ELECTRON-CAPTURES IN SUPERNOVAE
By
LeShawna Valdez

A THESIS
Submitted to
Michigan State University
in partial fulllment of the requirements
for the degree of
MASTER OF SCIENCE
Physics
2012

ABSTRACT
ELECTRON-CAPTURES IN SUPERNOVAE
By
LeShawna Valdez

A method to eectively compare electron-capture rates calculated from both theoretical
and experimental Gamow-Teller strength distributions has been created. In this thesis, the
method has been outlined and applied to the transition from

56

Fe to

56

Mn. Factors that

signicantly impact the electron-capture rates are the stellar temperature, the stellar density,
the excitation energy spectrum of the daughter nucleus, and the strength of the GamowTeller transitions. The ability of theoretical models to reproduce the locations (and to a lesser
degree, the strengths) of the low-lying states that can be captured into, is the most crucial
factor in producing reliable electron-capture rates. A good agreement was found between
Gamow-Teller transitions calculated using the GXPF1a interaction in the shell-model and
the recent data from a high resolution

56

3
Fe(t, He) experiment. Since calculations with the

KB3G interaction, which forms the basis for a weak reaction-rate library commonly used
in astrophysical simulations do worse, it is suggested that improvements can be made to
the current inputs for astrophysical models.
results from a

56

This conclusion is in agreement with recent

Ni(p, n) experiment where experimental Gamow-Teller strengths from

were compared with theory.

56

Ni

The procedure outlined in this thesis can be used to further

constrain theoretical Gamow-Teller strength calculations with new, successfully developed
charge-exchange probes which will be utilized at rare isotope beam facilities such as FRIB.

ACKNOWLEDGMENTS

First and foremost I want to thank my advisor, Remco Zegers. Words cannot express my
gratitude. You always supported me and I am so appreciative of that! Thank you for being
the best advisor out there! I learned so much and I feel spoiled rotten that I was able to
work with you. Second, I want to acknowledge my best friend Scott Bustabad for sticking
by me through thick and thin providing strength and encouragement. I also want to express
my deep thanks to Vladimir Zelevinski (Zele) for being such a good mentor and friend. I
also want to thank my committee members, Ed Brown and Chris Wrede, for your comments
and suggestions that make my thesis the best that is can be. Arthur Cole for the guidance
you gave me when I rst came to NSCL and his ongoing collaborative eort. Alex Brown
for providing me with NuShellX@MSU and always being willing to answer my questions.
Angelo Signoracci for the countless times you were willing to help me with NuShellX. Sanjib
Gupta for the electron-capture code and QRPA calculations. I thank the CE group for their
support. Mike Scott and Yoshi Shimbara for the data used in this thesis. Also, Mike Scott
for bringing such hilarity to our endeavors. Rhiannon Meharchand for all the mentoring and
help - amongst many other things.

Amanda Prinke for advice on my thesis work.

Carol

Guess for advice and encouragement early on in my graduate career (and all the tea). Jenna
Deaven for help with PAW and the DAQ. Wes Hitt for writing such a useful thesis in which
to refer. Jared Doster for simply being willing to listen. Masaki Sasano for always, always
helping me even when I felt my questions may be weird.
DAQ help and comments for gures in my thesis.

George Perdikakis for all the

Sam Austin for guidance with gures

and comments on my research. I want to acknowledge those at ASU who provided me with

iv

guidance and research experience: Rich Lebed and Ricardo Alarcon. Rich, thank you for
being such a good friend, mentor, and teacher. Ricardo, thank you for giving me such good
advice and for the opportunity to work with you. I also want to thank my many friends at
NSCL and BPS who provided me with laughs and fun trips on Mondays to the Dairy Store
for the grilled cheese-tomato soup combo without which the week would never be the same.
Last but denitely not least, my family for their never-ending encouragement.

v

TABLE OF CONTENTS

List of Figures
1

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

Introduction and Motivation
1.1

Supernovae

viii

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

1

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

1

1.1.1

Core-Collapse Supernovae

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

3

1.1.2

Thermonuclear Supernovae . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Electron-Capture and Charge-Exchange . . . . . . . . . . . . . . . . . . . . .

6

1.3

Factors in Electron-Capture Rate Calculations . . . . . . . . . . . . . . . . .

11

1.3.1

Temperature and Density of the Star

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

11

1.3.2

Excitation Energy in Daughter Nuclei . . . . . . . . . . . . . . . . . .

15

1.3.3

Gamow-Teller Transition Strength . . . . . . . . . . . . . . . . . . . .

15

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.4

2

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

20

2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.2

Codes Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.3

Input Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.4

Explanation Directory

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

26

2.5

Phaseoutles Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.6

Running the Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.7

3

Electron Capture Rate Calculation

Analyzing the Results

29

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

Obtaining Gamow-Teller Strengths

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

32

3.1

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

32

Theoretical Gamow-Teller Strengths . . . . . . . . . . . . . . . . . . . . . . .

34

3.2.1

Using NuShellX@MSU . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.2.1.1

Components of the Answer File . . . . . . . . . . . . . . . .

35

3.2.1.2

4

Experimental Gamow-Teller Strengths

3.2

After the Calculation is Complete . . . . . . . . . . . . . . .

40

Comparisons and Results

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

43

4.1

The Gamow-Teller Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4.2

The Electron-Capture Rate Calculations

49

vi

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

5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bibliography

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

vii

53
58

LIST OF FIGURES

Figure 1.1

Supernova remnant 1987a . . . . . . . . . . . . . . . . . . . . . . . . .

3

Figure 1.2

Supernova remnant 1572

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

5

Figure 1.3

Electron-Capture and Charge-Exchange . . . . . . . . . . . . . . . . .

7

Figure 1.4

Excitation Energies Probed by Charge-Exchange . . . . . . . . . . . .

8

Figure 1.5

Fermi Energy as a Function of Density and Electron Fraction . . . . .

10

Figure 1.6

Eects of Density and Temperature on Electron-Capture Rates.

12

Figure 1.7

. . .

Constant Temperature, Varying Density Comparison of Electron-Capture
Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Figure 1.8

Comparison of Electron-Capture Rates for Two Densities

16

Figure 1.9

Ratio of Gamow-Teller Strengths and Eects on Electron-Capture Rates 18

Figure 2.1

Nucleuslist Data File for the Electron-Capture Rate Calculation Program 22

Figure 2.2

Input File for the Electron-Capture Rate Calculator Program . . . . .

23

Figure 2.3

Parent File Read by The Electron-Capture Rate Calculation Program

24

Figure 2.4

Daughter File Read by The Electron-Capture Rate Calculation Program 25

Figure 2.5

Transitions File Read by The Electron-Capture Rate Calculation Program

Figure 2.6

. . . . . . .

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

26

Header File Read by The Electron-Capture Rate Calculation Program

26

viii

Figure 2.7

Starting PAW

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

Figure 2.8

PAW Command Prompt

Figure 2.9

Sample PAW Plot of

Figure 3.1

NuShellX@MSU Prompt

50

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

30

. . . . . . . . . .

31

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

35

Figure 3.2

Nushell Answer File . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Figure 3.3

NuShellX@MSU Answer File: Ground State Calculation . . . . . . . .

42

Figure 4.1

Nuclei in the Region Near

Figure 4.2

56

V Electron-Capture Results

30

56

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

44

Fe Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

56

Fe

56
Fe→
Mn Reaction.

Figure 4.3

B(GT) Spectrum for

Figure 4.4

Electron-Capture Rate Calculations:

ix

56

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

48

56
Fe→
Mn Reaction . . . . . .

51

Chapter 1

Introduction and Motivation

1.1 Supernovae
Interstellar clouds of gas and dust form the birthing region for stars. Compression of this
gas and dust due to gravity creates a protostar and as this protostar continues to contract,
the temperature and density increase. At low temperatures, the hydrogen nuclei repel each
as a result of Coulomb repulsion. However, if the nuclei were to get close enough the strong
nuclear force would take over and the nuclei would attract.

