . . . . . . . . . . . . . . . . . . .

71

3.8

36 Si

level schemes: renormalized interactions . . . . . . . . . . . . . .

77

3.9

36 Si

level schemes: comparison to experiment and empirical interactions 78

3.10 38 Si level schemes: comparison to experiment and empirical interactions 79
4.1

17 O

level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.2

Spectroscopic factors for 23 F . . . . . . . . . . . . . . . . . . . . . . .

90

4.3

Spectroscopic factors for 19 F . . . . . . . . . . . . . . . . . . . . . . .

91

4.4

Spectroscopic factors for N = 28 . . . . . . . . . . . . . . . . . . . . .

93

5.1

Energy differences with IOI interaction . . . . . . . . . . . . . . . . . 106

5.2

34 Si

5.3

Sn vs. N for even-Z isotopes . . . . . . . . . . . . . . . . . . . . . . . 110

5.4

Sn vs. N for odd-Z isotopes . . . . . . . . . . . . . . . . . . . . . . . 111

5.5

30 Si

5.6

32 Ne

level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.7

27 Ne

level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.8

31 Mg

level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.9

33 Na

level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

single particle energies . . . . . . . . . . . . . . . . . . . . . . . . 108

level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.10 Theoretical boundaries of the island of inversion region . . . . . . . . 119
x

5.11 33 Mg level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.12 33 Al level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.13 States populated in the β − decay of 33 Mg . . . . . . . . . . . . . . . 126
5.14 Neutron occupations for 30 Mg and 32 Mg . . . . . . . . . . . . . . . . 131
5.15 Comparison of 42 Si level schemes . . . . . . . . . . . . . . . . . . . . 134

xi

Chapter 1
Introduction
Because of the complexity of the nuclear many-body problem, the time-independent
Schrödinger equation
HΨ = EΨ

(1.1)

cannot be solved exactly for more than a few particles. However, over 3000 nuclei have been measured (and another 4000 are expected to exist), with nearly all
outside the range of exact ab initio methods. In order to provide theoretical estimates of nuclear properties, various methods have been developed to approximate
the time-independent Schrödinger equation. Configuration Interaction (CI) method
and Energy Density Functional (EDF) methods are two commonly used sets of nuclear structure techniques that expand the range of theoretical calculations beyond
the lightest-mass region of the nuclear chart.

1.1

Configuration Interaction Method

The Configuration Interaction method, including the nuclear shell model [1], [2], simplifies the nuclear problem by selecting an inert core of occupied single particle levels
with mass number Ac as a vacuum with zero point energy. The entire description of
occupied particles is contained in a model space of valence orbits outside the core,

1

reducing the problem from A-body to Aval -body, where Aval = A − Ac . One effect of
this reduction in the allowed configuration space (i.e. reduction in degrees of freedom)
requires the modification of the Hamiltonian. The interactions between nucleons in
the model space must take into account effects outside of the model space, both from
the core and from higher energy orbits which are not included in the valence space.
The utility of the CI method depends upon a single particle picture that simplifies
the nuclear many-body problem. In this model, the A nucleons produce a mean field
that binds them in the nucleus, and a one-body potential can be used to represent the
complicated effects of the nuclear interaction. Mathematically, the trick is to rewrite
the nuclear Hamiltonian
A

H =T +V =

A

t(ri ) +
i=1

v(ri , rj ) + . . . ,

(1.2)

i,j=1
i<j

where the ellipsis refers to many-body forces up to A−body, as

H = [T + Vmf ] + [V − Vmf ],

(1.3)

where the mean field potential Vmf is chosen as a single particle interaction such that
A

v(ri ).

Vmf =

(1.4)

i=1

The mean field Hamiltonian H0 = T + Vmf is usually selected so that the manybody system is reasonably approximated by the exact solution to the Schrödinger
equation. In this way, H1 = V − Vmf is small relative to H0 and can be treated as a
perturbation with well-established many-body perturbative techniques. The A-body
problem in the mean field is solved by separating the Schrödinger equation

H0 Ψ0 (r1 , r2 , . . . , rA ) = E0 Ψ0 (r1 , r2 , . . . , rA )

2

(1.5)

into A identical one-body equations

h(r)ψα (r) = α ψα (r),

(1.6)

where h(r) = t(r)+v(r), α is a representation of the complete set of quantum numbers
that solve the single particle states, and

Ψ0 (r1 , r2 , . . . , rA ) =

1
A ψα1 (r1 )ψα2 (r2 ) . . . ψαA (rA )
A

(1.7)

is a Slater determinant of single particle wavefunctions obtained from the antisymmetrization operator A. The energy is then given by
A
αi .

E0 =

(1.8)

i=1

A typical mean field potential includes a central term, the Coulomb interaction,
and a spin-orbit term as given by the one-body potential

v(r) = vHO (r) + vC (r) + vLS (r)L · S,

(1.9)

where the central term is a harmonic oscillator potential. A Woods-Saxon potential
vW S (r) is often used instead of vHO (r) without significant modifications. The potentials can be determined phenomenologically, or can be produced in a more microscopic

3

way. From Suhonen [3], for example,
1
vHO (r) = −V1 + mN ω 2 r2
2
−V0
vW S (r) =
r−R
1+e a
2
(0) r0 dvW S (r)
vLS (r) = VLS 2
dr
r
 1
r>R
Ze2  r
vC (r) =
4π 0  3−(r/R)2 r ≤ R

2R

(1.10)
(1.11)
(1.12)

(1.13)

−Z
with constants R = r0 A1/3 = 1.27A1/3 fm, a = 0.67 fm, V0 = (51 ± 33 N A )
(0)

MeV, with + for protons and − for neutrons, and VLS = −0.44. A comparison
between results with the Woods-Saxon and harmonic oscillator central potentials is
used to determine V1 and ω, by matching the depth and single particle spacing of
the two potentials, respectively. With the one-body potential of Eq. 1.9, the mean
field is given by Fig. 1.1. The solution of the one-body Schrödinger equation with
the given one-body potential results in the quantum numbers α = (n, , j, mj ) due to
the combination of central and spin-orbit interactions (see [4], or another standard
quantum mechanics textbook). A schematic description of the lowest single particle
states is shown for 132 Sn in Fig. 1.2, including a solution with only the harmonic
oscillator potential and the evolution of the single particle spacing when the spinorbit potential is added. The solution with the harmonic potential gives an energy
dependence on the main quantum number N = 2n + only, producing large gaps of
energy between oscillator shells (i.e., orbits with the same value of N ). A complete
description of the harmonic oscillator single particle basis can be found in [4] or [3],
but it is important to note three qualities:
(i) the degeneracy of the N th shell, including two spin states for each nucleon, is
given by g(N ) = (N + 1)(N + 2),
(ii) the parity is determined by (−1) , and

4

E
neutrons

protons

Figure 1.1: Schematic view of single particle orbits for protons and neutrons bound
in a one-body mean field potential.
(iii) in one shell,

takes the values N, N − 2, . . . , 0 for positive parity and N, N −

2, . . . , 1 for negative parity.
Hence, the N = 0 shell is given by the 0s orbit and has positive parity, the
N = 1 shell consists of the 0p orbits and has negative parity, the N = 2 shell
consists of the 0d and 1s orbits and has positive parity. As determined empirically by
Blomqvist and Molinari [5], the spacing of oscillator shells is dependent on mass and
can be approximated by ω = (45A−1/3 − 25A−2/3 ) MeV to reproduce experimental
measurements.
The characteristic gaps between certain numbers of neutrons and protons result
in “magic numbers” that require a relatively large amount of energy to excite a nucleon into the next highest orbit. For the harmonic oscillator potential, restricting to
one type of nucleon, the magic numbers are 2, 8, 20, 40, 70, and 112. As determined
experimentally, agreement is only achieved for lighter nuclei, as these magic numbers
should be 2, 8, 20, 28, 50, 82, and 126. The strong spin-orbit force in nuclear physics,
relative to the atomic case for instance, is essential for a treatment of the nuclear
problem, and results in splitting on the order of single particle energy differences as
seen in Fig. 1.2. With the choice of parameters listed above from [3], the experimental
magic numbers are reproduced.
5

20
neutrons
10

l(2j)
6

g7
d5
g9
126

112

0

5

E (MeV)

4

82

70

-10
3

50

40

28

-20

-30

20

20

8

2

p1i13
h9
f5 p3
f7
h11
s1
d3d5
g7
g9
p1
p3
f5
f7

8

s1
d3
d5
p1
p3

1
2

-40

0

s1

2

N = lmax
-50

Figure 1.2: Schematic view of the single particle states in 132 Sn with a harmonic
oscillator mean field and the splitting that occurs with the addition of a strong spinorbit potential. For interpretation of the references to color in this and all other
figures, the reader is referred to the electronic version of this dissertation.

6

The main approach of the CI method proceeds via a reduction to Aval particles by
selecting a core with magic numbers for protons and neutrons separately. The valence
space orbits should be chosen such that low-lying states of nuclei beyond the core
can be reproduced. Standard model spaces include the major oscillator shells, named
according to the composition of their orbits. Thus, the N = 0, 1, 2, 3 shells correspond
to the s, p, sd, pf model spaces, respectively. For instance, 16 O, with N = Z = 8,
represents a good core and has been used extensively for calculations from A = 16
to A = 40 within the sd model space. After selecting a core and model space, two
problems remain: finding an effective interaction and solving an eigenvalue problem in
the model space. Research through many decades has provided guidance. To solve the
eigenvalue problem, various shell model codes have been developed such as OXBASH
[6], ANTOINE [7], NUSHELL [8], and NUSHELLX [9].
The effective interaction determines the accuracy of the method, assuming an
appropriate core and model space and sufficient computing resources. The earliest
shell model calculations [10] used a simple square well potential between nucleons,
but fifty years of research have led to better interactions. The established shell model
codes require an interaction in the form of single particle energies (SPE) and two-body
matrix elements (TBME), given by ab V cd JT , where a, b, c, d are single particle
orbits, V is an effective two-body interaction, and J and T are the spin and isospin of
the coupled nucleons. Chapter 4 will discuss in detail the determination of SPE; for
standard model spaces with a stable core, the experimental energy difference between
the core with A particles and states in A + 1 nuclei provide reasonable SPE since the
lowest-energy experimental states can be approximated as single particle states. The
TBME should be determined, as discussed in [11] and [12], through a procedure that:
(i) utilizes a microscopic nucleon-nucleon interaction
(ii) overcomes the strongly repulsive short-range part of the interaction via renormalization group methods
(iii) employs many-body perturbative techniques to renormalize the interaction

7

Figure 1.3: A schematic diagram from [13] of the Hamiltonian obtained from an application of Wick’s theorem to a closed shell target. The black lines represent valence
particles and/or holes, while the red circles represent a summation over orbitals occupied in the target.
into the model space, accounting for configurations outside the model space.
Each component of this procedure will be covered later, in Sections 1.3, 1.4, and
1.5, respectively.
The immediate shortcoming of this approach is the exclusion of many-body forces
present in the nuclear Hamiltonian H. As has been found by many studies, notably
in [14], three-body forces are approximately one order of magnitude smaller than
two-body forces but contribute significantly to the total binding energy of nuclei. In
the typical procedure discussed above, the only inclusion of many-body forces occurs
implicitly through the experimentally-determined SPE. Truncating at the three-body
level, the Hamiltonian is represented diagrammatically by the leftmost column of Fig.
1.3. All three diagrams contain an effective zero-body component and a one-body
component, which lie within the dashed black box and can be described reasonably
via single particle energies, as discussed in Chapter 4. Two-body matrix elements,
as derived from a microscopic nucleon-nucleon interaction by the procedure above,
include only the explicit two-body component of Fig. 1.3 but can produce reasonable

8

results for few particles outside the core. As more valence particles are added, the
effect of many-body forces increases, along with the deviation from experiment. To
counteract this deviation, effective interactions have been derived in the standard
model spaces by parameterizing the SPE and TBME and by determining the nuclear
Hamiltonian through a fit to experimental levels in the model space, such that all
components of Fig. 1.3 except the explicit three-body term are effectively included.
Following this procedure, the USDB and USDA interactions were produced in the sd
model space, varying 66 parameters to fit over 600 energy levels with a root mean
square (rms) deviation of 130 keV and 170 keV, respectively [15]. Similar procedures
were performed by Cohen and Kurath in the p shell [16] and Honma et al. in the
pf shell [17]. While these effective interactions lose connection to the underlying
physics in the nuclear Hamiltonian, they account for many-body forces through their
modification of SPE and TBME. Calculations with a shell model code throughout
the standard model spaces agree with experiment on the order of the rms deviation
of the effective interactions. The most accuracy achieved to date is the 130 keV rms
deviation with the USDB interaction, while the Cohen and Kurath interaction has a
400 keV rms deviation [16].

1.2

Energy Density Functional Methods

The concept of the mean field has already been introduced. Hartree-Fock [18], [4] and
other self-consistent mean field methods apply the variational principle from quantum
mechanics to minimize the total energy of a system and construct the ground state
wavefunction. Density Functional Theory (DFT), as developed in condensed matter
physics and quantum chemistry [19], [20], is derived with a variational procedure in a
similar way to the Hartree-Fock method, except that the density is the basic variable.
DFT springs from the Hohenberg-Kohn theorems, which state that:
(i) the ground state properties of a many-particle system are uniquely determined

9

by the local density of the system, and
(ii) for a given Hamiltonian, a universal energy density functional exists that
defines the ground state when minimized.
Unfortunately, the theorems provide no insight on how to determine the functional.
DFT reduces the many-body problem to a system of non-interacting particles in an
effective potential, similar to the mean field methods discussed previously. Due to this
simplification, a single Slater determinant can be used to minimize the functional. The
Slater determinant is only a reference state, and therefore DFT is considered to be
independent of the wavefunction.
The problem is not as simple in nuclear physics. As particle number changes
from closed-shell to midshell, the ground state undergoes a phase transition due to
pairing, giving rise to collective modes. As a result, symmetry-breaking reference
states are implemented in the EDF minimization, which have not been proven to
satisfy the Hohenberg-Kohn theorems. For this reason, methods which follow a similar
procedure to DFT will be called single-reference EDF (SR-EDF), or more generally
EDF methods, in order to distinguish from DFT.
Such a formulation is attractive particularly in nuclear physics, since the nuclear
Hamiltonian is not known explicitly. The general prescription writes the energy

E[ρ] =

drE[ρ(r)]

(1.14)

where ρ is the one body density matrix with elements ρij given by
†

ψ | aj ai | ψ ,
or in the radial basis

(1.15)

A

| ψ(r) |2 .

ρ(r) =

(1.16)

i=1

Since the wavefunction can break symmetries such as particle number, the functional

10

can also include terms with the anomalous density- or its conjugate- with elements
κij given by
† †

ψ | aj ai | ψ .

(1.17)

Determining the form of the nuclear functional is a problem currently under investigation by groups throughout the world [21], [22], and is a goal of the U.S. Department of
Energy UNEDF collaboration [23], [24]. However, two common empirical approaches
developed by Decharge and Gogny [25] and Skyrme, Vautherin, and Brink [26], [27]
have been the basis for the majority of nuclear physics applications of EDF methods. In this work, only the latter approach of functionals, termed Skyrme EDFs, will
be used. The Skyrme EDF is given by local terms up to second order in ρ and its
derivatives around r = r for both isospin channels T as:
ρ

∆ρ

S
CT ρ2 + CT ST · ST + CT ρT ∆ρT
T

E[ρ] =
T =0,1

∆S
τ
ST
2
+ CT ST · ∆ST + CT (ρT τT − jT · jT ) + CT (ST · TT − JT )

+ CT J (ρT

jk
ijk JT · ri

·

+ ST ·

× j T ) + CT S (

· ST )2

(1.18)

ijk

where S is the spin density, r2 =
netic density, TT (r) =
i
2(

−

3
2
i=1 ri , τT (r)

=

· ρT (r, r ) |r=r is the ki-

· ST (r, r ) |r=r is the kinetic spin density, jT (r) =

i
)ρT (r, r ) |r=r is the current, and JT (r) = 2 (

−

) ⊗ ST (r, r ) |r=r

is the spin current.
Although complicated, the description can be simplified by connecting these coupling constants to the parameters of the Skyrme mean field interaction, as originally
derived in [26]. There is an explicit correspondence between the CT coupling constants and the so-called (t, x) parameterization of the Skyrme interaction, as related
by Stoitsov et al. [28]. In general, Skyrme EDFs are expressed in terms of the (t, x)

11

Table 1.1: Values of parameters for the Skx and Skxtb interactions.
Parameter
Skx
Skxtb
α
1/3
1/3
χw
0
0
χc
0
0
t0 -1445.3 -1446.8
t1
246.9
250.9
t2
-131.8
-133.0
t3 12103.9 12127.6
W0
148.6
153.1
x0
0.340
0.329
x1
0.580
0.518
x2
0.127
0.139
x3
0.030
0.018
parameters of the mean field interaction. The Skyrme interaction is given by
1
vskyrme = t0 (1 + x0 Pσ )δ + t1 (1 + x1 Pσ )(k∗ 2 δ + δk2 )
2
1
+ t2 (1 + x2 Pσ )k∗ · δk + t3 (1 + x3 Pσ )ρα (R)δ
6
+ iW0 (1 + χw Pτ )k∗ · δ(σ × k) + Vcoulomb
Wi,T {1 +

+ S12
i,T

3 e−ri
3
+ 2}
ri ri ri

(1.19)

1
where δ = δ(r − r ), R = (r + r )/2, k = 2i ( − ), k∗ is the adjoint of k, Pσ is
(S · r)2
the spin-exchange operator, and S12 = 2 3 2 − S2 is the tensor operator. The
r

last line of Eq. 1.19 is a particular parameterization of the tensor force, which is not
always included in the description of the Skyrme interaction.
Without the tensor force, the interaction has twelve parameters (ti , xi for i =
0, 1, 2, 3; α; W0 ; χw ; and χc , a parameter from the Coulomb interaction) that can be
fit to data. In general, the χw and χc parameters are fixed from physical considerations, and therefore ten parameters are varied. The parameters are given for the Skx
interaction [29] in Table 1.1, as an example.
Especially away from stability, the tensor force plays an important role in the
properties of nuclei. A Skyrme force including the tensor interaction was fit with the
12

Skx values as the initial parameters. The new parameterization, Skxtb [30], was refit
to the same data as Skx and is also displayed in Table 1.1.
With either parameterization, the energy can be found from a Hartree-Fock-like
minimization [18], which proceeds self-consistently in few iterations for spherical
closed-shell nuclei. The correspondence between the energy density functional and
the two-body interaction enables the minimization to be performed with respect to
either description. This is a particularly useful property of the Skyrme functional
which does not generally hold: while any Hamiltonian can be described in the ground
state by some energy density functional, a pure Hartree-Fock solution of the Hamiltonian need not reproduce the minimization of the EDF. This convenient property
of the Skyrme functionals will be exploited so that a so-called Skyrme Hartree-Fock
equation can be solved, without recourse to the functional form. When EDF methods are employed in this work, they will utilize the Skyrme interaction, solved in
coordinate space on a mesh.
While Skyrme EDFs are convenient and commonly used in the nuclear structure
community, a microscopic EDF that connects to the underlying two- and three-body
interactions is ultimately desired. The results in this work are dependent on single
particle energies calculated with Skyrme functionals, which are unreliable and deviate
significantly from experiment, as will be seen in Chapter 4. Improved single particle
properties could be implemented directly into the procedure described in Chapter 2,
and could be obtained from a microscopic description of the functional. The derivation
of microscopic EDFs is beyond the scope of this work, but is of great interest for the
applications in the following chapters.

1.3

Microscopic Nucleon-Nucleon Potentials

Several nucleon-nucleon potentials reproduce elastic scattering phase shifts of experimental data up to 350 MeV in a variety of channels and for all three types of in-

13

teraction (proton-proton, proton-neutron, and neutron-neutron). The potential is not
identical for the three cases; even after accounting for the primary cause of the difference due to the charge of the proton via the Coulomb force, small deviations between
the three cases still exist. The difference between neutron-neutron and proton-proton
scattering relates the charge asymmetry of the strong force, and is due to a chargesymmetry-breaking (CSB) interaction. Similarly, a difference between proton-neutron
scattering and the average of proton-proton and neutron-neutron scattering relates the
charge dependence of the interaction due to a charge-independence-breaking (CIB)
interaction.
The derived potentials all have a common feature: strong short-range tensor forces
and repulsion (the “hard core” of the nucleon-nucleon interaction) result in highly correlated many-body wavefunctions and highly non-perturbative few- and many-body
systems [31]. As a result, many-body perturbative approaches cannot be applied directly to microscopic nucleon-nucleon potentials to produce valence shell interactions,
as is done in chemistry for instance. There is not a single potential that can be derived to explain low-energy phemonema, as evidenced by the several potentials that
fit low-energy scattering data. In fact, there are an infinite number of potentials that
can be developed which deviate at higher energies but match the scattering data.
The differences in the two-body potentials result in unique many-body forces for each
potential. If the other degrees of freedom are included or calculations are limited to
low-energy behavior due to two-body forces, the choice of two-body potential will not
affect the results. Due to developments in many-body perturbative techniques, a twobody potential without a hard core is desired, such that model space interactions can
be produced for use in further calculations. As has been discussed in detail [31], any
potential can be evolved to reduce the high-momentum components (see Section 1.4).
As a result, the starting potential can be any derived microscopic nucleon-nucleon
potential, regardless of the hard core behavior, if renormalization group methods are
applied.

14

Results throughout this work will be obtained from the microscopic N N potentials
N3LO [32] and Argonne v18 [33]. The N3LO [32] interaction is derived from chiral
effective field theory (χEFT), beginning with the underlying quark physics expressed
through quantum chromodynamics (QCD). One advantage of χEFT is that two- and
three-body forces are generated on equal footing at a given order of perturbation
theory, such that the description can be continually improved by extending the calculation to higher orders of perturbation. As a low-momentum expansion based on a
sharp cutoff in momentum (500 MeV in [32]), χEFT provides an estimate of the theoretical uncertainty at a given order of perturbation by varying the cutoff. For N3LO,
the uncertainty in the two-body interaction due to contributions from higher orders
is 3% [32]. As the cutoff is decreased, the potential softens but the error increases.
The cutoff cannot be increased to reduce the error indefinitely, since a resolution scale
is set by the explicit treatment of long-range physics [31]. The 500 MeV cutoff is a
reasonable compromise between the pion mass which needs to be treated explicitly
and the rho meson mass which can be treated effectively through contact terms. The
current status of derivation is fourth order or next-to-next-to-next-to-leading order
(hence N3LO) for the N N potential, while the N N N potential has only been derived
at the N2LO level. In all, 29 parameters are varied to reproduce 4400 data points for
two-body elastic scattering with χ2 /Npt ≈ 1, where Npt is the number of data points.
The Argonne v18 potential [33] is composed of eighteen local operators expected
to contribute to the nucleon-nucleon interaction, such as central, spin-orbit, tensor,
etc. and includes fourteen charge-independent terms, a CIB interaction via three
charge-dependent terms, and a CSB interaction via one charge-asymmetric term.
The potential is solely a function of the separation variable r in each partial wave,
causing a strong short-range repulsion in S-waves and strong coupling between lowand high-momentum modes [31]. In all, 40 parameters are varied to reproduce 4300
scattering data with χ2 /Npt ≈ 1. A set of companion N N N potentials, called the
Illinois models, have been developed and provide good agreement for light nuclei with

15

exact ab initio methods [14].
As mentioned, many-body perturbative techniques cannot be applied to the Argonne v18 or N3LO potential directly, as the hard core and coupling between lowand high-momentum modes prevent convergent results. To produce an effective interaction in a reduced model space, the microscopic N N potential must be softened
with renormalization group methods.

1.4

Renormalization Group Methods

The renormalization of the free nucleon-nucleon (N N ) interaction into the nuclear
medium can only be calculated using many-body perturbative techniques if the highenergy degrees of freedom are decoupled from the microscopic potential. Renormalization group (RG) methods, an active area of nuclear structure research [31], [11], [34],
are designed to explain the decoupling and to transmit the relevant information to
the scale of interest through a finite number of parameters [35]. Results in this work
will implement two different RG methods, termed the G-matrix technique and the
vlowk approach.
The G-matrix technique applies Brueckner theory [36] to determine the ground
state of a many-body system from the perturbation due to the microscopic N N interaction. The energy shift in the ground state is obtained by a linked-cluster expansion
as expressed by the Goldstone linked-diagram theory [11]. The G-matrix is then defined relative to the microscopic N N potential V by the integral equation

G(ω) = V + V

Q
G(ω),
ω − H0

(1.20)

where H0 is the unperturbed Hamiltonian (without the potential), the Pauli operator
Q blocks intermediate states to prevent scattering into orbits occupied in the manybody system, and ω is the starting energy of the two nucleons. The problem can

16

be generalized if Q does not commute with H0 as assumed here. The expression
appears simple, but requires complicated many-body calculations for a solution. The
solution is dependent on a cutoff of active single particle states but can be evaluated
numerically through an iterative scheme; for full details, see [11].
The vlowk approach eliminates high-momentum modes in the microscopic N N
interaction through a sharp cutoff Λ in momentum space. In order to maintain consistency, the effects of the high-momentum modes must be incorporated into the
low-momentum behavior of the potential. Instead of merely truncating the potential, the high-momentum modes are integrated out. The condition of consistency is
achieved by demanding that the scattering T matrix for the initial potential vN N
remain unchanged as the cutoff is lowered, such that
∞v
2
N N (k , p)T (p, k) 2
p dp
P
π
k 2 − p2
0
Λ v Λ (k , p)T (p, k)
2
Λ
lowk
p2 dp
= vlowk (k , k) + P
π
k 2 − p2
0

T (k , k) = vN N (k , k) +

(1.21)

for all momenta k, k < Λ and
Λ
d Λ
2 vlowk (k , Λ)T (Λ, k)
vlowk (k , k) =
dΛ
π
1 − (k/Λ)2

(1.22)

from the requirement dT (k , k)/dΛ = 0 [31]. Since many microscopic potentials reproduce the low-energy scattering data but diverge for higher energies, they will collapse
to the same low-momentum potential as the cutoff is lowered to a sufficiently small
value (Λ ≈ 2.0 fm−1 ) [37]. The long-range pion exchange of the nuclear force is maintained in the vlowk interaction, while the short-range hard core has been integrated
out. The application of either of these RG methods converts the microscopic N N
potential into a potential amenable to perturbative techniques.

