11institutetext: 1Department of Physics, Institute of Science, Banaras Hindu
University Varanasi, Varanasi - 221 005, INDIA
2Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247
667, INDIA

# Shell-model study of $\beta^{+}$/EC-decay half-lives for $Z=21-30$ nuclei

Vikas Kumar1<EMAIL_ADDRESS>Praveen C. Srivastava2
<EMAIL_ADDRESS>
###### Abstract

In the present work, we have reported $\beta^{+}$/EC-decay half-lives for
$Z=21-30$ nuclei using large-scale shell-model. A recent study shows that some
proton-rich nuclei in this region belong to the island of inversion. We have
performed calculations for these nuclei using KB3G effective interaction,
while for Ni, Cu, and Zn nuclei we have used JUN45 effective interaction in
the $f_{5/2}pg_{9/2}$ model space. The calculated quenching factors for $fp$
and $f_{5/2}pg_{9/2}$ space using KB3G and JUN45 effective interactions are
also reported. Shell-model results of $\beta$-decay half-lives, excitation
energies, log$ft$ values, and branching fractions are discussed and compared
with the available experimental data. We have obtained a reasonable agreement
with the available data.

###### pacs:

21.60.CsShell model, 23.40.-s $\beta$-decay

##  1 Introduction

The recent development of radioactive ion facilities opens opportunities to
study nuclei away from the stability lines, these nuclei may decay by
different modes of beta decay. Thus, it is important to study these nuclei
theoretically using novel approaches. Recently, beta decay properties using ab
initio theory were reported in Ref. nature . It is still very challenging to
study nuclei in the entire region of the nuclear chart using an ab initio
approach, thus study of these nuclei using a large-scale shell model is very
important. A weak sub-shell effect at $N$=40 in the Cu isotopes is obtained in
M. L. Bissell . Recchia et al. F Recchia  reported sub-shell closure in Co
isotopes toward the $N$=40 and the new island of inversion. The recent studies
in vikas ; prc8161301 ; vikas1 ; prc8564305 ; vikas2 ; prc8614325 ; pc ;
prl102142501 ; prl115192501  show the coexistence of normal and intruder
configurations in neutron rich nuclei around $N$=40 shell gap. Due to the
emergence of a new island of inversion and sub-shell closure around $N$=40 the
nuclear structure study including $\beta$-decay properties of these nuclei is
very important Otsuka ; p3104 ; williams ; prc8254301 . Recently, shell-model
results for neutron capture rates for several nuclei including Cr and Ni
chains are reported in Ref. Sieja .

The experimental $\beta$-decay half-lives of 41,42Sc nuclei are reported in
41sc ; 42sc . The spectrum of $\gamma$-rays following the $\beta$-decay of
43Sc has been measured and reported in 43sc . Sarriguren et al. Sarriguren
studied stellar electron-capture rates for $fp$ shell nuclei using
quasiparticle random-phase approximation. In scvmnco , the half-lives of 42Sc,
46V, 50Mn, and 54Co nuclei are reported. The ground-state $\beta^{+}$-decay of
42Ti has been investigated and the half-life, branching ratios, and log$ft$
values are reported in 42ti . The beta decay of 43Ti to its mirror nucleus
43Sc has been studied up to 5 MeV excitation energy in 43ti . First time,
observations of non-analog $0^{+}$$\rightarrow$$0^{+}$ branches in 38mK, 46V,
50Mn, and 54Co are reported in 46v . The $\beta$-delayed proton radioactivity
of proton-rich nuclei 44Cr, 47Mn, 48,49Fe and 50Co are studied in 44v . In
Dossat et al 44v1 , a series of experiments at the SISSI/LISE3 facility of
GANIL was conducted for long time and they reported $\beta$-decay half-life
and their total $\beta$-delayed proton emission branching ratio of 26 nuclei
between 36Ca and 56Zn. Orrigo et al 48fe  reported the half-lives and the
total $\beta$-delayed proton emission branching ratios of three proton-rich
nuclei with $T_{z}$ = -2, namely 48Fe, 52Ni, and 56Zn, produced at GANIL. The
$\beta^{+}$-decay study of $T_{z}$ =-1 nuclei 54Ni, 50Fe, 46Cr, and 42Ti
produced in fragmentation reactions at GSI is reported by Molina et al 50fe .
The first time the $\beta$-decay half-life of 58Zn has been determined at the
ISOLDE on-line separator facility at CERN in 58zn . In 64cu , half-lives and
$\gamma$-ray intensities of 64Cu and 68Ga were measured at IFIN-HH.

