# Description of 93Nb stellar electron-capture rates by the Projected Shell
Model

Long-Jun Wang<EMAIL_ADDRESS>School of Physical Science and Technology,
Southwest University, Chongqing 400715, China  Liang Tan  School of Physical
Science and Technology, Southwest University, Chongqing 400715, China  Zhipan
Li  School of Physical Science and Technology, Southwest University, Chongqing
400715, China  Bingshui Gao  CAS Key Laboratory of High Precision Nuclear
Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  School of Nuclear Science and Technology, University of
Chinese Academy of Sciences, Beijing 100049, China  Yang Sun  School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

###### Abstract

Capture of electrons by nuclei is an important process in stellar environments
where excited nuclear states are thermally populated. However, accurate
treatment for excited configurations in electron capture (EC) rates has been
an unsolved problem for medium-heavy and heavy nuclei. In this work, we take
the 93Nb $\rightarrow$ 93Zr EC rates as the example to introduce the
Projected-Shell-Model (PSM) in which excited configurations are explicitly
included as multi-quasiparticle states. Applying the prevalent assumption that
the parent nucleus always stays in its ground state in stellar conditions, we
critically compare the obtained PSM results with the recently-measured Gamow-
Teller transition data, and with the previous calculations by the conventional
shell model and the quasiparticle random-phase approximation. We discuss
important ingredients that are required in theoretical models used for stellar
EC calculations, and demonstrate effects of the explicit inclusion of excited
nuclear states in EC rate calculations, especially when both electron density
and environment temperature are high.

††preprint: APS/123-QED

##  I introduction

The understanding of the dynamics of some astrophysical objects relies on
reliable information on the electron capture (EC) process [1, 2, 3, 4, 5, 6,
7, 8],

${}^{A}_{Z}\mathrm{X}+e^{-}\rightarrow\ ^{\ \ A}_{Z-1}\mathrm{Y}+\nu_{e},$ (1)

in which the parent nucleus (with proton number $Z$ and mass number $A$)
captures a relativistic electron and decays to the daughter nucleus by
emitting a neutrino. Due to the nature of the stellar EC reaction, it may lead
to three important consequences with regard to compact astrophysical objects:
reducing the degeneracy pressure of electrons, cooling the environment by
neutrino escaping, and driving nuclei to be more neutron-rich (see Ref. [8]
for the latest review). Especially, the information of EC process is
considered to be crucial for core-collapse supernovae [7, 9, 10, 11], cooling
of neutron star crust and ocean [12, 13], and the thermonuclear explosions of
accreting white dwarfs [14], etc.

Electron capture is one of the fundamental nuclear processes mediated by the
weak interaction. Stellar EC rates can be theoretically determined following
the pioneering works by Fuller, Fowler, and Newman (FFN) [1, 2, 3, 4], where
allowed nuclear transitions (especially the Gamow-Teller transitions in the
$\beta^{+}$ direction) dominate. In principle, the Gamow-Teller (GT)
transition-strength distributions can be measured with the modern experimental
technique with the charge-exchange (CE) reactions [15, 16, 11, 17], although
the energy resolution is relatively limited at present. One should however
keep in mind that stellar electron captures may differ substantially from
those which can be studied in laboratory today. For many relevant
astrophysical simulations such as the core-collapse supernovae and neutron
star crust, stellar EC rates for a large number of nuclei are required.
Furthermore, the contributions from excited states of parent nuclei are
indispensable [5, 18]. Therefore, for majority of the cases one has to rely on
theoretical calculations.

In the modern treatment, several nuclear-structure models have been applied to
study the GT transition and stellar EC reactions. For the $sd$-shell (with
$A=17-39$) [19, 20, 21] and some $pf$-shell ($A=45-65$) nuclei [22, 23, 24,
10], the large-scale shell-model (LSSM) diagonalization method is considered
as the optimal tool [7, 8]. While this method is exact within the adopted
model space, it becomes physically inappropriate when environment temperature
is high. This is because when highly excited nuclear states are involved in
the calculation, cross-shell configurations are inevitably needed, which is
technically difficult for the LSSM. For medium-heavy and heavy nuclei ($A>65$)
especially if they are deformed, alternative methods must be applied. The
popular methods include the hybrid model based on the shell model Monte Carlo
approach and the random-phase approximation [25, 26], and the quasiparticle
random-phase approximation (QRPA) with different energy density functionals
[27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. These methods differ from each other
in model space (the size of single-particles in the intrinsic mean field)
and/or in truncation on configuration space (the variety of many-body
correlations among valence nucleons beyond the mean field), as well as in
effective (two-body) interactions. While these methods usually do not have
problem for including a large model space, the way to build their
configuration space does matter for stellar EC calculations. This is because
even that the sumrule for the total GT transitions is satisfied, the GT
fragmentation as a function of excited states, which are closely related to
model configurations, can sensitively influence the stellar EC results.

The stellar EC rates for medium-heavy, odd-mass nuclei exhibit some unique
characteristics as compared to those in even-number nuclei. The main challenge
in odd-mass calculations lies in the fact of dense and variety in the
distribution of single-particle-type of states near the ground state.
Furthermore, unlike in even-number nuclei where the $0^{+}\rightarrow 1^{+}$
GT transition is the main concern, in odd-mass nuclei, spin and parity for
individual low-lying levels determine the GT distributions due to the
selection rule of allowed GT transition. For example, for the 93Nb
$\rightarrow$ 93Zr case, the ground state of the parent nucleus has the spin-
parity $9/2^{+}$ (see Fig. 1) and the GT transitions from such a ground state
would populate many final states with spin-parities of
$\\{7/2^{+},9/2^{+},11/2^{+}\\}$ of the daughter nucleus. Transitions from
individual excited states of the parent to the distributed states of the
daughter can result in a rich variety of GT patterns, for which the Brink-Axel
hypothesis [37, 38] may hardly be expected as a good approximation.

In the present work, we provide the calculation by the Projected Shell Model
for stellar EC rates of medium-heavy odd-mass nuclei, taking 93Nb
$\rightarrow$ 93Zr as the first example. When only transitions from the ground
state of the parent nucleus are considered, our PSM results are tested by the
very recent data and are compared to the results of the LSSM and the QRPA. For
more realistic stellar EC rates where transitions from all excited states of
the parent nucleus are taken into account, the contribution and effect of
excited states are studied, and discussed for different densities and
temperatures of the stellar environment. In Sec. II we briefly introduce our
PSM method for the calculation of stellar EC rates. In Sec. III we compare our
calculations for the EC reaction of 93Nb $\rightarrow$ 93Zr with the recent
data and with previous results of other nuclear models. We finally summarize
our work in Sec. IV.

##  II Theoretical framework

For high-density and high-temperature stellar environments, we assume that
parent nuclei are in a thermal equilibrium with occupation probability for
excited states following the Boltzmann distribution. The stellar EC rates (in
s-1) can be expressed as [1, 2, 3, 4]

$\displaystyle\lambda^{\text{EC}}=\frac{\ln
2}{K}\sum_{i}\frac{(2J_{i}+1)e^{-E_{i}/(k_{B}T)}}{G(Z,A,T)}\sum_{f}B_{if}\Phi^{\text{EC}}_{if},$
(2)

which explicitly takes into account all relevant transitions from initial
($i$) states of EC parent nucleus to final ($f$) states of daughter nucleus.
In Eq. (2), the first summation over $i$ represents the occupation probability
for the parent-nuclear states at specific stellar temperature $T$, where
$G(Z,A,T)=\sum_{i}(2J_{i}+1)\exp(-E_{i}/(k_{B}T))$ is the partition function
with $k_{B}$ being the Boltzmann constant. $J_{i}$ and $E_{i}$ appearing in
the $i$-summation in Eq. (2) are respectively the angular momenta and
excitation energies of the parent nucleus. The constant $K$ in Eq. (2) can be
determined from superallowed Fermi transitions, and $K=6146\pm 6$ s [39] is
adopted here. $\Phi^{\text{EC}}_{if}$ in the second summation over $f$ is the
phase space integral for individual nuclear transitions (with corresponding
reduced transition probability $B_{if}$),