17

1.5

Many-Body Perturbative Techniques

The discussion in this work will be limited to time-independent Rayleigh-Schrödinger
perturbation theory, although the time-dependent version, which still leads to an
energy-independent expansion, will be used in calculations. As discussed in Section
1.1, the Hamiltonian can be split into two terms H = H0 + H1 by introducing an
appropriate one-body mean field potential so that H1 is small in comparison to H0 .
The solution to H0 , where Vmf is usually chosen as a harmonic oscillator potential,
can be written as a product of single particle wavefunctions ψi (r). A model space of
single particle orbits is chosen separately for protons and neutrons with the dimension
D. Projection operators
D

| ψi ψi |

(1.23)

| ψi ψi |

P =

(1.24)

i=1

Q=
i∈D
/

defining the model space and excluded space, respectively, span the single particle
basis. The solution to the full Schrödinger equation can be projected into the model
space such that
P | Ψ =| ΨM ,

(1.25)

with the transformation back to the exact wavefunction given by the correlation
operator χ via [11]
| Ψ = (1 + χ) | ΨM .

(1.26)

The effective interaction in the model space is then given by

ˆ
Vef f (χ) = Q(ω) − P H1

18

1
χV (χ),
ω − QHQ ef f

(1.27)

(i)

(ii)

(iii)

(iv)

ˆ
Figure 1.4: One-body Goldstone diagrams included in the evaluation of the Q-box
to second order in perturbation theory: (i) Hartree-Fock, (iii) two-particle one-hole
(2p1h), and (iv) one-particle two-hole (1p2h). The kinetic diagram (ii) does not conˆ
tribute to the Q-box but affects the determination of TBME. Only the direct term
is depicted for each diagram, but the contribution from exchange diagrams is also
included in calculations.
where
ˆ
Q(ω) = P H1 P + P H1 Q

1
QH1 P
ω − QHQ

(1.28)

ˆ
ˆ
is the so-called Q-box and ω is the starting energy defined in Section 1.4. The Q-box
can be expanded into a perturbative form as

ˆ
Q(ω) = P H1 P + P (H1

Q
Q
Q
H1 + H1
H1
H + . . .)P.
ω − H0
ω − H0
ω − H0 1

ˆ
For a full derivation of the effective interaction and Q-box, see [11].

19

(1.29)

(i)

(ii)

(iii)

(iv)

ˆ
Figure 1.5: Two-body Goldstone diagrams included in the evaluation of the Q-box to
second order in perturbation theory: (i) first order, (ii) core polarization, (iii) particleparticle ladder, and (iv) four-particle two-hole (4p2h). Only the direct term is depicted
for each diagram, but the contribution from exchange diagrams is also included in
calculations.

20

The equation can be solved in many ways, but the focus here will employ Goldstone
ˆ
diagrams in an iterative scheme with the folded-diagram method. Given the exact Qbox, the eigenvalues are found from an iterative scheme where

ˆ
λ1 = ω + Qexact (ω),
∞

λn = λ1 +
m=1

(1.30)

ˆ
1 dm Q
m
m (λn−1 − ω)
m! dω

(1.31)

ˆ
from [11]. In practice, the exact Q-box cannot be determined, and time-dependent
ˆ
Rayleigh-Schrödinger theory is necessary to motivate the approximate Q-box through
ˆ
a folding operator which connects the Q-box to a wave operator through active model
space states. This derivation goes beyond the perfunctory description given here, but
can be found in [11] along with the solution to the effective interaction, given by
(n)

∞

ˆ
Vef f = Q +
m=1

ˆ
1 dm Q (n−1) m
(V
)
m! dω m ef f

(1.32)

(0)
ˆ
for a degenerate model space with Vef f = Q. A description of rules to produce and

evaluate linked-valence diagrams can be found in [38]. The one- and two-body diaˆ
grams for the Q-box up to second order are included in Figs. 1.4 and 1.5, respectively.
The Goldstone diagrams have been derived up to third order as represented in [11]
and can be implemented in the renormalization of a microscopic N N potential. In Fig.
1.5, the diagrams are: (i) first order, (ii) core polarization, (iii) particle-particle ladder,
and (iv) four-particle two-hole. Without the folding procedure, the effective interaction to second order is given by TBME evaluated as a sum over the four two-body
diagrams. The folding procedure essentially blocks some orbits from being accessed,
causing (in general) a reduction in the size of the TBME in a summation including
contributions to infinite order. The one-body diagrams in Fig. 1.4 produce effective
SPE, but experimental values provide better results. The model space interaction,
(n)

consisting of experimental SPE and TBME from Vef f can be used directly in a shell
21

model code.

1.6

Hybrid Methods

CI theory and EDF methods, as outlined above, have extremely different properties
and methods of calculation. The CI method falls under the scope of wavefunction procedures to solve Eq. 1.1, where a given Hamiltonian in matrix form is diagonalized to
give eigenvalues and eigenvectors, while EDF methods only require a reference state
wavefunction to minimize the energy density functional. The two procedures are desirable for different reasons. The CI method produces a fully correlated model space
calculation with simple wavefunctions, expressed for instance as a product of single
particle harmonic oscillator wavefunctions. The deviation from experiment depends
on the interaction but ranges from 125-400 keV in standard model spaces. It is the
most accurate method currently applicable beyond the lightest nuclei, or those near
closed shells. However, it suffers from limitations in mass and excitation energy, requires an effective interaction for each model space of interest, and has depended on
empirical interactions (i.e. effective SPE and TBME) for results in reasonable agreement with experiment. On the other hand, EDF methods use the full single particle
space instead of a reduced model space, and produce results for all nuclei with one
single parameterization of the functional. The drawbacks of EDF methods are that
the form of the functional is not known, the parameterization of a particular form is
not universal, dynamic correlations are missing in the standard Skyrme and Gogny
functionals, and a full level scheme cannot be calculated with these standard functionals. Recall that in Section 1.2 the discussion was limited to SR-EDF. The biggest
disadvantage of EDF methods is the large deviation with respect to experiment, given
by an rms deviation of 1.2 MeV for ground states with a Skyrme EDF [23], which
can be reduced to 0.6 MeV with additional terms in the functional [39]. Combined
with the lack of complete description of excited states, a comparison of theoretical

22

and experimental level schemes is difficult throughout most of the nuclear chart.
In the last few decades, there have been attempts [40], [41] to combine aspects
of CI and EDF methods to produce a more robust theory which achieves advantages
of both approaches. The ultimate goal seeks to maximize the advantages of each
method while minimizing their respective faults. As it happens, the two approaches
have distinct advantages and disadvantages which are not mutually attainable. In
general, progress has been made in past attempts by groups with expertise primarily
in EDF methods, resulting in new approaches lacking many of the advantages of the
CI method.
For instance, Bender et al. [40] have produced a symmetry-conserving variational
approach called Variation after Mean field Projection In Realistic model spaces (VAMPIR). Like EDF methods, a symmetry-breaking auxiliary wavefunction is varied to
determine the lowest-energy states for each combination of spin and parity quantum
numbers. The Hamiltonian in the form of TBME is taken from an effective interaction
in a standard shell model space; for instance, a modified version of the USD interaction is used in [40]. Correlations are included by expanding the wavefunction around
a symmetry-projected ground state via one or two quasiparticle excitations (for odd
and even nuclei, respectively) and by diagonalizing the interaction in the resultant
basis. Like the CI method, this allows for configuration-mixing, and the restricted calculations that have been performed so far reproduce CI calculations. The application
of the VAMPIR approach to heavy nuclei would extend the reach of configurationmixing diagonalizations beyond the limits of the CI method (A ≈ 100 as shown in
Fig. 1.6). However, the current formulation is dependent on a shell model effective
interaction, such that results do not improve upon CI calculations in its region of
applicability.
The opposite approach is utilized here, focusing instead on improved determinations of nuclear Hamiltonians as influenced by renormalization group and EDF
methods. The main benefit, due to the relative ease of determining nuclear inter-

23

actions from realistic nucleon-nucleon potentials for any model space of interest, is
improved accuracy for nuclei outside of standard shell model spaces. The effective
interactions will be used as input to the shell model code NUSHELLX. As a result,
an immediate limitation of the approach prevents heavy nuclei (A > 100) from being
calculated, except those near doubly magic nuclei like 100 Sn, 132 Sn, and 208 Pb.

1.7

Motivation

A chart of nuclides is shown in Fig. 1.6, overlaid by various nuclear structure techniques and their declared ranges of validity. Ab initio approaches, which solve the
many-body Schrödinger equation (nearly) exactly with microscopic potentials, are
valid only for the lightest nuclei due to computational limitations. The two approximate techniques cover the rest of the chart, with the CI method extending to A ≈ 100
and EDF methods not limited by mass. While the CI method is computationally applicable in the region shown on the graph, the standard CI method as described in
Section 1.1 does not produce satisfactory results throughout this region.

24

Figure 1.6: A chart of nuclides marked with regions of validity of various nuclear structure approaches. The figure is taken from
the UNEDF website [24].

25

Fig. 1.7 displays results in the sd shell with the standard USDB interaction [15].
The interaction is symmetric with respect to isospin, so only nuclei on the neutronrich side are shown. Good agreement between experiment and theory, on the order of
the rms deviation of 130 keV, is seen for all nuclei except Z ≈ 10, N ≈ 20. The advent
of rare isotope beams sparked experimental interest in this neutron-rich region, such
that a better theoretical description was desired. Brown and Richter [15] included
recent experimental data for neutron-rich exotic energy levels in the fitting procedure
to create the USDB and USDA interactions, so that they would produce better results
for neutron-rich nuclei. Since the connection to microscopic physics is lost in the fitting
procedure, the empirical interaction does not have predictive power for nuclei with
different shell structure than the experimental data used in the fit. The disagreement
for N = 19 and N = 20 is understood, because these nuclei are in the so-called island
of inversion region. The island of inversion is a region of the nuclear chart near the
supposed shell closure N = 20 where pf neutron orbits are filled preferentially over sd
orbits in ground states. See Section 1.1 and specifically the right side of Fig. 1.2 for a
description of the single particle orbits and shell structure. Experimental evidence [42]
confirms this picture of inversion, and similar “islands of shell-breaking” have been
found or postulated in other regions of the nuclear chart [43], [44]. The lack of inclusion
of pf orbits, which create additional correlation energy, results in underbinding in the
USDB calculation for the island of inversion region. Even with an improved interaction
fit to more neutron-rich data, these nuclei will not be calculated accurately in the sd
model space. Theoretical results using the standard CI method can often be improved
by producing better interactions in a standard model space (for instance, by fitting to
more experimental data), but cannot necessarily be applied accurately to all states in
the region. Another example of disagreement due to contributions from orbits outside
the model space is seen in 18 O for the first excited 0+ state at 3.63 MeV and the
lowest 1− state at 4.46 MeV. In the sd model space, which treats 16 O as the core
and only allows valence particles to fill the positive parity sd orbits, negative parity

26

22
2.0
1.5
1.0
0.5
0

20

Proton Number

18
16

-0.5

14

-1.0
-1.5

12
10
8
6
6

8

10

12

14

16

18

20

22

Neutron Number

Figure 1.7: A plot of energy differences EU SDB − EExp. in MeV calculated for nuclei
in the sd shell, as shown by Brown and Richter in Fig. 10 of [15].
states do not exist and the 1− state cannot be calculated. The 0+ state, while positive
parity, is a four-particle two-hole excitation from 16 O [45], meaning that two p shell
nucleons are excited into the sd shell. The theoretical capabilities can be expanded
through an increase in the size of the model space, which requires the creation of a
new effective interaction. For empirical interactions, the determination of an effective
interaction is a lengthy procedure that requires the compilation of data. Especially
with larger model spaces, available data does not span the space and computational
resources limit the viability of full calculations.

27

Legend: Eexp(2+) (MeV)

pf

sd

42Si

p

Figure 1.8: A chart of nuclides by 2+ energy of even-even nuclei, limited to the range of applicability of the CI method. Also
included are standard shell model spaces, with 42 Si highlighted as an exotic isotope outside of standard model spaces.

28

Calcium
Silicon

2 + Energy (MeV)

4
3
2
1
0

18

20

22

24

26

28

30

Neutron Number

Figure 1.9: Excitation energy of the first 2+ state in even-even silicon and calcium
isotopes as a function of neutron number. Notice the difference in behavior for the
supposed magic numbers of N = 20 and N = 28.
Another demonstration where reliable results are difficult to obtain with the CI
method is given by 42 Si. As can be seen from Fig. 1.8, 42 Si lies in a neutron-rich region
outside of standard model spaces, in the middle of the sd shell for proton orbits (yaxis) and in the middle of the pf shell for neutron orbits (x-axis). First calculations
for 42 Si were done in the so-called sdpf model space, with active sd proton orbits and
pf neutron orbits. As can be seen from Fig. 1.8, little experimental data is known in
this region. Deriving an empirical interaction for this model space cannot follow the
procedure discussed in Section 1.1 since the number of parameters (SPE and TBME)
is comparable to the number of experimental energy levels. Instead, the proton-proton
TBME can be taken from an empirical sd shell interaction, and the neutron-neutron
TBME can be taken from an empirical pf shell interaction, leaving only a cross-shell
proton-neutron interaction which can be determined phenomenologically or via RG
methods. An early sdpf interaction [46] was derived in this way, with improvements
made by Nummela et al. [47] as more experimental properties of exotic nuclei were
measured. However, such an approach fails due to a concept commonly referred to

29

as the evolution of shell structure, where the mean-field picture given by Fig. 1.2 no
longer reproduces experimental behavior away from stability. One quantity used to
determine the experimental magic numbers is depicted in Fig. 1.9. The plot displays
the excitation energy of the first 2+ state for both silicon and calcium isotopes from
N = 18 to N = 30. Nuclei with magic numbers of nucleons have a high 2+ energy
relative to their neighbors, since more energy is required to excite a particle below the
Fermi sea. The discussion earlier on magic numbers cited 20 and 28 as experimentally
determined values. As seen in Fig. 1.9, the peaked behavior is seen at the N = 20
shell closure for both 34 Si and 40 Ca, but the results diverge for N = 28. Whereas
48 Ca

displays the behavior of a good shell closure, the 2+ energies are similar for 40 Si

and 42 Si (44 Si has not been measured). In this way, shell structure “evolves” near the
driplines, where it is no longer energetically favorable to add a neutron or proton (the
excess nucleon of a particular type “drips” off the nucleus). As the neutron dripline
is approached for neutron-rich nuclei, a change in magic numbers has been observed.
Throughout this work, reference to the evolution of shell structure refers to a change in
experimental observables that suggest a single particle mean-field with shell closures
different from the harmonic oscillator plus spin-orbit mean-field potential of Fig. 1.2.
The cause of the deviation in 2+ energies in Fig. 1.9 could be a weakening of the
spin-orbit force (recall from Section 1.1 that N = 28 is the first shell closure caused
by the spin-orbit force), or a change in the spacing between the p and f orbits in the
N = 3 oscillator shell. Calculations with the sdpf interaction of Retamosa et al. [46]
are nearly 2 MeV too high for the 2+ state of 42 Si. Even with the improvements
made by Nummela et al. [47], the SDPF-NR interaction misses the excitation energy
of the 2+ state in 42 Si by 720 keV. The neutron-neutron TBME were taken from an
empirical fit in the pf shell, therefore mimicking the behavior of pf orbits for stable
and near-stable isotopes, like the calcium isotopes. The exotic behavior of the neutron
pf orbits must be taken into account in order to calculate nuclei in the sdpf shell such
as 42 Si. A recent work by Nowacki and Poves [48] developed a new interaction called

30

SDPF-U that empirically accounts for the shell structure of exotic nuclei by having
different TBME for Z ≥ 15 and Z ≤ 14 (i.e., for stable and unstable shell structure,
respectively). The piecewise TBME enable the SDPF-U interaction to reproduce the
2+ states in 42 Si and 48 Ca within 50 keV. This topic will be revisited and treated in
more detail in Chapter 3.
The goal of this work is to use hybrid methods, specifically an improved determination of effective model space interactions through a renormalization of microscopic
N N potentials based on EDF methods, to produce reliable calculations of nuclear
properties, especially for nuclei outside of standard shell model spaces which are accessible with current or planned experimental rare isotope facilities.

1.8

Methodology and Layout

Four nuclear structure techniques have been discussed in this introduction which
will be implemented into the renormalization procedure; namely, Configuration Interaction theory, Energy Density Functional methods, renormalization group methods, and many-body perturbative techniques. The development of such a procedure
would not have been possible without the previous work in these fields, or without the generous collaborative spirit in the nuclear structure community which led
to the author’s access to Fortran source codes. The renormalization of microscopic
nucleon-nucleon potentials into the nuclear medium, implementing both renormalization group methods and many-body perturbative techniques, and including various
ingredients (choice of potential, RG method, order of perturbation theory, etc.), had
been developed by Morten Hjorth-Jensen and is accessible [49]. Minor modifications
to the code were included in order to replace the standard harmonic oscillator radial wavefunctions with those derived from EDF methods and to output the effective
interaction into the format required for the shell model codes utilized in this work.
These codes, NUSHELLX [9] and OXBASH [6], produce CI calculations and were

31

obtained from Alex Brown. Both codes include a Skyrme Hartree-Fock solver, which
employs the equal filling approximation for odd nuclei (see Section 4.4). The renormalization codes [49] have been integrated into the newest version of NUSHELLX [9]
by Alex Brown via the “ham” executable, including the modifications developed by the
author. The main programming component of this work was the development of an
iterative fitting procedure to constrain single particle energies and other parameters,
as discussed in Section 5.1. The primary contribution of the author was the development of the renormalization procedure, in conjunction with Alex Brown, as well
as the applications to produce effective interactions for nuclear structure calculations
discussed throughout this dissertation.
The remainder of this work has the following layout:
(i) Chapter 2 describes the renormalization procedure, and includes an application
for comparison to known results in the sd shell. The ingredients of the procedure are
varied, and their dependence on the output two-body matrix elements is evaluated.
(ii) Chapter 3 compares the standard harmonic oscillator single particle basis to
the realistic Skyrme Hartree-Fock basis and to a mixed basis, consisting of Skyrme
single particle energies and harmonic oscillator single particle radial wavefunctions.
The effects of the basis on the renormalization procedure are evaluated in the sdpf
model space with two different target nuclei. This chapter has previously been published [50] with few minor modifications in this work.
(iii) Chapter 4 discusses the meaning of single particle energies, both in the uncorrelated and centroid sense, and distinguishes both types from one-nucleon separation
energies. Results are shown for both Configuration Interaction theory and Energy
Density Functional methods to illustrate the general behavior of single particle energies, to display their role in effective interactions and nuclear structure calculations,
and to emphasize the necessity of accurate single particle energies in effective interactions. Various methods to determine single particle energies for the renormalization
procedure are specified at the conclusion of the chapter. Previous publications have

32

included the results of Section 4.2, for the oxygen and fluorine isotopes [51] and for
the argon and calcium isotopes [52].
(iv) Applications of the renormalization procedure have produced effective interactions in the island of inversion region and near 42 Si, and results obtained with these
effective interactions are displayed and compared to experimental data in Chapter 5.
The theoretical calculations have been utilized to predict the behavior of nuclei beyond the experimental capabilities with current rare isotope facilities; for instance,
the neutron dripline has been determined from the effective interaction in the island
of inversion region in Section 5.1. These applications of the renormalization procedure
are presented for the first time in this work, but are expected to result in at least one
publication, if not more, in 2011 or 2012.
(v) Finally, Chapter 6 provides a conclusion and outlook, focusing on future applications of the renormalization procedure, the potential addition of three-body forces,
and further studies of single particle behavior, specifically for the Skyrme interaction.

33

Chapter 2
Renormalization Procedure
With the various techniques available for nuclear structure described in the introduction, the development of a hybrid method for calculations outside of standard model
spaces can be achieved. This chapter will detail the procedure to renormalize a microscopic N N potential into the nuclear medium. The dependence on various aspects
of the method, such as the choice of RG method, will be evaluated, and limitations
or potential failures of the procedure will be discussed.
The following steps compose the hybrid method, hereafter called the Hybrid
Renormalization Procedure (HRP).
(i) Choose a model space and target nucleus. The model space is determined by the
region of the nuclear chart of interest. For 28 Si, for instance, which is in the middle
of the sd shell, the sd model space would be chosen. In principle, the interaction
can be determined for any model space. Computational limitations regarding the
diagonalization of a matrix in the eventual CI calculation, which depends on the
dimension of the matrix that increases with the size of the model space, require a
judicious choice of model space as described in Section 1.1. While the core of the
CI calculation is determined by the model space, the target nucleus can be inside
or outside the model space with only one restriction: the renormalization requires a
target with a closed subshell structure in the mean field. The summation over orbits

34

in the Goldstone diagrams distinguishes between particle and hole orbits. Therefore,
partially filled subshells, which cannot be classified as particle or hole orbits, are not
allowed. Often the core and target will be identical.
(ii) Calculate properties of the target nucleus in Skyrme Hartree-Fock theory. The
properties of the target nucleus, for instance its binding energy, single particle radial wavefunctions, and single particle energy spectra for protons and neutrons, are
calculated with a Skyrme interaction.
(iii) Choose the ingredients of the renormalization. The selection of the microscopic
potential (Argonne v18 or N3LO), renormalization group method (G-matrix or vlowk ),
basis (to be covered later), order of perturbation theory (up to third order), and
number of excitations in ω must be set at this stage.
(iv) Evolve the microscopic potential with the RG method. As discussed in Section
1.4, the decoupling of high- and low-momentum modes can be achieved, allowing
many-body perturbative techniques to act on the low-momentum interaction.
(v) Renormalize into the model space of interest. The low-momentum interaction from step (iv), in the form of two-body matrix elements, provides the input for
Rayleigh-Schrödinger perturbation theory. The model space interaction, again in the
form of TBME, is determined from the calculation of Goldstone diagrams to the order
of perturbation theory and number of excitations specified in step (iii).
(vi) Perform calculations with a shell model code. Using the TBME from step (v)
and the SPE from step (ii), the effective interaction can be used as the input to a CI
calculation in available shell model codes with the model space specified in step (i).
Results can be compared to experiment.
(vii) If necessary, make improvements. If the theoretical results do not provide
sufficient agreement with respect to experiment, the procedure can be improved. Due
to the poor single particle energies calculated with Skyrme interactions, the SPE
of the effective interaction can be taken from experiment or treated as parameters.
See Chapters 4 and 5 for details. Other improvements are increases in the order of

35

perturbation theory, the number of excitations, or the size of the model space.

2.1

Bases for Renormalization

In step (iii) of the HRP, the choice of basis is an ingredient to the renormalization.
If the renormalization were performed to all orders and infinite ω, the results would
not depend on the basis. With limitations, the renormalization depends on the accuracy of the expansion in terms of a single particle basis. The standard basis is the
harmonic oscillator basis, due to its simple analytic form and common use throughout
nuclear structure, for example as the basis for shell model calculations. The harmonic
oscillator basis is sufficient for most applications, but deviates from the realistic case
away from stability. Realistic bases, such as the Woods-Saxon basis, reproduce experimental behavior to first order better, requiring fewer terms in the expansion.
Furthermore, the renormalization scheme converges in fewer iterations for a realistic
basis in comparison to the harmonic oscillator basis. The HRP utilizes a realistic
basis from step (ii) in the procedure, rather than a Woods-Saxon basis, in an effort
to maintain consistency and implement EDF methods as completely as possible.
While the basis derived from EDF methods is superior to the harmonic oscillator basis for renormalization, the harmonic oscillator basis is the standard for the
nuclear structure community. The basis for renormalization remains an ingredient in
the procedure so that a comparison of results with different bases can be achieved.
Chapter 3 is devoted to this topic, emphasizing the importance of a realistic basis
away from stability. Three bases for the renormalization are considered: harmonic oscillator single particle energies and wavefunctions (HO), Skyrme Hartree-Fock single
particle energies and wavefunctions (SHF), and Skyrme Hartree-Fock single particle
energies and harmonic oscillator single particle wavefunctions (MIX).
The MIX basis and HO basis give identical results to first order in perturbation
theory since they use identical wavefunctions. The second order Goldstone diagrams in

36

Fig. 1.5 are affected since the energies, which are different in the two procedures, enter
higher-order diagrams via energy denominators, as discussed in [11]. The comparison
between the MIX and HO bases displays the dependence of the two-body matrix
elements on the single particle energies, and will be discussed in Chapters 3 and 4.
The harmonic oscillator basis is given by the solution of the harmonic oscillator
mean field potential, as discussed in Section 1.1, without the spin-orbit potential. The
wavefunctions are given by
HO
ψnlm (r) = Rnl (r)Ylm (θ, φ),
l

l

(2.1)

and the valence orbits are bound.
Skyrme Hartree-Fock radial wavefunctions, once solved on a mesh as discussed in
Section 1.2, are implemented in the renormalization by using an expansion in terms
of the harmonic oscillator basis via:
SHF
ψnlj (r) =

HO
an Rnl (r)[Yl (θ, φ) ⊗ χs ]j ,

(2.2)

n

where a2 gives the percentage of a specific harmonic oscillator wavefunction comn
ponent in the Skyrme Hartree-Fock wavefunction. The Skyrme Hartree-Fock wavefunctions and single particle energies can only be determined for bound states. For
unbound orbits, the harmonic oscillator basis remains in use, but the Gram-Schmidt
process is used to ensure orthonormality of the single particle wavefunctions.
The Hybrid Renormalization Procedure, as stated in step (i), can be applied to
any model space and closed-subshell target nucleus of interest. A proof of principle
can be established by comparing to a standard model space, such as the sd model
space with 16 O as the core and target nucleus, which will also allow for a comparison
between different renormalization options, as shown throughout this chapter.

37

2.2

Application to sd Model Space

For the first application, the Hybrid Renormalization Procedure will be studied in a
standard model space. The steps will be listed explicitly once to clarify the HRP.
(i) Choose a model space and target nucleus. The model space is the sd shell and
the target nucleus is the doubly magic nucleus 16 O.
(ii) Calculate the target nucleus in Skyrme Hartree-Fock theory. Using the Skxtb
Skyrme interaction [30], the solution to Hartree-Fock equations is found for 16 O.
(iii) Choice of renormalization ingredients. For this application, the N3LO potential, vlowk RG method, and SHF basis will be used. Rayleigh-Schrödinger perturbation
theory will be applied to second order and 6 ω.
(iv) Evolve the microscopic potential with the RG method. Following the procedure
in Section 1.4, the N3LO potential is renormalized using vlowk .
(v) Renormalize into the model space of interest. Following the procedure in Section 1.5, the low-momentum interaction is renormalized to second order and 6 ω.
(vi) Perform calculations with a shell model code. NUSHELLX [9] is used to calculate 18 O in the sd model space with the effective interaction called HRP-SD, comprised
of TBME from step (v) and SPE from step (ii).
(vii) If necessary, make improvements. For an example, the procedure is repeated
with third order in perturbation theory, with the remaining ingredients of the procedure identical. See Section 4.3 for results with improved single particle energies.
The results for 18 O are shown in Fig. 2.1. The low-energy spectrum is reproduced
reasonably well, while the experimental level density above 3.5 MeV is much higher
than the theoretical level density. The negative parity experimental states, as well
as positive parity states with an even number of excitations from the p shell, cannot
be reproduced with the choice of the positive parity sd orbits outside the 16 O core
as discussed in Section 1.7. These excitations of the core occur around 4 MeV, such
that the description of valence neutrons outside the 16 O vacuum fails. Therefore, the

38

6

E (MeV)

5
4

4+

4+

3
2

2+

2+

1
0

0+ −12.19 MeV
Exp.