In the last few years, many experiments have been performed around the globe
using RIB facilities for the measurement of $\beta$-decay half-lives, log$ft$
values, branching ratios, etc. in $fp$ and $f_{5/2}pg_{9/2}$ shell nuclei,
many of them are neutron rich nuclei and belongs to the island of inversion. A
systematic theoretical estimate for $\beta^{+}$-decay half-lives of neutron
rich nuclei is needed despite the progress in the experimental side. Motivated
with these recent data, in the present work we have performed a systematic
shell-model (SM) study of $\beta$-decay half-lives, excitation energies,
log$ft$ values, and branching fractions for $Z=21-30$ nuclei. In the present
work, our SM calculations are based on allowed Fermi and GT-transitions.
Previously, SM calculations for the $\beta^{-}$-decays of $fp$ and $fpg$ shell
nuclei are reported by us in Ref. vikas .

This paper is organized as follows. In section 2, the theoretical formulas for
$\beta$ decay half-lives calculations are discussed. Shell model spaces,
effective interactions and the quenching factor calculations are reported in
section 3. In section 4, the theoretical results along with the experimental
data are discussed. Finally, in section 5 the summary and conclusions are
drawn.

##  2 Formalism for $\beta$-decay half-lives

In all three different types of $\beta$-decay processes, the mass number A of
the parent nucleus remains unchanged, only the atomic number Z changes by one
unit. The $\beta$-decay selection rules permit all those transitions which are
inside the energy window defined by $Q$-value from the ground state of the
parent nucleus to different excited states of the daughter nucleus. The
transition probability $T_{fi}$ is related to the half-life of $\beta$-decay
as

$t_{1/2}=\frac{ln2}{T_{fi}}.$ (1)

The resulting expression for total decay half-life of a combined $\beta^{+}$
and electron-capture (EC) transition, denoted by $\beta^{+}$/EC, is given by

$f_{0}t_{1/2}=\left[f_{0}^{(+)}+f_{0}^{(EC)}\right]t_{1/2}=\frac{\kappa}{[g^{2}_{A}*B(GT)+B(F)]},$
(2)

where, $g_{A}$ (= 1.270) represents the axial-vector coupling constant of the
weak interactions and $f_{0}$ represents the phase-space factor, sometimes
also called the Fermi integral. The B(F) and B(GT) are the Fermi and
Gamow–Teller reduced transition probabilities, respectively. the latest
updated value of $\kappa$ is taken from Patrignani

$\kappa\equiv\frac{2\pi^{3}\hbar^{7}ln2}{m^{5}_{e}c^{4}{(G_{F}cos{\theta}_{C})}^{2}}=6289s,$
(3)

where, the ${\theta}_{C}$ is the Cabibbo angle.

The Fermi reduced transition probability $B(F)$ is given by

$B(F)\equiv\frac{g^{2}_{V}}{{2J_{i}+1}}{|M_{F}|^{2}},$ (4)

where, $g_{V}$ (= 1.0) represents the vector coupling constant of the weak
interaction and $M_{F}$ is the Fermi matrix element.

The Gamow-Teller reduced transition probability $B(GT)$ is given by

$B(GT)={\langle{\sigma\tau}\rangle}^{2}.$ (5)

In the above expression, the nuclear matrix element for the Gamow-Teller
operator is given by

${\langle{\sigma\tau}\rangle}=\frac{{\langle{f}||\sum_{k}{\sigma^{k}\tau_{\pm}^{k}}||i\rangle}}{\sqrt{2J_{i}+1}},$
(6)

where initial and final states are represented by the quantum numbers $i$ and
$f$, respectively. $\pm$ refers to $\beta^{\pm}$ decay,
$\tau_{\pm}=\frac{1}{2}(\tau_{x}+i\tau_{y})$ with $\tau_{+}p$ = $n$,
$\tau_{-}n$ = $p$, and $J_{i}$ is the total angular momentum of the initial-
state. The sum in Eq. 6 runs over all the nucleons.

For $\beta^{\mp}$ decay, the phase-space factor is given by

$f_{0}^{(\mp)}=\int_{1}^{E_{0}}F_{0}({\pm}Z_{f},\epsilon)p\epsilon(E_{0}-\epsilon)^{2}\,d\epsilon,$
(7)

where, $F_{0}$ is called Fermi function and

$\epsilon\equiv\frac{E_{e}}{m_{e}c^{2}},E_{0}\equiv\frac{E_{i}-E_{f}}{m_{e}c^{2}},p\equiv\sqrt{\epsilon^{2}-1},$
(8)

where $E_{e}$ is the total energy of the emitted electron/positron and $E_{i}$
and $E_{f}$ are the energies of the initial and final nuclear state.
The phase-space factor for the electron capture suhonen  is given by