An increase in temperature

provides the energy needed for this attraction to begin. As the temperature increases, kinetic
energy increases eventually becoming greater than the coulomb barrier. The hydrogen nuclei
now have enough energy to overcome this barrier and attract one another leading to fusion
of nuclei. It is at this point that the star begins its life on the main sequence where it will
spend most of its life forming helium through hydrogen fusion [1]. Stars can fuse higher mass
nuclei and stars whose mass

M

8M

continue consuming heavier elements until the most

massive stars form a core of iron-group of nuclei through silicon burning. The members of

1

the iron-group dominating the late stellar core consist of

54

Fe,

56

56

Fe, and

nuclei have the highest binding energy per nucleon of all other nuclei with

62

Ni [1]. These

Ni having the

greatest. Reaching the iron-group means the end of nuclear fusion in the core of the star
since fusing iron would take away energy rather than provide it. This eventually sends the
star into the core-collapse supernova phase which is to be discussed in section 1.1.1. Stars
less massive than those ending their lives in core-collapse supernovae evolve o the main
sequence forming red giants and eventually white dwarfs. It is possible for these stars to end
their lives in explosions as well and is the topic of section 1.1.2. Both types of explosions are
important in that they are major sources of heavy elements in the universe.
It has taken centuries for observers to realize the importance of these catastrophic events.
The earliest record of such an event was labeled a guest star by ancient Chinese observers
[2]. Centuries later, the term nova was introduced by Tycho Brahe with the publication of
his book

De Nova Stella meaning Of the New Star, after observing the supernova that now

bears his name in 1572 [3]. In the 1930s, Fritz Zwicky introduced the term super-novae to
distinguish from more common novae. He, along with Walter Baade were the rst to dene
categories for these appearances: common novae, and super-novae [4]. Today, supernovae
themselves have been further categorized into two types, those containing hydrogen lines
being Type II and those without, Type I [3]. Subcategories are contained within the two
types but an important distinction is that Type Ia supernovae are thermonuclear while all
others are core-collapse supernovae. Although the exact dynamics of the explosion are not
well understood for either events, it has become clear that weak reactions, electron-capture
(EC) in particular, play an important role [5].

To understand EC's role in supernovae

dynamics, it is important to rst address the types of supernovae.

2

1.1.1

Core-Collapse Supernovae

Core-collapse supernovae take place in high mass stars (M

8M

) after the star has used up

its nuclear fuel for fusion in the core. At this point, thermal pressure is no longer supporting
the weight of the outer layers and electron-degeneracy pressure supports the star.
It has been shown that the maximum mass the degenerate core is able to withstand is
directly proportional to the square of the fraction of electrons in the core
Under

T ∼ 109

K and

ρYe ∼ 107 − 109 g·cm−3

2
MCH ∝ Ye

[6].

stellar conditions, ECs begin to take place

in high number thereby reducing the amount of weight electron-degeneracy pressure can
support and the layers fall inward.

This increases the density allowing more ECs to take

place, reducing the electron-degeneracy pressure and the layers fall inward once again. This
process continues until nuclear density is reached. At this point, the core rebounds and the
shock wave is initialized leading to the explosion of the star. The remnant of a core-collapse
supernova (SN 1987a) is shown in gure 1.1.

Figure 1.1:

The remnant of supernova 1987a.

tute/NASA [7].

Courtesy Space Telescope Science Insti-

For interpretation of the references to color in this and all other gures,

the reader is referred to the electronic version of this thesis.

3

During the pre- and post-collapse phases, ECs take place on a variety of

sdg−shell

nuclei [8], many of which under terrestrial conditions are unstable.

pf −

and

Moreover,

due to the high temperature ECs can take place on the thermally populated excited states.
Measuring all the relevant EC rates in this region would be impossible. This is particularly
due to the vast number of experiments it would involve and the impossibility of studying
the transition from excited states. Hence, it is crucial that theoretical models are developed
and tested against experiment [5].

1.1.2

Thermonuclear Supernovae

Although thermonuclear supernovae serve as standard candles in astronomy for determining cosmic distances due to the similarity of their light curves [1], they are even less well
understood than core-collapse supernovae. The current general hypothesis is that these supernovae begin in binary star systems.

Two progenitors have been suggested: two white

dwarfs that merge or a white dwarf and an accompanying red giant that accretes mass on
the dwarf [9]. The resulting explosion is similar in each case.
Stars that are

0.5M

M

8M

end up as white dwarfs. Fusion has ended in the

core and electron degeneracy pressure is left to support the star. As the mass of the dwarf
increases - either by accretion from a red giant or from a merger with another dwarf - the
temperature, pressure, and density increase. Slightly before the white dwarf nears the Chandrasekhar limit of approximately

1.4M

, carbon fusion is ignited in the core. Main sequence

stars are able to expand in order to control temperature and fusion; however, as a dwarf is in
a degenerate state it cannot expand. Therefore as fusion is ignited in the core, a deagration
front quickly moves to the surface. As it does so it converts about half of the mass to iron.

4

Figure 1.2: This is the remnant of the supernova spotted by Tycho Brahe in 1572. Here, it is
shown in the infrared and X-ray and was photographed using NASA's Spitzer and Chandra
observatories. SN1572 left behind a nebula type remnant that does not contain a neutron
star. Courtesy NASA/JPL-Caltech [11].

ECs take place on these nuclei, the degeneracy is lifted, and the dwarf quickly expands. This
provides enough energy to throw the star out of stability and nally, into an explosion. Type
Ia supernovae leave remnants in their wake such as that left by SN 1572, known as Tycho's
supernova shown in gure 1.2.
ECs are an important contributor to thermonuclear supernova models. These ECs depend both on the velocity of the deagration front and the central density of the dwarf [10].
Since ECs are taking place on the iron nuclei that were formed in the burning front, they
also aect the resultant nucleosynthesis. Therefore, knowing this resultant nucleosynthesis
and knowing accurate EC rates allow for important constraints on the central density and
the velocity of the ame front in supernova models [10].

Similar to the case of Type II,

models that generate EC rates must be tested against experiment for accuracy.

5

1.2 Electron-Capture and Charge-Exchange
During the EC process, an electron is captured by a proton in the nucleus forming a neutron
and releasing a neutrino. A diagram of this process is shown in gure 1.3. Charge-exchange
(CE) is an analogous reaction mediated by the strong force that connects the same initial
and nal states as EC

(∆L = 0, ∆S = 1) [12].

Therefore, the Gamow-Teller (GT) transition

strengths obtained from a CE experiment can be used as inputs into EC rate calculations.
An advantage of using CE to probe GT transition strengths to daughter excitation energies
is that it can probe to higher excitation energy since it is not limited by Q-value. Initially,
supernovae were believed to be insensitive to details of the EC rates and the associated GT
distributions. This was shown by Bethe et. al. [13] when applied to extreme conditions of
the collapse phase. However, when Fuller et. al. [14] applied EC rates to the less extreme
phases such as pre- and post-collapse, they determined that the weak reaction rates, in fact,
greatly aect the dynamics.
Determining weak reaction rates for masses relevant to these phases of the supernovae
was the goal of a series of papers published by Fuller, Fowler, and Newman [1417]. Although
the GT strength distributions that serve as inputs for these calculations turned out to be
too simple and more realistic models are now in use, their work formed the basis for the
techniques applied in modern simulations.

There are two important factors in EC rate

calculations:

•

GT strengths of the transition and

•

phase space.

As mentioned before, the GT strengths calculated by theoretical models must be tested
against experiment. CE experiments are the preferred tool for that purpose. It is important

6

(a) Charge-Exchange Reaction

(b) Electron-Capture

Figure 1.3: electron-capture is a weak reaction analogous to charge-exchange which is mediated by the strong force. The initial and nal states are the same in each reaction enabling
the use of charge-exchange to extract the Gamow-Teller strengths in electron-capture. a) A
triton is impinged on a target. A neutron is then exchanged for a proton resulting in the
3
release of He with a left over nucleus with less one proton but same mass number. b) An
electron is captured by a nucleus changing the proton number and releasing a neutrino.

to note that the CE experiments probe transitions to excited states in a daughter nucleus
from the ground state of the parent nucleus as shown in gure 1.4.

An EC rate can be

determined for each of these individual transitions with the total rate being the sum of the
individual rates.

λ = Σij λij .

(1.1)

The parent and daughter states are i and j, respectively. These individual rates are determined from [14],

λij = ln2
Here the half-life,

(f t)ij ,

fij (T, ρ, UF )
(f t)ij

.

(1.2)

is related to the nuclear matrix elements of a weak nuclear inter-

action,

B(F ) +

2
K/gV
gA 2
Bij (GT ) =
.
gV
(f t)ij
7

(1.3)

Figure 1.4: In an experiment, daughter states are probed using a charge-exchange reaction.
The dierence in the ground state of the mother nucleus and daughter nucleus is

The weak decay vector and axial-vector coupling constants,

gA
gV = −1.2694 ± 0.0028
B(F ) = 0

and

2
K/gV = 6143 ± 2

gV

and

gA ,

Qg.s. .

are related via

s [18]. For pure Gamow-Teller transitions,

and the half life becomes,

ft =

3812s
.
Bij (GT )

(1.4)

The individual rates depend on the temperature of the star (T ), the density (ρ), and the
electron chemical potential (U

F)

through the phase space integral

fij

used for calculating

electron-capture rates. For electrons it takes the form [14],

∞
fij (T, ρ, UF ) =

wl

w2 (qR + w)2 S− (T, UF (ρ))dw

with the total energy of the electron being
energy is

me c2

wl = qg.s.

and

mN

with

εe

and

qg.s. = Qg.s. /me c2 .

referres to nuclear mass excess.

8

w = εe /me c2 .