18 O

0+ −16.08 MeV
HRP − SD

18 O

Figure 2.1: Comparison of the level schemes for experiment (left) and for the interaction HRP-SD (right). The color of the lines displays the parity, red for positive and
blue for negative. The length of the line corresponds to the angular momentum of
the state, with a label each time a highest J value occurs, starting from the ground
state. Therefore, the ground state spin is always labeled. The ground state energy
of the nucleus relative to the core (16 O in this case) is included for both theory and
experiment, with the excitation energy of the states independent of the difference in
binding energy. Experimental states are taken from the ENSDF database [53] unless
otherwise cited. States in the ENSDF database without an assigned J π value are
plotted as black dots at the appropriate energies. The level schemes are displayed up
to 7 MeV, although the theoretical level scheme includes a finite number of states for
a given J π value (ten unless otherwise noted). As a result, nuclei with a high density
of states might not show the full theoretical level scheme through 7 MeV.

39

6

E (MeV)

5
4

4+

4+

3
2

2+

2+

1
0

0+ −12.19 MeV
Exp.

18 O

0+ −16.42 MeV
HRP − 3rd

18 O

Figure 2.2: Results in comparison to Fig. 2.1, extending the renormalization to third
order in perturbation theory. See the caption to Fig. 2.1 for details.
agreement between experiment and theory breaks down at high excitation energy.
The excitation energy of the 4+ state, as calculated with HRP-SD, is 265 keV
lower than experiment. The experimental two-particle 0+ state at 5.34 MeV occurs
at an excitation energy of 6.94 MeV. One possibility for improvement is to produce
an interaction HRP-3rd by repeating the HRP to third order in perturbation theory,
as discussed in step (vii) of this section. The results are shown in Fig. 2.2. The first
excited theoretical 0+ state now has an excitation energy of 6.37 MeV, 560 keV lower
in energy than in Fig. 2.1 but still one MeV higher than experiment. The 4+ state
is calculated 165 keV lower in energy, now with a deviation from experiment of only
100 keV. As the empirical interactions in the sd shell produce an rms deviation of
approximately 150 keV, as discussed in Section 1.1, the agreement with experiment
is now satisfactory. However, in both Figs. 2.1 and 2.2, the binding energy of 18 O
as determined from the Hybrid Renormalization Procedure is approximately 4 MeV
larger than experiment. The small improvement in the excitation energy spectrum
as a result of the extension to third order in perturbation theory fails to address a
more significant problem in the interaction: the single particle energies from the Skxtb

40

HRP−SD TBME (MeV)

4
neutron−neutron
orbits only
2

0

-2

-4

-4

-2

0

2

4

USDB TBME (MeV)
Figure 2.3: Comparison of the neutron-neutron TBME for the microscopic HRPSD interaction and the empirical USDB interaction. The solid line y = x denotes
equivalence between the matrix elements for the two interactions.
interaction are not sufficient for the desired accuracy. The necessary improvement to
the HRP in this case requires improved SPE, which necessitates a detailed discussion
on the meaning and behavior of SPE. This topic will be revisited in Chapter 4, where
a method to determine SPE for the effective interaction will be detailed.
The first nucleus outside the 16 O core which depends on the TBME of the interaction is 18 O. Thus, it is the simplest case for comparison. To determine the quality of
results throughout the sd shell, rather than perform lengthy calculations for all nuclei,
a comparison to the empirical USDB interaction is sufficient, since it produces good
results throughout the sd shell (see Fig. 1.7). Because the USDB interaction is symmetric with respect to isospin, while the renormalization procedure includes Coulomb,
charge-independence-breaking (CIB), and charge-symmetry-breaking (CSB) interac-

41

tions, neutron-neutron TBME are best-suited for comparison. Fig. 2.3 shows the
neutron-neutron TBME, plotted as HRP-SD vs. USDB. For perfect agreement between the two interactions, all TBME would lie along the line of equality (y = x).
Instead, it serves as a reasonable trendline, with scattering in both directions on the
order of 100 keV. One point, corresponding to the J = 0 d3/2 − d3/2 pairing matrix
element, disagrees by 1.3 MeV, which would have a significant effect on calculations
in the sd model space. It should be noted that the TBME are not expected to agree
for the two interactions, as the empirical interaction includes contributions from all
effects in the sd shell, such as many-body forces, while the HRP includes only a
renormalization of a microscopic N N potential. Since three-body forces contribute
at about an order of magnitude lower than two-body forces and the deviation between the empirical interaction and the microscopic interaction is on the order of
10%, the HRP produces results to the expected accuracy, and the proof of principle
is completed.
To avoid treating three-body forces explicitly, which would be very difficult in
the procedure delineated above, two approximate methods will be employed. When
possible, the nuclei of interest will be close in proximity to the target nucleus on the
chart of nuclides given in Fig. 1.8. Skyrme interactions are fit to experimental data
to reproduce properties of nuclei throughout the chart of nuclides. As a result, they
effectively include three-body forces, and the inclusion of single particle energies in the
SHF basis in the HRP contains contributions of many-body forces up to the effective
one-body level. As shown clearly in Fig. 7 of [54], the remaining residual three-body
force and the effective two-body component of the N N N interaction contribute at
another order of magnitude smaller than the effective one-body component. In fact,
this is the primary reason for the lack of consistency in the initial interactions, since
a Hartree-Fock calculation could be performed on the microscopic N N interaction
once the hard core has been softened with an RG method in step (iv) of the HRP.
However, a Hartree-Fock calculation with the low-momentum interaction would not

42

Argonne v18 microscopic interaction

8
6
4
2
0
-2
-4
-6
-8

-8

-6

-4

-2

0

2

4

6

8

N3LO microscopic interaction
Figure 2.4: Two-body matrix elements when the microscopic nucleon-nucleon interaction in an otherwise identical HRP is the N3LO interaction (x-axis) or the Argonne
v18 interaction (y-axis).
include the essential effective one-body component of the three-body (and higher)
terms of the interaction. If the calculated nuclei are similar to the target nucleus, the
three-body effects are approximately taken into account with the SHF basis.
The second method to treat three-body forces implicitly is only employed in this
work in Chapter 5. When nuclei far from the target are calculated, the three-body
forces are treated by parameterizing the monopole terms of the CI calculation. This
will be discussed in detail in Section 5.1.
At this point, the sd model space and 16 O target will be used to analyze the effects
of different parameters in the HRP on the output two-body matrix elements.

43

2.3

Dependence on Microscopic Interaction

The first ingredient of interest is the choice of the initial microscopic N N interaction.
The HRP was performed identically to Section 2.2, except with Argonne v18 as the
microscopic N N interaction. Fig. 2.4 shows the comparison of all TBME for this
new interaction to the HRP-SD interaction. Even though the N3LO and Argonne
v18 interactions have very different behavior at high energy, they both agree with
available scattering data up to 350 MeV as discussed in Sections 1.3 and 1.4. The
HRP decouples the high-momentum modes, such that the resulting low-momentum
interaction is nearly identical regardless of the initial microscopic N N potential. Fig.
2.4 is an application of the results shown in Fig. 17 of [31], which displays identical
low-momentum interactions when vlowk is applied to various phenomenological N N
potentials. Therefore, the choice of intial potential will not affect the results in this
work. The N3LO potential is preferred due to its connection to the underlying theory
of QCD and its expansion based on power counting in χEFT.

2.4

Dependence on RG method

The next ingredient of interest is the renormalization group method of choice. The
HRP was performed identically to Section 2.2, except with the G-matrix technique as
the RG method. Fig. 2.5 shows the comparison of all TBME for this new interaction
and the HRP-SD interaction. Again, the TBME lie along the line of equality with
small deviation. While there is more deviation than Fig. 2.4, the agreement is still
sufficient to assume that both RG methods accurately decouple the high-momentum
modes from the low-momentum modes, such that the low-momentum interaction
input for many-body perturbative techniques is acceptable. The vlowk renormalization
is preferred because the G-matrix technique takes approximately six times longer for
the application to the sd model space and depends on a somewhat arbitrary starting
energy.
44

8

G − matrix renormalization

6
4
2
0
-2
-4
-6
-8
-8

-6

-4

-2

0

2

4

6

8

vlow k renormalization
Figure 2.5: Two-body matrix elements for the vlowk renormalization group method
(x-axis) and for the G-matrix RG method (y-axis) in an otherwise identical HRP.

45

8
6

target nucleus

-2

28 Si

4

-4

2
0

-6
-8

-8

-6

-4
16 O

-2

0

2

4

6

8

target nucleus

Figure 2.6: Two-body matrix elements when the core in an otherwise identical HRP
is 16 O (x-axis) or 28 Si (y-axis).

2.5

Dependence on Target Nucleus

In order to compare the matrix elements, the model spaces must be identical. However,
the choice of target nucleus is available for comparison and is important to study since
multiple targets are available for each model space. The HRP-SD interaction was
developed in a similar way to standard CI techniques, with the target as the vacuum
in the CI calculations. If midshell nuclei are of interest, the nearest possible target,
fulfilling the closed-subshell requirement, is 28 Si. The HRP was performed identically
to Section 2.2, except with 28 Si as the target nucleus. In this application, the target
nucleus is not the same as the vacuum in the CI calculation, which is always 16 O
in the sd shell. Fig. 2.6 shows the comparison of all TBME for this new interaction
and the HRP-SD interaction. The deviation from the line of equality is still relatively
small, and occurs in both directions. While there is more deviation than in either of

46

the options studied already, it can be traced to the change in the SHF basis for the
different nuclei, which is partially due to three-body effects. At the same time, the
agreement is still sufficient to choose a target in the middle of the model space, rather
than the core, for calculations throughout the sd shell. As mentioned previously, the
preference for any nucleus of interest is the closest target, in order to account for
many-body effects as accurately as possible. It is important to note at this point
that the agreement between the interactions derived for these two targets in the sd
shell occurs because of the similarity in their behavior. Both targets are stable nuclei
with similar shell structure. Moving away from stability, shell structure evolves as
discussed in Section 1.7, even changing the experimental magic numbers. Results for
exotic targets, as well as the study of the effects of the different bases in the HRP,
will be discussed in Chapter 3.

2.6

Dependence on Skyrme Interaction

Another aspect of the renormalization to consider is the choice of Skyrme parameterization. As discussed in Section 1.2, two parameterizations of the Skyrme interaction
are Skx and Skxtb, with similar values for the parameters listed in Table 1.1. Since
the experimental data from the doubly magic 16 O nucleus constrained the fits for
Skx and Skxtb, and the difference in the two interactions is mainly due to the tensor force, which only has a significant effect on single particle energies away from
stability [55], the SPE for 16 O are nearly identical. Later applications will deal with
exotic nuclei and therefore Skx and Skxtb will present larger deviations, an effect
discussed in Chapters 4 and 5. For this application, a comparison to another Skyrme
interaction is of interest and is presented in Fig. 2.7. The HRP is followed identically
to Section 2.2 except with Sly4 [56] as the Skyrme interaction. Here the deviation
follows a trend towards reduction with Sly4 as the Skyrme interaction. The SPE with
the Sly4 interaction vary from the Skxtb and Skx values on the order of MeV because

47

8

Sly4 Skyrme interaction

6
4
2
0
-2
-4
-6
-8

-8

-6

-4

-2

0

2

4

6

8

Skxtb Skyrme interaction
Figure 2.7: Two-body matrix elements when the Skyrme interaction in an otherwise
identical HRP is the Skxtb interaction (x-axis) or the Sly4 interaction (y-axis).

48

8
6

Third order

4
2
0
-2
-4
-6
-8
-8

-6

-4

-2

0

2

4

6

8

Second order
Figure 2.8: Two-body matrix elements to second order (x-axis) or to third order (yaxis) in perturbation theory in an otherwise identical HRP.
the effective mass of the Sly4 interaction is reduced by 20%, which results in a lower
single particle density as discussed in Section 4.4. The single particle wavefunctions
will also be affected, and these changes in the basis reduce the TBME on the order
of 5-10%, as seen in Fig. 2.7. Therefore, the single particle energies and radial wavefunctions as determined from Skyrme parameterizations are an important aspect of
the calculation and will be treated in more detail in Chapters 3 and 4.

2.7

Dependence on Order of Perturbation Theory

As analyzed by Hjorth-Jensen et al., and displayed in Figs. 1 and 2 of [57], the renormalization in the HO basis does not converge through third order in perturbation theory, which is as far as the Goldstone diagram representation of Rayleigh-Schrödinger

49

perturbation theory has been derived for applications to nuclear structure. However,
the renormalization in a realistic Brueckner-Hartree-Fock basis nearly converges at
second order in perturbation theory, even though the results for 18 O and 18 F are in
worse agreement with experiment than the calculations with the HO basis. Their realistic basis neglects many-body forces, so the Hybrid Renormalization Procedure in
the SHF basis should produce better results with similar convergence properties. Fig.
2.8 compares the TBME for different orders of perturbation theory. The third order
increases the strength of the TBME, such that convergence has not been achieved.
The effect of the stronger interaction due to third order diagrams on 18 O can be seen
from a comparison of Figs. 2.1 and 2.2, which shows that the calculation does not depend strongly on the order of interaction. Second order will be treated as sufficiently
converged throughout this work.

2.8

Number of Excitations

To display the concept of excitations, Fig. 2.9 shows the expected ground state configurations of neutrons and protons for the 16 O target. Similar plots will be shown later,
so the salient features of the figure will be discussed. The target nucleus and Skyrme
interaction are displayed at the top of the figure. On the left, the proton single particle
orbits as determined from the Skyrme Hartree-Fock calculation are shown, labeled
by (2j). Occupied orbitals are marked by dots, with a total occupation of 8 protons
shown at the bottom of the figure. The occupation of a particular orbital is given by
(2j + 1), so each of the 0s1/2 , 0p3/2 , and 0p1/2 orbits is fully occupied. The black box
shows model space orbits, in this case the sd orbits. In the center of the figure, the
same situation is shown for neutrons. For 16 O with the Skxtb interaction, the results
for neutrons are similar to protons with the energy shifted down by approximately 4
MeV. On the right side of the figure, the results for the harmonic oscillator mean field
potential are shown, with ω determined by the mass (see Section 1.1). The absolute

50

20

16 O

sk28−skxtb
3

10

f7
p1
p3

d3

E (MeV)

0

-10

d3

s1
d5

s1
d5

2

p1
p1

p3

1
p3

-20
s1
s1

-30

0

-40
8
− parity
8
protons + parity neutrons
-50

Figure 2.9: Single particle orbits for 16 O, as calculated with the Skxtb interaction for
protons and neutrons and with a harmonic oscillator potential. Details are included
in the text.

51

20

16 O

sk28−skxtb
3

10

f7
p1
p3

d3

E (MeV)

0

-10

d3

s1
d5

s1
d5

2

p1
p1

p3

1
p3

-20
s1
s1

-30

0

-40
8
− parity
8
protons + parity neutrons
-50

Figure 2.10: A pair of protons are excited one oscillator shell to produce a configuration representing a 2 ω excitation of the Skyrme Hartree-Fock ground state of Fig.
2.9.

52

20

16 O

sk28−skxtb
3

10

f7
p1
p3

d3

E (MeV)

0

-10

d3

s1
d5

s1
d5

2

p1
p1

p3

1
p3

-20
s1
s1

-30

0

-40
8
− parity
8
protons + parity neutrons
-50

Figure 2.11: In this case, one neutron is excited two oscillator shells to produce a
configuration representing a 2 ω excitation of the Skyrme Hartree-Fock ground state
of Fig. 2.9.

53

energy of the harmonic oscillator orbits are chosen to correspond approximately to
the Skyrme results. The oscillator shell is listed by the main quantum number N . For
all three columns, the positive parity orbits are red and the negative parity orbits are
blue.
From Fig. 2.9, which shows the ground state configuration as determined in a
mean field calculation with Skyrme Hartree-Fock theory, excited configurations are
produced by promoting either protons or neutrons into higher orbits. An excitation
of one nucleon from the N = 1 p shell to the N = 2 sd shell corresponds to an
increase of one ω in the total energy of the configuration in the harmonic oscillator
potential. Figs. 2.10 and 2.11 display 2 ω configurations where, respectively, two
protons are excited into the next oscillator shell and one neutron is promoted into the
next oscillator shell of the same parity. A cutoff of 6 ω allows all excitations of protons
and neutrons which combine to 6 ω or less, requiring the N = 8 oscillator shell in
the calculation (from the greatest excitation possible of one model space nucleon).
For the number of excitations, the work of Shurpin et al. [58] shows that a folded
diagram procedure in perturbation theory with a self-consistent basis accelerates the
convergence of the interaction. Sommermann et al. [59] have shown that core polarization converges rapidly with modern potentials, such that 2 ω excitations are sufficient.
Fig. 2.12 displays a comparison of TBME for 4 ω and 6 ω excitations, while Fig. 2.13
compares 6 ω and 8 ω excitations. In each case, the values of the TBME increase
when more excitations are included. When the calculation is converged, an increase
in the number of excitations will not affect the TBME. Computational restrictions
prevent a determination of the number of excitations required for convergence. The
inclusion of more excitations will produce stronger interactions, but as will be seen in
Section 5.1, an overall normalization parameter will be introduced to account for the
approximately linear dependence on the ingredients of the renormalization. In Section
5.1, a reduction in the strength of the interaction is necessary. If enough excitations
were included for convergence, a smaller overall normalization factor would be neces-

54

8
6
4

6hω

2
0
-2
-4
-6
-8
-8

-6

-4

-2

0

2

4

6

8

4hω
Figure 2.12: Two-body matrix elements including 4 ω (x-axis) or 6 ω (y-axis) excitations in an otherwise identical HRP.

55

8
6
4

8 hω

2
0
-2
-4
-6
-8
-8

-6

-4

-2

0

2

4

6

8

6 hω
Figure 2.13: Two-body matrix elements including 6 ω (x-axis) or 8 ω (y-axis) excitations in an otherwise identical HRP.

56

sary. For computational reasons, excitations up to 6 ω will be included throughout
this work to produce reasonable results.

2.9

Issues

Regardless of the ingredients of the Hybrid Renormalization Procedure, there are
certain anomalies that arise in the procedure that can affect the accuracy of the output
two-body matrix elements. In general, the concern relates to divergences in the TBME
from calculations of Goldstone diagrams that contain energy denominators equal to
or nearly equal to zero. Several situations can lead to divergences, and modifications
to avoid these non-physical divergences must be considered. The methods employed
in this work are detailed so that the results can be replicated.
The energy denominator for the 2p1h diagram in Fig. 1.4 will diverge in the
harmonic oscillator basis if the hole line is 2 ω below the model space orbit and each
particle line is an ω excitation. The procedures to calculate diagrams are detailed
in [11] and [38]. For the example in the sd model space, the diagram for the d5/2 orbit
diverges in the HO basis if the particles are in the pf shell and the hole is in the 0s1/2
orbit. Even in the non-degenerate SHF basis, energy differences can approach zero,
based on nearly constant spin-orbit splitting in different oscillator shells, for example.
The folded diagram procedure is designed to overcome these types of divergences.
ˆ
Derivatives of the Q-box are evaluated to fifth order. The model space orbits
ˆ
are all set to the same valence energy such that the Q-box and its derivatives are
evaluated at the same energy for all contributing terms. The starting energy ω in each
diagram is set at twice the valence energy of the model space, where the valence energy
D

is determined from

(2j + 1) nlj . For convenience, since only energy differences

i=1

affect the results, a consistent starting energy of -10 MeV is applied in all cases.
The proton and neutron single particle spectra, as calculated in either the HO or
SHF basis, are shifted independently so that the valence energies are -5 MeV. As

57

ˆ
discussed in [11], calculating the Q-box to higher orders should reduce the dependence
on starting energies, but it can diverge due to intruder states outside of the model
space. These divergences could be avoided with a larger model space, but prohibitively
large calculations can result. Third order diagrams depend on the starting energy to a
greater degree than second order diagrams. The generation of divergences occurs more
readily when perturbative techniques are applied to third order, further supporting the
restriction to second order in Rayleigh-Schrödinger perturbation theory throughout
this work.
A large model space with orbits in multiple oscillator shells for one type of nucleon
will lead to divergences in this approach, since all model space orbits are set to a
constant valence energy. For instance, if the p and sd shells were included in a model
space with 16 O as the target, divergences would result. This can be seen from the
particle-particle ladder diagram in Fig. 1.5, for diagrams where p holes in the model
space are excited to sd particles in the model space with the same energy. Therefore,
when such a model space is necessary, the model space orbits will be placed in a
common oscillator shell. The choice of shell can be determined on a case-by-case
basis. This issue will only arise in Section 5.1 of this work and the specific treatment
will be discussed there.
In addition to the issues already discussed, the choice of vlowk as the renormalization group method requires the evaluation of additional parameters. Four parameters
enter the calculation, as seen from the definition of the vlowk procedure in Section
1.4. First, a cutoff Λ must be chosen. Second, a limit to the integration must be implemented, since the non-analytic integral cannot be solved from 0 to ∞. Finally, the
integrations, from zero momentum to the cutoff momentum and from the cutoff momentum to the integration limit, must be performed on a mesh in momentum space.
Therefore, a specific number of mesh points is chosen for each integration. The cutoff
should be Λ ≈ 2 fm−1 , as motivated in Section 1.4. The integration limit is set to
20 fm−1 , which is more than sufficient to account for the low-energy behavior of the

58

8
6

Λ = 2.2 fm−1

4
2
0
-2
-4
-6
-8
-8

-6

-4

-2
0
2
Λ = 2.0 fm−1

4

6

8

Figure 2.14: Comparison of the two-body matrix elements for a cutoff momentum of
Λ = 2.0 fm−1 (x-axis) or Λ = 2.2 fm−1 (y-axis) in an otherwise identical HRP.

59

microscopic N N potentials fit to 350 MeV. Sixty integration points are utilized for
both integrations. This produces a fine mesh at low-momentum (low-energy) and a
coarse mesh above the cutoff momentum. The TBME have been found to depend on
the number of integration points and the integration limit at the order of 10 eV. In
other words, the choices of these parameters do not affect the TBME, which are sufficiently converged to the order necessary for calculations at the keV level. However,
the cutoff does affect the results. Fig. 2.14 displays a comparison between the HRPSD interaction, where Λ = 2.0 fm−1 , and a new interaction which follows the same
HRP as Section 2.2 but employs a cutoff Λ = 2.2 fm−1 . The trend shows a reduction
in strength of the TBME as the cutoff is lowered, with significant deviation from the
line of equality. The cutoff dependence is related to the truncation of “induced” manybody forces [31], which contribute on the order of 15%. These induced many-body
forces are independent of the explicit three-body forces not taken into account with
a procedure based on the renormalization of a microscopic N N potential. See Fig.
35 of [31] for an example of the size of the various contributions as a function of the
cutoff. The cutoff dependence provides an estimate of a theoretical uncertainty from
the vlowk renormalization, but will not be studied in detail.

2.10

Standard Implementation

At this point, all ingredients of the Hybrid Renormalization Procedure have been
evaluated except for the choice of basis, which will be covered in detail in the next
chapter. A standard approach can be established for future implementations of the
HRP. In the standard implementation, the N3LO interaction will be converted to a
low-momentum interaction using vlowk with a sharp cutoff of Λ = 2.0 fm−1 , and will
then be renormalized with Rayleigh-Schrödinger perturbation theory to second order
and 6 ω with a realistic basis derived from the Skxtb interaction. In order to prevent
divergences, a common valence energy is applied to all model space orbits, equal to

60

half the starting energy of -10 MeV. When orbits of both positive and negative parity
are included in the model space for either protons or neutrons, all valence orbits of
that type of nucleon will be placed into a single oscillator shell.
Unless otherwise stated, the standard implementation will be assumed when discussing or referring to the HRP throughout the remainder of this work.

61

Chapter 3
Comparison of Bases
New facilities for rare isotope beams will push the experimental capabilities of nuclear
physics with radioactive beams to more unstable, shorter-lived nuclei. Properties of
these nuclei exhibiting different behavior than stable nuclei, like the evolution of shell
structure, are of significant interest for the next decades of research. The production of
model space interactions can be compared in stable and exotic regions of the nuclear
chart to examine the importance of refining theoretical approaches for unstable nuclei.
In the past, the conversion of a realistic N N interaction into an interaction in the
nuclear medium has typically employed renormalization methods in the harmonic oscillator basis for a doubly magic core. For more exotic closed-subshell nuclei, loosely
bound orbits often play a role. Loosely bound orbits particularly deviate from the
oscillator basis, by exhibiting a “long-tail” behavior with extended radial wavefunctions. The harmonic oscillator basis is therefore less applicable further from stability.
However, few calculations have been done with a realistic radial basis for unstable
nuclei with renormalized N N interactions.
Experimental interest in neutron-rich silicon isotopes and the failure of some shell
model Hamiltonians to reproduce data in the region have led to modifications in the
SDPF-NR interaction [47], which had been the standard for shell model calculations
in the sdpf model space. The new SDPF-U interaction [48] has different neutron-

62

neutron pairing matrix elements for Z ≥ 15 and Z ≤ 14 to account for the behavior
of pf neutron orbits relative to the number of valence protons, as discussed in Section
1.7. The Z ≤ 14 version of the interaction treats neutron-rich unstable nuclei that
exhibit different shell behavior than the less exotic nuclei in the Z ≥ 15 nuclei. The
interest in silicon isotopes and the nature of the SDPF-U interaction make 34 Si a
suitable choice for the renormalization procedure with a realistic basis. A similar
effect occurs for the neutron-rich carbon isotopes around the N = 14 closed subshell,
requiring a 25% reduction in the neutron-neutron two-body matrix elements from the
effective interactions derived for the oxygen isotopes [60].