$f^{(EC)}_{0}=2\pi({\alpha}Z_{i})^{3}(\epsilon_{0}+E_{0})^{2},$ (9)

where

$\epsilon_{0}\equiv 1-\frac{1}{2}({\alpha}Z_{i})^{2},$ (10)

and $\alpha$ is the fine-structure constant, ${\alpha}=\frac{1}{137}$. The
simple non-relativistic $s$-electron wave function was assumed in Eq. 10. This
approximation is valid when $\alpha Z_{i}$ $\ll$1\. For $Z_{i}$ $<$ 40 this
approximation holds good. In a non-relativistic approximation the Fermi
function $F_{0}$ can be written as Primakoff-Rosen approximation Primakoff

$F_{0}(Z_{f},\epsilon)\approx\frac{\epsilon}{p}F^{(PR)}_{0}(Z_{f}).$ (11)
$F^{(PR)}_{0}(Z_{f})=\frac{2\pi\alpha Z_{f}}{1-e^{-2\pi\alpha Z_{f}}}.$ (12)

This leads to the phase-space factor for $\beta^{+}$ \- decay

$f^{(+)}_{0}\approx\frac{1}{30}(E^{5}_{0}-10E^{2}_{0}+15E_{0}-6)F^{(PR)}_{0}(-Z_{f}).$
(13)

The endpoint energy $E_{0}$ can be extract by using the following relation

$E_{0}=\frac{Q_{EC}-m_{e}c^{2}}{m_{e}c^{2}}.$ (14)

The experimental $\beta$ -decay $Q_{EC}$ values are taken from ENSDF .

Usually, $ft$ values are large so expressed in terms of ‘log $ft$ values’. The
log$ft$ $\equiv$ log${}_{10}({f_{0}}t_{1/2}[s])$.

The total half-life can be calculated with the help of the partial half-life
($t_{i}$) of the daughter state $i$ using the following expression:

$t_{1/2}=\left({\sum_{i}{\frac{1}{t_{i}}}}\right)^{-1}.$ (15)

The expression for the partial half-life of the allowed $\beta$-decay is taken
from suhonen .

The branching ratio ${b_{r}}$ is related to partial half-life ${t_{i}}$ and
the total half-life $t_{1/2}$ of the allowed $\beta$-decay as

$t_{i}={\frac{t_{1/2}}{b_{r}}}.$ (16)

Figure 1:  The experimental versus theoretical matrix element $R(GT)$ values
are compared. SM calculations are based on the “free-nucleon” Gamow-Teller
operator. Each transition is represented by a point in the graph. The
theoretical and experimental values are taken on x, y coordinates,
respectively. The data points which are away from the average quenching factor
are indicated by ${}^{A}X$.

##  3 Shell model Hamiltonian and quenching factor

In the present work, for $Z=21-30$ nuclei we used KB3G kb3g  and JUN45 jun45
effective interactions for shell model calculations. The shell model code
NuShellX@MSU MSU-NSCL  is used for the diagonalization of energy matrices.

We have reported $\beta$-decay half-lives for nuclei in two different model
spaces using two different effective interactions KB3G and JUN45. The mass
dependence and original monopole changes in KB3 are known as KB3G effective
interaction kb3g . The idea behind this is to treat properly the $N=Z=28$
shell closure and its surroundings. The single-particle energies are taken to
be -8.6000, -6.6000, -4.6000, and -2.1000 MeV for the $f_{7/2}$, $p_{3/2}$,
$p_{1/2}$ and $f_{5/2}$ orbits, respectively for KB3G effective interaction.

Honma et al jun45  developed JUN45 effective interaction for $f_{5/2}pg_{9/2}$
model space. The JUN45 interaction is a realistic interaction that is based on
Bonn-C potential. Further, this interaction is fitted by 400 experimental data
(binding and excitation energies) with mass numbers A = 63 - 96. For JUN45
interaction the single-particle energies are taken to be -9.8280, -8.7087,
-7.8388, and -6.2617 MeV for the $p_{3/2}$, $f_{5/2}$, $p_{1/2}$, and
$g_{9/2}$ orbits, respectively. A large number of experimental data are taken
for fitting around $N=50$ shell closure. The full-fledged shell-model
calculations have been performed for all transitions.