Here

(1.5)

The EC threshold total

Qg.s. = mN (A X) − mN (A X) −
Z
Z−1

The Q-value of the reaction in units of

electron mass is [19]:

qR = QR /me c2 = Exj − Exi − Qg.s /me c2

(1.6)

Finally, the term

S− =

exp

U − UF
kT

−1
+1

(1.7)

is the Fermi-Dirac distribution function. In the distribution function, the kinetic energy is

U = (w − 1)me c2

and k is the Boltzmann constant.

At zero temperature, the chemical

potential is approximated by [14, 20],

UF = 0.511MeV

where

ρ6

energy is

is the density

ρ

divided by

εF = UF + me c2

= 1)

106 g/cm3

and a plot of

In order for EC to take place,
(i

1/2
−1
1.018 (ρ6 Ye )2/3 + 1

εF > QR .

εF

and

Ye

(1.8)

is the electron fraction. The Fermi

as a function of density is shown in gure 1.5.

Finally, putting these factors together and setting

since the parent nucleus is in the ground state the EC rates can be calculated with

the equation:

λg.s. =

B1j (GT ) ∞ 2
w (qR + w)2
6182s
wl
j

exp

U − UF
kT

−1
+1

.

(1.9)

EC rate calculations are complicated by the extreme conditions present in the pre- and
post-supernova phases and these factors will be discussed in the following section.

9

5

εF (MeV)

4

3

2

1

10

6

10

7

ρYe gcm-3

10

8

10

9

Figure 1.5: For electron-capture to take place in a star, the Fermi energy must be greater than
the Q-value of the reaction. This plot is shown for zero temperature. At higher densities,
the Fermi energy is higher, so transitions to higher excitation energies can contribute to the
electron-capture rate.

10

1.3 Factors in Electron-Capture Rate Calculations
In this section, the main factors that aect the EC rates in the stellar environment will be
discussed.

•

Temperature in the star

•

Density in the star

•

Excitation energy of daughter nuclei relative to Fermi energy

•

Gamow-Teller transition strength.

1.3.1

Temperature and Density of the Star

A pre-supernova star is at a temperature of

T ∼ 109

K. At low temperature (T

109

K)

the core is in a highly degenerate state, therefore all the electrons are in the lowest energy
state they can possibly occupy without violating Pauli's exclusion principle. As temperature
is increased, electrons are given thermal energy and begin to occupy higher energy states,
thereby lifting the degeneracy. This is visualized with an articially created GT spectrum
shown in gure 1.6(a). To create this gure, B(GT) for each transition was set to unity with
the ground state

Ex =

Q-value

set to zero. The placements for each GT transition strength were

1.5, 2.0, 2.5, and 3.0 MeV. EC rate calculations were performed with each transition

separately, i.e. assuming only one GT transition was active at a time. The result of each
calculation is shown in gure 1.6(b) where the logarithm of the EC rate is plotted against

9

stellar temperature (10

K). At low temperatures, the Fermi surface resides at the highest

electron energy occupied by a degenerate electron as indicated by the dashed red line. This
Fermi energy is a function of stellar density as shown in equation 1.2. As the temperature is
increased, the Fermi surface smears out and higher excitation energies can start to contribute
to EC as decribed schematically by the black curve.

11

B(GT)

1.2

degeneracy lifting (a)

εF

1
0.8
0.6

density

0.4
0.2
0
1

1.5

2

2.5

3

3.5

4

log10(λEC/s)

Ex (MeV)
8

-3

ρYe=10 gcm

0

(b)

-5

1.5MeV
2.0MeV
2.5MeV
3.0MeV
high degeneracy

-10
-15
-20
-25
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

9

Temperature (10 K)
=1.5 MeV,
8 g·cm−3 and low
At ρYe = 10

Figure 1.6: (a) An articially created B(GT) spectrum with four states at Ex
2.0 MeV, 2.5 MeV, and 3.0 MeV and B(GT)=1 for all states.

temperature the Fermi surface resides just above the 2 MeV level and a red dotted line was
placed at this energy to approximate which excitations are above and below the Fermi energy.
As temperature is increased the degeneracy is lifted. This is represented schematically with
the solid black line in order to visualize the eect of degeneracy lifting. (b) The logarithm of
the electron-capture rates as a function of temperature for

ρYe = 108 g·cm−3

for captures

into each of the states in (a). When the electron gas is degenerate, only states below the
Fermi surface have signicant rates.

At higher temperatures, electron-captures to states

above the Fermi energy start to contribute.

Captures into excited states that are above the Fermi surface (2.5 MeV and 3.0 MeV) will
contribute when the temperature is high enough to lift the degeneracy to a level of which
electrons are available to capture into those states.

Captures into the 2.5 MeV level will

contribute signicantly at lower temperatures than those into the 3.0 MeV level. For states
below the Fermi surface (1.5 MeV and 2.0 MeV) the increase in temperature hardly aects

12

log10(λEC/s)

10

T=.01 GK

(a)

T=10 GK

(b)

5
0
-5

-10
-15
10
5
0
-5

3.0MeV
6.0MeV
8.0MeV
20.MeV

-10
-15
0

2

4

6

8 10 12 14
-3
log(ρYe gcm )

Figure 1.7: Electron-capture rates were calculated for a constant B(GT)=1 and Qg.s
MeV. The logarithm of the electron-capture rate was plotted against the logarithm of
for (a)

T = .01

GK and (b)

T = 10

=0
ρYe

GK. In the former case, the Fermi energy does not

exceed excitation energy until a sucient increase in

ρYe .

εF > Ex (with Qg.s. = 0
ρYe for higher excitation
T = 10 GK, the Fermi surface

Once

MeV) there is a sharp increase in rates which occurs at higher
energies. When degeneracy has been lifted substantially at

is quite smeared out and all excitation energies are contributing to electron-capture, even at

ρYe .

low

3 MeV is shown as a solid green line and is the same energy that is used in gure

1.9.

the associated EC rates. At suciently high temperature, dierences in excitation energy
and Fermi energy are less noticeable due to the considerable amount of lifted degeneracy.
As previously mentioned, at low temperature the electrons are in a highly degenerate
state.

As

ρYe

in the star increases, the Fermi energy increases; therefore, EC to highly

excited states can contribute. This eect is visualized in gure 1.6(a) in the cyan shaded

13

region.

As

ρYe

increases, captures can take place into excitation energies that were once

above the Fermi energy. When

ρYe

increases, the Fermi energy increases allowing captures

into the state at 2.5 MeV to contribute. A further increase in

ρYe

would increase the Fermi

energy enough to allow captures into 3.0 MeV to nally contribute to EC. In gure 1.6(b),

ρYe = 108 g·cm−3

therefore only transitions to 1.5 MeV and 2.0 MeV contribute strongly

to EC at low temperature since they are below the cut-o set by the Fermi energy. At this

ρYe ,

only after a sucient increase in temperature would 2.5 MeV and 3.0 MeV begin to

contribute.
Figure 1.7 represents a combination of temperature and density eects. The logarithm
of the EC rate is plotted against the logarithm of
with articial input data.
Qg.s.

=

ρYe

from results of EC rate calculations

These calculations were performed with a constant B(GT)=1,

0 MeV, and excitation energies of 3.0, 6.0, 8.0, and 20.0 MeV, respectively.

At

low temperature (gure 1.7(a)), the electron gas is degenerate so only captures into states
below the Fermi energy set by
suciently increased,

εF

ρYe

signicantly contribute to the rate. Once the density has

will increase and EC into states at higher excitation energies can

contribute. This can be seen by the sharp increase in EC rates calculated for each excitation
energy in gure 1.7(a).
strongly at

Captures into the lowest excitation energy of 3.0 MeV contribute

ρYe ∼ 3 g·cm−3

(solid black line), whereas captures into the highest excitation

energy of 20 MeV only contribute at

ρYe ∼ 9 g·cm−3

(dot-dashed blue line).

At higher

temperature, this is complicated since degeneracy has been lifted so captures into higher
excitation energies can contribute, even at lower densities.
temperature of

T = 10

Figure 1.7(b) shows that at a

GK, the Fermi surface is smeared such that captures to states at all

excitation energies can signicantly contribute to EC.

14

1.3.2

Excitation Energy in Daughter Nuclei

As discussed in the previous section, the excitation energy of the daughter state relative
to the Fermi energy is a very important factor in the calculation of the EC rate.

This is

emphasized in gure 1.8. EC rates were calculated for a GT transition with B(GT)= 1 and
Qg.s

= 0

MeV for which the daughter nucleus excitation energies (horizontal axis of Fig.

1.8) were varied from 0.1 to 8.0 MeV. The rates for two densities are plotted against daughter
excitation energy at

T = 0.01

the Fermi energy resides at

GK. At a density of

∼ 2.5

ρYe = 108 g·cm−3 ,

MeV which is indeed where the drop o in rates occurs

for the black dotted line in gure 1.8. Figure 1.5 also shows that for
Fermi energy is

∼5

gure 1.5 shows that

ρYe = 109 g·cm−3 ,

the

MeV and that is location of the sharp decline in EC rates for the solid

blue line.
Although the sharp drop in the EC rate at the Fermi energy is not as dramatic at
nite temperatures since the Fermi surface is smeared out, it is clear that it is critical that
excitation energies are well-predicted by theoretical models for GT transition strengths. For
a level near the Fermi surface, even a small shift could result in a large change in the EC
rate. This is particularly true if there are no or only very weak transitions to states below the
Fermi surface. Likewise, when calculating EC rates from GT strength distributions extracted
from experimental data it is important that excitation energy distributions are determined
with high resolution and small systematical errors.