$\displaystyle\Phi^{\text{EC}}_{if}=\int_{\omega_{l}}^{\infty}\omega
p(Q_{if}+\omega)^{2}F(Z,\omega)S_{e}(\omega)d\omega,$ (3)

where $\omega$ and $p=\sqrt{\omega^{2}-1}$ (in $m_{e}c^{2}$ and $m_{e}c$,
respectively) label the total (rest mass and kinetic) energy and momentum of
the electron. For individual transitions, the total available energy (the
$Q_{if}$-value) is

$\displaystyle Q_{if}=\frac{1}{m_{e}c^{2}}(M_{p}-M_{d}+E_{i}-E_{f}),$ (4)

where $M_{p}$ ($M_{d}$) denotes the nuclear mass of the parent (daughter)
nucleus and $E_{f}$ is the excitation energy of the daughter nucleus.
$\omega_{l}$ labels the capture threshold energy (in $m_{e}c^{2}$), with
$\omega_{l}=1$ if $Q_{if}>-1$, or $\omega_{l}=|Q_{if}|$ if $Q_{if}<-1$. For a
given stellar temperature, the distribution functions read as

$\displaystyle
S_{e/p}(\omega)=\frac{1}{\exp{[(\omega\mp\mu_{e})/k_{B}T)]+1}},$ (5)

for electron (“$-$”) and positron (“$+$”), where the electron chemical
potential $\mu_{e}$ could be determined with the provided stellar baryon
density $\rho$, the electron-to-baryon ratio $Y_{e}$ through

$\displaystyle\rho
Y_{e}=\frac{1}{\pi^{2}N_{A}}\Big{(}\frac{m_{e}c}{\hbar}\Big{)}^{3}\int_{0}^{\infty}(S_{e}-S_{p})p^{2}dp,$
(6)

with $N_{A}$ the Avogadro’s number. In Eq. (3), $F(Z,\omega)$ is the Fermi
function that reflects the Coulomb distortion of the electron wave function
near the nucleus [1, 23].

In calculating stellar EC rates, the central task is the evaluation of
$B_{if}$ in the second summation over $j$ in Eq. (2). In the present work we
consider the stellar environments with the typical densities $\rho
Y_{e}=10^{7},10^{9}$, and $10^{11}$ g/cm3, and temperatures $T=1-15$ GK
($10^{9}$ K). Under these conditions, the allowed Gamow-Teller transitions
dominate. The reduced transition probability, $B(\text{GT})$, being defined
such that the strength associated with the decay of the free neutron has
$B(\text{GT})=3$, can be measured experimentally with the modern CE reactions
[11], or calculated by nuclear-structure models as

$\displaystyle
B_{if}(\text{GT}^{+})=\Big{(}\frac{g_{A}}{g_{V}}\Big{)}^{2}_{\text{eff}}\frac{\big{\langle}\Psi^{n_{f}}_{J_{f}}\big{\|}\sum_{k}\hat{\sigma}^{k}\hat{\tau}^{k}_{+}\big{\|}\Psi^{n_{i}}_{J_{i}}\big{\rangle}^{2}}{2J_{i}+1}$
(7)

where $(g_{A}/g_{V})_{\text{eff}}$ is the effective ratio of axial and vector
coupling constants,

$\displaystyle\Big{(}\frac{g_{A}}{g_{V}}\Big{)}_{\text{eff}}=f_{\text{quench}}\Big{(}\frac{g_{A}}{g_{V}}\Big{)}_{\text{bare}},$
(8)

which usually differs from the bare value
$(g_{A}/g_{V})_{\text{bare}}=-1.2599(25)$ [39] or $=-1.27641(45)$ [40] by a
quenching factor. In the present 93Nb $\rightarrow$ 93Zr work we take
$f_{\text{quench}}=0.9$ as suggested in Refs. [41, 42, 43, 44] for medium-
heavy nuclei. The GT transition operator in Eq. (7) keeps only the one-body
current contributions as in the chiral effective field theory [45, 41, 46,
42], which includes the Pauli spin operator $\hat{\sigma}$ and the isospin
raising operator $\hat{\tau}_{+}$ for electron captures.

Description of the nuclear many-body states, $\Psi^{n}_{J}$ in Eq. (7), is
purely a structure problem, and is precisely the place where nuclear structure
enters into the discussion. It is desired that calculations for excited states
in medium-heavy nuclei apply modern many-body techniques to obtain properly
the wave functions with good angular momentum $J$, as this index appears
explicitly in Eq. (7). In the present work such wave functions are described
in the PSM by the projection technique [47, 48, 49, 50, 51, 52], i.e.,

$\displaystyle|\Psi^{n}_{JM}\rangle=\sum_{K\kappa}F^{n}_{JK\kappa}\hat{P}^{J}_{MK}|\Phi_{\kappa}\rangle,$
(9)

where the angular-momentum projection operator $\hat{P}^{J}_{MK}$ (see Refs.
[53, 54] for details) restores the broken rotational symmetry in the intrinsic
frame and transforms the description of the wave functions from the intrinsic
to the laboratory frame. In the present work, the employed multi-quasiparticle
(qp) configuration space, $|\Phi_{\kappa}\rangle$ in Eq. (9), is large enough,
which includes up to 7-qp states, i.e.

$\displaystyle\Big{\\{}$
$\displaystyle\hat{a}^{\dagger}_{\nu_{i}}|\Phi\rangle,\
\hat{a}^{\dagger}_{\nu_{i}}\hat{a}^{\dagger}_{\nu_{j}}\hat{a}^{\dagger}_{\nu_{k}}|\Phi\rangle,\
\hat{a}^{\dagger}_{\nu_{i}}\hat{a}^{\dagger}_{\pi_{j}}\hat{a}^{\dagger}_{\pi_{k}}|\Phi\rangle,$
$\displaystyle\hat{a}^{\dagger}_{\nu_{i}}\hat{a}^{\dagger}_{\nu_{j}}\hat{a}^{\dagger}_{\nu_{k}}\hat{a}^{\dagger}_{\pi_{l}}\hat{a}^{\dagger}_{\pi_{m}}|\Phi\rangle,\
\hat{a}^{\dagger}_{\nu_{i}}\hat{a}^{\dagger}_{\nu_{j}}\hat{a}^{\dagger}_{\nu_{k}}\hat{a}^{\dagger}_{\nu_{l}}\hat{a}^{\dagger}_{\nu_{m}}\hat{a}^{\dagger}_{\pi_{n}}\hat{a}^{\dagger}_{\pi_{o}}|\Phi\rangle,$
$\displaystyle\hat{a}^{\dagger}_{\nu_{i}}\hat{a}^{\dagger}_{\nu_{j}}\hat{a}^{\dagger}_{\nu_{k}}\hat{a}^{\dagger}_{\pi_{l}}\hat{a}^{\dagger}_{\pi_{m}}\hat{a}^{\dagger}_{\pi_{n}}\hat{a}^{\dagger}_{\pi_{o}}|\Phi\rangle.\Big{\\}}$
(10)

for odd-neutron nuclei and

$\displaystyle\Big{\\{}$
$\displaystyle\hat{a}^{\dagger}_{\pi_{i}}|\Phi\rangle,\
\hat{a}^{\dagger}_{\pi_{i}}\hat{a}^{\dagger}_{\pi_{j}}\hat{a}^{\dagger}_{\pi_{k}}|\Phi\rangle,\
\hat{a}^{\dagger}_{\pi_{i}}\hat{a}^{\dagger}_{\nu_{j}}\hat{a}^{\dagger}_{\nu_{k}}|\Phi\rangle,$
$\displaystyle\hat{a}^{\dagger}_{\pi_{i}}\hat{a}^{\dagger}_{\pi_{j}}\hat{a}^{\dagger}_{\pi_{k}}\hat{a}^{\dagger}_{\nu_{l}}\hat{a}^{\dagger}_{\nu_{m}}|\Phi\rangle,\
\hat{a}^{\dagger}_{\pi_{i}}\hat{a}^{\dagger}_{\pi_{j}}\hat{a}^{\dagger}_{\pi_{k}}\hat{a}^{\dagger}_{\pi_{l}}\hat{a}^{\dagger}_{\pi_{m}}\hat{a}^{\dagger}_{\nu_{n}}\hat{a}^{\dagger}_{\nu_{o}}|\Phi\rangle,$
$\displaystyle\hat{a}^{\dagger}_{\pi_{i}}\hat{a}^{\dagger}_{\pi_{j}}\hat{a}^{\dagger}_{\pi_{k}}\hat{a}^{\dagger}_{\nu_{l}}\hat{a}^{\dagger}_{\nu_{m}}\hat{a}^{\dagger}_{\nu_{n}}\hat{a}^{\dagger}_{\nu_{o}}|\Phi\rangle.\Big{\\}}$
(11)