The matrix element $M(GT)$ brownrgt  can be defined in terms of reduced
transition probability $B(GT)$, by following

$M(GT)=[(2J_{i}+1)B(GT)]^{1/2},$ (17)

where $J_{i}$ represents the total angular momentum of the initial state. The
matrix element $M(GT)$ is normalized to the “expected” total strength W in
order to get effective axial-vector coupling constant $g_{A}$, W is defined as

$W=\left\\{\begin{array}[]{@{}ll@{}}|g_{A}/g_{V}|[(2J_{i}+1)3|N_{i}-Z_{i}|]^{1/2},&forN_{i}\neq
Z_{i},\\\
|g_{A}/g_{V}|[(2J_{f}+1)3|N_{f}-Z_{f}|]^{1/2},&forN_{i}=Z_{i},\end{array}\right.$
(18)

The matrix elements $R(GT)$ are defined as

$R(GT)=M(GT)/W.$ (19)

The theoretical versus experimental $R(GT)$ values are plotted in Fig. 1. From
table 2, the experimental log$ft$ values are used to get the $R(GT)_{Expt}$.

The theoretically calculated Gamow-Teller strengths on the basis of model
independent Ikeda sum rule “3(N-Z)” are larger than observed values, so we
needed a quenching factor (q) for a particular model space. The average of all
the ratios between experimental and theoretical $R(GT)$ values give the
quenching factor (q) for a given model space. The straight line in Fig. 1
gives the average quenching factor. In the present work, we have obtained two
different quenching factors: $q$ = 0.719 $\pm$ 0.050 for $pf$ space using KB3G
interaction, and $q$ = 0.743 $\pm$ 0.030 for $f_{5/2}pg_{9/2}$ space using
JUN45 interaction. Fig. 1 shows that a few data points are notably away from
the straight line, we marked those data points by ${}^{A}X$. There are four
data points in $fp$ space and eight data points in $f_{5/2}pg_{9/2}$ space
which are away from the straight line. After excluding these data points we
get the quenching factor $q$ = 0.669 $\pm$ 0.020 and $q$ = 0.768 $\pm$ 0.030
for $pf$ and $f_{5/2}pg_{9/2}$ space, respectively.