1.3.3

Gamow-Teller Transition Strength

It is clear from equation 1.9 that EC rates are proportional to B(GT). This is visualized
in gure 1.9. EC rates were calculated for three cases; for all three, the excitation energy

15

log(λEC s-1)

20

T=.01GK

0

-20

-40

-60

log(ρYe g cm-3) = 8
log(ρYe g cm-3) = 9

-80

-100

2

3

4

5

6

7

8

Ex (MeV)
Figure 1.8:

Qg.s. = 0
108 g·cm−3

Electron-capture rates were calculated for various excitation energies with
MeV and constant B(GT)=
and

ρYe = 109 g·cm−3 )

for

1. Two
T = 0.01

relavant densities were chosen (ρYe

=

GK. The sharp cut-o in the electron-

capture rates for the two densities occur at approximately 2.5 and 5.0 MeV respectively, in
agreement with gure 1.5.

16

of the daughter state was set to

Ex = 3

MeV (with Qg.s

=0

MeV) but the B(GT)s were

set to 1, 2, and 3, respectively. In the gure, the ratio of B(GT)=

2

and 3 to B(GT)=

1

are plotted. As expected, the ratios are independent of density and are simply equal to the
ratios of the corresponding B(GT)s. Therefore, errors introduced in the calculated EC rates
are proportional to the uncertainties in the theoretical B(GT) strengths, or the uncertainties
in the extraction of B(GT) for the experimentally obtained strengths.
It has been shown that there are four factors that alter the EC rates:
excitation energy, and GT strength.

ρYe ,

temperature,

Accurate EC rate calculations are highly dependent

on the placement of the excitation energies of the daughter nucleus relative to the Fermi
energy for given stellar density.

Near the Fermi surface, small shifts in the calculated or

experimentally extracted excitation energy can change the rate associated with the capture
into that state by large factors. The lower the stellar temperature, the higher the uncertainty
will be that is induced by the misplacement of states near the Fermi surface. Therefore, if
there are only a few or no states available for weak transitions below the Fermi energy,
the total EC rate would be highly dependent on the states located near the Fermi energy.
However, if there is a signicant amount of strength found below the Fermi energy, the
EC rate would mostly be determined by transitions into these states. The uncertainty in
the EC rate calculation would then be determined by a combination of both uncertainties
due to calculated/experimental B(GT) strengths and the placement of daughter excitation
energy states. It is concluded that it is crucial that theory accurately predicts the location
of daughter states and to a somewhat lesser extent, the strength associated with captures
into those states.

17

λEC/λEC

3.25
3

2.75
2.5
2.25
2
1.75
1.5
1.25
1
0.75

9

9.5

10 10.5 11 11.5 12 12.5 13 13.5 14
log(ρYe g cm-3)

Figure 1.9: Electron-capture rates were calculated for B(GT)= 1, 2, and 3 at
and Qg.s

= 0

Ex =

3 MeV

MeV. It can be seen that the ratio of the rates is simply the ratio of the

Gamow-Teller strengths.

18

1.4 Outline
The case has been made that theoretical Gamow-Teller transition strengths need to be
accurately predicted in order to calculate reliable electron-capture rates.
which to test the theory against experiment will be presented.

The means by

The step-by-step process

of calculating electron-capture rates for both theoretical and experimental Gamow-Teller
strengths will be described in chapter 2. The procedure to use the shell-model to theoretically
predict GT strengths will be described in chapter 3.
determined Gamow-Teller strengths for the reaction
previous

56

56

A comparison study using newly
3
56
Fe(t, He)
Mn and results from a

56
Fe(n, p)
Mn experiment along with theory will be investigated in chapter 4.

Finally, the thesis is concluded with remarks in regards to the results in chapter 5.

19

Chapter 2

Electron Capture Rate Calculation

2.1 Overview
The EC rates are calculated using a program based on the method proposed by Fuller,
Fowler, and Newman in their 1980 paper [14] mentioned in chapter 1. The program, nucleus_iterator.f , is written in Fortran by Sanjib S. Gupta in 2006 [21] modied for the use
with a continuous chemical potential calculated from an electron-positron equation of state
routine [22] instead of the original discrete grid provided by the program. The EC rates are
calculated using equation 1.9. There are ve input les:

•

nucleuslist.dat : contains Z and A of parents

•

cap_daughters_ZA : contains daughter transition energies and strengths

•

cap_header_ZA : contains information about the reaction

•

cap_parents_ZA : contains parent nucleus information

•

cap_transitions_ZA : contains transition strength information
20

where Z and A are the three-digit atomic and mass number.

All Z and A numbers must

be three-digits with zeroes lling in the spaces in front for one and two-digit numbers.
For example, if Z=1 and A=2, the extension would be 001002.

The output le is named

phaseout_ZA which contains the EC rates at various temperatures and densities. All input
and output les are contained within the subdirectories of the ecrates_temp folder.
The ecrates_temp folder is located within the ceclub portion of the projects folder
on intranet (i.e. /projects/ceclub/ecrates_temp.) When creating an ecrate directory for
the rst time, it is important to copy this directory and change the label temp to the
user name. This will identify the user of the directory and decipher it from other CE group
member's working ecrate directories. The les contained within this directory have been
prepared to move to a new ecrates_user directory without having to adjust how les are
called. There four subdirectories with xed names:

•

codes.dir

•

expl.dir

•

input.dir

•

phaseoutles.dir

The contents of these directories will be discussed in detail in following sections.

2.2 Codes Directory
The codes.dir contains the EC rate program nucleus_iterator.f  along with the referenced
programs.

These les do not need modied.

The codes.dir directory also contains nu-

cleuslist.dat as seen in gure 2.1, which is used by the program to determine the parent

21

Figure 2.1: This is the structure of nucleuslist.dat. It contains the atomic and mass number
of the parent nucleus and can be found within the codes directory.

nucleus.

The lecreator program discussed in section 2.3 modies this le and it does

not need changed by hand. Details on how to run the program nucleus_iterator.f  are in
section 2.6.

The program accepts GT strengths and energies, and outputs EC rates and

neutrino-loss rates. It uses a standard temperature grid of .01 GK - 100 GK and
from 10 g·cm

−3

- 10

are necessary and

14 g·cm−3 .

ρYe

ρYe

ranges

The temperatures can be found in gmp_temp if changes

is located within a do-loop in nucleus_iterator.f .

If changes are

made to the do-loop, recompiling is necessary and is described in section 2.6.

2.3 Input Directory
As the name implies, input.dir contains the input les for the program. These are the four
les mentioned in section 2.2 that begin with cap_. To ease le creation, there exists a
program titled lecreator.f  that creates from one input le the aforementioned les along
with the fth mentioned in section 2.1, nucleuslist.dat. A sample inputle is located in
the input directory and is shown in gure 2.2(a).
The rst row contains the relevant information about the mother and daughter nuclei.
An explanation of what each value is can be found in gure 2.2(b). The parent excitation
energy is always 0.000 MeV since the parent nucleus is in the ground state; furthermore,
with one parent state the spin will be 0.0.

Z and A are the atomic and mass number of

the parent nucleus. The Q-value is the energy dierence between the parent ground state

22

(a) Input File Data

(b) Input File Denition
Figure 2.2: This is the structure of inputle which creates the ve les necessary to run
the electron-capture program. a) This is an example of data that was placed in inputle.
b) This is an explanation of each of the values in inputle. It requires mother and daughter
nucleus energies and spin, number of mother and daughter states, Q-value of the transition,
proton and mass number, and B(GT) values.

23

Figure 2.3: The parent le contains the excitation energy in MeV and the spin of the parent
nucleus.

and the daughter ground state less the rest energy of the electron, calculated with nuclear
masses,

Qg.s. = Mm − Md − me

in MeV. Figure 1.4 gives a schematic of this process. The

number of daughter and parent states is the total number of states for each nucleus. If there
is one parent state, the number of transitions will equal the number of daughter states.
The second row begins listing the transition information as shown in gure 2.2(b). After
running the program, the four cap_ les located in input.dir are created. Eight example
cap_ folders are located in the ecrates_temp input directory.

Half of those les are

experiment and the other half are theory. The dierences in theory and experiment will be
discussed in chapter 3.

•

Parents File
The cap_parents_ZA le contains the information about the initial nucleus. This is
where the le creator program places the parent excitation energy and parent spin in
one row by two columns as seen in gure 2.3.

•

Daughters File
The le creator program places all daughter excitation energies from the input le into
a single column in cap_daughters_ZA as shown in gure 2.4.