3.1

Application to sdpf model space

A deeper understanding of the need for multiple interactions in the sdpf model space,
as seen by the form of SDPF-U, can be gained by performing the renormalization for
the same model space in multiple ways. The active orbits in the sdpf model space are
the sd proton orbits and pf neutron orbits. The Hybrid Renormalization Procedure
is followed for two different target nuclei in all three bases, producing a total of six
interactions. The two target nuclei are the stable 40 Ca doubly magic nucleus and
the neutron-rich 34 Si semi-magic nucleus. Figs. 3.1 and 3.2 display the single particle
bases for 40 Ca and 34 Si, respectively. Furthermore, the single particle energies of the
SHF basis, using the Skxtb interaction, are presented in Table 3.1 for both target
nuclei. For an SHF state that is unbound, the radial wavefunction is approximated
by a state artificially bound by 200 keV, obtained by multiplying the SHF central
potential by a factor larger than unity. The energy of the unbound state is estimated
by taking the expectation value of this bound state wavefunction in the original SHF
potential.
In the SHF basis, the calculation of single particle energies shows that the proton
orbits are shifted down in energy for 34 Si relative to 40 Ca, while the neutron orbits

63

20

40 Ca

sk28−skxtb
10

g9
d3
d5
g9
f5
p1
p3

f5
p1
p3

E (MeV)

0

-10

-20

-30

f7
d3
s1

4

3

f7

d5

d3
s1

p1
p3

2

d5
p1
p3

s1

s1
-40

1

0

20
− parity
20
protons + parity neutrons
-50

Figure 3.1: Single particle orbits for 40 Ca as calculated with the Skxtb interaction.
See Fig. 2.9 and discussion in Section 2.8 for details.

64

20

34 Si

sk28−skxtb
d5
g9

10

E (MeV)

0

-10

g9
f5
p1
p3

f5
p1
p3
f7
d3
s1
d5

1

s1

s1

2

p1
p3

f7
d3

3

0

d5
-20

p1
p3

-30
s1
-40

14
− parity
20
protons + parity neutrons
-50

Figure 3.2: Single particle orbits for 34 Si as calculated with the Skxtb interaction. See
Fig. 2.9 and discussion in Section 2.8 for details.

65

Table 3.1: Single particle energies from the solution to the Skyrme Hartree-Fock mean
field for 34 Si and 40 Ca using the Skxtb interaction. Values in bold are in the model
space.
n j

34 Si

34 Si

40 Ca

40 Ca

0s1/2
0p3/2
0p1/2
0d5/2
0d3/2
1s1/2
0f7/2
0f5/2
1p3/2
1p1/2
0g9/2
0g7/2

proton
-37.73
-27.60
-22.39
-17.29
-9.08
-13.49
-5.97
3.70
-1.06
1.49
6.39
18.26

neutron
-32.79
-23.10
-21.74
-13.07
-9.03
-10.04
-2.62
3.33
-0.40
-0.27
9.22
18.23

proton
-30.49
-22.14
-19.03
-12.79
-7.23
-8.31
-2.68
4.81
1.44
3.27
8.63
18.76

neutron
-38.18
-29.70
-26.67
-20.20
-14.65
-15.75
-9.89
-2.43
-5.48
-3.66
1.15
10.28

Table 3.2: Single particle energies for 34 Si and 40 Ca in the harmonic oscillator basis.
The energy shift is chosen so that the valence energy is identical in both bases. Values
in bold are in the model space.
n j

34 Si

34 Si

40 Ca

40 Ca

0s1/2
0p3/2
0p1/2
0d5/2
0d3/2
1s1/2
0f7/2
0f5/2
1p3/2
1p1/2
0g9/2
0g7/2

proton
-36.93
-25.42
-25.42
-13.91
-13.91
-13.91
-2.40
-2.40
-2.40
-2.40
9.11
9.11

neutron
-34.59
-23.09
-23.09
-11.58
-11.58
-11.58
-0.07
-0.07
-0.07
-0.07
11.44
11.44

proton
-32.22
-21.20
-21.20
-10.18
-10.18
-10.18
0.84
0.84
0.84
0.84
11.86
11.86

neutron
-39.21
-28.19
-28.19
-17.17
-17.17
-17.17
-6.15
-6.15
-6.15
-6.15
4.87
4.87

66

5

Energy (MeV)

f5/2
p 1/2
p 3/2
f7/2

0

f5/2
p 1/2
p 3/2

-5

f7/2

-10

40 Ca

34 Si

-15

Neutron Orbits in SHF Basis
Figure 3.3: Single particle neutron orbits for 40 Ca and 34 Si as calculated with the
Skxtb interaction. The same valence orbits are shifted to higher energy for the exotic,
neutron-rich 34 Si relative to 40 Ca, such that the valence orbit radial wavefunctions
are no longer well-described by harmonic oscillator wavefunctions.

34 Si

neutron radial wavefunctions

r R(r)

p 1/2

f7/2

r R(r)

p 3/2

f5/2

0.0

0.0

HO
SHF
0

2

4

6

8 10 0

radius (fm)

2

4

6

8 10

radius (fm)

Figure 3.4: Comparison of the single particle radial wavefunctions for 34 Si in the HO
and SHF bases.

67

are shifted up. For the valence neutrons, this shift results in a switch from four orbits
bound by 5.4 MeV on average for 40 Ca to four orbits centered at 0.0 MeV for 34 Si. The
shift is highlighted in Fig. 3.3, which shows the single particle neutron orbits in the
SHF basis for 40 Ca and 34 Si. This change, specifically the loosely bound energies of
the p3/2 and p1/2 orbits, has a significant effect on the wavefunctions, as seen in Fig.
3.4. The harmonic oscillator radial wavefunctions fall off rapidly at large distances,
while the realistic radial wavefunctions exhibit long tail behavior, extending beyond
10 fm. The effects of the shape of the wavefunction will be discussed in more detail
later.
For comparison, the single particle energies used in the HO basis are given in Table
3.2. The Blomqvist-Molinari formula [5] ω = (45A−1/3 −25A−2/3 ) MeV gives 11.508
MeV for A = 34 and 11.021 MeV for A = 40. The absolute value of the harmonic
oscillator basis is irrelevant, as only energy differences come into the diagrams in
Rayleigh-Schrödinger perturbation theory. For a better comparison to the SHF basis,
D

(2j+1) α

the absolute value is chosen separately for protons and neutrons such that
1

is identical in the HO and SHF bases, where D, the number of valence orbits, is three
for protons and four for neutrons and α is the energy of the single particle orbit given
by the α = n, l, j quantum numbers.
Fig. 3.5 shows a comparison of the pf matrix elements in MeV when the Hybrid
Renormalization Procedure is followed for both target nuclei in the HO basis. The
J = 0 pairing matrix elements are singled out in color due to their importance in
ground state properties of even-even nuclei. The TBME in Fig. 3.5 deviate slightly
from the line of equality but agree well with each other. Therefore, the choice of
target nucleus, whether 34 Si or 40 Ca, has little effect on the matrix elements in the
HO basis. In the explanation of Nowacki and Poves [48], the SDPF-U interaction
has different neutron-neutron pairing matrix elements for Z ≥ 15 and for Z ≤ 14
to account for 2p2h excitations of the core correctly, depending on whether 34 Si or
40 Ca

should be considered the core. The SDPF-U neutron-neutron pairing matrix

68

1

neutron−neutron
orbits

-1

34 Si

HO basis

0

J> 0
J= 0 ff − ff
J= 0 ff− pp
J= 0 pp− pp

-2

-3

-3

-2

-1
40 Ca

0

1

HO basis

Figure 3.5: Comparison of pf neutron-neutron matrix elements (in MeV) for the renormalization procedure in the HO basis for the two target nuclei. Black dots correspond
to matrix elements with J > 0, while the J = 0 matrix elements are split into three
groups: f f − f f (crosses), f f − pp (diamonds), and pp − pp (plus signs). The solid
line y = x denotes where the matrix elements would be identical.

69

1

neutron−neutron
orbits

40 Ca

SHF basis

0

-1

J> 0
J= 0 ff − ff
J= 0 ff− pp
J= 0 pp− pp

-2

-3

-3

-2

-1
40 Ca

0

1

HO basis

Figure 3.6: Comparison of pf neutron-neutron matrix elements (in MeV) for the
renormalization procedure for 40 Ca in the HO and SHF bases. Black dots correspond
to matrix elements with J > 0, while the J = 0 matrix elements are split into three
groups: f f − f f (crosses), f f − pp (diamonds), and pp − pp (plus signs). The solid
line y = x denotes where the matrix elements would be identical.

70

1

neutron−neutron
orbits

34 Si

SHF basis

0

-1

J> 0
J= 0 ff − ff
J= 0 ff− pp
J= 0 pp− pp

-2

-3

-3

-2
40 Ca

-1

0

1

SHF basis

Figure 3.7: Comparison of pf neutron-neutron matrix elements (in MeV) for the
renormalization procedure in the SHF basis for the two target nuclei. Black dots
correspond to matrix elements with J > 0, while the J = 0 matrix elements are split
into three groups: f f − f f (crosses), f f − pp (diamonds), and pp − pp (plus signs).
The solid line y = x denotes where the matrix elements would be identical.

71

elements are reduced by 300 keV for Z ≤ 14 in order to produce results in better agreement with experimental data. However, the change in target in the Hybrid
Renormalization Procedure mimics the change in core for the SDPF-U interaction,
and yet the TBME do not show a reduction in the HO basis. While the occupation
of proton orbits has changed, the neutron-neutron pairing matrix elements are not
affected. Diagrams involving excitation of protons are unlinked, and are not included
in Rayleigh-Schrödinger perturbation theory. Still, the reduction in pairing matrix
elements is empirically necessary and should result from a microscopic treatment of
the interaction.
For the stable 40 Ca target, where the valence orbits are all bound by multiple MeV,
the HO basis reproduces the realistic nucleus well as expected from the discussion in
Section 1.1. The results in Fig. 3.6 are consistent for 40 Ca regardless of the choice of
basis. However, in Fig. 3.7, where the comparison is for both target nuclei in the SHF
basis, a reduction in the strength of the interaction for 34 Si is observed. For instance,
the f7/2 f7/2 V f7/2 p3/2 matrix element coupled to J = 2 is -0.642 MeV with 40 Ca
as the target but -0.419 MeV with 34 Si as the target in Fig. 3.7. This 35% reduction
in the matrix element responsible for deformation in neutron-rich silicon isotopes has
a significant effect on the behavior of nuclei outside the 34 Si core. The reduction with
34 Si

as the target nucleus seen generally in Fig. 3.7 is due to the weakly bound nature

of the pf neutron orbits.
In the SHF basis, the f7/2 orbit is bound by 2.6 MeV, and its radial wavefunction
agrees reasonably well with the harmonic oscillator wavefunction as seen in Fig. 3.4.
The Skyrme wavefunction is expanded in the harmonic oscillator basis up to n =
nmax

nmax and the an coefficients are renormalized to ensure that
n=0

a2 = 1. For the
n

Hybrid Renormalization Procedure, orbits up to (2n + l) = 9 are included, which
gives nmax = 3 for the f7/2 and f5/2 orbits and nmax = 4 for the p3/2 and p1/2
orbits. These values include over 99% of the strength for the f orbits, but only 93%
and 92% for the p3/2 and p1/2 orbits respectively. A first order calculation can be

72

done to nmax = 6 for all orbits, which gives 100%, 98%, and 97% for the f , p3/2 , and
p1/2 expansions respectively.
HO
With this procedure, 99% of the f7/2 orbit is represented by the R03 waveHO
HO
function, but the 1% represented by R23 and R33 has a significant effect at large

radii. The p3/2 and p1/2 orbits are only bound by 400 and 269 keV, respectively. The
HO
expected harmonic oscillator component R11 only makes up 80% and 78% of the

respective radial wavefunctions. Higher n orbits which extend farther away from the
center of the nucleus contribute the remaining strength. The f5/2 orbit is unbound
by three MeV, but the solution for the Skyrme radial wavefunction is determined by
assuming that the orbit is bound by 200 keV. With this method, 97% of the realistic
HO
radial wavefunction is given by the R03 wavefunction. Single particle radial wave-

functions of valence space neutron orbits, shown in Fig. 3.4 in both the HO and SHF
basis, enter directly in the evaluation of TBME. The long tail behavior of the realistic
basis is evident, as the wavefunctions have significant strength beyond 8 fm unlike
the oscillator wavefunctions, especially for the loosely bound p orbits.
The J = 0 matrix elements in Fig. 3.7 deviate more from the line of equality, i.e.
the pairing matrix elements are most reduced for 34 Si when the N3LO interaction is
renormalized in the SHF basis. The pairing matrix elements in Fig. 3.7 are reduced
for the 34 Si target by 214 keV on average, relative to the case with 40 Ca as the target,
in comparison to the 300 keV reduction of the SDPF-U interaction. The reduction
of 214 keV is due solely to the change in occupation of the d5/2 proton orbit, as the
target nucleus is the only change in the HRP for the two interactions in Fig. 3.7. However, the change in occupation affects not only the single particle energies and radial
wavefunctions, but also the available orbits in second-order diagrams. The change in
single particle radial wavefunctions plays the most significant role, as suggested by
Figs. 3.5 and 3.7, but the contribution of each effect can be analyzed.
Table 3.3 isolates a few matrix elements and compares the total matrix elements
and their various components for both target nuclei in all three bases. Three diagrams

73

Table 3.3: First order, particle-particle ladder, core polarization, four-particle twohole, and total matrix elements in MeV of the form aa V bb J=0 for different renormalization procedures.
34 Si

a

b

f7/2

f7/2

f7/2

p3/2

f7/2

p1/2

p3/2

p3/2

p3/2

p1/2

first
2p-ladder
core pol.
4p2h
total
first
2p-ladder
core pol.
4p2h
total
first
2p-ladder
core pol.
4p2h
total
first
2p-ladder
core pol.
4p2h
total
first
2p-ladder
core pol.
4p2h
total

HO
-0.906
-0.409
-0.449
-0.376
-1.855
-0.518
-0.148
-0.121
-0.123
-0.800
-0.585
-0.042
-0.055
-0.057
-0.665
-1.109
-0.233
-0.037
-0.085
-1.319
-1.540
-0.060
0.068
-0.048
-1.456

MIX
-0.906
-0.414
-0.377
-0.442
-1.869
-0.518
-0.157
-0.118
-0.141
-0.822
-0.585
-0.050
-0.041
-0.058
-0.663
-1.109
-0.242
0.001
-0.093
-1.313
-1.540
-0.069
0.082
-0.051
-1.462

74

40 Ca

SHF
-0.807
-0.422
-0.417
-0.435
-1.824
-0.322
-0.119
-0.074
-0.104
-0.552
-0.421
-0.039
-0.043
-0.037
-0.492
-0.776
-0.165
0.000
-0.057
-0.931
-0.857
-0.078
0.045
-0.026
-0.863

HO
-0.938
-0.409
-0.637
-0.374
-1.957
-0.518
-0.152
-0.282
-0.126
-0.903
-0.572
-0.049
-0.212
-0.062
-0.767
-1.096
-0.233
-0.021
-0.090
-1.267
-1.478
-0.068
-0.038
-0.053
-1.469

MIX
-0.938
-0.418
-0.649
-0.434
-1.982
-0.518
-0.164
-0.309
-0.142
-0.926
-0.572
-0.062
-0.236
-0.060
-0.779
-1.096
-0.237
-0.005
-0.098
-1.252
-1.478
-0.081
-0.069
-0.054
-1.488

SHF
-0.870
-0.450
-0.768
-0.499
-2.084
-0.368
-0.138
-0.283
-0.143
-0.749
-0.452
-0.038
-0.231
-0.052
-0.642
-1.082
-0.242
-0.037
-0.109
-1.278
-1.382
-0.077
-0.080
-0.056
-1.409

contribute at second order, denoted as core polarization, particle-particle ladder, and
four-particle two-hole, shown diagrammatically in Fig. 1.5. The total matrix element
is not a simple summation of the first and second order diagrams, as the folded
diagram procedure tends to reduce the matrix element via the blocking of orbitals
[11]. The reduction for total matrix elements involving p orbits in the SHF basis
with 34 Si as the core is dramatic (≈ 30%) and is primarily due to the extension of
wavefunction strength to large distances. Kuo et al. [61] noted a reduction of core
polarization in the harmonic oscillator basis and used different oscillator parameters
to account for the core nucleons and valence nucleons separately in halo nuclei. While
34 Si

is not a halo nucleus, the loosely bound p orbits behave in much the same

way as the valence nucleons in a halo nucleus. The reduction in core polarization
is seen going from the 40 Ca target to the 34 Si target in any basis in Table 3.3,
although the size of the polarization is reduced for nucleons far from the core. As
noted in [61], the core interacts less with nucleons far away, so the excitations of
the core are reduced. The core polarization for matrix elements solely involving p
orbits is under 100 keV. Nowacki and Poves [48] attributed the reduction in neutronneutron pairing matrix elements for the empirical SDPF-U interaction to a decrease
in core polarization for the 34 Si target. Table 3.3 shows that the core polarization
can be reduced significantly without a proportional reduction in the total matrix
element. For instance, the f7/2 f7/2 V f7/2 f7/2 matrix element is only reduced by
5% from 40 Ca to 34 Si in the HO basis even though the core polarization is reduced
by 30%. In the SHF basis, which takes into account the realistic wavefunction, the
total matrix element is reduced by 13% while the core polarization is reduced by 46%.
For this matrix element, the reduction due to core polarization is 188 and 351 keV
in the HO and SHF basis, while the first order reduction due only to the change in
wavefunction is 32 and 63 keV. For the p3/2 p3/2 V p3/2 p3/2 matrix element, and
other TBME involving only loosely bound p orbits, the core polarization becomes very
small, skewing percentage comparisons. However, the reduction in core polarization

75

is negligible relative to the wavefunction contribution of 306 keV in the first order
diagram for the SHF basis. There is no reduction for the first order diagram in the HO
basis. Ogawa et al. [62] produce seemingly consistent results, identifying a 10%-30%
reduction in nuclear interaction matrix elements involving loosely bound orbits using
a realistic Woods-Saxon basis. However, they were limited to comparisons of ratios of
matrix elements and did not include core polarization. Core polarization suppression
and reduction due to spread of the wavefunctions are both important effects which
should be included, as well as the other diagrams which contribute at second order.
The full treatment of the renormalization in a realistic basis, as developed here, is
necessary for accurate results. With the improvements based on the HRP, calculations
for neutron-rich silicon isotopes can be performed directly.

3.2

Calculations for

36

Si and

38

Si

The effect of the different interactions on nuclear structure calculations can be evaluated as neutrons are added to 34 Si. In order to obtain a consistent starting point, the
proton-proton and proton-neutron matrix elements of SDPF-U have been used, with
proton SPE that reproduce those obtained by SDPF-U. Because SDPF-U overbinds
35 Si,

the SDPF-U neutron SPE have been increased by 660 keV. The six interactions

use neutron SPE that reproduce the values of this modified SDPF-U interaction.
The only difference in the six interactions used in the calculations are the neutronneutron matrix elements. Calculations have been done in the sdpf model space with
the shell model code NUSHELLX [9]. Fig. 3.8 shows the lowest J = 0, 2, 4, 6 states in
36 Si,

relative to 34 Si. The HO basis and the MIX basis deviate by no more than 20

keV for the same target nucleus. However, the SHF basis noticeably shifts the states,
with the largest effect being 300 keV less binding in the ground state with 34 Si as the
target nucleus. The binding energy of 36 Si changes by nearly 500 keV depending on
which renormalization procedure is used. Furthermore, the level schemes for 36 Si are

76

-4
6+

Energy relative to 34Si (MeV)

-5
4+
-6

-7

2+

-8

-9

0+
36 Si

-10

34 Si

34 Si

34 Si

40 Ca 40 Ca 40 Ca

HO MIX SHF HO MIX SHF
Figure 3.8: Calculations of the lowest-energy states for J = 0, 2, 4, 6 in 36 Si relative
to 34 Si from the renormalization procedure for 34 Si and 40 Ca, in the HO, MIX, and
SHF bases for both target nuclei. Crosses are used for calculations with 40 Ca as the
target nucleus.

77

-4
6+

Energy relative to 34Si (MeV)

-5
4+
-6

-7

2+

-8

0+

-9
36 Si

-10

Z>

Z<

SDPF−U

34 Si 40 Ca 34 Si 40 Ca

Exp.

SHF SHF HO HO

Figure 3.9: Calculations of the lowest-energy states for J = 0, 2, 4, 6 in 36 Si relative to
34 Si using the empirical SDPF-U interaction and the renormalization procedure for
both 34 Si and 40 Ca as target nuclei, using the SHF and HO bases. Experimental data
is shown for comparison, with a new mass from [63]. Crosses are used for calculations
with 40 Ca as the target nucleus.
more spread out for the crosses where 40 Ca is chosen as the target nucleus.
Fig. 3.9 shows the same states in 36 Si relative to 34 Si, but now the comparison
includes the SDPF-U calculations and experimental data. The MIX basis results
are not included since they are so similar to the HO basis calculations. The level
scheme for 36 Si is more spread out for the Z ≥ 15 SDPF-U calculation than for the
Z ≤ 14 calculation, in agreement with the results discussed above. Calculations for
each method are in reasonable agreement with the comparable SDPF-U calculation;

78

Energy relative to 34Si (MeV)

-14

-15
2+
-16

-17

0+
38 Si

-18
Z>

Z<

SDPF−U

34 Si 40 Ca 34 Si 40 Ca

Exp.

SHF SHF HO HO

Figure 3.10: Calculations for the lowest-energy states for J = 0 and J = 2 in 38 Si
relative to 34 Si using neutron-neutron matrix elements from SDPF-U and the renormalization procedure for both 34 Si and 40 Ca as target nuclei, in the SHF and HO
bases. Crosses are used for calculations with 40 Ca as the target nucleus.

79

the 0+ state differs most with about 300 keV more binding compared to the respective
empirical interaction for each core. The experimental binding energy relative to 34 Si
is taken from a new mass measurement of 36 Si which is 140 keV higher in energy than
previously measured [63]. The excitation energies of the Z ≤ 14 SDPF-U calculation
are comparable to experiment, as expected from an interaction fit specifically to
neutron-rich silicon isotopes. While no interaction reproduces the experimental data
very well, general trends can be seen. The calculations with 40 Ca as the target nucleus
depicted by crosses result in level schemes that are too spread out in comparison to
the experimental data. The reduction in the strength of the interaction for 34 Si using
the SHF basis results in a reduction of the energy of the states in 36 Si, especially for
the ground state (the pairing matrix elements were most reduced). The rms deviation
between experiment and theory with 34 Si as the target nucleus in the SHF basis is
223 keV for the four states shown. One reason for this deviation is the lack of three
body forces in the procedure. The inclusion of the N N N interaction, at least via
the effective two-body part, is important for a higher level of accuracy. Additionally,
the chosen SPE may contribute to the deviation, which would be better constrained
if all the single particle states in 35 Si were known experimentally. For exotic nuclei,
the calculated single particle state is often above the neutron separation energy and
determination of the experimental single particle states may not be possible with
current facilities. Thus it is essential to improve energy density functionals such that
they provide reliable single particle energies.
As more nucleons are added, the disagreement between the various models increases as seen in Fig. 3.10, which plots the level scheme of 38 Si relative to 34 Si. Only
the 0+ and 2+ states are shown since the 4+ and 6+ states are not known experimentally, but the binding energy is best reproduced by the calculations with 34 Si in the
SHF basis. As noted in the 36 Si case, the excitation energy of the 2+ state is too high
in the SHF basis but is reproduced well by the Z ≤ 14 SDPF-U calculation for 38 Si.
While the theoretical calculations of the binding energy vary by more than 750 keV

80

for 36 Si in Fig. 3.9, the effect gets magnified as more particles are added. The binding
energy of 38 Si varies by 1.8 MeV for twice the number of valence nucleons. Accurately accounting for the two-body interaction in the exotic medium is essential for
calculations of exotic nuclei. The renormalization of a microscopic nucleon-nucleon
interaction into the nuclear medium with a realistic basis and an appropriate target nucleus offers an improvement in the description of exotic nuclei, resulting in a
decrease in the strength of the interaction and less binding in exotic nuclear systems.
The renormalization was performed in three different bases: harmonic oscillator,
Skyrme Hartree-Fock, and a mix of harmonic oscillator and Skyrme Hartree-Fock.
The choice of basis can significantly affect the value of matrix elements, as shown
in the comparisons of pf neutron-neutron matrix elements for the stable 40 Ca and
the neutron-rich 34 Si nuclei. The difference primarily results from the long tail of the
radial wavefunctions relative to the harmonic oscillator wavefunction, especially for
valence orbits bound by only a few hundred keV. The loosely bound orbits cause a
reduction in the overall strength of the interaction, an effect that becomes magnified
as full scale shell model calculations are performed. Accounting for the properties
of the orbits by using a realistic basis is essential for an accurate description of the
nuclear interaction in exotic nuclei as determined by the renormalization of an N N
interaction, but N N N forces must be included for accuracy at the level of 100 keV.

81

Chapter 4
Single Particle Energies
Single particle energies enter the Hybrid Renormalization Procedure in two ways:
through the basis in the calculation of Goldstone diagrams and through the effective
interaction in the CI calculation. The dependence on SPE via Goldstone diagrams is
minimal, as evidenced by the calculations in Chapter 3. Tables 3.1 and 3.2 display SPE
for 34 Si and 40 Ca in the SHF basis and HO basis, respectively. Although the energies
deviate on the order of MeV in the two different bases, the two-body matrix elements
resulting from the calculation of Goldstone diagrams in Rayleigh-Schrödinger perturbation theory are not significantly affected by the energy differences. The MIX basis,
which utilizes harmonic oscillator wavefunctions and SHF SPE, can be compared to
the HO basis to quantify the contribution due to changes of the SPE on the order of
MeV. In Table 3.3, the greatest deviation between a total matrix element in the HO
and MIX bases for the same target is 25 keV. For a comparison with different targets,
the core polarization diagram of Fig. 1.5 must be excluded since the occupation of
proton orbits affects the size of the core polarization. For the remaining components of
the two-body matrix elements (first order, particle-particle ladder, and four-particle
two-hole), the deviations are still on the order of tens of keV, even though small
changes in the HO wavefunctions for the two targets contribute as well.
The second form of SPE dependence, as a part of the CI calculation, will influence

82

the results to a much larger degree. Therefore, it will be essential to evaluate the
reliance on SPE as accurately as possible.