Table 1: Computed phase space factors for $\beta^{+}$/EC decay (considered only those nuclei in which the EC phase space factors are significant), log$(f^{(+)}_{0}+f^{(EC)}_{0})$ values, together with experimental $Q$ values taken from ENSDF . ${}^{A}Z_{i}(J^{\pi})$ | ${}^{A}Z_{f}(J^{\pi})$ |  $Q^{(EXPT.)}$ (MeV) | $f^{(+)}_{0}$ | $f^{(EC)}_{0}$ | log$(f^{(+)}_{0}+f^{(EC)}_{0})$
---|---|---|---|---|---
43Sc($7/2^{-}$) |  43Ca($7/2_{1}^{-}$) | 2.220 | 7.137 | 0.425 | 0.878
|  43Ca($5/2_{1}^{-}$) | 1.848 | 1.779 | 0.294 | 0.316
45Ti($7/2^{-}$) |  45Sc($7/2_{1}^{-}$) | 2.062 | 4.063 | 0.421 | 0.652
|  45Sc($5/2_{1}^{-}$) | 1.519 | 0.288 | 0.228 | -0.287
|  45Sc($5/2_{2}^{-}$) | 1.342 | 0.066 | 0.178 | -0.614
|  45Sc($7/2_{2}^{-}$) | 1.088 | 0.0005 | 0.116 | -0.933
53Fe($7/2^{-}$) |  53Mn($7/2_{1}^{-}$) | 3.742 | 174.33 | 2.292 | 2.247
|  53Mn($5/2_{1}^{-}$) | 3.364 | 92.39 | 1.851 | 1.974
|  53Mn($9/2_{1}^{-}$) | 2.122 | 4.510 | 0.734 | 0.720
|  53Mn($5/2_{2}^{-}$) | 1.468 | 0.179 | 0.350 | -0.277
55Co($7/2^{-}$) |  55Fe($5/2_{1}^{-}$) | 2.521 | 14.546 | 1.161 | 1.196
|  55Fe($7/2_{1}^{-}$) | 2.135 | 4.581 | 0.831 | 0.733
|  55Fe($7/2_{2}^{-}$) | 2.043 | 3.313 | 0.761 | 0.610
|  55Fe($5/2_{2}^{-}$) | 1.308 | 0.039 | 0.310 | -0.457
|  55Fe($9/2_{1}^{-}$) | 1.240 | 0.016 | 0.279 | -0.530
|  55Fe($9/2_{1}^{-}$) | 1.151 | 0.003 | 0.240 | -0.615
57Co($7/2^{-}$) |  57Fe($5/2_{1}^{-}$) | 0.699 | – | 0.087 | -1.060
58Co($2^{+}$) |  58Fe($2_{1}^{+}$) | 1.497 | 0.216 | 0.407 | -0.206
|  58Fe($2_{2}^{+}$) | 0.633 | – | 0.071 | -1.148
56Ni($0^{+}$) |  56Co($1^{+}$) | 0.415 | – | 0.034 | -1.468
57Ni($3/2^{-}$) |  57Co($3/2_{1}^{-}$) | 1.886 | 1.736 | 0.722 | 0.391
|  57Co($1/2_{1}^{-}$) | 1.759 | 0.976 | 0.628 | 0.205
|  57Co($3/2_{2}^{-}$) | 1.506 | 0.224 | 0.459 | -0.166
|  57Co($5/2_{1}^{-}$) | 1.344 | 0.057 | 0.365 | -0.375
61Cu($3/2^{-}$) |  61Ni($3/2_{1}^{-}$) | 2.238 | 6.087 | 1.131 | 0.858
|  61Ni($5/2_{1}^{-}$) | 2.171 | 4.900 | 1.064 | 0.775
|  61Ni($1/2_{1}^{-}$) | 1.955 | 2.241 | 0.862 | 0.491
|  61Ni($1/2_{2}^{-}$) | 1.582 | 0.359 | 0.563 | -0.035
|  61Ni($5/2_{2}^{-}$) | 1.329 | 0.048 | 0.396 | -0.353
|  61Ni($3/2_{2}^{-}$) | 1.138 | 0.002 | 0.290 | -0.535
|  61Ni($5/2_{3}^{-}$) | 1.105 | 0.0007 | 0.273 | -0.563
|  61Ni($3/2_{3}^{-}$) | 1.052 | 0.00003 | 0.247 | -0.607
64Cu($1^{+}$) |  64Ni($0_{1}^{+}$) | 1.675 | 0.617 | 0.631 | 0.096
60Zn($0^{+}$) |  60Cu($1_{1}^{+}$) | 4.109 | 269.56 | 4.240 | 2.437
|  60Cu($1_{2}^{+}$) | 3.806 | 172.51 | 3.636 | 2.245
|  60Cu($1_{3}^{+}$) | 3.501 | 105.18 | 3.075 | 2.034
62Zn($0^{+}$) |  62Cu($1_{1}^{+}$) | 1.620 | 0.439 | 0.653 | 0.038
|  62Cu($1_{2}^{+}$) | 1.072 | 0.0002 | 0.284 | -0.547
|  62Cu($1_{3}^{+}$) | 0.983 | – | 0.238 | -0.623
65Zn($5/2^{-}$) |  65Cu($3/2_{1}^{-}$) | 1.352 | 0.059 | 0.453 | -0.291