•

Transitions File
The cap_transitions_ZA le contains three columns.

24

Figure 2.4: The daughter le contains the excitation energy in MeV of all the daughters
states.

 Total number of parent states (usually 1)
 List of daughter states
 log(f t) for each transition

Figure 2.5 gives an example of what the cap_transitions_ZA le looks like. In equation in chapter 1,  f t are the nuclear matrix elements for a weak reaction. The lecreator.f  calculates the logarithm of this number and places it in the cap_transtions_ZA
le.

•

Header File
Finally, the cap_header_ZA le shown in gure 2.6 contains one column which is
from top down:

 number of parent states
 number of daughter states
 atomic number of parent
 mass number of parent
25

Figure 2.5: The transitions le contains the parent state number, daughter state number,
and log(f t) for each transition.

Figure 2.6: The header le contains one column with the number of parent states, number
of daughter states, atomic and mass number of the parent nucleus,

Qg.s ,

and the number of

transitions.



Qg.s.

 number of transitions

2.4 Explanation Directory
The expl.dir directory contains information about the electron-capture program. It is not
necessary to modify any of the les within this folder.

The notes about the program are

contained here along with a readme le discussing the le structure presented in section
2.3.

26

2.5 Phaseoutles Directory
The phaseoutles.dir directory contains the result of the calculation. The program produces
a le labeled phaseout_ZA as mentioned in 2.1. It is made up of four columns from left
to right:

• log10 (ρYe g·cm−3 )
•

Temperature (GK)

• log10 (λec s−1 )
• log 10(λν s−1 )
where the rst,

log10 (ρYe g·cm−3 ), is the density of the star.

Following are the temperature

of the star, the logarithm of the EC rates, and the logarithm of the neutrino energy loss
rates. Here, the neutrino energy loss rate is the energy loss associated with the transition
from parent state to daughter state in units of MeV

s−1

[14].

2.6 Running the Program
When the programs are rst used, they need to be compiled. Compiling of the le creation
programs must be done in the input directory. In this directory, the gfortran compiler is
to be used with the ag `-o'.

> gfortran lecreator.f -o lecreator
> gfortran lecreator_theory.f -o theory_lecreator

This creates executables named lecreator and theory_lecreator which will be used
to run the le creation programs. The program lecreator_theory.f  takes into account a

27

quenching factor for GT transitions and must be used to create the le for B(GT) distributions calculated in shell-model theory. This is discussed in chapter 3. When compiling
the nucleus_iterator.f  program it is necessary to include three les total during compilation that are called by the program. The programs, eos_wrap and helm.f  are included
with nucleus_iterator.f  to account for the continuous chemical potential. This creates an
executable ecrates that when run, will calculate the electron-capture rates. To compile:

> gfortran nucleus_iterator.f eos_wrap.f helm.f -o ecrates

After both programs have been compiled, it is not necessary to compile them again unless changes have been made to the program. An important note is that the program will
look for the cap_ les in the form cap_header_ZA, etc. If the program is being run for
the same nucleus but for both experiment and theory, it is necessary to add an extension

after the program has run to the old cap_ les to prevent them from being overwritten.
For example, if the EC rates have been calculated for GT strengths obtained from an experiment on

45

Sc, the cap_ les will need an extension before running either le creator

program again. An example of how this could be done is to rename cap_parents_021045
to cap_parents_021045_exp. For further reference, the ecrates_temp_input.dir folder
contains example cap_ folders. The following steps will need to be taken every time the
program is run.

1. In the input.dir directory, modify the inputle.

2. In the input.dir directory, run lecreator or theory_lecreator.

3. From the codes.dir directory, run ecrates.

28

There should now exist a phaseout_ZA le in phaseoutles.dir for this parent nucleus.
A common error that EC rate calculation program will encounter is a run time error when
it reaches a load_cap_ le.

This happens when the cap_ le already exists for that

nucleus. After adding an extension to the old le, this error will be resolved. The EC rate
calculation program will overwrite phaseout les so it is important here as well to add
extensions to run a particular reaction more than once.

2.7 Analyzing the Results
Any type of plotting program can be used to view the results such as Physics Analysis
Workstation (PAW) [23], gnuplot [24], or ROOT [25].

Included in ecrates_temp is an

example of analysis using PAW. The PAW website contains tutorials and references that can
be used in combination with the example for a user new to PAW (see Ref. [23]).
PAW les are those with the extension .kumac.
is phasoutgraphs_023050_demo_exp.kumac.

The example located in the folder

Executing a .kumac le with PAW will

create either a Post Script le or Encapsulated Post Script le depending on the what was set
by the user within the .kumac. To execute the .kumac from ecrates_temp\phaseoutles:

> PAW

> Workstation type (?=HELP) <CR>=1 : PRESS <ENTER>

PAW > exec phaseoutgraphs_023050_demo_exp

This command will produce a gure such as that shown in gure 2.9. Here, the logarithm
of the EC rate was plotted against temperature for xed

29

ρYe .

Only a range of

ρYe

that are

Figure 2.7: Typing the PAW command brings the user to the welcome screen.

Figure 2.8: After pressing <enter>, this PAW command prompt is shown.

relevant in late stellar evolution are shown [5, 26]. With the EC rate calculation program
described, it is now necessary to understand how to get GT strengths to input into the
program.

30

0
-1
-2

log(λEC s-1)

-3
-4
-5
-6
ρYe=105 g/cm3
ρYe=106 g/cm3
ρYe=107 g/cm3
ρYe=108 g/cm3
ρYe=109 g/cm3

-7
-8
-9
-10

1

2

3
4
5
6
9
Temperature (10 K)

7

8

Figure 2.9: This gure is an example of a PAW output for electron-capture on

50

Ti. The

y-axis is the log of the electron-capture rate and the x-axis is the temperature. The densities
are held constant and are represented by each line.

31

Chapter 3

Obtaining Gamow-Teller Strengths

3.1 Experimental Gamow-Teller Strengths
EC rates are an important input for astrophysical simulations of supernovae as described
in chapter 1.

A large number of isotopes play a role, including many that are unstable.

In addition, due to the relatively high stellar temperatures, excited states can be thermally
populated and EC can take place on these states.

Therefore, it is impossible to measure

all GT strength distributions of relevance for EC in stellar environments and theoretically
calculated GT strength distributions must be heavily relied upon. Because stellar evolution
models are quite sensitive to the EC rates, it is important that the theoretical rates are
benchmarked against experiment. Since GT strengths extracted from
are only available in a limited

Q−value

β−decay

experiments

window, CE reactions have become the preferred

tool to measure the relevant GT strength distributions. The extraction of GT strengths from
CE experiments is based on the proportionality between B(GT) and the dierential cross
section at vanishing linear momentum transfer

32

dσ
dΩ q=0

for GT transitions at intermediate

energies [27]:

dσ
= σGT B(GT ).
ˆ
dΩ q=0
The GT unit cross section

σGT
ˆ

(3.1)

can either be directly calibrated (if the ground-state to

ground-state transition has a known B(GT) from a

β -decay

half-life measurement) or it

can be determined from a mass-dependent relationship such as that developed in Ref. [28].
This proportionality is remarkable since the same information on transitions mediated by
the weak nuclear force (EC and

β−decay)

can be obtained from the CE reaction, which is

mediated by the strong nuclear force. In addition to the interest for nuclear astrophysics,
GT transition strengths have successfully been determined in CE experiments for a variety of other purposes: nuclear structure, neutrino physics, and giant resonances and their
macroscopic properties.
The proportionality of Eq. 3.1 was originally established for nucleon-induced CE
and

(n, p)

(p, n)

reactions, but has been shown to also hold for CE experiments with composite

probes such as

(d,2 He)

[29],

(3 He, t),

and

(t,3 He)

strengths of relevance for EC in supernovae the

[18]. For the purpose of extracting GT

(n, p), (d,2 He),

and

(t,3 He)

reactions have

been commonly used.
Use of the

(n, p) reaction has the advantage that the reaction mechanism is the simplest,

but it suers from relatively poor resolution (∼

1

2
MeV). With the (d, He) reaction, resolu-

tions of 100-200 keV can be achieved, but the reaction mechanism is complex and there are
2
presently no readily available facilities where (d, He) experiments can be performed. With
the

(t,3 He) reaction,

resolution of 250-400 keV can be achieved and the reaction mechanism

has been well studied using the inverse

(3 He, t)

reaction [18]. The present experimental sit-

uation can be summarized as follows. GT strengths for

33

pf −shell

nuclei have been extracted

from many

2
experiments and to a lesser extent from (d, He) experiments.

(n, p)

experiments are relatively new and only a few cases have been studied.

(t,3 He)

experiments are planned.

(t,3 He)

However, further

In addition, rst eorts to extract GT strengths from

rare isotopes for astrophysical purposes have been undertaken. Recently, the GT strength
distribution from

56

To extract the

Ni was extracted from a

∆L = 0

(p, n)

experiment in inverse kinematics [30].