for odd-proton nuclei. In Eqs. (II) and (II), $|\Phi\rangle$ is the qp vacuum
state, and $\hat{a}^{\dagger}_{\nu}$ ($\hat{a}^{\dagger}_{\pi}$) the
corresponding neutron (proton) qp creation operators, which are obtained by
the Nilsson+BCS mean-field calculations. For the detailed construction of the
wave function and the philosophy of the model, we refer the recent review
article [51].

The Hamiltonian is diagonalized by solving the Hill-Wheeler-Griffin equation
for the case of a non-orthonormal basis as in the PSM, which determines the
expansion coefficients $F^{n}_{JK\kappa}$ in Eq. (9). In this work, the
separable two-body Hamiltonian with explicit two-body GT force as well as the
same numerical parameter as in Refs. [52, 18] are adopted. The Nilsson
parameters are taken from Ref. [55] and the quadrupole deformation is adopted
as $\varepsilon_{2}=0.1$ for both nuclei.

The present PSM calculation for $A=93$ nuclei considers a large model space,
allowing the nucleons active in four major harmonic-oscillator shells
($N=2,3,4,5$ with 16O as the inert core). Such a large model space ensures
that both the neutron and proton Fermi levels lie roughly in the middle of the
single-particle space, so that the nucleons have sufficient freedom to be
excited. However, the unique feature for the PSM, which differ conceptually
from all other existing structure models for stellar EC rates, is the
algorithm for the construction of the configuration space. As shown in Eqs.
(II) and (II), we classify the configuration space by the $(2n+1)$-qp states
(with $n$ being integers), corresponding to $n$ broken-pairs in each
configuration. For example, the term
$\hat{a}^{\dagger}_{\nu_{i}}\hat{a}^{\dagger}_{\pi_{j}}\hat{a}^{\dagger}_{\pi_{k}}|\Phi\rangle$
in Eq. (II) describes a 3-qp configuration in an odd-neutron nucleus with one
broken proton-pair added to the single-neutron state. This algorithm for
constructing the PSM configuration space using the broken-pair states as
building blocks was proposed to study nuclear chaoticity at highly excited
regions [56, 57], and is supported by the Oslo experiments for nuclear level
density, where, as excitation energies go up, clear evidences for step-wise
pair breakings are found [58]. The observed several peaks in the 93Nb
$\rightarrow$ 93Zr GT distribution [11] seem to indicate also the pair
breaking mechanism in the daughter nucleus although the current experimental
resolution may not allow a quantitative interpretation.

##  III results and discussions

With the method introduced in the previous section, we calculate stellar EC
rates taking the 93Nb $\rightarrow$ 93Zr case as the example to discuss
features of the PSM results. The discussion begins with the level structure of
the parent and daughter nuclei and the characteristic GT strength
distribution, before the EC rates at different densities and temperatures are
presented. The theoretical results from the PSM are illustrated in parallel
with those from two other models, LSSM and QRPA, and compared with the recent
experimental data in Ref. [11] measured at MSU.

###  III.1 Level structure in parent and daughter nuclei

Level structures in parent and daughter nuclei directly influence calculated
EC rates as their labels of the nuclear states appear explicitly in the two
summations for $\lambda^{\rm EC}$ in Eq. (2). Especially, the level
distribution determines patterns of GT strengths because a GT strength
distribution is expressed as the function of excitation energies of the
daughter nucleus. Therefore, for a model that is used for calculation of EC
rates, it is required that it can correctly describe basic features of the
level structure.

Figure 1:  (Color online) The calculated energy levels for 93Nb and 93Zr, and
compared with experimental data [59]. Figure 2:  (Color online) The GT
strength distribution $B(\mathrm{GT}^{+})$ (upper panels) and the cumulative
sum of the $B(\mathrm{GT}^{+})$ (lower panels) for the transitions from 93Nb
to 93Zr as a function of the excitation energy of the daughter nucleus 93Zr.
For transitions from the ground state (g.s.) of 93Nb (panels (a) and (b)),
calculations by three nuclear models are compared with each other and the
recent data [11]. For transitions from different excited states of 93Nb,
predictions of the projected shell model are shown in panels (c) and (d). See
the text for details.

In Fig. 1, we show the PSM calculation for the EC parent (93Nb, with $Z=41$
and $N=52$) and the daughter nucleus (93Zr, with $Z=40$ and $N=53$) for energy
levels up to $E_{i/f}\lesssim 1.2$ MeV, and compare them with the known
experimental data [59]. It is seen that for 93Nb, the energy levels are
described reasonably well by the calculation. The spin-parity of the ground-
state is correctly reproduced as $9/2^{+}$, with the corresponding wave
function found to be a strong mixture of proton 1-qp states originating from
the proton $g_{9/2}$ orbital. The very low-lying first excited negative-parity
$1/2^{-}$ state is also well described. Its wave function is found to be
rather pure, with the main configuration as $\pi 1/2^{-}[301]$ (the Nilsson
notation) originating from the proton $p_{1/2}$ orbital. The very different
structures between the $9/2^{+}$ ground state and the $1/2^{-}$ first excited
state can explain the fact that the $1/2^{-}$ state is a well-known isomer
with a half-life of about 16 yr. Such a low-lying state could be populated
readily at low stellar temperature $T\sim 1$ GK (corresponding to about 90
keV), and may potentially affect the stellar EC rates of 93Nb even at low
temperatures. In Fig. 1, the energy separation between this $1/2^{-}$ and the
higher-lying states are also reproduced successfully. The implication of the
existence of this energy gap in 93Nb is that contributions to the EC rates
from the excited states could be suddenly enhanced at the stellar temperature
$T\approx 6$ GK (corresponding roughly to 550 keV). The effect will be
discussed later in Fig. 4.

Figure 3:  (Color online) The individual GT strength distribution
$B(\mathrm{GT}^{+})$ for the transitions from different states of the EC
parent nucleus 93Nb to states of the daughter nucleus 93Zr as a function of
its excitation energies, as calculated by the PSM.

The EC daughter nucleus 93Zr is described by the present calculation only at a
qualitative level, as shown in Fig. 1. The energy of the experimental ground
state $5/2^{+}$ is calculated to be higher than the first excited state
$3/2^{+}$, although the energy gap between the $3/2^{+}$ and the higher-lying
excited states is roughly reproduced. For this odd neutron nucleus with
$N=53$, the last neutron occupies the $\\{d_{5/2},g_{7/2},s_{1/2}\\}$
spherical mean-field orbitals, which can not be ideally described by models
based on deformed mean fields such as the PSM. As we shall discuss in the next
section, the relatively poor description of 93Zr will directly affect the GT
strength distribution as differences in excitation energies of the daughter
nucleus 93Zr will shift GT strength distribution peaks.

###  III.2 GT strength distribution

In Fig. 2(a), we present our calculated GT strength distribution
$B(\mathrm{GT}^{+})$ for the transitions from the ground state $9/2^{+}$ of
93Nb to all obtained $\\{7/2^{+},9/2^{+},11/2^{+}\\}$ states of 93Zr as
functions of the excitation energy $E_{f}$ (see red solid curve in Fig. 2(a)).
Our results are compared with the recent experimental data (black dots with
the shaded area indicating statistical errors) [11], and with the calculations
from the LSSM and QRPA. In the comparison with experiment, all the theoretical
results are plotted in 0.5-MeV-wide bins as the data analysis does.