Table 2: The theoretical excitation energies, log$ft$ values, and branching ratios of $\beta^{+}$-decays of the concerned nuclei are compared with the experimental values. Where the experimental ground state energy and parity $J^{\pi}$ of the parent nucleus is uncertain is indicated by an asterisk. In the last column, the references of the experimental data are given. |  |  | Ex. energy (keV) |  | log$ft$ value |  | Branching ($\%$) |  |
---|---|---|---|---|---|---|---|---|---
AZ${}_{i}(J^{\pi})$ |  AZ${}_{f}(J^{\pi})$ |  | Theo. | Expt. |  | Theo. | Expt. |  | Theo. | Expt. |  | Ref.
41Sc(${7/2}^{-}$) |  41Ca($7/2^{-}$) |  | 0 | 0 |  | 3.768 | 3.452 |  | 99.97 | 99.96 |  | 41sc
|  41Ca($5/2^{-}$) |  | 6500 | 2575 |  | 3.643 | 5.830 |  | 0.027 | 0.023 |  |
42Sc($0^{+}$) |  42Ca($1^{+}$) |  | 9068 | – |  | 3.969 | – |  | 37.32 | – |  |
43Sc($7/2^{-}$) |  43Ca($7/2^{-}$) |  | 0 | 0 |  | 5.240 | 5.04 |  | 69.67 | 77.54 |  | ENSDF
|  43Ca($5/2^{-}$) |  | 216 | 373 |  | 5.113 | 4.98 |  | 25.57 | 22.53 |  |
|  |  | 3425 | 1931 |  | 5.527 | 5.68 |  | 4.75 | 0.02 |  |
42Ti($0^{+}$) |  42Sc($1^{+}$) |  | 342 | 611 |  | 3.216 | 3.495 |  | 80.23 | 56.06 |  | 42ti
|  42Sc($1^{+}$) |  | 4111 | 1888 |  | 4.694 | 4.80 |  | 0.03 | 0.41 |  |
43Ti($7/2^{-}$) |  43Sc($7/2^{-}$) |  | 0 | 0 |  | 3.888 | 3.554 |  | 88.03 | 90.28 |  | 43ti
|  |  | 3115 | 1408 |  | 6.145 | 5.130 |  | 0.015 | 0.67 |  |
|  43Sc($5/2^{-}$) |  | 2162 | 845 |  | 3.833 | 4.78 |  | 11.93 | 2.6 |  |
|  |  | 3740 | 2288 |  | 6.763 | 3.85 |  | 0.001 | 4.6 |  |
|  |  | 4309 | 2335 |  | 5.589 | 4.91 |  | 0.004 | 0.38 |  |
45Ti($7/2^{-}$) |  45Sc($7/2^{-}$) |  | 0 | 0 |  | 4.598 | 4.591 |  | 89.94 | 99.69 |  | 45ti
|  |  | 2519 | 1408 |  | 5.777 | 5.78 |  | 0.12 | 0.09 |  |
|  45Sc($5/2^{-}$) |  | 1533 | 720 |  | 6.064 | 6.26 |  | 0.35 | 0.14 |  |
|  45Sc($9/2^{-}$) |  | 1727 | 1662 |  | 5.275 | 5.55 |  | 0.001 | 0.06 |  |
44V($(2)^{*+}$) |  44Ti($2^{+}$) |  | 1300 | 1083 |  | 4.497 | 4.70 |  | 70.21 | 32 |  | 44v
|  44Ti($2^{+}$) |  | 3360 | 2530 |  | 4.628 | 4.56 |  | 21.72 | 23 |  |
45V($7/2^{-}$) |  45Ti($7/2^{-}$) |  | 0 | 0 |  | 4.305 | 3.64 |  | 89.89 | 95.6 |  | 45v
|  45Ti($5/2^{-}$) |  | 12 | 40 |  | 5.701 | 5.0 |  | 3.57 | 4.3 |  |
46V($0^{+}$) |  46Ti($1^{+}$) |  | 3809 | 4315 |  | 4.953 | 5.0 |  | 6.62 | 0.01 |  | 46v
47V($3/2^{-}$) |  47Ti($5/2^{-}$) |  | 0 | 0 |  | 4.786 | 4.901 |  | 99.67 | 99.55 |  | 47v
|  |  | 1950 | 2166 |  | 5.836 | 6.25 |  | 0.02 | 0.01 |  |
|  47Ti($3/2^{-}$) |  | 1338 | 1550 |  | 6.576 | 6.08 |  | 0.02 | 0.04 |  |
|  |  | 1922 | 2163 |  | 5.294 | 5.36 |  | 0.10 | 0.07 |  |
|  |  | 2294 | 2548 |  | 5.913 | 5.77 |  | 0.01 | 0.01 |  |
|  47Ti($1/2^{-}$) |  | 1848 | 1794 |  | 5.195 | 5.10 |  | 0.15 | 0.28 |  |
|  |  | 2844 | 2793 |  | 4.812 | 5.18 |  | 0.002 | 0.003 |  |