GT strength from CE experiments, it must be separated from

contributions of excitations associated with higher multi-pole transfers through a multi-pole
decomposition analysis (MDA). This method is based on the fact that the

∆L = 0

distribution is strongly forward peaked, unlike the distributions associated with
citations. After the MDA, the extracted

angular

∆L > 0

ex-

∆L = 0 cross sections can be extrapolated to q = 0

based on the results of a distorted-wave born-approximation (DWBA) [31]. Thereafter, they
can be converted to B(GT)s using equation 3.1.

3.2 Theoretical Gamow-Teller Strengths
There are two theoretical methods used in this work to compare calculated Gamow-Teller
strengths with experiment:

•

by using the shell-model code NuShellX@MSU [32] and

•

Quasi-particle Random Phase Approximation (QRPA).

The QRPA calculations were obtained from Ref. [33, 34]. NuShellX@MSU calculations were
performed in the

pf −model

space using two Hamiltonians:

[37, 38] and will be described in the following section.

34

KB3G [35, 36] and GXPF1a

Figure 3.1: After double-clicking login-aaa, to begin NuShellX@MSU, type shell in the
prompt.

3.2.1

Using NuShellX@MSU

NuShellX@MSU can be obtained from Ref. [32].

Contained within the NuShellX@MSU

package is a help.pdf  le which outlines the procedure for NuShellX@MSU installation.
After NuShellX@MSU has been installed, accessing login.aaa will allow the user to log
into the program. By typing shell into the prompt shown in gure 3.1, the program will
begin by requesting the name of the batch le to be created. By responding with a batch
le name, the user is choosing to answer the questions that follow in a step-by-step manner.
However, it is possible to answer the questions beforehand with the creation of an answer le
with the extension .ans. An answer le for the reaction

45

45
Sc→
Ca is shown in gure 3.2.

By responding to the question regarding the batch le name with the name of the answer
le, the program will skip to the end of the question section and provide the user with a
batch le name. At this point, by typing the name of the batch le into the prompt, the
NuShellX@MSU calculation will begin.

3.2.1.1

Components of the Answer File

This process describes the creation of an answer le to calculate Gamow-Teller transition
strengths using NuShellX@MSU. The rst two sections of the le create the wave functions

35

Figure 3.2: The series of questions asked by the program can be answered with the creation
of an answer le.

36

for the parent and daughter nuclei. The following sections of the le provide information for
calculating the transition amplitudes for the wave functions. The structure of the example
answer le shown in gure 3.2 is described.

•

lpe, 1: The rst section is for the parent nucleus information. The term lpe noties
NuShellX@MSU to create wave functions. The number is set to one because there is
one parent ground state.

•

fp: The fp is the model space name.

•

n: Generally, there are no restrictions to the model space so this is set to n for no.

•

gx1a: The interaction Hamiltonian name in this case is gx1a which is also referred
to outside of the answer le as GXPF1a.

For weak-reaction rates, the interaction

Hamiltonian KB3G is also used and is labeled kb3g in the answer le.

•

21: Number of protons in the parent nucleus.

•

45: Total number of nucleons in the parent nucleus.

•

3.5, 3.5, 0.0,: Minimum and maximum total angular momentum (J) of the wave
functions to be generated. The third number is the total angular momentum step size.
Since the parent nucleus will be restricted to the ground state, the minimum will equal
the maximum and the step will be zero.

Ref. [39] can be used to look up the total

angular momentum of the ground state.

•

1: The parity of the parent ground state. For positive parity enter 0, for negative
enter 1. The parity can be referenced using Ref. [39].

•

lpe, 100: NuShellX@MSU is notied that wave functions are to be created for a
daughter nucleus. The choice 100 is used to create 100 eigen functions.

•

20: Number of protons in the daughter nucleus.
37

•

45: Number of nuclei in daughter nucleus.

•

2.5, 4.5, 1.0,: Minimum and maximum total angular momentum of the daughter
nucleus that can be probed through a GT transition.
size of the total angular momentum.
ground state is

Ji

•

If the total angular momentum of the parent

then the possible values of

Jf − 1 ≤ Ji ≤ Jf + 1.

If

Ji = 0

then

The last number is the step

Jf

can be found with the condition that

Jf = 0.

1: Parity of the daughter states. In a GT transition, parity is constant so this will be
the same as the ground state of the parent nucleus.

•

den: This noties NuShellX@MSU that the overlap will be calculated between two
wave functions.

•

t: This tells NuShellX@MSU to calculate a one-body transition density.

•

f51517: Initial wave function name. NuShellX@MSU uses a six letter naming scheme:
abcdef. a) The

f p-model

space is f  in the naming convention. b) When the f  is

followed by either a 5 or 4, the le will refer to GXPF1a or KB3G, respectively.
c) Number of valence protons.
1 for

−).

d) Number of valence nucleons.

f ) Two times the parent total angular momentum

e) Parity (0 for

(2Ji ).

+,

A more in depth

explanation of the naming procedure can be found in the login.bat le included with
the NuShellX@MSU installation pack.

•

1: Total number of initial states. There is one parent state, thus this is unity.

•

f50515: Final wave function name. This is the lowest total angular momentum wave
function for the daughter nucleus. The name is determined with the same procedure
as above except part f ) will be two times the lowest total angular momentum of the
daughter nucleus.

38

•

100: Number of nal states. In this answer le, it is requested that NuShellX@MSU
calculate up to 100 states in the daughter nucleus for this total angular momentum
value.

•

3.5, 3.5, 0.0,: Minimum and maximum total angular momentum for parent state.

•

2.5, 2.5, 1.0,: Minimum and maximum total angular momentum for daughter state
with the minimum total angular momentum. The last number is the step size for total
angular momentum.

•

n: No restrictions to the model space.

•

den: NuShellX@MSU is notied another overlap is to be calculated.

•

t: Overlap for one-body transition density.

•

f51517: Initial state name.

•

1: Initial number of states for the parent nucleus.

•

f50517: Final state name for next step in total angular momentum.

•

100: Number of nal states for this angular momentum.

•

3.5, 3.5, 0.0,: Total angular momentum for the parent ground state.

•

3.5, 3.5, 1.0,: Total angular momentum for the daughter state. The next step in total
angular momentum is chosen.

•

n: No restrictions to the model space.

•

den: The nal density overlap will be calculated.

•

t: Overlap for one-body transition density.

•

f51517: Parent ground state wave function name.

•

1: Initial number of states for the parent nucleus.

39

•

f50519: Daughter state wave function name for the highest total angular momentum
in the daughter nucleus.

•

100: Number of nal states for this angular momentum.

•

3.5, 3.5, 0.0,: Total angular momentum for the parent ground state.

•

4.5, 4.5, 1.0,: Total angular momentum for the daughter state. This is the maximum
total angular momentum for the daughter nucleus.

•

n: No restrictions to the model space.

•

st: Stop

With the questions answered the program creates a batch le that can be run to calculate
the GT transition strengths.

Typing the batch le name into the prompt will start the

calculation. The computation time can be quite long since the number of active nucleons
can be large in the

pf -shell.

Answering yes to model space truncations above can allow

the user to manually input the number of holes in each shell. This will in eect decrease the
computing time however will limit the accuracy of the calculated GT strengths [40].

3.2.1.2

After the Calculation is Complete

The program places the GT strengths in a .bgt le. The GT strength information is placed
into columns of: energy (MeV), B(GT), and

Σ

B(GT). The states are grouped by the total

angular momentum of the daughter nucleus. Within each group, they are sorted by increasing excitation energy.
If the ground state to ground state transition is a GT transition, the entire daughter excitation energy spectrum has to be shifted down in order to place the lowest excitation energy
at 0.0 MeV. If, however, the ground state to ground state transition is not a GT transition,

40

the lowest lying excitation energy corresponding to a GT transition in the daughter nucleus
has to be calculated with NuShellX@MSU. For example, in the reaction
ground state of
is

4+

46

Ti has total angular momentum and parity

0+ .

46

46
Ti→
Sc, the

The ground state of

46

Sc

hence, it is not a GT transition. The rst state that can be probed through a GT

transition is a

1+

state at energy 978.2 keV according to the Evaluated Nuclear Structure

Data File (ENSDF) [39]. It is possible to use this value to shift the energy levels of

46

Sc,

however to isolate theory from experiment as much as possible for the purpose of comparison,
NuShellX is used to calculate this energy.
An answer le used for this calculation is shown in gure 3.3. It contains a single section for the creation of the wave function for the daughter nucleus. Therefore, the proton
number and nucleon number will be that of the daughter nucleus and the total angular momentum range will start from the lowest (0.0 for integer total angular momentum and 0.5
for half integer total angular momentum) to a total angular momentum value that exceeds
the maximum probed by the transition.