4.1

Definition of Single Particle Energy

Unfortunately, the term “single particle energy” is used throughout the nuclear structure community to describe several concepts, and so far in this work has not been
treated carefully. As a result, a division into two categories, effective and uncorrelated SPE, is necessary. Uncorrelated SPE are defined as the solution to the mean
field Hamiltonian in Eq. 1.6, where the energies represented by α are completely
independent. As seen from Eqs. 1.5 and 1.8, the energy difference between a nucleus
with mass number A and a nucleus with mass number A + 1 is αA+1 . For instance,
E(17 O) − E(16 O) = 0d
assuming the standard filling of orbits as depicted in Fig.
5/2
2.9. Uncorrelated SPE do not correspond to an experimental quantity, because the
single particle picture is only an approximate representation of physical nuclei. Uncorrelated SPE in the CI method are defined as the difference in energy between a
single-reference calculation for the nucleus with mass number A and a single-reference
calculation for the A + 1 nucleus with the additional particle in a particular orbit. For
example, in the sd shell with 16 O as the core, the calculation of 17 O with the USDB
interaction is displayed in Fig. 4.1. The 5/2+ ground state and the 1/2+ and 3/2+
excited states are the only possible levels in the theoretical calculation since there is
only one particle in the valence space. The selection of the sd model space enforces the
single-reference nature of the 16 O and 17 O wavefunctions, such that no correlations
from configuration mixing are included in the calculation. The energies of the states
represent uncorrelated SPE for the model space orbits. As discussed in Section 1.1,
effective interactions in standard model spaces often implement experimental energy
differences as SPE. In particular, the USDB effective interaction treats the SPE as
parameters, and reproduces the lowest experimental states in 17 O as seen in Fig. 4.1.

83

9/2−

5

E (MeV)

USDB

Exp.

6

4
3
2
1
0
17 O

5/2+
− 4.14 MeV

17 O

5/2+
− 3.93 MeV

Figure 4.1: Comparison of the 17 O level schemes for experiment and for the USDB
interaction. See the caption to Fig. 2.1 for details.
The accuracy of treating the ground state of 17 O as a 0d5/2 neutron coupled to
a single Slater determinant, occupying the N = 0 and N = 1 oscillator shells, can
be evaluated from experimental considerations. Spectroscopic factors for the addition
and removal of a nucleon can be defined generally as
†
+
Sα =| ΨA+1 | aα | ΨA |2

(4.1)

−
Sα =| ΨA−1 | aα | ΨA |2 ,

(4.2)

where the particle number has been singled out in the wavefunction. For a nucleus
with even mass number A, the spectroscopic factors will be referred to generally as
S, with the distinction between addition and removal apparent from the final state.
To evaluate the treatment of 16 O as a single Slater determinant, the spectroscopic
factor S + is of interest, with A = 16 and α = 0d5/2 . A spectroscopic factor of unity
would denote the accuracy of the single particle picture, such that the experimental
energy would provide an uncorrelated SPE α . However, physical states never achieve
a unit spectroscopic factor as defined in Eq. 4.1. A neutron transfer reaction, such as

84

17 O(p,d)16 O,

will probe the behavior of the “last” neutron in 17 O and can be used

to determine spectroscopic factors. For this reaction, the spectroscopic factor of the
ground state to ground state transition ranges from 0.74 to 0.99, with a proposed
best value of 0.81 [64]. In Section 1.1, the mean field description of nuclei motivated
the description of 16 O as a good core, such that 16 O could be treated as a vacuum.
Even though the spectroscopic factor shows that other configurations contribute significantly, calculations throughout the sd shell confirm the validity of the shell model
approximation. Local and short-range correlations contribute in nuclei; local correlations can be quantified by the behavior of valence particles in the sd model space,
while short-range correlations are missing and contribute to physical properties at
higher excitation energies (> 5 MeV as estimated from the energy to break the 16 O
core). Configuration mixing must be taken into account to produce local correlations
and to understand properties such as the fragmentation of spectroscopic strength.
Uncorrelated SPE for a Skyrme interaction are determined by the solution of the
Skyrme Hartree-Fock equation for a particular nucleus. These uncorrelated SPE are
determined in the potential well of the spherical mean field of the nucleus. As a result,
the solution to one nucleus provides the entire single particle spectrum. The Skyrme
SPE cited in the previous chapters were uncorrelated SPE determined in Skyrme
Hartree-Fock theory.
An effective single particle energy (ESPE) is provided by the prescription of
Baranger [65], who defines a centroid energy as
+
−
−
+
(Eo − Ef )C 2 Sα + (2J + 1)(Ef − Eo )C 2 Sα

f

f

α

=

f
−
+
C 2 Sα + (2J + 1)C 2 Sα

.

(4.3)

f

f

f

For the 16 O 0d5/2 neutron ESPE, a sum over final states f in 17 O (S + ) and 15 O
−
(S − ) is necessary, where Eo is the energy of 16 O, Ef are the energies of 5/2+ states
+
in 15 O, Ef are the energies of 5/2+ states in 17 O, and C is the isospin Clebsch-

85

Gordan coefficient [4]. Both particle and hole strength are included in Eq. 4.3. The
denominator gives the total available occupation (2J + 1) of a given orbit α. The infinite summation can be prohibitive if the spectroscopic strength is fragmented across
many states. Additionally, experimental restrictions on the detection of states at high
excitation energies and on the determination of the principal quantum number for
the spectroscopic factor prevent the full spectroscopic strength from being measured.
In physical states, the local and short-range correlations cannot be isolated from one
another. Spectroscopic strength can also be difficult to calculate theoretically, particularly with EDF methods which produce neither the explicit wavefunctions nor
the necessary excited states. Configuration Interaction theory can be employed to
determine theoretical ESPE. For the 16 O 0d5/2 ESPE in the sd model space, the hole
states are outside the model space and the complete theoretical spectrum is given in
Fig. 4.1. In this particular case, for one particle outside the core in the shell model,
the summation over final states is eliminated and the effective and uncorrelated SPE
are identical. The ESPE of model space orbits can be determined even for midshell
nuclei, providing information solely on the local correlations.
The term single particle energy is used in scientific literature to denote other
concepts beyond the uncorrelated SPE of Eq. 1.6 and the ESPE of Eq. 4.3. An
example that will be referenced throughout this work is more accurately denoted the
one-nucleon separation energy. The one-nucleon addition energy for particles outside
a core with A nucleons is given by
+
Eκ = ΨA+1 | H | ΨA+1 − ΨA | H | ΨA ,
κ
κ

(4.4)

where κ denotes the quantum numbers of the A + 1-body state of interest. A similar
−
equation can be derived for the one-nucleon removal energy Eκ of holes from the
+
−
core. The separation energies for a nucleus are given by Eκ and Eκ . Experimen-

tal one-nucleon separation energies, as given by experimental binding energy differ-

86

ences, include all correlations. In the lowest-state approximation, the experimental
one-nucleon separation energy from an even-even nucleus with ground state 0+ to an
π
odd-A nucleus with Jk = j (−1) represents the ESPE of the orbit with α = (n, , j).

Only in a purely single particle picture, i.e. assuming a spectroscopic factor of one,
would the energy difference between the two physical states exactly determine the
ESPE of the orbit α. The lowest-state approximation for single particle energies is
evaluated in Sections 4.2 and 5.1.
For uncorrelated SPE, a nearly identical equation can be written, substituting
H0 for H and Ψ0 for Ψ, where the Slater determinant Ψ0 is the solution to H0 in
Eq. 1.5. Theoretically, one-nucleon separation energies are determined by making an
approximation to the nuclear Hamiltonian. For instance, with the nuclear potential in
the form of the Skyrme interaction, one-nucleon separation energies can be determined
from Skyrme Hartree-Fock solutions to three nuclei with mass number A, A − 1,
and A + 1, where the wavefunction Ψ is represented by a single Slater determinant.
Similarly, the truncation to a reduced model space, where Ψ and H represent the
many-body wavefunction and Hamiltonian outside the core, enables the calculation
of one-nucleon separation energies with Configuration Interaction theory.

4.2

Comparison of Effective SPE and One-Nucleon
Separation Energies

A comparison between ESPE and one-nucleon separation energies is instructive, especially since the distinction is not generally made in the literature. Michimasa et
al., in a study of 23 F, found that the low-lying states were not reproduced accurately
by the USD interaction [66]. They conclude that “this may be attributed to an increase in the energy difference of the single-particle energies,” and found reasonable
agreement with experiment by changing the SPE parameters in the USD effective
interaction. They cited an increase in spin-orbit splitting for the 0d5/2 and 0d3/2
87

Table 4.1: Comparison of spectroscopic factors C 2 S for experiment and for three sd
Hamiltonians in the proton capture reaction on 22 O.
E (MeV) J π
Exp.
USD USDA USDB
+ n/a
0.00
5/2
0.80 0.77
0.78
+ 0.36+0.10 0.67
2.27
1/2
0.64
0.65
−0.16
+ 0.24+0.07 0.47
4.06
3/2
0.56
0.47
−0.09
proton orbits relative to the case at 17 F. Their conclusion was based on a comparison between the lowest-energy experimental states with large spectroscopic factors in
the proton transfer reaction 4 He(22 O,23 Fγ)3 H and the single particle states in 17 F,
consisting of one proton outside the 16 O vacuum. Though they did not make the distinction, Michimasa et al. were comparing one-nucleon separation energies instead of
the appropriate ESPE. Since the physical states in 23 F are not represented accurately
by single particle states, a better comparison between 17 F and 23 F is achieved with
effective SPE from Eq. 4.3.
Table 4.1 includes the experimental and theoretical values for the strong proton
transfer spectroscopic factors for three sd shell Hamiltonians. (In Fig. 5 of [66], the
authors list values of (2J + 1)C 2 S for two experimental levels and for the USD calculations. However, the theoretical values listed are actually for (2J + 1)S; C 2 = 6/7 for
this reaction.) The three Hamiltonians give similar results, with at most 20% difference for the 3/2+ state. The ratio of the experimental and theoretical spectroscopic
factors of about 0.50 is consistent with the global reduction factor observed in nucleon
transfer and nucleon knockout reactions shown in Fig. 2 of [67]. This reduction factor
is attributed to configuration mixing beyond the sd shell, including those due to the
short-range and tensor two-nucleon correlations [68].
Fig. 4.2 shows summed proton transfer C 2 Sf values as a function of Ef − Eo for
USDB, where Ef are the energies of states in 23 F and Eo is the energy of the 22 O
ground state. The calculated energy difference between 22 O and 23 F includes the 3.48
MeV Coulomb energy correction for Z = 9 relative to Z = 8 [15]. This plot shows
that the lowest-energy state for each spin only contains 50-80% of the total strength.
88

The remaining strength is fragmented over many final states up to 15 MeV higher in
energy. The large fragments of strength near Ef − Eo = 0 − 5 MeV correspond to the
strength going to the T ≥ 7/2 isospin states that are the isobaric analogues of the
low-lying states of 23 O.
The centroid single particle energies given by Eq. 4.3 are shown on the right-hand
side of Table 4.2. The S − terms do not contribute because nitrogen is outside of
the sd model space. The centroid energies are significantly higher than the loweststate energies, with energy shifts of four MeV for the 0d3/2 and 1s1/2 orbits with all
interactions.
The best experimental information available for the fragmentation of proton single
particle strength in the fluorine isotopes is from the 18 O(d,n)19 F experiment of [69].
The spectroscopic factors from this experiment are compared to the calculation with
USDB in Fig. 4.3. The fragmentation to T ≥ 5/2 states is large, as evidenced by
the behavior of the lowest T = 5/2 state, which is the strong 5/2+ state at 7.54
MeV (at Ef − Eo = −0.46 MeV in Fig. 4.3). The experimental data stops at an
excitation energy of 14 MeV where the experimental lines stop in Fig. 4.3. Most of
the theoretical strength is below 14 MeV excitation with the exception of a large value
for 3/2+ near 14 MeV. Thus some d3/2 strength may be missed. The overall pattern
of experimental strength distributions is in excellent agreement with theory except
that the d3/2 strength above 10 MeV in excitation (above Ef − Eo = 2 MeV in Fig.
4.3) is more fragmented in experiment than theory, presumably due to mixing with
intruder states coming from two nucleons excited from the p shell to the sd shell or
from the sd to the pf shell.
For 23 F, a comparison of USDA and USDB results gives an estimate of the theoretical error in the centroid energy of 0.1, 0.4 and 0.4 MeV for 0d5/2 , 1s1/2 and
0d3/2 , respectively. The 0d5/2 − 0d3/2 proton spin-orbit splitting for 23 F is thus estimated to be 6.2(4) MeV. With USDB, the spin-orbit splitting can be found from
the proton-drip line, 6.0 MeV in 17 F, to the neutron drip line, 6.9 MeV in 29 F. Thus,

89

Table 4.2: Proton separation energies for 23 F relative to 22 O based on the lowest state
for each spin and the centroid energy from Eq. 4.3 for each n, , j value.
lowest-energy state
centroid energy
n j
Exp.
USD
USDA USDB USD
USDA USDB
0d5/2
1s1/2
0d3/2

-13.20
-10.93
-9.14

-13.05
-11.27
-9.55

-13.28
-11.03
-9.59

-13.26
-11.27
-9.28

-11.16
-7.58
-5.04

-11.21
-6.92
-5.46

-11.31
-7.34
-5.09

1.0

5/2+

0.5

ε = − 11.31

C2Sf

0.0
1.0

3/2+

0.5

ε = − 5.09
0.0
1.0

1/2+

0.5

ε = − 7.34
0.0
-20

-10

0

10

D (MeV)
Figure 4.2: C 2 Sf vs. D= Ef − Eo for 23 F as the product of a proton transfer reaction
using the USDB Hamiltonian. The dotted lines mark the centroid energies.

90

ε = − 3.70

1.0

ε = − 4.29

0.8
0.6

Exp.

USDB

0.4

5/2+

0.2

5/2+
ε = 1.59

1.0

C2Sf

0.8
0.6

Exp.

USDB

0.4

3/2+

0.2

3/2+

ε = − 2.51

1.0

ε = − 2.99

0.8
0.6

Exp.

USDB

0.4

1/2+

0.2
-8

-4

0

4

D (MeV)

8

1/2+
-8

-4

0

4

8

D (MeV)

Figure 4.3: C 2 Sf vs. D= Ef − Eo for experimental [69] and USDB calculations for
19 F as the product of a proton transfer reaction. The dotted lines mark the centroid
energies. The 3/2+ experimental centroid energy is not listed because the strength was
not measured above Ef − Eo = 6 MeV, near the theoretical state at Ef − Eo = 5.54
with a large spectroscopic factor.

91

including the uncertainty of the calculation, the spin-orbit splitting is nearly constant
as a function of neutron number when comparing ESPE from the centroid energy
instead of the lowest-state energy from an approximate single particle picture.
Similarly, Gaudefroy et al. discuss the behavior of the spin-orbit splitting in 49 Ca
and 47 Ar based upon one-nucleon separation energies given by lowest-energy states
observed in the 46 Ar(d,p)47 Ar reaction [70]. They deduce a 45(10)% reduction in the
p3/2 − p1/2 spin-orbit splitting for Ar (Z = 18) compared to Ca (Z = 20). However,
there is significant fragmentation of the pf -shell spectroscopic strength between 45 Ar
and 47 Ar which must be taken into account to evaluate the shifts of ESPE and to
determine the spin-orbit splitting.
The calculations for 47,48,49 Ca and 45,46,47 Ar were carried out using OXBASH
[6] and the sdpf interaction from Nummela et al. [47], where proton excitations
are restricted to the sd shell and neutron excitations are restricted to the pf shell.
The same calculation was carried out in [70] to produce a level scheme for 47 Ar
in reasonably good agreement with experiment. In both 47 Ar and 49 Ca the first
excited state has J π = 1/2− with experimental (theoretical) excitation energies of
2.02 (1.70) MeV for 49 Ca and 1.13(8) (1.28) MeV for 47 Ar. Gaudefroy et al. deduce
a reduction of spin-orbit splitting around the N = 28 shell closure as a result of this
decrease in excitation energy. Taking into account both particle and hole strength
through Eq. 4.3 for the p1/2 and p3/2 orbits, the reduction does not occur. For all
final nuclei, 200 states were included; this is enough to exhaust over 99% of the
(2J + 1) sum-rule strength for all isotopes of interest. As seen in Fig. 4.4, the lowestenergy states in 49 Ca account for 95% of the total spectroscopic strength for the p1/2
and p3/2 orbits, whereas the lowest-energy states in 47 Ar account for only 80% and
65%, respectively, of the total strength. The change of the spin-orbit splitting is of
interest: δ so = [ (Ar, p1/2 ) − (Ar, p3/2 )] − [ (Ca, p1/2 ) − (Ca, p3/2 )]. Given that
there is some difference between experiment and theory for the energies of the lowest
3/2− and 1/2− states in 47 Ar and 49 Ca (as noted above), the experimental shift of

92

C2 Sf

4
3

3/2 −

49 Ca

3/2 −

47 Ar

2
1
0

45 Ar

47 Ca

2.5

C2 Sf

2.0
1.5

49 Ca

1/2 −

1/2 −

47 Ar

1.0
0.5
0.0

45 Ar

47 Ca
-12 -6

0

6

D (MeV)

-12 -6

0

6

D (MeV)

Figure 4.4: C 2 Sf vs. D= Ef − Eo from the initial ground states in 48 Ca and 46 Ar
to 1/2− and 3/2− states via the addition or subtraction of a neutron. The large
fragmentation of strength for 46 Ar, not evident in 48 Ca, results in deviation between
the ESPE and the energy of the lowest state in 47 Ar.
−0.89(13) MeV provides a starting point as the difference in experimental one-nucleon
separation energies. A theoretical correction due to fragmentation, +0.88 MeV, is
determined by the theoretical shift from the lowest-state energy to the centroid energy.
This correction is added to the change in one-nucleon separation energies to obtain
δ so = −0.01(13) MeV. In another procedure, the experimental binding energies and
theoretical excitation energies are entered directly into Eq. 4.3 to obtain δ so =
+0.09(13) MeV. In either case, the result shows no change in the spin-orbit splitting
within the uncertainty in the calculation. Again, a comparison of ESPE rather than
the lowest state of a given value of total angular momentum is necessary to determine
spin-orbit splitting. The splitting is constant for changes in proton number at N = 28,
in agreement with the result for fluorine isotopes as a function of neutron number.

93

4.3

SPE Dependence of CI Calculations

As seen in Fig. 2.1, the binding energy of 18 O with the HRP-SD interaction is 3.9
MeV larger than experiment. Extending the calculation to third order in perturbation
theory increases the deviation to 4.2 MeV, although it improves the description of the
4+ state as mentioned in Section 2.2. The two interactions employ identical single
particle energies, determined from the Skyrme Hartree-Fock solution to the mean field
for 16 O with the Skxtb interaction. If the one-nucleon separation energies of the Skxtb
calculation as determined by Eq. 4.4 were implemented in the HRP-SD interaction,
the result would not change significantly. Table 4.3 compares the uncorrelated Skxtb
single particle energies, along with the ESPE from the empirical USDB interaction
and the lowest experimental 5/2+ , 3/2+ , 1/2+ states in 17 O. From the definitions in
Section 4.1, the experimental states provide neither uncorrelated or effective SPE,
but are the appropriate input for an effective interaction in the sd shell with 16 O as
the core. Unlike the CI calculation, the physical states cannot simply be represented
by an sd neutron coupled to a configuration of 16 O with filled N = 0 and N = 1 oscillator shells and instead include all correlations. The spectroscopic factors for these
states are not equal to one, indicating that 16 O can only approximately be treated as
a core. Without accounting for the full fragmentation of the single particle strength,
which is not known experimentally, the experimental ESPE cannot be determined.
The USDB interaction, as fit to data throughout the sd shell, can provide an estimate
for the single particle energies necessary to reproduce local correlations in the sd shell.
A comparison to the values determined from experimental states in 17 O suggests that
only the 0d3/2 orbit has significant modification, with an increase in energy from 0.94
to 2.11 MeV. The uncorrelated SPE for the Skxtb interaction differ from both the
USDB ESPE and the experimental states for all three orbits. Since the USDB ESPE
differ significantly from the results with the Skyrme interaction, the binding energy
for 18 O with the HRP-SD interaction deviates significantly from experiment. If the

94

Table 4.3: A comparison of the lowest-energy experimental states in 17 O for each spin,
the neutron SPE component of the effective USDB interaction, and the uncorrelated
SPE of the Skxtb interaction from the Skyrme Hartree-Fock solution to the mean
field in 16 O.
n j
Exp. USDB Skxtb SPE
0d5/2
1s1/2
0d3/2

-4.14
-3.27
0.94

-3.93
-3.21
2.11

-6.22
-3.73
0.30

two-body matrix elements from the HRP-SD interaction are combined with the USDB
ESPE (experimental one-nucleon separation energies), the binding energy of 18 O relative to 16 O is 11.75 (12.24) MeV, in reasonable agreement with the experimental
value of 12.19 MeV. If instead the TBME derived from a calculation to third order in
perturbation theory are used, the resultant binding energy is 12.18 (12.65) MeV. In
other words, the Skxtb interaction produces a 0d5/2 uncorrelated SPE approximately
two MeV too low in energy, which results in a four MeV increase in the binding energy
of 18 O. As more valence particles are added, the agreement with experiment worsens.
Single particle energies also contribute to the CI calculation through the amount
of configuration mixing, which is a function of the energy differences or gaps between
model space SPE. This nonlinear effect depends on the complicated matrix diagonalization, and will not be evaluated quantitatively. It is essential for the effective
interaction to contain accurate SPE in order to produce reliable results throughout
the model space. As discussed in Section 1.1, lowest-energy experimental states and
uncorrelated SPE often provide the single particle energies of effective interactions.
For standard model spaces, this approximation is reasonable, as seen in the two leftmost columns of Table 4.3. However, the approximation is not viable for the HRP
as described in Chapter 2 if the target nucleus does not have large energy gaps at
the Fermi surface, i.e. if the target is not a magic nucleus. The uncorrelated SPE of
Skyrme forces, which have been used as the one-body component of effective interactions thus far, must be evaluated in greater detail for midshell nuclei.

95

Table 4.4: Uncorrelated neutron single particle energies in MeV for 16 O with the Skx,
Skxtb, Sly4, and MSk7 interactions.
n j
Skx Skxtb
Sly4 MSk7
0s1/2
0p3/2
0p1/2
0d5/2
0d3/2
1s1/2
0f7/2
0f5/2
1p3/2
1p1/2

-30.20
-18.38
-13.45
-6.16
0.24
-3.72
7.07
15.44
5.31
6.50

-30.11
-18.36
-13.34
-6.22
0.30
-3.73
6.94
15.47
5.27
6.48

-34.52
-19.23
-13.48
-5.79
0.61
-3.37
6.32
13.83
4.34
5.50

-28.95
-18.14
-12.54
-6.82
0.29
-4.38
5.65
14.68
4.52
6.00

Table 4.5: Uncorrelated neutron single particle energies in MeV for 22 O with the Skx,
Skxtb, Sly4, and MSk7 interactions.
n j
Skx Skxtb
Sly4 MSk7
0s1/2
0p3/2
0p1/2
0d5/2
0d3/2
1s1/2
0f7/2
0f5/2
1p3/2
1p1/2

4.4

-28.85
-16.94
-12.73
-5.96
0.24
-4.81
5.69
14.42
4.15
5.01

-28.71
-17.50
-11.05
-6.85
1.42
-4.84
4.65
15.70
3.93
5.64

-37.97
-20.72
-15.42
-6.26
0.73
-4.10
5.79
14.61
3.86
4.81

-29.16
-17.35
-12.16
-6.49
1.03
-5.05
4.91
15.45
3.70
4.80

Skyrme Single Particle Energies

The Hybrid Renormalization Procedure is designed to connect to the underlying
physics and to avoid unnecessary parameters. However, an effective interaction requires reliable single particle energies in order to reproduce low-energy excitations
throughout the model space. For a target nucleus outside of the core of a model
space, an approximate method to determine single particle energies introduces as
many parameters as model space orbits and fits to experimental data in the model

96

space. This procedure is implemented for the application of the HRP to the island
of inversion region in Section 5.1. The parameters are unnecessary if Skyrme interactions are able to provide uncorrelated SPE or one-nucleon separation energies that
are consistently close to the experimental ESPE. The ten lowest uncorrelated SPE
for the doubly magic 16 O and the exotic closed subshell 22 O are shown in Tables
4.4 and 4.5, respectively, for different paramaterizations of the Skyrme interaction.
In addition to the Skx and Skxtb interactions discussed in detail in Section 1.2, SPE
are obtained from the Sly4 [56] and MSk7 [71] interactions. For the exotic case, the
difference between the Skx and Skxtb results in Table 4.5 display the effect of the
tensor force, which can be greater than one MeV as seen from the 0d3/2 orbit, for instance. For orbits in the valence space of these nuclei (the sd model space), the values
change by about one MeV for the four parameterizations of the Skyrme interaction.
As noted in Section 4.3, this corresponds to a change of two MeV in the binding
energy of 18 O. For practical purposes, since a reproduction of the low-energy states
of nuclei in a given model space on the hundred keV level is desired, the Skyrme SPE
are unsatisfactory.
The level density of the single particle spectrum in Skyrme Hartree-Fock calculations is dependent on the effective mass of the Skyrme interaction, which differs for
the four parameterizations in Tables 4.4 and 4.5. As a result, the uncorrelated SPE
for deeply bound orbits can vary significantly, by nearly ten MeV for the 0s1/2 orbit
in 22 O. An effective mass approximately equal to the mass of a nucleon, as in the Skx
and Skxtb interactions, can provide an accurate single particle spectrum for 208 Pb,
while a reduction of 20% as in Sly4 is necessary to reproduce the spectra in light
nuclei [29]. However, it is impossible to reproduce experimental one-nucleon separation energies for both 208 Pb and light nuclei with the Skyrme interaction [29]. Since
an energy density functional should describe all nuclei equally, the mass dependence
indicates that the Skyrme interaction does not correspond to the realistic microscopic
functional. With a microscopic energy density functional that has an effective mass

97

equal to the mass of a nucleon, the one-nucleon separation energies are expected to
describe the experimental ESPE, and therefore will not necessarily agree with the
experimental one-nucleon separation energies. To reproduce the experimental energy
spectrum and spectroscopic behavior, multi-reference EDF methods must be applied
on top of the SR-EDF calculation. Such calculations are outside the scope of this
work, but the single particle properties of both light and heavy nuclei are not reproduced by any current functionals, whether single- or multiple-reference EDF methods
are employed.
The one-nucleon separation energies of the Skyrme interaction should have been
preferentially chosen over the uncorrelated SPE as the input to effective interactions. Even though the Skyrme interaction is described as a mean field interaction,
the one-nucleon separation energies are only approximately given by Eq. 1.8. The
density-dependent term in Eq. 1.19 represents a zero-range three-body force and
produces correlations which vary based on the parameterization [72]. To solve the
Skyrme Hartree-Fock equation for odd nuclei, the symmetry related to time-reversal
invariance must be broken. To calculate the one-nucleon separation energies with a
time-reversal invariant Skyrme Hartree-Fock code, the equal filling approximation is
implemented, where an average over time-reversal partners is considered. The energy
in this approximation will be greater than the solution which breaks time-reversal
invariance, but the accuracy of the equal filling approximation is not currently known
and must be analyzed in more detail. From preliminary investigations with the equal
filling approximation, the one-nucleon separation energies of different Skyrme parameterizations have a smaller variance than the uncorrelated SPE, another beneficial
result. However, the results from either method deviate significantly from experiment
for light nuclei. A complete evaluation of the one-nucleon separation energies and
their relation to uncorrelated SPE for stable and exotic nuclei with EDF methods is
a topic for future research. The implementation of Skyrme separation energies into
the effective interactions in Chapter 3 and Sections 2.2 and 5.2 should be performed

98

(see Chapter 6) but is not expected to require modification to the conclusions drawn
in this work.
Tables 4.4 and 4.5 display a variation of uncorrelated SPE of approximately one
MeV in the valence space for the different parameterizations. However, there is no
guarantee that the range of values agrees with the physical ESPE. The 0d5/2 neutron
SPE from Table 4.4, which vary from -5.79 to -6.82 MeV, differ from the experimental one-nucleon separation energy and the USDB SPE from Table 4.3, -4.14 and
-3.93 MeV respectively, by as much as 2.9 MeV. It should be noted that light nuclei in general are not reproduced by the Skyrme functionals, which better describe
heavier nuclei. Until a functional is derived that improves upon the Skyrme single
particle spectra, an uncertainty of at least one MeV in SPE must be accepted when
experimental data does not provide approximate values.
The difficulty in obtaining ESPE and the unreliability of uncorrelated Skyrme
SPE motivate several methods to determine SPE for effective interactions derived
from the Hybrid Renormalization Procedure:
(i) Select a core such that the lowest-energy experimental states are good representations of single particle states as discussed in Section 4.2. For 17 O, the three lowest
positive parity states in Fig. 4.1 have large spectroscopic factors [73] with J π values that correspond to one valence nucleon outside of the core. The energy of these
lowest states should approximate the physical ESPE, which cannot be determined
experimentally.
(ii) As in the case of empirical interactions like USDB, treat the SPE as parameters and fit to available data in the region. In this way, the SPE are constrained to
reproduce values close to the ESPE by the data.
(iii) Directly utilize results from Skyrme Hartree-Fock theory with caution, realizing that the deviation from the ESPE can be more than one MeV.
The remainder of this dissertation is devoted to applications of the HRP, with a
focus on comparison to recent experimental quantities of interest. Effective interac-

99

tions will consist of two-body matrix elements calculated from the HRP as described
in Chapter 2, with single particle energies determined from one of the three methods
above in order to provide the most reliable comparison to experimental data.
Subsequent to the completion of this dissertation, a forthcoming work by Duguet
et al. [74] was brought to the author’s attention. Two important results relevant to
this work are:
(i) One-nucleon separation energies determined via EDF methods can be identified
with SR-EDF ESPE. As discussed above, but now based on a formal derivation, the
one-nucleon separation energies from the Skyrme interaction should supplant the
Skyrme uncorrelated SPE in the SPE component of effective interactions derived
with the HRP.
(ii) The treatment of ESPE in Eq. 4.3 is an approximation. The meaningful definition of ESPE requires the computation of a centroid matrix, as described in the work
of Baranger [65], which reduces to Eq. 4.3 only in the diagonal basis of the matrix.
The off-diagonal terms in the bases constructed in this work have been neglected. The
effect on the ESPE as a result is unknown and must be studied in more detail.
See Section 6.3 for a discussion of possible future work based on these results.