For descriptions of the medium-heavy nuclei 93Nb and 93Zr, the cross-shell
correlations in a large model space and multi-particle-multi-hole (multi-qp)
correlations in large configuration space are expected to be important. In the
LSSM calculation, a small model space with the
$\\{0f_{5/2},1p_{3/2},1p_{1/2},0g_{9/2}\\}$ orbitals for protons and the
$\\{0g_{7/2},1d_{5/2},1d_{3/2},2s_{1/2},0h_{11/2}\\}$ orbitals for neutrons is
adopted, assuming a 78Ni core [11]. To make the calculation feasible, the
configuration space is further truncated to allow only up to three protons in
the $0g_{9/2}$ orbital and no neutrons in the $0h_{11/2}$ orbital. To remedy
these truncations, the calculated GT strengths $B(\mathrm{GT}^{+})$ were
renormalized (shrunk) by a phenomenological hindrance factor $h=5.43$ [11]. It
is seen from Fig. 2 (a) that with the above calculation conditions, most of
the calculated $B(\mathrm{GT}^{+})$ by the LSSM concentrate in two peaks
roughly at $E_{f}\approx 2.5$ MeV and 4 MeV. No $B(\mathrm{GT}^{+})$ strengths
appear for $E_{f}\gtrsim 6$ MeV and the experimentally-observed fragmentation
at higher excitations is not reproduced, which can be clearly seen in Fig. 2
(b) where the cumulative (running) sums of $B(\mathrm{GT}^{+})$ are plotted.

For the QRPA, a larger model space is adopted for calculation, with the
quenching factor $f_{\text{quench}}\approx 0.8$ [11]. However, its
configuration space is constructed with phonon-like structures that are summed
by individual quasiparticle states. This is probably the reason that leads to
a resonance-like GT peak at $E_{f}\approx 2.5$ MeV, which almost exhausts the
$B(\mathrm{GT}^{+})$ at the low-energy region, leaving little strengths for
necessary fragmentations, as illustrated in Fig. 2 (a) and (b).

By comparison, the PSM employs a large model space (with four major harmonic-
oscillator shells for both neutrons and protons) and a large configuration
space (with up to 7-qp states with the configurations listed in Eqs. (II,
II)). For the present calculation from 93Nb’s $9/2^{+}$ ground-state
transition, for example, we include about 1000 daughter states for each of the
$\\{7/2^{+},9/2^{+},11/2^{+}\\}$ states in 93Zr, which is almost doubled as
compared with the number of states in the LSSM calculation [11]. The very
distributed daughter states enable a more realistic description of the broad
fragmentation of the experimental $B(\mathrm{GT}^{+})$ strengths for the
entire excitation-energy region. The fragmented GT distribution can be
realized in Fig. 2(a), with the four GT peaks observed in experiment are
qualitatively reproduced by the PSM calculation. We note that the predicted
locations for the first two peaks are shifted to higher energies, which is
likely attributed to the problem in the description of level structure in
93Zr, as discussed in Fig. 1. We also note that each of our 0.5-MeV-wide bin
in Fig. 2(a) may contain many individual transitions to the daughter states.
In Fig. 3, we plot the original $B(\mathrm{GT}^{+})$ values from the PSM
calculation without being grouped in bins. The plot in Fig. 3 (a) indicates
the calculated GT transitions from the ground state of 93Nb. Apart from the
few strong strengths at low energies, at about 4 MeV and 5-6 MeV we find
concentrations, corresponding to the $B(\mathrm{GT}^{+})$ peaks in Fig. 2 (a).
From 8 MeV up, many relatively weak strengths exist, which may bring important
contribution to the behavior of the stellar EC at high temperatures, as we
shall discuss later.

We emphasize an important feature from the PSM results, which is in a clear
distinction from the LSSM and QRPA, that from $E_{f}\gtrsim 8$ MeV, the PSM
predicts an increasing trend for $\sum B(\mathrm{GT}^{+})$ where the other
calculations suggest vanishing strengths. This increasing trend in $\sum
B(\mathrm{GT}^{+})$ predicted by the PSM can be attributed to the contribution
from the higher-order of qp configurations. Although individual strengths of
the allowed GT transitions from the parent ground state to these states may
not be significant, the number of such higher-order of qp configurations is
large and increases with the excitation energy, and therefore, the total
contribution may not be small. At this excitation energy and beyond where the
uncertainty of experimental $B(\mathrm{GT}^{+})$ gets larger due to the
difficulty to separate the GT strengths from the isovector spin giant monopole
resonance, the current data analysis stopped [11].

With growing excitation energies, the cumulative sum of $B(\mathrm{GT}^{+})$
can be viewed in 2(b). Among the theories, the LSSM (with renormalizations for
the results) gives the best comparison with data up to about 6 MeV in
excitation, and afterwards, begins to departure from the data curve because no
new strengths further contribute. The discrepancy for the LSSM in the high-
excitation region is clearly due to the lack of cross-shell correlations in
its limited model space. The QRPA produces too large $\sum B(\mathrm{GT}^{+})$
at the low-energy region but too little at the high-energy region. The reason
leading to this result is clearly due to the resonance-like behavior at 2.5
MeV in the QRPA. The PSM gives qualitatively correct trend with increasing
$\sum B(\mathrm{GT}^{+})$, but the problem for the PSM result is that the
increase is delayed to high energies due to inappropriate distribution of the
energy levels in 93Zr, as discussed before. At 10 MeV, the PSM and QRPA
calculations almost matches the experimental result, but begin to show an
important distinction: The QRPA curve seems to keep flat while a robustly
increasing trend starting from 6 MeV is predicted by the PSM. This difference
occurs because of the applied algorithms for building configuration space in
models. As the energy goes up, higher-order qp states in the PSM start to
contribute to $\sum B(\mathrm{GT}^{+})$ until the GT sumrule is reached, while
no such mechanism exists in the QRPA. Note that 5-qp (7-qp) configurations for
odd-mass nuclei in the PSM calculation correspond conceptually to the
2-particle-2-hole (3-particle-3-hole) excitations in the language of the QRPA.

In stellar environments with finite temperatures, nuclei can have considerable
probability to be thermally populated in excited nuclear states, which would
in principle contribute to stellar EC rates. The contributions depend on
strengths of the GT strength distributions as well as the precise excitation
energies where the GT strengths locate. If no experimental data are available
and no explicit theoretical calculations are performed, one assumes the Brink-
Axel hypothesis. In Fig. 2 (c) and (d) we show respectively the
$B(\mathrm{GT}^{+})$ distributions and their cumulative sums for the
transitions from three representative excited states (with different
excitation energies and spin-parity values) of the parent nucleus 93Nb. For
the very low-lying first $1/2^{-}$ state with experimental value at 30.77 keV,
the calculated $B(\mathrm{GT}^{+})$ is found to be negligible at
$E_{f}\lesssim 5.0$ MeV (seen more clearly in Fig. 3(b)). This is because that
the $1/2^{-}$ state has a relatively pure wave function with the main
configuration as $\pi 1/2^{-}[301]$ from the proton $p_{1/2}$ orbital, so that
the connection by the GT operator with negative-parity states of 93Zr can only
be the type that mixes strongly with the neutron $p_{1/2}$ orbital. Such
states lie in higher energies across the neutron $N=50$ shell gap. Figure 3(b)
illustrates the situation.

On the other hand, the wave function of the first $5/2^{+}$ state in 93Nb (see
Fig. 1) contains a strong mixture from many 1-qp and 3-qp configurations that
involves the mean-field levels originating from the proton $g_{9/2}$ orbital.
This orbital can be readily connected by the GT operator either with the
neutron $g_{7/2}$ orbital at small energies, or with the neutron $g_{9/2}$
orbital at large energies crossing the $N=50$ shell gap. The
$B(\mathrm{GT}^{+})$ is then found to be sizable in both lower-lying
($E_{f}\approx 1$ MeV) and higher-lying ($E_{f}>4$ MeV) regions (see Fig.
3(c)), with a broad fragmentation due to the latter configurations, as shown
in Fig. 2(c).