This is to ensure the program will calculate the

desired excitation energy state since it needs to include the ground state and the rst GT
state. After running NuShellX@MSU with the batch le created with this answer le, a level
diagram will be constructed. This level diagram will be named with the same heading as the
batch le but with the .eps extension. The location of the rst excited state belonging to
a GT transition can be determined from this .eps le. The entire GT spectrum will then
need to be shifted in order to place the rst excited state in the daughter nucleus at this
energy.
A quenching factor of
NuShellX@MSU in the

0.742

needs to be applied to the GT strengths calculated by

pf -model space to account for degrees of freedom not included within

41

Figure 3.3: The answer le used in the generation of the low lying excitation energy spectrum
for the daughter nucleus.

the model space [41]. This factor is a constant for nuclei within a major oscillator shell (e.g.
fp). With both experimental and theoretical GT strengths secured, EC rates can be calculated and compared.

42

Chapter 4

Comparisons and Results

With the motivation to compare GT strengths calculated with theoretical models with those
extracted from experiments in order to better constrain inputs for astrophysical models, a
study has been completed based on the GT transition strengths from
is located in the

pf −shell

56

Fe to

56

Mn.

56

Fe

and is a member of a group of nuclei that plays a signicant role

in core-collapse and thermonuclear supernovae. Select nuclei in this region with half-lives
greater than

103

s are shown in gure 4.1. By investigating

56

Fe, one can thus gain further

insight into the ability of theoretical models to predict GT strength distributions used for
calculating weak reaction rates of relevance for stellar evolution.
Two experimental probes have been used to extract the GT strengths for the transition of
56

Fe to

56

3
56
3
Mn: the (t, He) and (n, p) reactions. The
Fe(t, He) experiment was completed

at Michigan State University's National Superconducting Cyclotron Laboratory with a beam
energy of 115 MeV/u [42, 43]. The resolution achieved was 400 keV. It is important to note
that the results of this experiment are preliminary. The

56

Fe(n, p) experiment took place

at TRIUMF with a beam energy of 198 MeV [44]. The resolution achieved was 1.3 MeV. A

43

Figure 4.1: Select nuclei in the

pf -shell near the valley of stability including 56 Fe are shown.

The colors represent the half-life of the nucleus.

second experiment has been completed with the

56

Fe(n, p) probe, however, the resolution

was 2.8 MeV [45]. Therefore, the data from Ref. [44] was chosen for this work simply due
to the better resolution. The shell model calculations were performed in accordance with
the procedure outlined in chapter 3 using NuShellX@MSU. The QRPA calculations were
calculated by Sanjib Gupta. Each GT strength distribution will be discussed as well as the
resulting EC rates.

4.1 The Gamow-Teller Strengths
The GT transition strengths used in this work are shown in gure 4.3.

Figures 4.3(a-e)

show GT strength distributions as a function of excitation energy for the
56

56

Fe(n, p), the

3
Fe(t, He) experiments and the QRPA, KB3G, and GXPF1a, theoretical calculations.

Figure 4.3(f ) shows the running sums (with excitation energy (MeV)) of the GT strength

44

Figure 4.2: The ground state and low-lying

1+

states of

56

Mn.

distributions shown in Fig. 4.3(a-e).

(a)

56

Fe(n, p)

Following Ref. [44] the GT strengths are given in 1-MeV wide bins. For the purpose of the
EC rate calculation, the entire GT strength in each bin was placed at the central excitation
energy of the bin. The data obtained from Ref. [44] contained GT strength (B(GT)=0.136)
below

Ex = 0.0

rst bin (Ex

MeV. For this work, this GT strength was added to the GT strength in the

= 0.0

MeV).

(b)

56

3
Fe(t, He)

With the better resolution of 400 keV, two distinct low-lying transitions to
could be observed in the

56

Ex = 1.1677

diagram in gure 4.2.

states

3
Fe(t, He) experiment. By comparing the location of these states

to those known from other experiments, they were identied as the
MeV and

1+

MeV, respectively [39].

1+

states at

Ex = .1105

These two states are shown in the level

The GT strength at higher excitation energies, extracted using an

MDA, were placed into 250 keV-wide bins.

45

(c) T heory : QRP A

Following the recommendation by the authors, the GT strengths calculated using QRPA
do not have a quenching factor applied [34]. To maintain the same vertical scale as the other
gures, the GT strengths have been multiplied by a factor of

1+

state found in the QRPA calculation resides at

is also

∼2

1/3

Ex = 2.52

in the gure. The lowest

MeV. The summed strength

times larger than that predicted by the shell-model calculations and found by

experiment.

(d) and (e) T heory : KB3G and GXP F 1a

The GT strengths were calculated using the shell-model in the full

f p-model space with-

out truncations. The interaction Hamiltonians KB3G and GXPF1a were used. Both GT
strength distributions have been quenched by a factor of
Since the ground state of

56

.742

as mentioned in chapter 3.

Mn cannot be populated by a GT transition, the low-lying

portion of the excitation energy spectrum was calculated for each Hamiltonian, to ensure
that the rst

1+

state is located at the appropriate excitation energy relative to the

3+

ground state discussed in section 3.2.1.2. The GT transition strength calculated with the
KB3G interaction (Fig. 4.3(d)) produces stronger rst and second excited states than that
calculated with the GXPF1a interaction (Fig. 4.3(e)), although the total summed strength
is about the same for both shell-model calculations.

(e) Running Sums and Comparisons

46

A running sum of GT transition strength has been plotted for each GT distribution over
the excitation energy range from 2 MeV to 10 MeV. The two experimental results are shown
in red with the solid line representing the
56

56

Fe(n, p) and the dotted line representing the

3
Fe(t, He). The two experimental strength distributions appear quite dierent but that

is largely due to the dierences in energy resolution since the summed strengths up to 10
MeV are nearly identical. The results from the theoretical calculations are shown in green
dashed (QRPA), dotted (KB3G), and dot-dashed (GXPF1a). The summed strength up to
10 MeV for both shell-model calculations is less than that observed experimentally by about
20%.

This could partially be due to the fact that some of the quenched GT strength at

low-excitation energy has moved to intermediate-excitation energies (5-10 MeV). The QRPA
calculations over predict the total amount of GT strength observed experimentally by a factor of 2.
The shell-model calculations performed with the GXPF1a interaction match the high
resolution

56

3
Fe(t, He) data well; the locations and summed strength of the rst two tran-

sitions are nearly identical (for GXPF1a:
for

56

3
Fe(t, He):

Ex1 = 0.1105

MeV and

Ex1 = 0.1300
Ex2 = 1.100

MeV and

Ex2 = 1.2110

MeV,

MeV). The shell-model calculation

with the KB3G interaction does slightly poorer; the two low-lying states are too strong and
shifted upward by about 400 keV (Ex1

= 0.5450

MeV and

Ex2 = 1.4640

MeV). Both sets

of shell-model calculations predict strength up to about 6 MeV. The QRPA calculations
reproduce the data very poorly; the total strength is too high and the rst excited state is
placed at an excitation energy about 2 MeV too high (Ex

= 2.5200

MeV). The reliability

of each theoretical GT strength distribution for the purpose of predicting EC rates will be
tested in the next section.

47

Figure 4.3: GT transition strengths for

56

data, (c) a QRPA calculation (scaled by

56
56
56
3
Fe→
Mn for (a)
Fe(n, p) data, (b)
Fe(t, He)

1/3),

(d) a shell-model calculation using the KB3G

interaction Hamiltonian, (e) a shell-model calculation using the GXPF1a interaction Hamiltonian, and (f ) a running sum of B(GT) strengths against daughter excitation energy for
each of the above cases.

48

4.2 The Electron-Capture Rate Calculations
EC rates were calculated with each of the GT strength distributions in Fig.
weak reaction

56

− , ν )56 Mn using the procedure outlined in chapter 2.
e

Fe(e

4.3 for the

The EC rates of

relevance for late stellar evolution have been represented in gure 4.4 [26]. Figure 4.4(a) is
the result of the EC rate calculations for
g·cm

−3 .

ρYe = 107 g·cm−3 and gure 4.4(b) is for ρYe = 109

Shown in solid and dashed red lines are the logarithm of the EC rate calculations

based on the

56

Fe(n, p) and

56

3
Fe(t, He) data, respectively.

Shown in solid, dashed, and

dot-dashed green lines are the logarithms of the EC rates calculated based on the QRPA and
shell-model calculations using the GXPF1a and KB3G interactions, respectively. The four
factors aecting EC rate calculations described in chapter 1 are evident in the calculations
performed using the GT strength distributions of gure 4.3.

(a) ρYe = 107
At
1.5).

ρYe = 107 g·cm−3 ,
Since

· cm−3

the Fermi energy is approximately located at 1.223 MeV (see Eq.