100

Chapter 5
Applications
As discussed in Section 1.7, standard Configuration Interaction techniques struggle
to produce accurate results outside of standard model spaces in the region of applicability, shown in Fig. 1.6. The island of inversion region and neutron-rich silicon
isotopes were identified as cases of recent experimental interest which are appropriate
for the Hybrid Renormalization Procedure. Extensive calculations will be performed
with interactions derived from the HRP in order to display the utility and accuracy
of the method.

5.1

Island of Inversion Region

The first evidence of the island of inversion region was noted by Thibault et al.
during a study of the masses of neutron-rich sodium isotopes [42]. Particularly for
31 Na,

which should display extra stability in comparison to nearby isotopes due to

the closure of the sd neutron shell at N = 20, the measured binding energy was
larger than the shell model predictions. The occupation of pf orbits in the ground
state increases the binding energy in the form of additional correlation energy. The
inversion of the standard filling of neutron orbits near N = 20 has since been identified
for several nearby nuclei, and this island of inversion is currently under investigation

101

in several experimental facilities.
The island of inversion exists at the edge of nuclear structure research with exotic beams, so the boundaries of the region have not yet been determined. The inversion has been observed in nuclei with both N ≥ 19 and Z ≤ 13, but the extent of ground states in this region with inverted configurations is unknown. The
high-N and low-Z boundaries have not been investigated at all experimentally. Following the Hybrid Renormalization Procedure of Chapter 2, the first step requires
the selection of a model space and target nucleus near the region of interest. The
island of inversion region, as determined by an evaluation of the available experimental evidence, is centered around 31 Na. The selected model space to calculate
nuclei in this region is composed of the 0d5/2 , 0d3/2 , 1s1/2 proton orbits and of the
0d3/2 , 1s1/2 , 0f7/2 , 1p3/2 , 1p1/2 neutron orbits. The entire sd and pf shells cannot be
included for the neutron orbits due to computational restrictions on the diagonalization of the Hamiltonian. Therefore, the 0d5/2 and 0f5/2 orbits, the most bound and
unbound orbits respectively, are excluded from the model space. Calculations in the
sd shell suggest that excitations from the 0d5/2 orbit affect level schemes on the order
of 250 keV around N = 18, while the six MeV spin-orbit splitting between the 0f7/2
and 0f5/2 orbits causes a minimal contribution from the exclusion of the 0f5/2 orbit
up to N = 28. In order to prevent divergences, as discussed in 2.9, all model space
orbits were placed in the N = 2 oscillator shell for the renormalization procedure.
Additionally, the excluded ν0d5/2 orbit was placed in the N = 1 oscillator shell. All
other orbits were in their standard oscillator shells given by N = 2n + .
The core of the model space, 22 O, is selected as the target nucleus for the Hybrid
Renormalization Procedure. Even though it is further away from 31 Na than 34 Si or
28 O,

the core is more conducive to a straightforward treatment of three-body forces

through the addition of monopole terms to the Hamiltonian. As shown by Otsuka et
al. in Fig. 2 of [75], the main contribution of the three-body force can be represented
by a repulsive two-body monopole interaction between two valence nucleons and one

102

nucleon in the core. Additionally, 22 O has approximately the correct asymmetry for
the region (N/Z = 1.8, in comparison to N/Z = 2.1 for 31 Na).
With the standard implementation of the Hybrid Renormalization Procedure in
this model space with 22 O as the target, the low-energy behavior of nuclei in the model
space is not reproduced. Two problems with the HRP, previously discussed in Section
2.2, are the Skyrme single particle energies and lack of three-body forces. Following
step (vii) of the HRP and the discussion in Chapter 4, three-body forces and SPE are
parameterized. Twelve free parameters are included: SPE for each orbit (eight total),
and valence monopole terms for a proton-proton, neutron-neutron, proton-neutron,
C

ij
and three-nucleon interaction. The two-body monopoles are given by 1+δ Oi (Oj −
ij

δij ), where i, j denote the type of nucleon (proton or neutron) and Oi is the occupation
number of valence nucleons of type i. The three-body monopole is symmetric with
C

respect to isospin and is given by 63 Aval (Aval − 1)(Aval − 2). With these twelve
parameters, the 2+ energies of even-even nuclei in the model space could not be
reproduced to the desired accuracy. Therefore, a phenomenological normalization of
the two-body matrix elements was included as a thirteenth parameter, serving to
reduce the overall strength of the interaction by approximately 10%. In Chapter 3,
a reduction in the overall strength of the interaction was observed due to loosely
bound orbits. With 22 O as the target, there are no loosely bound orbits, whereas a
target nucleus near the island of inversion region like 28 O or 34 Si has loosely bound
neutron orbits with the Skxtb interaction. The overall normalization can represent
an empirical way to account for the reduction in the strength of the interaction due
to these loosely bound orbits.
With thirteen parameters, a fit to data is required. Even though 22 O, the core of
the model space, was selected as the target in the HRP, the single particle properties
in the island of inversion region are more comparable to those of the midshell nucleus
34 Si.

Therefore, the known states with one particle added to or removed from 34 Si are

included in the fit to constrain the single particle energy parameters. From the ENSDF

103

database [53], the lowest two states in 33 Al (both 5/2+ ), the lowest three states in
35 P

(1/2+ , 3/2+ , 5/2+ ), and the lowest four states in 33 Si (3/2+ ,1/2+ , 7/2− ,3/2− )

are known. Nuclei whose spin and parity have been determined from “weak” rules [76]
are assumed to be accurate throughout this section. The three lowest states in 35 Si
(7/2− ,3/2− ,3/2+ ) were determined in [47], but the 1/2− state is also necessary to
constrain the 1p1/2 SPE. The value is estimated from a systematic study of spinorbit splitting in N = 20 isotones with the Skxtb interaction; the result is 350 keV
higher than a recently identified state in 35 Si. The spin and parity have not been
determined in the preliminary results, but a full analysis of the experimental data is
still in progress [77]. The overall normalization is constrained by including the 0+ and
2+ states of 34 Si and two-particle or two-hole excitations from 34 Si. The monopoles
are constrained by including data from the lightest isotopes to the heaviest isotopes of
interest, but states near the island of inversion are also included to better reproduce
the primary region of interest. The ground states of the light nuclei 24 O (0+ ) and 27 F
(5/2+ ), the three lowest known states of the heavy nuclei 37 Cl (3/2+ ,1/2+ ,5/2+ ) and
38 S (0+ ,2+ ,4+ ), and the island of inversion nuclei 31 Na (3/2+ ,5/2+ ), 30 Ne (0+ ,2+ ,4+ )

[78], 30 Mg (0+ ,2+ ,0+ ) [79], and 31 Mg (1/2+ ,3/2+ ,3/2− ,7/2− ) [80] are also included.
These 43 states are implemented in an iterative fitting procedure to constrain the
thirteen parameters.
The parameters are included in Table 5.1. The small three-body monopole supports the conclusion of Otsuka et al. [75] that the three-body force can be represented
by repulsive two-body monopole interactions. The rms deviation to the 43 experimental states is 370 keV, larger than the standard CI Hamiltonians in the sdpf space by
approximately a factor of two. An effective interaction applicable throughout the island of inversion region, called IOI, is composed of the four monopole terms, the eight
parameterized SPE and the two-body matrix elements from the HRP multiplied by
the overall normalization.
One hundred nuclei were calculated from proton number Z = 8 to Z = 17 and

104

Table 5.1: Values of parameters for the IOI interaction obtained from a fit to 43 experimental states near the island of inversion region of the nuclear chart. The first
eight parameters are single particle energies, the next four are coefficients in monopole
terms, and the final parameter is a normalization coefficient for the two-body matrix
elements. The initial values of the first eight parameters are uncorrelated SPE obtained from the Skyrme Hartree-Fock solution to the mean field for 34 Si with the
Skxtb interaction, while the final values are uncorrelated SPE from a single-reference
CI calculation, as discussed in Section 4.1. Typically, the one-nucleon separation energies for 34 Si with the Skxtb interaction would be chosen as the initial values as
discussed in Section 4.4, but the output of the fit, i.e. the final SPE, should not be
affected. Other than the unitless normalization, all entries are in MeV.
Parameter Initial Final
π0d5/2 SPE -17.29 -18.43
π0d3/2 SPE
-9.08 -10.94
π1s1/2 SPE -13.49 -12.38
ν0d3/2 SPE
-9.03
-7.87
ν1s1/2 SPE -10.04
-7.60
ν0f7/2 SPE
-2.62
-2.71
ν1p3/2 SPE
-0.40
-1.61
ν1p1/2 SPE
-0.27
1.01
Cpp
0.00 0.515
Cpn
0.00 0.020
Cnn
0.00 0.288
C3
0.00 -0.008
overall normalization
1.00 0.918

105

Legend
In Fit

Cl

16

1.6

S

15

1.2

P

14

0.8

Si

13

0.4

Al

12

0.0

Mg

11

− 0.4

Na

10

− 0.8

Ne

9

− 1.2

8

Proton Number

17

− 1.6

F
O

15 16 17 18 19 20 21 22 23 24

Neutron Number
Figure 5.1: A plot of energy differences EIOI − EExp. in MeV calculated for one
hundred nuclei in or near the island of inversion region. The black boxes denote
nuclei which constrain the parameters in the IOI interaction. Experimental masses
are taken from [81], excluding non-experimental masses estimated by extrapolation.

106

from neutron number N = 15 to N = 24. The first quantity of interest is the binding energy of these nuclei relative to experiment. Fig. 5.1 displays the calculated
energy differences, with experimental masses taken from [81]. However, it must be
noted that the fit to data was performed with the older, published masses from [82].
Specifically, the binding energy of 24 O decreased by 521 keV from [82] to [81] such
that EIOI − EExp. = 1.090 MeV in Fig. 5.1, even though 24 O was included in the
fit. As expected, the 370 keV rms deviation provides an approximate uncertainty for
calculations in this model space, with the binding energy differences in Fig. 5.1 consistent within this uncertainty in most cases. The exceptions are the N = 15 and
some N = 16 isotones, suggesting that excitations from the 0d5/2 orbit are essential
for the description of these nuclei, and the magnesium isotopes (Z = 12). Standard
and inverted configurations coexist for the magnesium chain near N = 20. It is unclear why the N ≥ 21 isotopes are not reproduced, considering that the sodium and
aluminum isotopes are reproduced to sufficient accuracy.
The rms deviation of 370 keV is sufficient for detailed studies in the island of inversion region. The capabilities of interactions derived with the Hybrid Renormalization
Procedure and the structure of nuclei near the island of inversion can be explored
by studying local trends, level schemes of specific nuclei, and reactions of particular
interest.

5.1.1

Systematic Trends

The discussion in Chapter 4 led to the parameterization of single particle energies.
A comparison of SPE in the model space is depicted in Fig. 5.2. The uncorrelated
single particle energies obtained from the Skx and Skxtb interactions for the unstable
34 Si

nucleus display the effect of the tensor interaction. The contribution from the

tensor force can be larger than one MeV, as seen from the proton and neutron 0d3/2
orbits. As stated in Section 4.4, the one-nucleon separation energies, which are better
representations of ESPE with SR-EDF methods, do not vary as much as the uncor107

34 Si

Single Particle Energies
neutrons
protons Uncorrelated SPE ESPE Lowest State
p 1/2
p 3/2

Energy (MeV)

0

p 1/2
p 3/2
f7/2

f7/2
-5

εf

-10

-15

d3/2
s1/2
d3/2
s1/2

d3/2
s1/2
d3/2
s1/2
εf

d5/2
d5/2

-20

Skx

Skxtb

IOI IOI

IOI

Exp

Figure 5.2: Single particle energies for 34 Si evaluated with different methods as described in Section 4.1. On the right, the lowest state approximation is compared for
experiment and for calculations with the IOI interaction. On the left, uncorrelated
SPE are shown as determined from the Skx and Skxtb interactions and the IOI interaction, as previously presented in Table 5.1. In the middle, the centroid or effective
SPE of Eq. 4.3 is determined for the IOI interaction by including at least 99% of the
strength of each orbit. The Fermi energy f for 34 Si is represented by a dotted line
for neutron orbits (red) and proton orbits (blue).

108

related SPE in the equal filling approximation. The contribution of the tensor force
might be smaller than evident from Fig. 5.2 as a result. The uncorrelated Skyrme
SPE significantly deviate from the lowest experimental states. The IOI uncorrelated
SPE from the fitting procedure are in much better agreement with the experimental
states, which are not the effective SPE but are good approximations of single particle
states for 34 Si. As mentioned in Chapter 4, microscopic energy density functionals
which focus on reproducing single particle properties should improve upon the Skyrme
SPE, providing values consistently close to experiment. Until then, the experimental
states or parameterized SPE are preferred. The lowest states with the IOI interaction
are in good agreement with experiment, and the amount of fragmentation can be
inferred from the shift between the lowest state and ESPE for the IOI results. Only
the p1/2 orbit has significant fragmentation, implying that 34 Si exhibits the behavior
of a magic nucleus.
The neutron dripline can only be reached for the lightest isotopes with current
experimental facilities, and has therefore only been established with certainty up to
Z = 8. The neutron separation energy Sn (A Z) = BE(A Z) − BE(A−1 Z) becomes
negative at the first unbound nucleus. The odd-even behavior of the separation energy
is prevalent due to the pairing interaction in nuclear physics. The separation energy
is plotted as a function of neutron number for the ten isotopic chains in Figs. 5.3 and
5.4. The isotopic chains with an even number of protons are displayed in Fig. 5.3,
and the agreement between experiment and calculations with the IOI interaction is
excellent. For the oxygen isotopes, the binding energies for N ≥ 18 are determined
by an extrapolation from nearby masses and suggest that 24 O resides on the neutron
dripline. Even though the theoretical Sn is positive for 28 O, albeit by less than the rms
deviation, 28 O is not bound. The two-neutron separation energy S2n is negative, such
that the ground state decays by two-neutron emission to 26 O. For 26 O, however, the
IOI interaction predicts a bound state since both Sn and S2n are positive, with values
1.27 MeV and 0.68 MeV respectively. The experimental values from the extrapolated

109

Even Z Isotopic Chains
IOI
Exp.

14
12
10

Sn

8
6
4

S
Si

2
0
-2

Ne Mg

O
16

18

20

22

24

26

28

30

32

34

Neutron Number
Figure 5.3: Neutron separation energy vs. neutron number for the isotopic chains with
an even number of protons for both experimental data [81] and calculations with the
IOI interaction. Non-experimental masses estimated by extrapolation and experimental masses with uncertainties greater than or equal to the rms deviation of the IOI
interaction are excluded from the plot. The value of the two-neutron separation energy
S2n for unbound nuclei is represented by a black dot.

110

Odd Z Isotopic Chains
IOI
Exp.

14
12
10

Sn

8
6
4

Cl
P

2
0
-2

F
16

18

20

22

24

26

28

30

Na
32

Al
34

Neutron Number
Figure 5.4: Neutron separation energy vs. neutron number for the isotopic chains
with an odd number of protons for both experimental data [81] and calculations
with the IOI interaction. Non-experimental masses estimated by extrapolation and
experimental masses with uncertainties greater than or equal to the rms deviation of
the IOI interaction are excluded from the plot.The value of the two-neutron separation
energy S2n for unbound nuclei is represented by a black dot.

111

masses are 0.28 MeV and -0.50 MeV, and experimental evidence suggests that 26 O
is unbound [83], [84]. Although 24 O is included in the fit to constrain the monopole
terms for the lightest isotopes in the model space, the single particle energies are fit
to data around 34 Si. The neutron 0d3/2 SPE is 6.3 MeV higher than the 1s1/2 SPE at
22 O

with Skxtb, leading to the shell gap at N = 16 that is reflected in the stability of

24 O

relative to its neighbors. However, for 34 Si, the 0d3/2 SPE is only one MeV higher

in energy than the 1s1/2 SPE. The neutron separation energy of the oxygen isotopes,
and therefore the neutron dripline, may not be reproduced with the 34 Si SPE. The
evolution of SPE for calculations away from the nuclei in the fit will be discussed in
Section 5.2. Another possibility, presented by Otsuka et al. [75], requires three-body
forces for the accurate description of the neutron dripline for oxygen isotopes.
Another feature of Fig. 5.3 that is not reproduced in the calculation is the behavior of the N = 21 isotones. In the calculation, the spacing between the five nuclei
is relatively large and consistent with the behavior throughout the rest of the model
space. However, the experimental Sn for 33 Mg and 35 Si are nearly identical. Both nuclei have less than 50 keV uncertainty in their binding energies, but a remeasurement
of both masses, or at least 33 Mg, is recommended. The effect is not reproduced by
the IOI interaction but is interesting for future study.
The calculated dripline can be read off from the plot, where the last bound nucleus is 26 O, 34 Ne, 44 Mg, and 46 Si. All sulfur nuclei in the model space are bound,
terminating at the closure of the 1p1/2 orbit at N = 34 for 50 S. The excluded 0f5/2
orbit may be important as the N = 34 subshell closure is approached.
The isotopic chains with an odd number of protons are displayed in Fig. 5.4.
The agreement between experiment and theory is not quite as good as in Fig. 5.3,
but the trends are still reproduced. The fluorine isotopic chain has been measured
up to N = 22. The odd-odd isotopes are unbound starting with 28 F, but 31 F, the
most neutron-rich isotope observed to date, is bound with respect to one and two
neutron decay [85]. The theoretical predictions with the IOI interaction reproduce the

112

experimental trend for the neutron separation energy, with only N = 17 deviating
significantly. The binding energy of 26 F agrees with experiment within 120 keV, but
25 F

and 24 F are underbound by 0.73 MeV and 1.14 MeV, respectively. Excitations

from the 0d5/2 orbit are likely the cause, which do not affect the predictions for the
more neutron-rich isotopes. The neutron dripline for the fluorine isotopes has not been
determined experimentally, but the particle stability of 31 F has been identified [85].
The ground state of 31 F is unbound by two-neutron emission in the calculation with
the IOI interaction, on the order of the rms deviation. The next odd-even fluorine
isotope, 33 F, has a neutron separation energy of 70 keV and a two-neutron separation
energy of -1.57 MeV with the IOI interaction. The ground state is unbound, and
therefore the observed 31 F nucleus is predicted to reside at the neutron dripline. The
dripline is not reached in this model space for the chlorine and phosphorus isotopes,
but occurs at 37 Na and 45 Al. In addition to the caution as N = 34 is approached,
the dripline assignment is tentative since 39 Na is unbound to two-neutron emission
by less than the rms deviation.

5.1.2

Level Schemes of Representative Nuclei

To present the predictive power of the IOI interaction, level schemes of representative
nuclei in the model space will be compared to experimental data.
While 22 O is the target of the Hybrid Renormalization Procedure for the reasons
discussed at the beginning of Section 5.1, the single particle energies are fit to states
near 34 Si in order to better approximate their properties in the island of inversion.
Relative to 34 Si, the valley of stability can be reached via the removal of four neutrons
or by the addition of two protons. Since states in 36 S were included in the fit, the stable
isotope 30 Si provides a comparison of experimental and theoretical level schemes
in Fig. 5.5. The IOI interaction underbinds 30 Si by 1.03 MeV, which again can be
attributed to the missing excitations from the 0d5/2 orbit. The excitation energy is
consistent within the rms deviation for most states, and the level density is similar
113

6
5

4+
3+

4+

E (MeV)

4
3
2

2+

2+

1
0

0+ −93.59 MeV
30 Si
Exp.

0+ −92.56 MeV
30 Si
IOI

Figure 5.5: Comparison of experimental and theoretical level schemes for 30 Si up to
6 MeV. See the caption to Fig. 2.1 for further explanation.

6
5

E (MeV)

4

5+

3

4+

2
1
0

2+
0+ −51.16 MeV
32 Ne

Exp.

2+
0+ −50.76 MeV
32 Ne
IOI

Figure 5.6: Comparison of experimental and theoretical level schemes for 32 Ne up to
6 MeV. Four states of each J π are included in the figure. See the caption to Fig. 2.1
for further explanation.

114

6

IOI

Exp.

5

E (MeV)

4

11/2−
3
2

7/2−

7/2−

1
0

3/2+
−41.01 MeV

27 Ne

3/2+
−41.47 MeV

27 Ne

Figure 5.7: Comparison of experimental and theoretical level schemes for 27 Ne up to
6 MeV. Experimental data from [86] is included in addition to the ENSDF database.
Four states of each J π are included in the figure. See the caption to Fig. 2.1 for further
explanation.
up to five MeV. The experimental level density is greater at higher energies, with a
negative parity state at 5.5 MeV. The first negative parity state occurs at 7.8 MeV
with the IOI interaction, suggesting that excitations of the 22 O vacuum must be
included to reproduce states above 5 MeV.
If four protons are removed from 34 Si instead of four neutrons, the 30 Ne nucleus,
which lies in the island of inversion region, is reached. However, the 0+ , 2+ , and 4+
states in 30 Ne, i.e. all currently known states, were included in the fit. Therefore,
the experimental and theoretical level schemes of 32 Ne, including an additional two
neutrons, are compared in Fig. 5.6. The binding energy for theory and experiment
differ by approximately the rms deviation of the fit. Only the ground state and first
excited state are known, which are the only bound states with the IOI interaction.
The unbound states have a high level density with the 4+ state of primary interest for
comparison to the known states in 30 Ne, occurring 450 (437) keV above the theoretical
(experimental) neutron separation barrier. On the other side of the N = 20 shell
closure, 27 Ne is shown in Fig. 5.7. Only the ground state is included in the ENSDF
115

database [53], but three excited states have recently been detected experimentally. The
negative parity states correspond to single particle excitations into the pf shell. Brown
et al. [86] report that the gap between the sd and pf neutron orbits is reduced since
calculations with the WBP interaction, including the sd and pf orbits, are unable
to reproduce experiment. They conclude that “it would be interesting to develop a
new shell model interaction that would succeed in reducing the effective gap between
the 0d3/2 orbital and the 1p − 0f shell in a natural way, without the need for adhoc changes.” The IOI interaction does not require any modification to reproduce
nuclei throughout the island of inversion region. The theoretical binding energy is in
reasonable agreement with experiment, and the level scheme is reproduced. The 3/2−
and 1/2+ states are inverted in the theoretical calculation, but are only separated by
120 keV experimentally. The energies of the negative parity states provide insight
into the effective gap between the 0d3/2 orbital and the pf shell, such that the IOI
interaction reasonably reproduces the gap.
Experimental evidence places 31 Mg [80] and 33 Na [87] in the island of inversion.
The theoretical binding energies are in good agreement with experiment, including
the rms deviation. The experimental level density for 31 Mg in Fig. 5.8 is higher than
predicted at low energy, and the ground state spin is not reproduced in the calculation
with the IOI interaction. This 1/2+ ground state has three sd neutron holes [80],
and therefore two neutrons occupy the pf shell. The theoretical wavefunction, which
is calculated in the harmonic oscillator basis in the CI calculation, reproduces this
two-particle three-hole (2p3h) behavior for the 1/2+ state from a 2 ω excitation as
discussed in Section 2.8. The absolute energy of the 1/2+ state is underbound by
360 keV, which is within the rms deviation of the fit. However, coupled with the
overbinding of the 7/2− 1p2h state and a high density of states at low energy, the
1/2+ state occurs theoretically as the third excited state instead of the ground state.
Even though the ground state resides in the island of inversion in both cases, the
3/2+ state is a 1h state and occurs at an experimental (theoretical) excitation energy

116

6
5

IOI

Exp.