The wave function of the first high-spin $13/2^{+}$ state shows qualitatively
a similar configuration mixing as in the first $5/2^{+}$ state. The difference
is that the former tends to mix configurations that involve the high-$K$
components of the proton $g_{9/2}$ orbital, i.e. $\pi 9/2^{+}[404]$ and $\pi
7/2^{+}[413]$, while the latter mixes configurations with low-$K$ components.
Concretely, the wave function of this $13/2^{+}$ state consists of $\sim 16\%$
of $\pi 1/2^{+}[440]$, $\sim 13\%$ of $\pi 3/2^{+}[431]$, $\sim 10\%$ of $\pi
5/2^{+}[422]$, $\sim 11\%$ of $\pi 3/2^{+}[431]\otimes 5/2^{+}[422]\otimes
5/2^{+}[422]$ with the $K=3/2$ configuration, and many other 3-qp and 5-qp
configurations (some of which includes the $\pi 7/2[413]$ 1-qp configuration).
The $B(\mathrm{GT}^{+})$ from the first $13/2^{+}$ state is found to be much
smaller in strength than the first $5/2^{+}$ state in the entire excitation-
energy region, as seen from Fig. 2 (d) as well as Fig. 3 (c) and (d).

###  III.3 Stellar electron-capture rates

Figure 4:  (Color online) The electron-capture rates for 93Nb $\rightarrow$
93Zr as a function of the temperature $T_{9}$ (GK) at different stellar
densities $\rho Y_{e}=10^{7},10^{9}$ and $10^{11}$ g/cm3. For the case where
the parent nucleus 93Nb is supposed to stay in the ground state, calculations
from three nuclear models are compared with each other and with the recent
data [11] in panels (a-c), with the common legend indicated in (a). For the
realistic case that all low-lying states of 93Nb could be populated in the
finite-temperature condition, calculations of the projected shell model with
and without the excited states are shown in panels (d-f), with the common
legend indicated in (d). See the text for details. Figure 5:  (Color online)
The phase space integral $\Phi^{\text{EC}}_{1f}$ for transitions from the
ground state of 93Nb to states of 93Zr at different stellar densities and
temperatures, as a function of the excitation energy $E_{f}$. Note that the
figure is in logarithmic scale and the scale is different in each panel.

With the calculated GT strength distributions, we can now study the 93Nb
stellar EC rates at different densities and temperatures. From the second
summation over $f$ in Eq. (2), it becomes evident that in order for a daughter
state labeled $f$ to contribute significantly to the total
$\lambda^{\text{EC}}$, it is required that the $f$-state necessarily has a
large $B_{if}$. At the same time, the phase space must be open for the
$f$-state to contribute, which requires a sizeable $\Phi_{if}^{\rm EC}$.

We first discuss a popular treatment for electron capture processes occurring
in astrophysical problems. In such a simplified treatment, the parent nucleus
93Nb is supposed to stay in its ground state all the time regardless of
thermal excitations. This simplification thus completely neglects detailed
thermal populations of the parent-nucleus states, and corresponds to replacing
the $i$-summation in Eq. (2) by a unity and setting $i\equiv 1$ in the second
summation. We may call this treatment zero-temperature approximation for
parent nuclei. The resulting EC rates as functions of the temperature $T_{9}$
(GK) at three typical stellar densities are shown in Fig. 4 (a), (b) and (c).
Among the three density conditions, $\rho Y_{e}=10^{7}$ g/cm3 corresponds to
the condition of the core during the silicon-burning stage in a pre-supernova
star, $\rho Y_{e}=10^{9}$ g/cm3 represents the condition for the pre-collapse
of the core for massive stars, and $\rho Y_{e}=10^{11}$ g/cm3 labels the onset
of the core collapse. Calculations from the three different nuclear models are
compared with each other, and with the recent experimental values [11].

As the ground-state to ground-state EC $Q$-value is as small as
$Q_{11}=-0.602$ MeV in the present example, the contribution to stellar EC
rates from transitions to excited states of the daughter nucleus 93Zr would be
considerable even at low stellar densities and temperatures. For the low-
density case with $\rho Y_{e}=10^{7}$ g/cm3, the electron chemical potential
$\mu_{e}\approx 1$ MeV. From Eq. (2) and the corresponding phase space
integral $\Phi^{\text{EC}}_{1f}$ distributions in Fig. 5 (a), at low
temperatures of $T_{9}\lesssim 4$ GK, transitions to the low-lying states of
93Zr with $E_{f}\lesssim 1$ MeV are important. As the QRPA shows no
$B(\mathrm{GT}^{+})$ strength in such low-excitation region (see Fig. 2(b)),
the calculated EC rates are strongly underestimated as seen from Fig. 4(a).
With increasing temperature, the electron Fermi surface becomes more diffuse
and the corresponding phase space integral $\Phi^{\text{EC}}_{1f}$ increases
rapidly as seen from Fig. 5 (a). The transitions to higher excited states of
93Zr within $1\lesssim E_{f}\lesssim 2$ MeV begin to contribute. Within
experimental uncertainties, the derived EC rates from the three nuclear models
(LSSM, QRPA, and PSM) are all consistent with data for $T_{9}>4$ GK.

For the $\rho Y_{e}=10^{9}$ g/cm3 case, the electron chemical potential
increases to $\mu_{e}\approx 5$ MeV. The contributions of GT transitions to
the excited states of 93Zr with $E_{f}\lesssim 5.5$ MeV would play a role in
the stellar EC rates. From the $\Phi^{\text{EC}}_{1f}$ distributions in Fig. 5
(b), the states with higher excitations in 93Zr would be involved effectively
when $T_{9}$ goes high. The EC rates are found not as sensitive to the details
of the GT strength distribution as in the low-density case, but largely depend
on both the GT strength distribution and the integrated GT strength for
$E_{f}\lesssim 5.5$ MeV. As a consequence, differences in calculated results
from different models can be as large as several times. It was seen from Fig.
2(b) that for this energy region, the QRPA overestimates the
$B(\mathrm{GT}^{+})$, the PSM underestimates that, and the SM (with
renormalizations for the GT results) agrees better with the data. The derived
EC rates by the SM and PSM are compared well with data within the
uncertainties, but for the QRPA, the calculation is a little bit
overestimated, as seen from Fig. 4(b). Unfortunately, due to large
uncertainties, the current experimental data are not able to discriminate the
models.

For the high density case with $\rho Y_{e}=10^{11}$ g/cm3, the electron
chemical potential increases to about 20 MeV. As seen from Fig. 5(c), the
phase space for this density condition opens for all excited states that are
included in the plot, with $\Phi^{\text{EC}}_{1f}$ (nearly) exponentially
decreasing with $E_{f}$ and essentially being independent of stellar
temperatures. The derived stellar EC rates then mainly depends on the
integrated GT strength. As seen from Fig. 4(c), the calculated
$\lambda^{\text{EC}}$ by all the three models fall consistently within data
uncertainties, with no temperature-dependence. One should keep in mind that
these $\lambda^{\text{EC}}$ results are obtained with the zero-temperature
approximation, i.e. when the assumption that electrons are captured always by
the ground state of the parent 93Nb is applied.