45Cr(($7/2^{-}$)*) |  45V($7/2^{-}$) |  | 0 | 0 |  | 4.599 | – |  | 40.06 | – |  | 45cr
|  |  | 2425 | 4800 |  | 5.250 | 3.68 |  | 2.83 | 19.6 |  |
|  45V($9/2^{-}$) |  | 1565 | 1322 |  | 4.685 | – |  | 16.11 | – |  |
47Cr($3/2^{-}$) |  47V($3/2^{-}$) |  | 0 | 0 |  | 4.535 | 3.70 |  | 76.11 | 96.1 |  | 47cr
|  47V($5/2^{-}$) |  | 3 | 87 |  | 5.056 | 5.1 |  | 22.88 | 3.9 |  |
48Mn($4^{+}$) |  48Cr($4^{+}$) |  | 2700 | 1858 |  | 5.665 | 5.4 |  | 8.540 | 5.9 |  | 48mn
|  |  | 4932 | 4428 |  | 4.352 | 4.6 |  | 68.32 | 10.0 |  |
|  |  | 5394 | 5792 |  | 7.032 | 3.49 |  | 0.11 | 58.3 |  |
49Mn($5/2^{-}$) |  49Cr($5/2^{-}$) |  | 0 | 0 |  | 4.388 | 3.68 |  | 86.42 | 91.8 |  | ENSDF
|  49Cr($7/2^{-}$) |  | 280 | 272 |  | 5.109 | 4.8 |  | 13.43 | 5.8 |  |
|  |  | 2299 | 2504 |  | 6.324 | 4.3 |  | 0.14 | 2.3 |  |
50Mn($0^{+}$) |  50Cr($1^{+}$) |  | 3539 | 3628 |  | 5.004 | 5.14 |  | 6.41 | 0.056 |  | 46v
|  |  | 4814 | 4998 |  | 6.254 | 5.90 |  | 0.13 | 0.0007 |  |
51Mn($5/2^{-}$) |  51Cr($7/2^{-}$) |  | 0 | 0 |  | 5.315 | 5.297 |  | 99.73 | 99.62 |  | 51mn
|  |  | 1307 | 1557 |  | 7.099 | 7.09 |  | 0.06 | 0.007 |  |
|  |  | 2321 | 2312 |  | 5.678 | 5.662 |  | 0.05 | 0.09 |  |
|  51Cr($5/2^{-}$) |  | 1168 | 1353 |  | 7.178 | 7.32 |  | 0.06 | 0.05 |  |
|  |  | 1971 | 2001 |  | 6.214 | 6.314 |  | 0.05 | 0.03 |  |
48Fe($0^{+}$) |  48Mn($1^{+}$) |  | 90 | 403 |  | 4.085 | 3.9 |  | 33.75 | 42 |  | 48fe
|  |  | 1702 | 3204 |  | 5.033 | 4.8 |  | 2.10 | 1.0 |  |
|  |  | 2123 | 3495 |  | 4.056 | 4.5 |  | 16.9 | 1.8 |  |
|  |  | 2823 | 3619 |  | 4.197 | 4.7 |  | 9.13 | 0.9 |  |
|  |  | 3187 | 3713 |  | 6.974 | 4.6 |  | 0.01 | 1.3 |  |
|  |  | 3291 | 4299 |  | 4.986 | 4.4 |  | 1.20 | 1.2 |  |
|  |  | 3523 | 4399 |  | 4.626 | 4.5 |  | 2.49 | 0.9 |  |
|  |  | 3728 | 4517 |  | 4.404 | 4.3 |  | 3.77 | 1.3 |  |
|  |  | 3931 | 4755 |  | 4.374 | 4.4 |  | 3.68 | 0.8 |  |
49Fe(($7/2^{-}$)*) |  49Mn($5/2^{-}$) |  | 0 | 0 |  | 5.985 | – |  | 4.79 | – |  | 49fe
|  |  | 1873 | 3959 |  | 5.349 | 5.97 |  | 9.10 | 1.2 |  |
Table 2: Continuation. |  |  | Ex. energy (keV) |  | log$ft$ value |  | Branching ($\%$) |  |
---|---|---|---|---|---|---|---|---|---
AZ${}_{i}(J^{\pi})$ |  AZ${}_{f}(J^{\pi})$ |  | Theo. | Expt. |  | Theo. | Expt. |  | Theo. | Expt. |  | Ref.
|  49Mn($7/2^{-}$) |  | 279 | 261 |  | 4.900 | – |  | 52.02 | – |  |
|  |  | 2298 | 4381 |  | 7.576 | 5.83 |  | 0.04 | 1.4 |  |
|  |  | 2525 | 4814 |  | 6.923 | 4.36 |  | 0.18 | 34.5 |  |
50Fe($0^{+}$) |  50Mn($1^{+}$) |  | 426 | 651 |  | 3.967 | 3.81 |  | 39.63 | 22.5 |  | 50fe
|  |  | 2415 | 2403 |  | 4.352 | 4.36 |  | 3.53 | 1.47 |  |
|  |  | 3119 | 2684 |  | 4.543 | 4.56 |  | 1.15 | 0.70 |  |
|  |  | 3302 | 3380 |  | 4.401 | 4.14 |  | 1.32 | 0.84 |  |
|  |  | 3539 | 3643 |  | 5.002 | 4.80 |  | 0.25 | 0.15 |  |
|  |  | 3787 | 4012 |  | 5.245 | 5.10 |  | 0.10 | 0.04 |  |