E (MeV)

4
3

11/2−

2
1
0

7/2−
3/2+
1/2+
31Mg −82.01 MeV

31Mg

7/2−
−82.15 MeV

Figure 5.8: Comparison of experimental and theoretical level schemes for 31 Mg up to
6 MeV. The lowest four states are included in the fit to data for the IOI interaction.
Experimental data from [80] is included in addition to the ENSDF database. The
angular momentum is unassigned for the state at 1.15 MeV since the two sources
have inconsistent values. Four states of each J π are included in the figure. See the
caption to Fig. 2.1 for further explanation.

6
5

Exp.

IOI

E (MeV)

4
3

11/2+
2

9/2+
7/2+

1
0

5/2+
3/2+
33 Na −70.83 MeV 33 Na −70.37 MeV

Figure 5.9: Comparison of experimental and theoretical level schemes for 33 Na up to
6 MeV. Experimental data from [87] is included in addition to the ENSDF database.
Four states of each J π are included in the figure. See the caption to Fig. 2.1 for further
explanation.

117

of 51 (106) keV. The boundaries of the island of inversion region, as determined in
Section 5.1.3, must be treated with caution due to the coexistence of standard and
inverted configurations below the rms deviation of 370 keV.
Doornenbal et al. [87] have identified the ground state and first excited state
in the island of inversion nucleus 33 Na, and argue from systematics in the sodium
isotopic chain that the states should have 3/2+ and 5/2+ in some order. They also
argue that the second excited state should be above 700 keV in excitation energy.
The predictions with the IOI interaction display this behavior, with the 3/2+ state as
the ground state. Because the 5/2+ excited state at 92 keV has an excitation energy
smaller than the rms deviation, a definitive statement about the spin of the ground
state cannot be made.
One issue that arises in the comparison of level schemes in the island of inversion
is the dearth of experimental data; the few excited states that have been measured
rarely have assigned spin and parity. The five nuclei described in this section have
excited states for comparison to calculations and are representative of the behavior
of level schemes throughout the model space. The comparison between experimental
data and calculations with the IOI interaction for all 100 nuclei can be accessed at [88].

5.1.3

Ground State Occupations

The island of inversion is composed of the nuclei near 31 Na where pf orbits are preferentially filled before the lower-energy sd shell is occupied completely. Because the
physical states and calculations with a diagonalization of a model space interaction
like the IOI interaction are composed of a linear combination of many configurations,
the occupations of the neutron orbits are not integral. As a result, the boundaries of
the island of inversion are dependent on a minimum value of excited neutrons. An
excitation of 1 ω from the sd shell to the pf shell, as defined in Section 2.8, is necessary for a single configuration to be considered inverted. The cutoff on the average
excitation in ω is selected as 1.00 to determine the boundaries of the island of inver118

Legend

17

Stable
2.0

15

1.75

14

1.5

13

1.25

12

1.0

11

0.75

10

0.5

9

0.25

8

Proton Number

16

0.0

15 16 17 18 19 20 21 22 23 24

Neutron Number
Figure 5.10: A plot of ω excitations for one hundred nuclei, as determined from the
ground state wavefunctions calculated with the IOI interaction. The wavefunctions
are composed of linear combinations of configurations, each with an integral number
of excitations as defined in Section 2.8. The normalization condition Ψ | Ψ = 1
leads to non-integral values for each nucleus. The island of inversion is composed of
nuclei with more than 1.00 ω excitation. Stable isotopes are outlined in black.

119

sion. For N ≤ 20, the occupation of the pf orbitals is identical to the ω excitation
for a ground state wavefunction. This does not provide a perfect description, as can
be seen from the ground state of 33 Al. The primary component of the wavefunction
corresponds to the standard filling of orbitals, resulting in a completely filled sd shell
at N = 20. This component contributes 47% of the total probability density of the
neutron occupation, but the other 53% is composed of hundreds of configurations,
primarily with 2p2h excitations into the pf shell. The ground state of 33 Al has an
average occupation of 1.10 pf neutrons, or an average excitation of 1.10 ω. Since
this value exceeds the minimum value of 1.00 ω excitation, 33 Al would be considered
inside the island of inversion even though the primary component of the ground state
has 0p0h structure. Experimental evidence has confirmed that the 33 Al ground state
exhibits mixed 0p0h − 2p2h structure for the neutron configurations, such that Tripathi et al. consider 33 Al inside the island of inversion region [89]. As the presence of
standard and inverted states within the rms deviation of the ground state demanded
in Section 5.1.2, the minimum value of 1.00 ω excitation requires caution. Further
investigation of the wavefunction for each individual isotope of interest is necessary
if definitive statements about the boundary of the island of inversion are desired.
An approximate theoretical boundary is displayed graphically in Fig. 5.10, which
plots the number of ω excitations as a function of proton and neutron number. While
34 Si

is partially composed of configurations with 2 ω, no isotopes with Z ≥ 14 lie

in the island of inversion region. The valley of stability, marked by the black boxes,
has almost no contribution from configurations with an inversion in the occupation
of neutron orbits. Similarly, all nuclei with N ≤ 18 exhibit the standard occupation
structure. While the N = 24 isotones 32 O and 33 F have nearly two holes in the sd
neutron orbits, neither of these isotopes are bound, as seen in Figs. 5.3 and 5.4. In
fact, no bound states in oxygen exhibit inversion. The results in Fig. 5.10 suggest that
the island of inversion extends further than the limits of experimental investigation
thus far, and that nearly all nuclei with Z ≤ 13 and N ≥ 19 exhibit inversion.

120

An evaluation of the pf occupation in the fluorine isotopes with N ≥ 19 and for
magnesium, sodium, and neon isotopes with N ≥ 22 would be interesting with rare
isotope facilities.

5.1.4

β decay

The primary mode of decay for the calculated ground states occurs via β emission.
The nuclei near the island of inversion approach the valley of stability through β −
decay, where a neutron is converted into a proton, an electron, and an antineutrino.
The Q-value of the decay is determined by the difference in binding energy between
the daughter and parent nuclei, in addition to 782 keV for the difference in mass
between the neutron and the hydrogen atom. Because the reaction depends on the
Q-value to the fifth power, experimental binding energies will be employed instead of
the values determined with the IOI interaction. The partial half-life for the decay to
a particular state in the daughter is given by

f t1/2 =

C
,
B(F ) + (gA /gV )2 B(GT )

(5.1)

where f is a phase-space factor that can be calculated, C is a constant that has been
determined experimentally as C = 6177, and B(F ) and B(GT ) are Fermi and GamowTeller transitions, respectively. A complete description of β decay is contained in [12].
At the quark level, gA = −gV , but the value is generally modified for nuclear structure
calculations to reproduce the decay of the neutron. From Wilkinson, | gA /gV | =
1.261(8) [90].
The decays of exotic nuclei around N = 20 are at the limits of experimental
investigation with rare isotope beams. In order to optimize the comparison between
theoretical and experimental results, the decay of a more stable nucleus around N =
20 with known transition strengths must be calculated. As 35 P fulfills the requirements
and has been previously studied with sd shell interactions [91], it will provide a test

121

case of the IOI interaction.
Experimentally, the transition to the ground state of 35 P has not been measured,
but only inferred. An upper limit of 9% on the branching ratio to the ground state
is included in the ENSDF database [53]. The branching ratios of the two measured
transitions, to the 1/2+ state at 1.57 MeV and to the 3/2+ state at 2.94 MeV, are
99.5% and 0.5%, respectively [53]. The calculation with the standard USDB interaction for the β − decay of the 1/2+ ground state of 35 P occurs to the lowest three
states in 35 S. The 1/2+ state in 35 S at 1.68 MeV is the primary state populated,
with a branching ratio of 90.7%. The 3/2+ ground state of 35 S has a branching ratio
of 7.8%, and the remaining strength populates the 3/2+ excited state at 2.80 MeV.
The standard interaction produces results that are in reasonable agreement with the
experimental transitions, especially with the unknown ground-state to ground-state
transition. The half-life of 35 P, as calculated with USDB, is 62.9 s, in comparison
to the experimental value of 47.3(7) s. As the results in Section 2.2 have shown, the
Hybrid Renormalization Procedure produces interactions which are comparable to
the best Hamiltonians derived empirically. Although the model spaces are different
for the IOI and USDB interactions, calculations with the two interactions might be
naively expected to produce similar results.
The calculations with the IOI interaction populate only the lowest two states,
with branching ratios of 18.9% and 81.1% to the 3/2+ ground state and 1/2+ excited
state at 1.46 MeV in 35 S. The higher branching ratio to the ground state is important
for the evaluation of the half-life of 35 P, resulting in a much shorter half-life of 18.3s,
compared to the USDB and experimental results.
The empirical sd shell interactions require a quenching factor to reproduce experiment, implemented through a reduction of 30% of the ratio of the axial-vector
to vector coupling constants gA /gV . In other words, β decays in the sd shell require
approximately the quark value of gA /gV . This reduction, which is necessary to approximately reproduce β decays throughout the sd shell with the USDB interaction,

122

was included for the calculations above with both the USDB and IOI interactions
in order to incorporate effects of higher-order nucleon configuration mixing and from
the presence of the delta isobar in the nuclear wavefunctions [92]. However, the 30%
reduction of gA /gV with the IOI interaction results in too short of a lifetime. To
approximately reproduce the experimental lifetime, the necessary reduction factor
must be larger, with additional quenching due to the exclusion of the 0d5/2 from
the model space. Transition strength at higher energies in 35 S, to isobaric analogue
states of 35 P, are missing without the entire sd neutron shell. The reduction factor is
phenomenologically determined for the IOI interaction by approximately reproducing
the half-life of 35 P. With a reduction factor of 55%, the calculated half-life is 45.9 s, in
reasonable agreement with the experimental value of 47.3(7) s. This reduction factor
is necessary for all calculations with the IOI interaction and will be applied to the
β − decay of 33 Mg.
The decay of 33 Mg was studied by Tripathi et al. [89] and is of particular interest
because of conflicting experimental evidence regarding the parity of the ground state.
From the β decay, the ground state has been identified as 3/2+ . A measurement
of 33 Mg combining laser spectroscopy with nuclear magnetic resonance techniques
unambiguously identified the ground state spin as J = 3/2, while assigning a negative
parity from a measurement of the magnetic dipole moment [93]. Theoretically, the
level scheme of 33 Mg, and particularly the β decay transitions and magnetic moments
of the 3/2+ and 3/2− states, can be compared to the experimental data to provide
insight into the inconsistent assignment of the ground state parity.
The level schemes for 33 Mg and the daughter of its β − decay, 33 Al, are included
in Figs. 5.11 and 5.12, respectively. The spins and parities of the known levels are
mostly unassigned, including the ground state of 33 Mg since the values from available
experimental data are inconsistent. In the calculation, the ground state is 3/2− , but
four states occur within the rms deviation of the fit. As a result, it is necessary
to consider the lowest four states in the calculation. The 7/2− state at 330 keV is

123

6

IOI

Exp.

5

E (MeV)

4
3

11/2−
9/2−

2
1
0

7/2−
3/2−
33 Mg −90.05 MeV 33 Mg −88.90 MeV

Figure 5.11: Comparison of experimental and theoretical level schemes for 33 Mg up
to 6 MeV. Four states of each J π are included in the figure. See the caption to Fig.
2.1 for further explanation.

6

IOI

Exp.

5

E (MeV)

4
3

11/2−

2

9/2+
7/2+

1
0
33 Al

5/2+
−102.69 MeV

33 Al

5/2+
−102.55 MeV

Figure 5.12: Comparison of experimental and theoretical level schemes for 33 Al up to
6 MeV. Experimental data from [89] is included in addition to the ENSDF database.
Four states of each J π are included in the figure. See the caption to Fig. 2.1 for further
explanation.

124

ruled out by the identification of J = 3/2 in [93], but the remaining three states can
all reasonably be expected to match the experimental ground state. To distinguish
between the theoretical 3/2− ground state, the excited 3/2− state at 186 keV, and the
3/2+ state at 316 keV, the β − decay and magnetic moment of each state have been
calculated with the IOI interaction. It is unusual for two states with the same J π to
occur at such low energy, especially with similar wavefunctions as in this case. The
main component of each wavefunction (≈ 45%) is 3p3h in nature, with two neutron
holes in the 0d3/2 orbit, a pair of 0f7/2 neutrons and an unpaired neutron in the 1p3/2
orbit to produce J π = 3/2− .
Tripathi et al. measure the β − decay of 33 Mg from γ transitions in the 33 Al
daughter [89]. As a result, the Gamow-Teller strength above the neutron separation
energy at 5.54(11) MeV was not measured. The branch to the ground state is also not
measured directly, as it does not γ decay, but was inferred from the remaining strength
after accounting for neutron emission. The half-life of 33 Mg was determined from a fit
to the decay curve, with t1/2 = 89(1) ms [89]. With the IOI interaction, the β decay of
the three lowest states in 33 Mg and the γ decay scheme of 33 Al were calculated with
NUSHELLX to compare to the experimental results. Since the ground state of 33 Mg
has J = 3/2, the β decay can only occur to states in 33 Al with J = 1/2, 3/2, 5/2.
As can be seen in Fig. 5.12, the negative parity states begin at 2.68 MeV in
33 Al.

Conservation of parity in the Gamow-Teller decay is a theoretical constraint,

such that both 3/2− states in 33 Mg decay only to negative parity states in 33 Al.
The majority of the strength populates states above five MeV for both 3/2− states.
A significant neutron emission probability Pn is calculated, with Pn = 25.8% and
Pn = 49.6% for the ground state and excited 3/2− state respectively, in comparison
to the experimental value of Pn = 14(2)% [89]. Furthermore, the respective β decay
half-lives are 444 ms and 580 ms, in comparison to 89(1) ms. The 3/2+ state at 316
keV has a half-life of 49 ms, with Pn = 7.4% and a branching ratio of 42.5% to the
ground state, in comparison to the inferred branch of 37(8)% experimentally. The

125

BR(%) log ft

Excitation Energy (MeV)

6
5
4
3
2

log ft

I(%)

4.7(2)
4.9(1)
5.0(1)
5.9(5)

15(5)
13(2)
12(3)
2(2)

2

5.1

5/2 +
5/2 +

5
6

5.0
5.0

3/2 +
3/2 +

5.5(2) 7(2)
5.7(2) 6(2)
−5(3)

2 5.8
5 5.4
2 states
+
3/2
4 5.6
+
5/2
13 5.2

5.2(1) 37(8)

5/2 +

1
0

5/2 +

42 4.9

Levels in 33Al
Figure 5.13: Comparison of experimental and theoretical level schemes for 33 Al as
determined from the β − decay of the lowest 3/2+ state in 33 Mg. The experimental
levels and properties are taken from Fig. 3 of [89]. All theoretical states below the
neutron separation energy with a beta decay branching ratio of at least 2% are plotted.
The “2 states” at 2.54 MeV are: (i) a 5/2+ state with a branching ratio of 3% and a
log f t value of 5.5, and (ii) a 1/2+ state with a branching ratio of 4% and a log f t
value of 5.5.
reduction in the neutron emission probability compared to experiment is due to the
missing strength at higher energies from the exclusion of the d5/2 neutron orbit.
Figure 5.13 compares the decay schemes of the 3/2+ state in 33 Mg for experiment
and theory by analyzing the states in 33 Al. Although it is not shown graphically, the
gamma decay observed in the 33 Al states has been compared to the theoretical E2 and
M1 transitions to aid in the comparison of levels. For instance, there are three states
in the theoretical decay that could match the 2.37 MeV experimental state. Neither
of the two states at 2.54 MeV decays to the ground state, while the only measured
decay from the experimental state populates the ground state. The theoretical 3/2+

126

state at 2.73 MeV has a 93% branch to the ground state in the gamma decay, and the
remaining 7% decays to a 0.96 MeV state that is not populated by β decay, but could
correspond to the 0.76 MeV state in the ENSDF database. Besides the two states at
2.54 MeV, the remaining states can be matched to the data. The experimental state
at 1.62 MeV has a negative intensity because it is populated through gamma decay of
the higher states. Theoretically, this state is assigned to the 5/2+ state at 1.26 MeV,
even though it has a relatively large branching ratio, because of its population in the
de-excitation of the higher-energy states. The highest three theoretical states in Fig.
5.13 have smaller branching ratios than expected from the intensity of measured γ
transitions. In particular, the theoretical transition to the state at 5.60 MeV is much
reduced, while the log f t value and theoretical energy are 2σ higher than experiment.
The theoretical energy places the state right at the neutron decay threshold. The
state is nevertheless matched to the experimental state at 4.73 MeV because two
of the three γ transitions are reproduced, and because there are no other viable
theoretical states. Again, the exclusion of the d5/2 neutron orbit could contribute
to the disagreement. The reproduction of the experimental data provides convincing
evidence that the experiment by Tripathi et al. measured the decay of the 3/2+ state
in 33 Mg. They assigned the ground state parity as positive based on the experimental
results, and suggested that the magnetic moment could be negative if the 3/2+ state
had a predominant 4p3h configuration. However, the theoretical wavefunction of the
3/2+ state is mainly 2p1h in nature.
The nuclear magnetic moment was measured as µ = −0.7456(5)µN , which led to
an assigned ground state spin of 3/2− [93]. For the 3/2+ state, which matches the β −
decay, the magnetic moment has the opposite parity with µ = 0.882µN . The 3/2−
states have the correct parity, with µ = −1.236µN and µ = −0.815µN for the ground
state and excited state at 186 keV, respectively. The predominant configuration for
both 3/2− states has two neutron holes in the sd shell. The magnetic moment is in
much better agreement with the excited 3/2− state. The electric quadrupole moment

127

has not been measured experimentally, but could be used to further differentiate
between the states. The quadrupole moments for the three states are 3.96 (-3.48,13.69)
e2 fm2 for the ground state (3/2− excited state, 3/2+ state). The comparatively large
value of the 3/2+ state and the opposite sign of the 3/2− states should aid in the
identification of the ground state spin through a measurement of the ground state
quadrupole moment. Based on the current status of experimental data, an isomer
may be present with a lifetime similar to that of the ground state, which could have
led to the inconsistent determination of the ground state parity.
The gamma decay from the 3/2+ to 3/2− state, or vice versa, is strongly dependent
on the energy difference between the states. Since all three states are within the rms
deviation, no conclusions can be drawn regarding the gamma decay half-life. In the
event that the energy difference is small, on the order of one keV, the beta decay
half-life could be even shorter than the gamma decay. An upper limit of the halflife is therefore given by the beta decay half-lives for the three states, about half a
second for each 3/2− state and 49 ms for the 3/2+ state. The beta decay scheme and
magnetic moment measurement have identified the ground state as 3/2+ and 3/2− ,
respectively, but either experiment could have measured the properties of an isomer.
The theoretical calculation with the IOI interaction produces a 3/2− ground state,
but also excited 3/2+ and 3/2− states within the rms deviation. The beta decay
scheme of the 3/2+ state agrees well with the experimental decay, while the magnetic
moment of the excited 3/2− is in reasonable agreement with the experimental value.
An experiment that can resolve the isomer from the ground state, even though the
half-lives may be on the same order of magnitude, would be required to assign the
ground state spin in 33 Mg.

5.1.5

26

Ne(d,p)27 Ne

Although β − decay is the primary mode of decay for ground states, other types of
reactions are also of interest in the island of inversion region. Spectroscopic factors
128

Table 5.2: Comparison between experimental [86] and theoretical spectroscopic factors
for states in 27 Ne.
Jπ

EExp.

EIOI

C 2 SExp.

C 2 SIOI

3/2+
3/2−
1/2+
7/2−

0.000
0.765
0.885
1.714

0.000
0.960
0.318
1.670

0.42(22)
0.64(33)
0.17(14)
0.35(10)

0.58
0.63
0.41
0.46

from transfer reactions, as defined in Section 4.1, are of particular interest as a probe of
the evolution of shell structure, evaluated by the strength of single particle states. The
level scheme of 27 Ne has already been shown in Fig. 5.7, including states obtained from
the 26 Ne(d,p)27 Ne reaction [86]. The spectroscopic factors can be calculated directly
with the IOI interaction through an overlap of the wavefunctions. Table 5.2 displays
the experimental and theoretical results for excitation energies and spectroscopic
factors of states in 27 Ne populated by the transfer reaction.
From the table, it can be seen that only the 1/2+ state is inconsistent in the
comparison, once the rms deviation of 370 keV and the experimental uncertainty in
the spectroscopic factors are taken into account. Short-range correlations are reduced
for exotic nuclei, such that the global reduction factor between theory and experiment
for spectroscopic factors [67] is not as pronounced for 27 Ne [94]. The other states rival
the modified empirical interaction WBP-M, tuned by Brown et al. [86] specifically
for 27 Ne. As seen in [86], the original WBP Hamiltonian derived by Warburton and
Brown [95] does not reproduce the low-energy negative parity states, suggesting that
the gap between the sd and pf neutron orbits is too large. With a more microscopic
approach utilizing the Hybrid Renormalization Procedure, the shell structure evolves
away from stability without fine-tuning. Even though the results are not consistent
with experiment, the 1/2+ state is within a deviation of 2σ for both the energy and the
spectroscopic factor. The overall agreement between experiment and calculations with
the IOI interaction is satisfactory and is representative of the accuracy of structure

129

information determined by reactions with exotic nuclei. To evaluate the description
of reaction mechanisms, a more complicated reaction with recent experimental data
is necessary.

5.1.6

30

Mg(t,p)32 Mg

The coexistence of sd and pf neutron configurations in nuclei in the island of inversion
region provides information regarding the stability of the N = 20 closure. The reaction
code FRESCO [96] has been utilized to calculate the cross sections of the two-neutron
transfer reaction 30 Mg(t,p)32 Mg and to evaluate the behavior of the N = 20 shell
closure for the magnesium isotopes.
Figure 5.14 displays schematically the occupation of neutron orbits for the twoneutron transfer reaction, with the negligible contribution of the 1p1/2 orbit not
included. The 0+ ground state of 30 Mg, which lies outside the island of inversion
as seen in Fig. 5.10, can be represented as two neutron holes in the 0d3/2 and 1s1/2
orbits. The two-nucleon transfer to 0+ states in 32 Mg, assuming a direct transfer,
requires two neutrons coupled to zero total angular momentum. In scenario A, sd
orbits are populated in the transfer and the sd shell is filled. In scenario B, pf orbits
are populated, which results in a 2p2h configuration in 32 Mg. As seen in Fig. 5.10,
inverted configurations are necessary to describe the 0+ ground state of 32 Mg, such
that both scenarios must be included to properly describe the two-nucleon transfer
reaction.
Global optical potentials from Perey and Perey [97] are used to describe the
projectile-target and ejectile-remnant systems, as well as the core-core (30 Mg+p)
interaction. Two-body coefficients of fractional parentage are determined from an
overlap of the 30 Mg and 32 Mg wavefunctions with NUSHELLX [9]. The transfer mechanism is assumed to consist of one step, i.e. a simultaneous transfer of a di-neutron.
Sequential transfers of a single neutron should be suppressed due to the large negative
Q-value of the 30 Mg(t,d)31 Mg reaction. The reaction calculations depend strongly on
130

0f7/2, 1p 3/2
1s1/2, 0d3/2
30 Mg

32 Mg

32 Mg

(A)

(B)

Figure 5.14: A schematic representation of the two-neutron transfer in the
30 Mg(t,p)32 Mg reaction. The ground state configuration of 30 Mg consists of two holes
in the sd shell, with the small contribution of 2p4h configurations neglected. Two scenarios are possible for the transfer of a di-neutron: sd orbits are populated and the
nominal N = 20 shell closure is filled (A), or pf orbits are populated, resulting in a
2p2h configuration for 32 Mg (B).
the Q-value; since the experimental binding energies are not reproduced by the IOI interaction within the rms deviation for all the states of interest, the input to FRESCO
will include experimental energies.
In the simplest calculation, the USDB interaction operating in the sd model space
can be used to calculate two-nucleon transfers from 30 Mg to 32 Mg. In this case,
scenario A of Fig. 5.14 is the only possible reaction mechanism. The ground state
of 32 Mg in the sd model space is missing correlations from the pf shell as described
in Section 1.7 and displayed in Fig. 1.7. The 0+ state occurs above nine MeV in
2
excitation energy, since the protons are deeply bound for neutron-rich nuclei and
neutron excitations are not possible. The reaction cross section for the ground state
to ground state transition is 1.70 mb, in comparison to the experimental value of
10.5(7) mb [98]. The exclusion of pf orbits prevents an adequate description of the
reaction process.
With the IOI interaction, which allows both scenarios of Fig. 5.14, the ground state
to ground state transition has a reaction cross section of 9.69 mb, in good agreement
with the experimental value. The average occupation of the pf orbits in the ground
state is 1.79, with the primary configuration (≈ 50% of the total) composed of two

131

neutron holes in the 0d3/2 orbit and two particles in the 0f7/2 orbit. The standard
filling of neutron orbits, given by scenario A, represents only 16% of the ground
state wavefunction. The same configuration, however, is the primary configuration in
the 0+ wavefunction, representing 64% of the wavefunction. The excitation energy
2
of the 0+ state is 2.19 MeV with the IOI interaction, while a recent experiment by
2
Wimmer et al. [98] has identified the 0+ excited state in 32 Mg at 1.06 MeV. With
the experimental excitation energy, the reaction cross section is 0.21 mb. With the
theoretical excitation energy, however, the reaction cross section is 1.32 mb. Since
the shape coexisting 0+ state at 1.06 MeV predominantly has a configuration with a
2
fully-occupied sd shell, a comparable cross section to the USDB case might naively
be expected. This result is obtained only with the theoretical excitation energy.
The mixing with the 2p2h configurations in the 0+ state are important in the
2
determination of the cross section. The inclusion of pf orbits can result in constructive
or destructive interference in the calculation of the cross section. With the coefficients
of fractional parentage from the IOI interaction, the sd and pf coefficients are in
phase and cause destructive interference. A calculation with the same magnitude
of the coefficients, but with the opposite phase for the sd coefficients, results in a
0.48 mb cross section with the experimental excitation energy. This value is still an
order magnitude smaller than the experimental cross section of 6.5(5) mb [98], and
smaller than the result with the USDB ground state, but emphasizes that all orbits
contributing to the low-energy behavior of the nucleus must be included in the model
space.
Wimmer et al. calculate a theoretical cross section for the excited state based
on the ground state wavefunction of the USD interaction in the sd model space. The
result is a factor of two smaller than measured [98], while a similar result here with the
USDB interaction is a factor of 3.8 smaller than measured. The difference in effective
interaction between USDB and USD affects the cross section on the 2% level. The
optical potentials, coupling scheme, and other parameters of the reaction calculation

132

significantly affect the results. Global optical potentials and a simplistic description
of the two-nucleon transfer were approximations which could be improved upon in
future work and are likely the cause for the deviation between these results and those
from Wimmer et al. From the results obtained in this work, the reaction cross section
to the ground state is in good agreement with experiment but the cross section to
the excited 0+ state is significantly reduced relative to experiment. The reaction
calculation and the amount of mixing depend on the energy difference between the
0 ω and 2 ω configurations for 32 Mg. Since the excitation energy of the sd-like 0+
2
state is over one MeV higher than experiment, the amount of mixing is likely reduced
relative to the physical state. The similar experimental reaction cross sections to the
ground state and excited 0+ state, 10.5(7) and 6.5(5) mb respectively, suggest that
the necessary mixing between 0 ω and 2 ω configurations is greater than calculated
with the IOI interaction.