εF > QR = Ex − Qg.s = 4.2072

degeneracy is lifted at

g

T ∼ 109

MeV, EC can only take place because the

K. As discussed in chapter 1 the EC rate in this situation

will strongly depend on the location of the lowest-lying daughter states relative to the Fermi
level.
Since, due to the poor resolution, some strength extracted from the

(n, p)

data appears

at very low excitation energy, the EC rate is relatively strong (overproduced by a factor of
32) compared to the EC rates calculated from the high resolution

56

3
Fe(t, He) data. The

EC rate based on the QRPA calculation is much too low (under produced by a factor of
100) because the rst daughter state appears at much higher excitation energy. The EC rate

49

based on the shell-model calculations match that based on the

56

3
Fe(t, He) data quite well.

The EC rate based on the KB3G calculations are slightly lower, although the rst daughter
states are stronger than those seen in the data and in the calculation with the GXPF1a
interaction. The reason is that the daughter states appear at higher excitation energies and
as discussed in chapter 1, the rates are more sensitive to the location than the magnitude
of the strengths.

As the temperature is increased, the degeneracy is lifted more and EC

rates increasingly occur to higher lying states. As a result, the EC rates converge at high
temperature (T

∼ 1010

K).

(b) ρYe = 109
The Fermi energy for
(with

ρYe = 109 g·cm−3

g

· cm−3

is 4.67 MeV. Since

εF > QR = Ex − Qg.s.

Qg.s = −4.2027 MeV) for the rst excited states in both sets of experimental data and

the shell-model calculations with GXPF1a, the rate is strongly dominated by EC into that
daughter state and thus, the dierences between the calculated rates are relatively small.
Although the rst daughter state of the calculation with the KB3G interaction is just above
the Fermi energy, it is close enough to have a relatively strong EC rate due to the smearing
of the Fermi level at temperatures

∼ 109

K.

At low temperature, the EC rates calculated using the results from QRPA are dramatically under produced by a factor of

∼ 3000.

The location of the rst excited state in the

GT strength distribution obtained from the QRPA calculation results in a reaction
that is much greater than the Fermi energy (Q

R = 6.7272

MeV) so temperature smearing

eects have a large impact on the EC rates. The rate of EC is increased by
magnitude as temperature increases from

T = 2

GK to

Q−value

T = 10

∼5

orders of

GK. Note that the total

rates are almost 7 orders of magnitude higher than those at the lower densities. At higher

50

log10(λEC s )

0
3

(a)

9

-1

7

3

(b)

ρYe=10 g/cm

-2
-4
-6
-8

-10
-12
-14
-16

ρYe=10 g/cm

0
-1
-2
-3
-4
-5

QRPA
GXPF1a
KB3G

56

data Fe(n,p)
56
3
data Fe(t, He)

-6
2

3

4

5

6

7

8

9

10

9

Temperature(10 K)
Figure 4.4: Electron-capture rate calculations performed for

56

Fe(e

imental and theoretical data plotted against core temperature.

107 g/cm3

and (b) EC rates for

ρYe = 109 g/cm3 .

− , ν )56 Mn
e

with exper-

(a) EC rates for

ρYe =

temperatures, the Fermi surface smears out more and the rates slowly increase further, and
converge.
It is concluded that the shell-model calculations with the GXPF1a interaction best match
the high-resolution

56

3
Fe(t, He) data, and consequently provide the best estimates when

calculating EC rates. Since the EC rates based on the KB family of interactions (including

51

KB3G) form the basis for the standard weak-reaction rates library used in many astrophysical calculations [46], it is clear that further improvements to that set of rates can still be
made. This result is very similar to that made based on the results from the recent

56

Ni(p, n)

experiment [30]. Finally, since the QRPA calculations clearly fail in producing accurate EC
rates, it is not advised to use these strength distributions in astrophysical calculations where
accuracy is essential if other theoretical techniques are available.

52

Chapter 5

Conclusion

Since weak reactions play a crucial role in the dynamics of core-collapse supernovae and the
production of iron group nuclei through thermonuclear supernovae, accurately predicting
these rates has great implications in understanding these stellar processes and the production mechanism of a vast number of nuclei in the universe. Calculating EC rates requires
GT strengths acquired through both theoretical calculations and experiments. Due to the
impossibility to experimentally probe all nuclei relevant in supernova models, theoretically
obtained GT strengths have been heavily relied upon, most recently the family of KB interactions in the shell-model. It was shown that EC rates are highly sensitive to results of the
GT strength calculations, therefore, it is crucial to constrain these theoretical calculations
in order to condently calculate EC rates. A method used to compare theoretical and experimental GT distributions and the EC rates calculated with those distributions has been
presented in this thesis.
In chapter 1, it was shown that there are four factors that aect EC rate calculations:
stellar temperature, stellar density (ρYe ), the excitation energy of the daughter nucleus in
53

a weak reaction, and GT transition strengths.

Temperature,

ρYe ,

and excitation energy

were shown to have the strongest inuence since they aect rates exponentially whereas the
magnitude of the GT transition strengths only aect the rates linearly. The placement of
the low-lying states in the daughter nucleus relative to the Fermi energy are the dominate
factor, especially at low temperatures. Small shifts in excitation energy can alter the EC
rates by orders of magnitude in that case. Since the Fermi energy relies on the density of
the core,

ρYe

is also a critical parameter. The temperature aects the level of degeneracy;

at higher temperatures, states at higher excitation energies in the daughter nucleus become
available for EC even if located above the Fermi energy.
In order to better understand these factors and test the accuracy of predicted GT strength
distributions, a study of transitions from

56

Fe to

56

Mn was performed. The choice of this

reaction was due, in part, to the important role EC plays on medium-heavy nuclei in late stellar evolution. The GT strength distributions were obtained from two experimental sources:
56

Fe(n, p) data and

56

3
Fe(t, He) data. The experimental results were compared with three

sets of theoretical results: QRPA calculations and shell-model calculations using the KB3G
and GXPF1a interactions. Comparisons are discussed in chapter 4 and the results show a
good agreement between the calculations with the GXPF1a interaction and the high resolution

56

3
Fe(t, He) experimental data. It was shown that this was largely due to the close

proximity of the rst excited states in both GT distributions. Conversely, the results of the
QRPA calculation had the least agreement since the excitation energy of the rst excited
state that can be populated is located

∼2

MeV higher than the rst excited state found in

any other GT spectrum.
EC rate calculations were then performed on each of these GT distributions. With the

54

rst excited states of the GT distributions using the GXPF1a interaction and

56

3
Fe(t, He)

data appearing in such close proximity in the GT spectrum, the EC rates are similarly close
in proximity. The energy dierences in the GT distributions in both the

56

Fe(n, p) data and

those calculated with QRPA were exacerbated in the results of the EC rate calculation. This
further demonstrates the importance of an accurate location for the rst excited state in the
daughter nucleus. It was also conrmed that once the Fermi energy exceeds the rst excited
state, the inuence that temperature has on an EC rate calculation is reduced. This was
observed in the EC rate calculations for

ρYe = 109 g·cm−3

in which the rst excited state

for the GT strengths calculated with the GXPF1a interaction and obtained through both the
56

Fe

(t,3 He)

and

56

Fe(n, p) experiments were below the Fermi energy. Thus, the tempera-

ture was seen to have little eect on the calculated EC rates. The GT strengths calculated
with the KB3G interaction show still a slight EC rate increase as temperature in increased
since the rst excited state was only slightly above the Fermi energy. However, the strengths
calculated with QRPA still varied signicantly with temperature increase. Whereas, the EC
rate calculations for

ρYe = 107 g·cm−3

in which the rst excited state for all GT distribu-

tions was above the Fermi energy all increased substantially as temperature was increased.
It is concluded that the shell-model GXPF1a interaction does remarkably well in predicting the GT strength distribution for the reaction

56

56
Fe→
Mn. As stated before, most

astrophysical models have relied on the KB family of interactions. The results of this study
show that further improvements can still be made to the current inputs for these models
since, in this case, GXPF1a predicts a more accurate GT strength distribution. It is further concluded that for this reaction, the GT strengths predicted by QRPA failed to model
the high resolution data.

A broader study involving many other nuclei in the

55

pf -shell

is

ongoing. If these QRPA calculations are inaccurate for many other cases as they were for
56

Fe, it is better to refrain from using weak rates based on this set of QRPA calculations for

astrophysical purposes. Finally, high resolution data are the preferred tool for testing weak
reaction calculations, where available. However, low resolution data is still a useful tool in
constraining theoretical calculations of GT strength distributions.
One of the main aims of this work was to streamline the processes used for analyzing
and comparing the strength distributions and EC rates from CE experiments and theoretical
models. A number of tools were developed for that purpose and these are currently used to
perform a broader study, focusing on all nuclei in the

pf −shell

for which strengths from CE

data are available. With the advent of experimental techniques to probe GT strength distributions in rare isotopes (see Ref. [30]), and the construction of next generation rare isotope
beams such as the Facility for Rare Isotope Beams (FRIB), there is a realistic opportunity
to constrain weak reaction rates to the level required for modern astrophysical simulations.
The procedures dened in this thesis will be helpful in assessing the quality of existing and
future theoretical calculations based on these forthcoming experiments.

56

BIBLIOGRAPHY

57

BIBLIOGRAPHY

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Historical Supernovae and their Remnants.

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Consequences of nuclear electron capture in

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