When the $i$-summation for the parent-nucleus states is fully taken into
account in the stellar EC rates expression in Eq. (2), thermal populations of
different nuclear states may vary considerably with stellar temperatures, and
we thus call it the finite-temperature environment. This can be clearly seen
in Eq. (2) through the $i$-summation content
$\sum_{i}(2J_{i}+1)e^{-E_{i}/(k_{B}T)}/G(Z,A,T)$, which depend explicitly on
$E_{i}$, $J_{i}$ and $T$. In general, as temperature goes up, excited states
are successively populated while the occupation probability of the ground
state drops down. To see clearly how population variations in the nuclear
states affect the EC rates, we present in Fig. 4 (d-f) the PSM calculations
for EC rates with and without the contributions of excited states of 93Nb.
When only transitions from the $9/2^{+}$ ground state of 93Nb are considered
in the finite-temperature environment (i.e. we extract only the $i=1$ term
from the summation), the occupation probability of the ground state decreases
with temperature, and the corresponding EC rates are smaller when compared
with the calculations in Fig. 4 (a-c) where the parent nucleus is supposed to
stay in its ground state all the time. For the $\rho Y_{e}=10^{7}$ g/cm3 case
with small $\mu_{e}$, EC rates climb rapidly with temperature and the effect
of the thermal occupation is not very significant if we compare Fig. 4 (d)
with (a). For the $\rho Y_{e}=10^{9}$ g/cm3 case with large $\mu_{e}$, the EC
rates in Fig. 4 (b) only show moderate increase with the temperature. The
effect of the thermal occupation of the ground state leads to a clear decrease
of the EC rates with temperature, if we compare the black-dotted curve in Fig.
4 (e) and the PSM results in Fig. 4 (b). Similar large differences can also be
seen when comparing Fig. 4 (f) and (c).

With all the excited states of the parent nucleus 93Nb fully taken into
account in the $i$-summation in Eq. (2), one can see how the EC rates are
changed by comparing the two curves in Fig. 4 (d-f). At low temperatures with
$T_{9}\lesssim 3$ GK, only the first excited $1/2^{-}$ state in 93Nb is
populated effectively. Since this state has negligible $B(\mathrm{GT}^{+})$ at
$E_{f}\lesssim 5.5$ MeV (see Fig. 2(d)) and a small $2J+1$ factor (see Eq.
(2)), the effect of thermal population of excited parent-nucleus states is
found to be very small (namely, the two curves in Fig. 4 (d-f) in this
temperature region are very close to each other). This conclusion has been
obtained in a recent simulation by Misch et al. [60]. At higher temperatures
with $T_{9}\gtrsim 6$ GK (corresponding roughly to 550 keV), many excited
states with sizable $2J+1$ factors begin to be populated effectively (see Fig.
1). The $B(\mathrm{GT}^{+})$ strengths of these excited states are comparable
to that of the ground state (see Fig. 2), and the effects on EC rates are thus
significant. The realistic stellar EC rates (and the effects of excited
parent-nucleus states) are found to increase rapidly with temperature. At
$T_{9}\approx 15$ GK, the effect of excited parent-nucleus states is found to
be more than one order of magnitude when comparing the two curves in Fig. 4
(d-f).

Finally, it is illustrative to compare the two calculated EC rates for 93Nb
from the PSM: those in the zero-temperature approximation in Fig. 4(a-c) (red
solid lines) and those in the full calculations in Fig. 4(d-f) (labeled as
‘from all states’). It is seen that similar curves are obtained in the two
cases, except when the stellar temperatures are very high (with $T_{9}\gtrsim
10$ GK) the full calculation gives slightly higher EC rates (which is
illustrated clearly in Fig. 4 (c) and (f)). This may suggest that the zero-
temperature approximation used for stellar EC rates is reasonably good.
However, we caution that this conclusion should be taken as special for the
present 93Nb $\rightarrow$ 93Zr case because for the EC parent nucleus 93Nb,
the very-low first $1/2^{-}$ state has negligible $B(\mathrm{GT}^{+})$ values,
and furthermore, there exists a large energy gap between this $1/2^{-}$ state
and the other excited states, which diminishes their contributions to the EC
rates.

##  IV summary

To summarize, electron capture (EC) rates are important nuclear inputs for
understanding many astrophysical phenomena such as the core-collapse
supernovae, the Urca cooling of neutron star crust, etc. In such stellar
environments with high temperature and density, the contribution of excited
nuclear states to the EC rates is indispensable, and their effects should be
considered. Astrophysical simulations usually take one of the two approaches
to the distribution of nuclear states: either they assume a thermal
equilibrium distribution (e.g. a finite-temperature environment with the
Boltzmann distribution), or they use only the ground state properties (the
zero-temperature approximation). With new development of nuclear many-body
techniques, explicit treatment of all excited states in theoretical
calculations become possible. This allows one to compare the novel method with
the traditional ones before it can be applied in astrophysical simulations.

In the present article, we introduced a Projected-Shell-Model calculation for
stellar EC rates in medium-heavy odd-mass nuclei by taking 93Nb $\rightarrow$
93Zr as the example. In the zero-temperature approximation with only the
ground state of the parent nucleus considered, the calculated Gamow-Teller
transition strength distribution $B(\mathrm{GT}^{+})$ were compared with the
recent experimental data [11] and discussed together with the large-scale
shell model calculation and the quasiparticle random-phase approximation. As
seen in Fig. 2 (a) and (b), theoretical $B(\mathrm{GT}^{+})$ from different
models give large deviations from each other, showing even no qualitative
similarities. We have discussed possible reasons that lead to the
characteristic GT pattern for each model. We further discussed the derived
stellar EC rates at different temperatures and densities. We found that for
the present example, although different nuclear models give very different GT
strength distributions, the derived stellar EC rates are all qualitatively
consistent with the data within the statistical errors and uncertainties for
most cases. In the full calculations by the Projected Shell Model with all
excited parent-nucleus states taking into account exactly, similar EC rates
are obtained as compared to those from the zero-temperature approximation in
which only the ground state properties of the parent nucleus is considered.
This seems to suggest equivalence of the two approaches to including the
effects of excited nuclear states.

The insensitivity of the stellar EC rates found in the present work, to
detailed nuclear structure and to different approaches to treat the
contribution of nuclear states, needs further investigation. In the present
93Nb case, the electron-capture $Q$ value is close to zero, and when the Fermi
energy is sufficiently high, the details of the GT strength distribution do
not have a strong impact on the derived EC rates. We however caution that this
conclusion for 93Nb should not be taken as general since in principle the EC
rates are expected to depend on detailed nuclear-structure properties such as
the energy levels, $B(\mathrm{GT}^{+})$, $Q$-values, and phase space factors,
etc. For example, the EC rates of 59Co in the full calculations are by one
order of magnitude larger than those from the zero-temperature approximation
in most temperature-density conditions [18]. Experimental data with improved
resolution, which can discriminate theoretical models, are very much desired.

###### Acknowledgements.

This work is supported by the Fundamental Research Funds for the Central
Universities (with Grant No. SWU019013), by the National Natural Science
Foundation of China (Grant Nos. 11905175, 11875225, and U1932206), by the
National Key Program for S$\&$T Research and Development (Grant No.
2016YFA0400501), and by the Venture $\&$ Innovation Support Program for
Chongqing Overseas Returnees (with Grant No. cx2019056).