5.2

42

Si

Another nucleus of interest for the Hybrid Renormalization Procedure is 42 Si, due
to the evolution of the N = 28 shell closure away from stability. Bastin et al. [99]
measured the 2+ energy of 42 Si at 770(19) keV, indicating the disappearance of the
shell closure as displayed in Fig. 1.9. Because 42 Si is outside of standard model spaces
and has only recently been reproduced with the empirical SDPF-U interaction [48]
discussed in Chapter 3, the HRP provides a more microscopic description which can
be compared directly to experimental data. Again, the sd proton orbits are active,
and the sd and pf neutron orbits could be of interest. As discussed in the beginning
of Section 5.1, the matrix diagonalization restricts the number of neutron orbits. Two
separate valence spaces will be employed, using the model space from the previous
section and the sdpf model space of Chapter 3.
In the model space from Section 5.1, the IOI interaction can be applied directly

133

Excitation Energy (keV)

2500
2250
2000
1750
1500
1250
1000
750
500
250
0

0+

4+

4+

2+
4+
0+
2+

0+
SDPF−U

2+

0+

2+
0+

0+

2+
0+

2+

0+
4+

4+
2+
0+

0+

(i)
42 Si

2+

0+

(ii)

2+

0+
Exp.

(iii)

Level Scheme

Figure 5.15: A comparison of level schemes for 42 Si from experiment, from the empirical SDPF-U interaction for Z ≤ 14, and with three interactions derived from the
HRP: the IOI interaction (i), the IOI interaction with SPE modified by Eq. 5.2 (ii),
and an interaction in the sdpf model space with 42 Si as the target (iii).
to 42 Si, composed of eight neutrons on top of the 34 Si target. The monopole terms
account for evolution to heavier and more unstable isotopes, but reduce the predictive
power outside of the experimental data included in the fit. Figure 5.2 showed the
inaccuracy of Skyrme SPE in this model space for 34 Si and emphasized the necessity
of parameterizing the SPE. If the Skxtb interaction reproduces the evolution of SPE
from 34 Si to 42 Si, the appropriate SPE for the 42 Si calculation are modified from the
Skyrme SPE by
42 Si
mod

42

34

34

Si
Si
Si
= Skxtb + f it − Skxtb .

(5.2)

Two calculations will be performed in this model space: one with the IOI interaction,
and another where the SPE of the IOI interaction have been modified by Eq. 5.2.
In the sdpf model space, the closed-subshell 42 Si nucleus is the target and the HRP
is applied to determine two-body matrix elements and Skyrme single particle energies.
With the limited data around 42 Si, SPE cannot be determined from experimental

134

levels that approximate single particle states or by a fit to data; Skyrme uncorrelated
SPE are utilized as discussed in Section 4.4.
The level schemes for 42 Si are shown in Fig. 5.15 for the different methods. In addition to the three HRP calculations, level schemes for experiment [99] and an empirical
sdpf interaction [48] are provided for comparison. Thus far, the experimental studies
have only detected the ground state and the first excited 2+ state. Theoretically, the
first excited 0+ varies by 1.8 MeV. With the interaction from the application of the
HRP in the sdpf model space (iii), the 0+ state is higher in energy than the 4+ state.
2
1
On the other hand, the 0+ state is the first excited state with both the empirical
2
SDPF-U interaction and the IOI interaction from Section 5.1. The energy of the 0+
2
state in 42 Si can provide information regarding the mixing between shape coexisting
states and the stability of the N = 28 shell closure, following the discussion in Section
5.1.6.
The IOI interaction (i) produces a high level density relative to the other theoretical approaches, including an interaction with the same TBME but modified SPE (ii).
Accounting for the evolution of Skyrme single particle energies from 34 Si to 42 Si also
results in a lower 2+ energy, in better agreement with experiment. For the microscopic
interaction in the sdpf model space (iii), a low density of states is observed. While
excitations of the sd orbits cannot occur, such that the level density is expected to be
lower than in (i) and (ii), the active orbits are identical to the SDPF-U calculation.
The microscopic approach does not reproduce the low-lying 0+ state of the empir2
ical interaction. More experimental data is needed for definitive conclusions on the
importance of excitations of the sd neutrons and on the possibility of improving single particle energies for exotic isotopes by modifying Skyrme SPE with Eq. 5.2. The
identification of the 4+ and 0+ states, which will occur with rare isotope facilities,
1
2
can provide valuable insight for the application of the HRP far from stability.

135

Chapter 6
Outlook and Conclusions
The Hybrid Renormalization Procedure has been applied to regions of recent experimental interest. The agreement with available experimental data suggests the
viability of the method, such that it can be extended to other regions of the nuclear chart. However, the deviation from experiment indicates that there are still
improvements necessary. As discussed in Chapter 4, one improvement relates to the
inconsistency and inaccuracy of Skyrme single particle energies. Microscopic energy
density functionals, which are in the early stages of development, should focus on
accurately accounting for the single particle properties of nuclei. The need for SPE
parameters, as implemented in Section 5.1, could be eliminated with better functionals. Progress on any research method utilized by the HRP, i.e. renormalization group
methods, many-body perturbative techniques, Configuration Interaction theory, and
Energy Density Functional methods, will reduce the uncertainties in derived effective
interactions and produce more reliable results.
Advancements in nuclear structure research external to this work can be continually implemented to enhance the HRP. Concurrently, the HRP can be improved
by accounting for the effect of many-body forces. As discussed in Section 2.2, the
HRP attempts to approximately include three-body effects in two ways: through the
Skyrme interaction and through the introduction of monopole parameters as in Sec-

136

tion 5.1. The effective two-body component of the three-body force is not included
accurately in either case, but could be implemented in the HRP directly through a
modification of the two-body matrix elements. The importance of this effective twobody component can be estimated from Fig. 2.3 since the USDB interaction implicitly
includes the contribution from three-body forces while the HRP-SD interaction does
not. Therefore, a treatment of three-body forces at the effective two-body level could
significantly improve the effective interaction.

6.1

Three-body Forces

Truncating at three-body forces, the Hamiltonian was represented diagrammatically
in Fig. 1.3. Single particle energies derived from EDF methods, for instance the onenucleon separation energies calculated in Skyrme Hartree-Fock theory, include contributions from all terms within the dashed black box. Two-body matrix elements in
the Hybrid Renormalization Procedure currently include only the explicit two-body
component as derived from a microscopic nucleon-nucleon interaction. However, the
effective two-body component of the three-body diagram, as determined from Skyrme
Hartree-Fock theory [13], can be added through the monopole terms of the effective
interaction given by
(2J + 1) ab | V | ab JT
¯
Vab = J

.

(6.1)

(2J + 1)
J

The monopole terms of the effective interaction, as derived from a microscopic nucleon¯N
nucleon interaction with the HRP, are denoted Vab N . To account for the three-body
forces at the effective two-body level, the monopole can instead be taken from calculations with EDF methods for two particles coupled to the target nucleus. For a
target nuclues T , the monopole is
¯ EDF = E(T + a + b) − E(T ) − a − b ,
Vab
137

(6.2)

where i are one-nucleon addition energies of particle orbits outside the core, obtained
with the equal filling approximation in an SR-EDF calculation as discussed in Section
4.4, and E(T +a+b) is the energy of the configuration for a closed-shell target plus two
nucleons constrained to be in orbitals a and b in a single reference EDF calculation
¯ EDF and V N N represents
¯
restricted to spherical symmetry. The difference between Vab
ab
the effective two-body component of the three-body force. All valence TBME can be
modified by
ef f
¯N
¯ EDF
ab | V | ab JT = ab | V | ab HRP − Vab N + Vab
JT

(6.3)

to include the three-body force via the monopole two-body interaction.
Brown et al. [13] have applied this procedure to a 208 Pb target to produce level
schemes in good agreement with experiment. An important effect of the procedure,
shown in Fig. 7 of [13], is the reproduction of experimental binding energies as many
particles are added. In order for a direct comparison between single-reference calculations with CI and EDF methods, the single particle energies in the effective interaction
must be taken from the one-nucleon separation energies as determined by the energy
¯ EDF
density functional that is used to calculate Vab , rather than from experimental
states or from a fit to data. The difference between the two calculations with a single
Slater determinant can then be represented as the monopole of the explicit three-body
term in Fig. 1.3, with matrix elements on the order of 1-2 keV [13]. The diagonalization of the full model space Hamiltonian results in correlations due to configuration
mixing, which increases the binding energy for nuclei away from the closed shell.
The inclusion of three-body forces through a monopole two-body interaction has
been shown to have a 5.3 MeV effect for ten protons outside of 208 Pb [13]. In Section
5.1, the monopole parameters contributed 11.1 MeV for 34 Si, which is composed of
twelve nucleons outside the core. For a three-body interaction with identical strength
in both regions, a contribution of 9.7 MeV is expected for 34 Si due to the additional

138

permutations allowed for twelve particles. The monopole parameters provide approximately the expected strength due to three-body forces, but are not understood on a
fundamental level. The parameters could be eliminated with an appropriate treatment
of the effective two-body component of the three-body force, through an implementation of Eq. 6.3 in the Hybrid Renormalization Procedure. The application to the
island of inversion region could be revisited, or calculations in other regions of interest
could be performed.

6.2

Other Regions of Interest

Ab initio approaches are applicable for A < 16 and provide a nearly exact solution
starting from microscopic potentials. Standard Configuration Interaction methods
treat 16 O as a vacuum and produce results throughout the sd shell. For instance, the
USDB interaction can be used throughout the sd shell with an rms deviation around
130 keV. As seen in Section 2.2, the Hybrid Renormalization Procedure can be applied
to nuclei in the sd shell to provide results in reasonable agreement with experiment.
However, due to the lack of three body forces, the results will not surpass those of
empirical interactions like USDB. The form of the HRP-SD and USDB interactions is
identical, as they both consist of single particle energies and two-body matrix elements
in the model space. The USDB interaction supplies the best fit two-body Hamiltonian,
thereby incorporating effects from many-body forces and orbits outside of the model
space. A microscopic treatment of the many-body problem, as presented with the
HRP, cannot improve upon the description of nuclei with the USDB interaction in
the sd model space, but can provide insight into the contributions of three-body
forces, for example.
The lightest nuclei that are particularly of interest for the HRP, since they cannot
be calculated easily with other methods, lie on the neutron-rich side of the chart of
nuclides. For instance, as mentioned in Chapter 3, the neutron-rich carbon isotopes

139

require a reduction of 25% in the neutron-neutron interaction relative to the more
stable oxygen isotopes. As in Chapter 3, the HRP might identify a microscopic origin
of this reduction due to loosely bound valence orbits with long tails in their realistic
radial wavefunctions.
By mass, the next neutron-rich isotopes outside of standard model spaces occur
around N = 20. The HRP was applied in Chapter 5 to explain the island of inversion
region, but the produced interaction was applied to one hundred nuclei, including
the light neutron-rich isotopes around N = 20. In Section 5.2, the behavior of the
N = 28 subshell was also probed through an investigation of 42 Si. Future applications
of the HRP are to other “islands of shell-breaking” where, like the island of inversion,
higher-energy orbits are filled preferentially to the standard ordering of orbits from
Fig. 1.2. The next region of the chart of nuclides with similar behavior to the island
of inversion occurs near the neutron-rich nickel isotopes. A fully occupied pf shell
for 68 Ni results in magicity which is broken for 66 Fe when two protons are removed.
Positive parity orbits, namely the g9/2 neutron orbit, are occupied in some ground
states with N = 40. This region could be studied with the HRP, treating 68 Ni as
the target in the same manner that 34 Si was the target for the island of inversion
region: as an exotic nucleus with good closed-subshell properties outside of the island
of shell-breaking. The computing power required for the 68 Ni region is significantly
larger than the island of inversion region, but calculations are tractable with modern
parallelized resources.
Proton-rich nuclei outside of standard model spaces are another possibility for an
application of the HRP. Unfortunately, less experimental data is known on this side
of stability, but nuclei that occupy sd neutron orbits and pf proton orbits could be
compared to their mirror nuclei in the sdpf model space defined in Section 1.7. In this
case, 34 Ca would be the target and results could be compared to the known isotopes
from Z = 20 to Z = 24 with N ≤ 19.
Finally, a study of heavy nuclei near the doubly magic 100 Sn, 132 Sn, and 208 Pb

140

cores would provide a more microscopic prediction of collectivity in the tin and lead
isotopic chains, of recent experimental [100] and theoretical [101] interest. The onset
and reduction in collectivity as a function of neutron number offers insight into the
shell structure and the strength of pairing matrix elements. A realistic calculation like
the HRP that accounts for these effects directly would better explain the experimental
results, especially with the inclusion of three-body effects from Section 6.1.

6.3

Single Particle Energies

As discussed in Section 4.4, the uncorrelated SPE of Skyrme interactions have been
implemented in the Hybrid Renormalization Procedure to produce the SPE component of effective interactions. However, the one-nucleon separation energies of Skyrme
interactions, which require the solution of Skyrme Hartree-Fock equations for three
different nuclei, provide better indications of experimental behavior and should have
been implemented in effective interactions derived from the HRP. The calculations
in Chapter 3 and Sections 2.2 and 5.2 should be examined with the single particle
energies of the effective interactions derived from the one-nucleon separation energies
of the Skxtb interaction. Unless the uncertainty due to the equal filling approximation
is quantified and is small relative to the global uncertainty of separation energies with
the Skyrme interaction, a Skyrme Hartree-Fock solution for odd nuclei which breaks
time-reversal symmetry is necessary. With either uncorrelated SPE or one-nucleon
separation energies from EDF methods, the dependence of results on the parameterization of the Skyrme interaction should also be examined; in particular, the role of
the tensor force, determined by a comparison of Skx and Skxtb for instance, should
be understood from stability to exotic nuclei.
The centroid energy as defined in Eq. 4.3 is a simplification of the centroid energy
described by Baranger [65], since the off-diagonal centroid matrix elements are neglected. The effective single particle energies obtained from a diagonalization of the

141

centroid matrix can be compared to the results presented in Chapters 4 and 5 to evaluate the dependence on the simplified treatment of the centroid energy. A systematic
study of ESPE and one-nucleon separation energies with various nuclear structure
techniques would provide guidance regarding the uncertainties in single particle energies and the derivation of energy density functionals.
In a large model space, the SPE of the effective interaction should be determined
from the ESPE of the target nucleus. For nuclei within the model space that are many
particle or hole excitations from the target, the ESPE can be significantly different
than the SPE of the effective interaction. In other words, the single particle structure
can evolve as a function of proton or neutron number. In Section 5.2, the evolution of
uncorrelated Skyrme SPE from 34 Si to 42 Si was assumed to reproduce the necessary
change in the experimental ESPE from 34 Si to 42 Si, resulting in a modified interaction
denoted (ii) in Fig. 5.15. The resultant calculations for 42 Si are affected greatly by
the evolution of SPE, as seen in the comparison of (i) and (ii) in Fig. 5.15. The lack
of experimental data and the lack of further applications of SPE modified by Eq. 5.2
prevent a definitive conclusion regarding the best way to calculate nuclei far from the
target nucleus in a large model space.

6.4

Conclusion

The extension of experimental capabilities to more exotic nuclei with new or planned
rare isotope facilities has emphasized the necessity of reliable theoretical calculations
for nuclei outside of standard model spaces. Hybrid methods, which combine properties of the Configuration Interaction method and Energy Density Functional methods,
can incorporate the generality of EDF methods with the accuracy of the CI method.
A hybrid method has been developed to convert a microscopic nucleon-nucleon potential into a low-momentum interaction via renormalization group methods, namely
the vlowk procedure. Many-body perturbative techniques can then renormalize the

142

low-momentum interaction into a reduced model space using Rayleigh-Schrödinger
perturbation theory. An effective interaction, composed of single particle energies obtained from Skyrme Hartree-Fock theory and two-body matrix elements from the
renormalization procedure, can be implemented into full-scale Configuration Interaction calculations to determine theoretical properties of interest. A proof of principle
was confirmed for this Hybrid Renormalization Procedure in a standard model space;
the HRP results in the sd shell with 16 O as the target nucleus agree with the empirical USDB Hamiltonian to the expected accuracy, accounting for the restriction to a
microscopic two-body force.
One ingredient of the HRP is the basis, with three choices studied: the harmonic
oscillator basis, standard for many applications in nuclear physics; a realistic basis
derived from Energy Density Functional methods; and a mixed basis, using harmonic
oscillator radial wavefunctions and Energy Density Functional single particle energies. Due to recent experimental and theoretical interest in the sdpf shell and the
neutron-rich silicon isotopes, six interactions were derived in the sdpf shell using all
combinations of the three bases and two target nuclei, the doubly magic 40 Ca nucleus and the exotic 34 Si nucleus. In the harmonic oscillator and mixed bases, the
interaction is nearly identical regardless of the choice of target. However, the 34 Si
target reduces the strength of the interaction in the realistic basis as a result of
the extended wavefunctions for loosely bound model space orbits. Calculations for
neutron-rich silicon isotopes display the importance of the accurate determination of
effective interactions away from stability. Accounting for the evolution of shell structure through a realistic basis is essential for an accurate description of exotic nuclei,
but three-body forces must also be included for accuracy at the level of 100 keV.
Single particle energies are an important component of the Hybrid Renormalization Procedure, but are not determined accurately in Skyrme Hartree-Fock theory,
particularly for light nuclei. Centroid energies from Eq. 4.3, whether determined from
experiment or theory, are desired for the target nucleus since they better represent

143

the single particle properties of interest throughout a given model space. As centroid energies cannot be easily obtained, the SPE in the effective interaction can be
parameterized to improve upon the Skyrme Hartree-Fock description.
The Hybrid Renormalization Procedure was performed to generate an effective
interaction for calculations in the island of inversion region. With twelve parameters
to improve upon the Skyrme Hartree-Fock treatment of single particle energies and
three-body forces, and an overall normalization of the two-body matrix elements, the
level schemes of one hundred nuclei were calculated. Binding energies and low-energy
states are in good agreement with available data when the 370 keV rms deviation of the
interaction is taken into consideration. The boundaries of the island of inversion are
determined theoretically through the evaluation of the ground state wavefunctions,
and suggest that the island of inversion is larger than observed thus far, extending
to both more neutron-rich nuclei and nuclei with fewer protons. Beta decay transitions from 35 Si and 33 Mg, and one- and two-neutron transfer on 26 Ne and 30 Mg,
respectively, have also been studied and compared to recent experimental data. The
level schemes of 42 Si with different applications of the HRP have been compared to
experiment and to the result with an empirical interaction. The measurement of the
0+ and 4+ energies, which vary significantly in the theoretical approaches, will enable
2
1
the evaluation of different techniques to calculate exotic isotopes via the HRP.
The Hybrid Renormalization Procedure can be improved by implementing threebody forces at the effective two-body level. Other aspects of the Hybrid Renormalization Procedure, particularly the single particle energies derived from Energy Density
Functional methods but also renormalization group methods and many-body perturbative techniques, are of interest for future study. Progress in these areas can be
implemented directly into the Hybrid Renormalization Procedure along with threebody forces to provide better convergence properties and effective interactions.
The extensive results obtained in the island of inversion region pertain to one application of the Hybrid Renormalization Procedure. Theoretical predictions for other

144

nuclei outside of standard model spaces can be produced in future applications.

145

Appendix

146

Appendix A
Abbreviations and Definitions of
Selected Terms
α - representation of the quantum numbers of a single orbit
A - mass number
Aval - number of valence particles outside of a core
Argonne v18 - a microscopic N N potential composed of eighteen local operators
BE - binding energy of a system
BR - branching ratio
Cij - coefficients for the two-body monopole parameters in the IOI interaction
C3 - coefficient for the three-body monopole parameter in the IOI interaction
C 2 S - spectroscopic factor, where C is the isospin Clebsch-Gordan coefficient and S
is defined in Eq. 4.1
CI - Configuration Interaction; CI theory is an approximate method to calculate the
Schrödinger Equation 1.1 that operates in a reduced model space
CIB - charge-independence breaking; denotes a strong force where the average of the
proton-proton and neutron-neutron interactions is not identical to the proton-neutron
interaction
CSB - charge-symmetry-breaking; denotes a strong force where the proton-proton

147

interaction is not identical to the neutron-neutron interaction
d - deuteron
D - dimension of the model space
DF T - Density Functional Theory; DFT attempts to write the energy of a nucleus as
a functional of the local density of the system, and to solve for the energy of a system
through the minimization of that functional
- see ESPE
E - energy of a system
E2 - electric quadrupole transition; see [12] or [3] for description
EDF - Energy Density Functional; the energy written as a functional of the density
of a system. EDF methods are similar to DFT, except that the Hohenberg-Kohn
theorems are not satisfied by the nuclear functionals which break symmetries present
in the nuclear system
ENSDF - Evaluated Nuclear Structure Data File, an online database containing nuclear properties such as excitation energies and log f t values
ESPE - effective single particle energy or centroid, as defined by Eq. 4.3 and denoted
by
Exp. - experiment
f t - a phase space factor f multiplied by the half-life t for nuclear β decays; often
given as log f t, where the logarithm has base 10.
gA , gV - coupling constants for the axial-vector, vector component of the weak interaction
G-matrix - an RG method that solves or approximates Eq. 1.20
ω - energy scale in nuclear physics based on the solution of a harmonic oscillator
mean field, where empirically ω = (45A−1/3 − 25A−2/3 ) MeV; an ω excitation
refers to an increase in the main quantum number N by one
H - Hamiltonian
H0 - the mean field component of the nuclear Hamiltonian

148

H1 - the remainder of the nuclear Hamiltonian, which can be treated as a perturbation
from H0
HO - harmonic oscillator
HRP - Hybrid Renormalization Procedure; defined in Chapter 2
HRP-SD - an interaction in the sd model space calculated for a 16 O target in the
standard implementation of the HRP
HRP-3rd - identical to HRP-SD, except that diagrams are calculated to third order
in Rayleigh-Schrödinger perturbation theory
IOI - an effective interaction derived from the HRP to calculate nuclei in and near
the island of inversion region; see Section 5.1 for details
j - total angular momentum of a single particle
J - total angular momentum of a system
J π - total angular momentum and parity of a system
k - representation of the quantum numbers of a many-body system
Λ - the cutoff in momentum space for the vlowk renormalization procedure
- orbital angular momentum
mj - projection of j along the z-axis
M1 - magnetic dipole transition; see [12] or [3] for description
MIX - a basis which is composed of HO radial wavefunctions and SHF single particle
energies
MSk7 - parameterization of the Skyrme interaction without a tensor force
ν - neutron
n - neutron
n - radial quantum number
npnh - number of particle and hole excitations from a reference state, typically a
Slater determinant with the natural ordering of orbits
N - can refer either to neutron number or main quantum number for a harmonic
oscillator potential, where N = 2n + ; the disctinction should be clear from the

149

usage in the text
N N - nucleon-nucleon; microscopic N N interactions or potentials describe the interaction between two free nucleons (proton or neutron) and are fit to scattering data
N N N - nucleon-nucleon-nucleon; microscopic N N N interactions or potentials describe the interaction between three free nucleons and are dependent on the choice of
N N interaction
N2LO,N3LO - next-to-next-to-leading order, next-to-next-to-next-to-leading order;
the order of perturbation theory in χEFT; the N3LO interaction is a chiral microscopic
N N potential at the N3LO level fit to elastic scattering data
π - proton
p - proton
p - model space composed of all N = 1 single particle orbits (0p3/2 , 0p1/2 )
pf - model space composed of all N = 3 single particle orbits (0f7/2 , 0f5/2 , 1p3/2 , 1p1/2 )
P, Q - projection operators into (P ) and out of (Q) the model space
Q-value - mass or binding energy differences between the reactants and products of
a reaction
ˆ
Q(ω) or Q-box - the sum of all irreducible and valence-linked diagrams with at least
one H1 vertex; see [11] for a complete description
QCD - quantum chromodynamics
ρ - the local density of a system
RG - Renormalization Group; RG methods are used to convert a microscopic N N
potential with a hard core into a softer potential where the high- and low-momentum
modes are decoupled
σ - standard deviation
s - model space composed of all N = 0 single particle orbits (0s1/2 )
sd - model space composed of all N = 2 single particle orbits (0d5/2 , 0d3/2 , 1s1/2 )
sdpf - the model space composed of the sd proton orbits and pf neutron orbits
Sn - one-neutron separation energy

150

S2n - two-neutron separation energy
SDPF-NR,SDPF-U - empirical interactions in the sdpf model space
SHF - Skyrme Hartree-Fock
Skx, Skxtb - parameterizations of the Skyrme interaction without, with a tensor force
Sly4 - parameterization of the Skyrme interaction with a tensor force
SPE - single particle energy
SR-EDF - Single Reference EDF; EDF methods where a single Slater determinant is
used as an auxiliary wavefunction for the minimization of the energy density functional
T - in general, T refers to the isospin of a system; in Section 1.4, it refers to the
scattering matrix while in Section 6.1 it refers to a target nucleus
TBME - two-body matrix elements
µ - the nuclear magnetic moment, often reported in terms of the nuclear magneton
µN
UNEDF - Universal Nuclear Energy Density Functional; a collaboration funded by
the U.S. Department of Energy to determin a microscopic EDF for nuclear physics
USD,USDA,USDB - empirical two-body interactions in the sd model space
vlowk - an RG method that approximates Eq. 1.21
ω - when separate from , the starting energy of a two-body diagram in RayleighSchrödinger perturbation theo
WBP,WBP-M - empirical interactions in a model space composed of all orbits through
the main quantum number N = 3
χEFT - chiral effective field theory
Z - proton number

151

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