## References

  * Fuller  _et al._ [1980] G. M. Fuller, W. A. Fowler, and M. J. Newman, Stellar weak-interaction rates for $sd$-shell nuclei. I. nuclear matrix element systematics with application to ${}^{26}\text{Al}$ and selected nuclei of imprtance to the supernova problem, Astrophys. J. Suppl. 42, 447 (1980).
  * Fuller  _et al._ [1982a] G. M. Fuller, W. A. Fowler, and M. J. Newman, Stellar weak-interaction rates for intermediate-mass nuclei. II., Astrophys. J. 252, 715 (1982a).
  * Fuller  _et al._ [1982b] G. M. Fuller, W. A. Fowler, and M. J. Newman, Stellar weak-interaction rates for intermediate-mass nuclei. III., Astrophys. J. Suppl. 48, 279 (1982b).
  * Fuller  _et al._ [1985] G. M. Fuller, W. A. Fowler, and M. J. Newman, Stellar weak-interaction rates for intermediate-mass nuclei. ${IV}$., Astrophys. J. 293, 1 (1985).
  * Heger  _et al._ [2001] A. Heger, K. Langanke, G. Martínez-Pinedo, and S. E. Woosley, Presupernova collapse models with improved weak-interaction rates, Phys. Rev. Lett. 86, 1678 (2001).
  * Hix  _et al._ [2003] W. R. Hix, O. E. B. Messer, A. Mezzacappa, M. Liebendörfer, J. Sampaio, K. Langanke, D. J. Dean, and G. Martínez-Pinedo, Consequences of nuclear electron capture in core collapse supernovae, Phys. Rev. Lett. 91, 201102 (2003).
  * Langanke and Martínez-Pinedo [2003] K. Langanke and G. Martínez-Pinedo, Nuclear weak-interaction processes in stars, Rev. Mod. Phys. 75, 819 (2003).
  * Langanke  _et al._ [2021] K. Langanke, G. Martínez-Pinedo, and R. G. T. Zegers, Electron capture in stars, Rep. Prog. Phys. 84, 066301 (2021).
  * Janka  _et al._ [2007] H.-T. Janka, K. Langanke, A. Marek, G. Martínez-Pinedo, and B. Müller, Theory of core-collapse supernovae, Phys. Rep. 442, 38 (2007).
  * Cole  _et al._ [2012] A. L. Cole, T. S. Anderson, R. G. T. Zegers, S. M. Austin, B. A. Brown, L. Valdez, S. Gupta, G. W. Hitt, and O. Fawwaz, Gamow-teller strengths and electron-capture rates for $pf$-shell nuclei of relevance for late stellar evolution, Phys. Rev. C 86, 015809 (2012).
  * Gao  _et al._ [2020] B. Gao, R. G. T. Zegers, J. C. Zamora, D. Bazin, B. A. Brown, P. Bender, H. L. Crawford, J. Engel, A. Falduto, A. Gade, P. Gastis, T. Ginter, C. J. Guess, S. Lipschutz, A. O. Macchiavelli, K. Miki, E. M. Ney, B. Longfellow, S. Noji, J. Pereira, J. Schmitt, C. Sullivan, R. Titus, and D. Weisshaar, Gamow-teller transitions to ${}^{93}\mathrm{Zr}$ via the ${}^{93}\mathrm{Nb}(t,^{3}\mathrm{He}+\gamma$) reaction at 115 mev/u and its application to the stellar electron-capture rates, Phys. Rev. C 101, 014308 (2020).
  * Schatz  _et al._ [2014] H. Schatz, S. Gupta, P. Möller, M. Beard, E. F. Brown, A. T. Deibel, L. R. Gasques, W. R. Hix, L. Keek, R. Lau, A. W. Steiner, and M. Wiescher, Strong neutrino cooling by cycles of electron capture and beta decay in neutron star crusts, Nature 505, 62 (2014).
  * [13] L.-J. Wang, L. Tan, Z. Li, G. W. Misch, and Y. Sun, Phys. Rev. Lett. 127, 172702 (2021).
  * Iwamoto  _et al._ [1999] K. Iwamoto, F. Brachwitz, K. Nomoto, N. Kishimoto, H. Umeda, W. R. Hix, and F.-K. Thielemann, Nucleosynthesis in chandrasekhar mass models for type ia supernovae and constraints on progenitor systems and burning-front propagation, Astrophys. J. Suppl. 125, 439 (1999).
  * Zegers  _et al._ [2008] R. G. T. Zegers, E. F. Brown, H. Akimune, S. M. Austin, A. M. v. d. Berg, B. A. Brown, D. A. Chamulak, Y. Fujita, M. Fujiwara, S. Galès, M. N. Harakeh, H. Hashimoto, R. Hayami, G. W. Hitt, M. Itoh, T. Kawabata, K. Kawase, M. Kinoshita, K. Nakanishi, S. Nakayama, S. Okumura, Y. Shimbara, M. Uchida, H. Ueno, T. Yamagata, and M. Yosoi, Gamow-teller strength for the analog transitions to the first $t=1/2,{J}^{\pi}=3/{2}^{-}$ states in ${}^{13}\mathrm{C}$ and ${}^{13}\mathrm{N}$ and the implications for type ia supernovae, Phys. Rev. C 77, 024307 (2008).
  * Fujita  _et al._ [2011] Y. Fujita, B. Rubio, and W. Gelletly, Spin–isospin excitations probed by strong, weak and electro-magnetic interactions, Prog. Part. Nucl. Phys. 66, 549 (2011).
  * Gao  _et al._ [2021] B. Gao, S. Giraud, K. A. Li, A. Sieverding, R. G. T. Zegers, X. Tang, J. Ash, Y. Ayyad-Limonge, D. Bazin, S. Biswas, B. A. Brown, J. Chen, M. DeNudt, P. Farris, J. M. Gabler, A. Gade, T. Ginter, M. Grinder, A. Heger, C. Hultquist, A. M. Hill, H. Iwasaki, E. Kwan, J. Li, B. Longfellow, C. Maher, F. Ndayisabye, S. Noji, J. Pereira, C. Qi, J. Rebenstock, A. Revel, D. Rhodes, A. Sanchez, J. Schmitt, C. Sumithrarachchi, B. H. Sun, and D. Weisshaar, New ${}^{59}\mathrm{Fe}$ stellar decay rate with implications for the ${}^{60}\mathrm{Fe}$ radioactivity in massive stars, Phys. Rev. Lett. 126, 152701 (2021).
  * Tan  _et al._ [2020] L. Tan, Y.-X. Liu, L.-J. Wang, Z. Li, and Y. Sun, A novel method for stellar electron-capture rates of excited nuclear states, Phys. Lett. B 805, 135432 (2020).
  * Oda  _et al._ [1994] T. Oda, M. Hino, K. Muto, M. Takahara, and K. Sato, Rate tables for the weak processes of $sd$-shell nuclei in stellar matter, At. Data Nucl. Data Tables 56, 231 (1994).
  * Martínez-Pinedo  _et al._ [2014] G. Martínez-Pinedo, Y. H. Lam, K. Langanke, R. G. T. Zegers, and C. Sullivan, Astrophysical weak-interaction rates for selected $a=20$ and $a=24$ nuclei, Phys. Rev. C 89, 045806 (2014).
  * Misch  _et al._ [2014] G. W. Misch, G. M. Fuller, and B. A. Brown, Modification of the brink-axel hypothesis for high-temperature nuclear weak interactions, Phys. Rev. C 90, 065808 (2014).
  * Caurier  _et al._ [1999] E. Caurier, K. Langanke, G. Martínez-Pinedo, and F. Nowacki, Shell-model calculations of stellar weak interaction rates. i. gamow-teller distributions and spectra of nuclei in the mass range a = 45–65, Nucl. Phys. A 653, 439 (1999).
  * Langanke and Martínez-Pinedo [2000] K. Langanke and G. Martínez-Pinedo, Shell-model calculations of stellar weak interaction rates: Ii., Nucl. Phys. A 673, 481 (2000).
  * Langanke and Martínez-Pinedo [2001] K. Langanke and G. Martínez-Pinedo, Rates tables for the weak processes of pf-shell nuclei in stellar environments, At. Data Nucl. Data Tables 79, 1 (2001).
  * Langanke  _et al._ [2001] K. Langanke, E. Kolbe, and D. J. Dean, Unblocking of the gamow-teller strength in stellar electron capture on neutron-rich germanium isotopes, Phys. Rev. C 63, 032801(R) (2001).
  * Langanke  _et al._ [2003] K. Langanke, G. Martínez-Pinedo, J. M. Sampaio, D. J. Dean, W. R. Hix, O. E. B. Messer, A. Mezzacappa, M. Liebendörfer, H.-T. Janka, and M. Rampp, Electron capture rates on nuclei and implications for stellar core collapse, Phys. Rev. Lett. 90, 241102 (2003).
  * Engel  _et al._ [1999] J. Engel, M. Bender, J. Dobaczewski, W. Nazarewicz, and R. Surman, $\beta$ decay rates of r-process waiting-point nuclei in a self-consistent approach, Phys. Rev. C 60, 014302 (1999).
  * Paar  _et al._ [2009] N. Paar, G. Colò, E. Khan, and D. Vretenar, Calculation of stellar electron-capture cross sections on nuclei based on microscopic skyrme functionals, Phys. Rev. C 80, 055801 (2009).
  * Bai  _et al._ [2010] C. L. Bai, H. Q. Zhang, H. Sagawa, X. Z. Zhang, G. Colò, and F. R. Xu, Effect of the tensor force on the charge exchange spin-dipole excitations of ${}^{208}\mathrm{Pb}$, Phys. Rev. Lett. 105, 072501 (2010).
  * Dzhioev  _et al._ [2010] A. A. Dzhioev, A. I. Vdovin, V. Y. Ponomarev, J. Wambach, K. Langanke, and G. Martínez-Pinedo, Gamow-teller strength distributions at finite temperatures and electron capture in stellar environments, Phys. Rev. C 81, 015804 (2010).
  * Niu  _et al._ [2011] Y. F. Niu, N. Paar, D. Vretenar, and J. Meng, Stellar electron-capture rates calculated with the finite-temperature relativistic random-phase approximation, Phys. Rev. C 83, 045807 (2011).
  * Fang  _et al._ [2013] D.-L. Fang, B. A. Brown, and T. Suzuki, $\beta$ decay properties for neutron-rich kr-tc isotopes from deformed pn-quasiparticle random-phase approximation calculations with realistic forces, Phys. Rev. C 88, 024314 (2013).
  * Niu  _et al._ [2013] Z. M. Niu, Y. F. Niu, H. Z. Liang, W. H. Long, Nikšić, D. Vretenar, and J. Meng, $\beta$-decay half-lives of neutron-rich nuclei and matter flow in the r-process, Phys. Lett. B 723, 172 (2013).
  * Sarriguren [2013] P. Sarriguren, Stellar electron-capture rates in $pf$-shell nuclei from quasiparticle-random-phase approximatoin calculations, Phys. Rev. C 87, 045801 (2013).
  * Robin and Litvinova [2019] C. Robin and E. Litvinova, Time-reversed particle-vibration loops and nuclear gamow-teller response, Phys. Rev. Lett. 123, 202501 (2019).
  * Ejiri  _et al._ [2019] H. Ejiri, J. Suhonen, and K. Zuber, Neutrino–nuclear responses for astro-neutrinos, single beta decays and double beta decays, Phys. Rep. 797, 1 (2019).
  * Brink [1955] D. M. Brink, Ph.D. thesis, Oxford University (1955).
  * Axel [1962] P. Axel, Electric dipole ground-state transition width strength function and 7-mev photon interactions, Phys. Rev. 126, 671 (1962).
  * Haxton and Henley [1995] W. C. Haxton and E. M. Henley,  _Symmetries and fundamental interactions in nuclei_ (World Scientific (Singapore), 1995).
  * [40] B. Märkisch, H. Mest, H. Saul, et al., Phys. Rev. Lett. 122, 242501 (2019).
  * Menéndez  _et al._ [2011] J. Menéndez, D. Gazit, and A. Schwenk, Chiral two-body currents in nuclei: gamow-teller transitions and neutrinoless double-beta decay, Phys. Rev. Lett. 107, 062501 (2011).
  * Wang  _et al._ [2018a] L.-J. Wang, J. Engel, and J. M. Yao, Quenching of nuclear matrix elements for $0\nu\beta\beta$ decay by chiral two-body currents, Phys. Rev. C 98, 031301(R) (2018a).
  * Gysbers  _et al._ [2019] P. Gysbers, G. Hagen, J. D. Holt, G. R. Jansen, T. D. Morris, P. Navrátil, T. Papenbrock, S. Quaglioni, A. Schwenk, S. R. Stroberg, and K. A. Wendt, Discrepancy between experimental and theoretical beta-decay rates resolved from first principles, Nature Physics 15, 428 (2019).
  * Bodenstein-Dresler  _et al._ [2020] L. Bodenstein-Dresler, Y. Chu, D. Gehre, C. Gößling, A. Heimbold, C. Herrmann, R. Hodak, J. Kostensalo, K. Kröninger, J. Küttler, C. Nitsch, T. Quante, E. Rukhadze, I. Stekl, J. Suhonen, J. Tebrügge, R. Temminghoff, J. Volkmer, S. Zatschler, and K. Zuber, Quenching of $g_{\mathrm{a}}$ deduced from the $\beta$-spectrum shape of ${}^{113}\mathrm{Cd}$ measured with the $\mathrm{COBRA}$ experiment, Phys. Lett. B 800, 135092 (2020).
  * Epelbaum  _et al._ [2009] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Modern theory of nuclear forces, Rev. Mod. Phys. 81, 1773 (2009).
  * Klos  _et al._ [2017] P. Klos, A. Carbone, K. Hebeler, J. Menéndez, and A. Schwenk, Uncertainties in constraining low-energy constants from ${}^{3}\text{H}$ $\beta$ decay, Eur. Phys. J. A. 53, 168 (2017).
  * Sun and Feng [1996] Y. Sun and D. H. Feng, High spin spectroscopy with the projected shell model, Phys. Rep. 264, 375 (1996).
  * Gao  _et al._ [2006] Z.-C. Gao, Y. Sun, and Y.-S. Chen, Shell model method for gamow-teller transitions in heavy, deformed nuclei, Phys. Rev. C 74, 054303 (2006).
  * Wang  _et al._ [2014] L.-J. Wang, F.-Q. Chen, T. Mizusaki, M. Oi, and Y. Sun, Toward extremes of angular momentum: Application of the pfaffian algorithm in realistic calculations, Phys. Rev. C 90, 011303(R) (2014).
  * Wang  _et al._ [2016] L.-J. Wang, Y. Sun, T. Mizusaki, M. Oi, and S. K. Ghorui, Reduction of collectivity at very high spins in 134Nd: Expanding the projected-shell-model basis up to 10-quasiparticle states, Phys. Rev. C 93, 034322 (2016).
  * Sun [2016] Y. Sun, Projection techniques to approach the nuclear many-body problem, Phys. Scr. 91, 043005 (2016).
  * Wang  _et al._ [2018b] L.-J. Wang, Y. Sun, and S. K. Ghorui, Shell-model method for gamow-teller transitions in heavy deformed odd-mass nuclei, Phys. Rev. C 97, 044302 (2018b).
  * Ring and Schuck [1980] P. Ring and P. Schuck,  _The nuclear many-body problem_ (Springer-Verlag, 1980).
  * Hara and Sun [1995] K. Hara and Y. Sun, Projected shell model and high-spin spectroscopy, Int. J. Mod. Phys. E 4, 637 (1995).
  * [55] A. K. Jain et al., Rev. Mod. Phys. 62, 393 (1990).
  * Wang  _et al._ [2019] L.-J. Wang, J. Dong, F.-Q. Chen, and Y. Sun, Projected shell model analysis of structural evolution and chaoticity in fast-rotating nuclei, J. Phys. G 46, 105102 (2019).
  * Wang  _et al._ [2020] L.-J. Wang, F.-Q. Chen, and Y. Sun, Basis-dependent measures and analysis uncertainties in nuclear chaoticity, Phys. Lett. B 808, 135676 (2020).
  * Guttormsen  _et al._ [2015] M. Guttormsen, M. Aiche, F. L. Bello Garrote, L. A. Bernstein, D. L. Bleuel, Y. Byun, Q. Ducasse, T. K. Eriksen, F. Giacoppo, A. Görgen, F. Gunsing, T. W. Hagen, B. Jurado, M. Klintefjord, A. C. Larsen, L. Lebois, B. Leniau, H. T. Nyhus, T. Renstrøm, S. J. Rose, E. Sahin, S. Siem, T. G. Tornyi, G. M. Tveten, A. Voinov, M. Wiedeking, and J. Wilson, Experimental level densities of atomic nuclei, Eur. Phys. J. A. 51, 170 (2015).
  * Baglin [2011] C. M. Baglin, Nuclear data sheets for $\mathrm{A}=93$, Nuclear Data Sheets 112, 1163 (2011).
  * Misch  _et al._ [shed] G. W. Misch, T. M. Sprouse, M. R. Mumpower, A. Couture, C. L. Fryer, B. S. Meyer, and Y. Sun, Symmetry 13, 1831 (2021).