|  |  | 3898 | 4316 |  | 4.114 | 4.60 |  | 1.28 | 0.08 |  |
51Fe($5/2^{-}$) |  51Mn($5/2^{-}$) |  | 0 | 0 |  | 4.323 | 3.653 |  | 82.69 | 93.7 |  | 51fe
|  51Mn($7/2^{-}$) |  | 265 | 237 |  | 5.074 | 4.86 |  | 12.26 | 5.0 |  |
|  51Mn($3/2^{-}$) |  | 1854 | 1825 |  | 5.627 | 5.32 |  | 0.98 | 0.49 |  |
|  |  | 2063 | 2140 |  | 5.323 | 5.51 |  | 1.63 | 0.24 |  |
|  |  | 2943 | 2914 |  | 5.189 | 5.54 |  | 1.50 | 0.10 |  |
|  |  | 3301 | 3555 |  | 5.185 | 5.00 |  | 0.91 | 0.16 |  |
53Fe($7/2^{-}$) |  53Mn($7/2^{-}$) |  | 0 | 0 |  | 5.132 | 5.22 |  | 72.27 | 55.95 |  | 53fe
|  |  | 2409 | 2685 |  | 7.877 | 5.10 |  | 0.0002 | 0.01 |  |
|  53Mn($5/2^{-}$) |  | 385 | 378 |  | 5.276 | 5.06 |  | 26.84 | 42.09 |  |
|  |  | 2055 | 2273 |  | 6.321 | 4.90 |  | 0.03 | 0.38 |  |
|  |  | 2937 | 3126 |  | 6.515 | 4.50 |  | 0.002 | 0.14 |  |
|  53Mn($9/2^{-}$) |  | 1815 | 1619 |  | 6.974 | 5.30 |  | 0.01 | 1.0 |  |
|  |  | 3007 | 2946 |  | 6.673 | 5.10 |  | 0.002 | 0.05 |  |
|  |  | 3465 | 3248 |  | 5.933 | 4.80 |  | 0.001 | 0.04 |  |
50Co(($6^{+}$)*) |  50Fe($6^{+}$) |  | 3151 | 3159 |  | 7.032 | 4.78 |  | 0.87 | 15 |  | 50co
|  |  | 3739 | 8458 |  | 4.889 | 3.32 |  | 99.12 | 42.1 |  |
52Co(($6^{+}$)*) |  52Fe($6^{+}$) |  | 4394 | 4326 |  | 4.331 | – |  | 58.02 | – |  | 52co
|  |  | 4936 | 5655 |  | 4.355 | 3.4 |  | 41.97 | – |  |
53Co(($7/2^{-}$)*) |  53Fe($7/2^{-}$) |  | 0 | 0 |  | 4.266 | 3.62 |  | 87.42 | 94.49 |  | 53co
|  53Fe($9/2^{-}$) |  | 1459 | 1328 |  | 4.652 | 4.44 |  | 12.57 | 5.6 |  |
55Co($7/2^{-}$) |  55Fe($5/2^{-}$) |  | 946 | 931 |  | 5.923 | 6.25 |  | 67.22 | 51.6 |  | ENSDF
|  |  | 2110 | 2144 |  | 6.346 | 6.69 |  | 0.56 | 0.55 |  |
|  |  | 2688 | 2578 |  | 7.877 | 7.44 |  | 0.05 | 0.04 |  |
|  55Fe($7/2^{-}$) |  | 1398 | 1316 |  | 6.114 | 6.725 |  | 14.95 | 5.6 |  |
|  |  | 1692 | 1408 |  | 5.979 | 5.785 |  | 15.31 | 36.3 |  |
|  55Fe($9/2^{-}$) |  | 2314 | 2212 |  | 5.937 | 6.110 |  | 1.22 | 1.86 |  |
|  |  | 2592 | 2301 |  | 6.309 | 5.780 |  | 0.17 | 3.4 |  |
57Co($7/2^{-}$) |  57Fe($5/2^{-}$) |  | 0 | 136 |  | 6.286 | 6.450 |  | 77.25 | 99.80 |  | ENSDF
|  |  | 981 | 706 |  | 7.877 | 7.70 |  | 22.74 | 0.17 |  |
58Co($2^{+}$) |  58Fe($2^{+}$) |  | 802 | 810 |  | 6.386 | 6.61 |  | 98.19 | 98.8 |  | ENSDF
|  |  | 1789 | 1674 |  | 7.178 | 7.69 |  | 1.80 | 1.21 |  |
54Ni($0^{+}$) |  54Co($1^{+}$) |  | 871 | 936 |  | 3.899 | 3.84 |  | 46.07 | 19.8 |  | 54ni
|  |  | 2075 | 2424 |  | 5.397 | 5.43 |  | 0.66 | 0.16 |  |
|  |  | 2715 | 3376 |  | 5.555 | 4.67 |  | 0.28 | 0.37 |  |
|  |  | 3934 | 3889 |  | 5.226 | 4.43 |  | 0.21 | 0.37 |  |
|  |  | 4547 | 4293 |  | 5.250 | 4.46 |  | 0.10 | 0.21 |  |
|  |  | 4824 | 4323 |  | 4.500 | 4.70 |  | 0.42 | 0.11 |  |
|  |  | 4993 | 4543 |  | 4.912 | 4.46 |  | 0.13 | 0.15 |  |
|  |  | 5095 | 4822 |  | 3.994 | 4.37 |  | 0.97 | 0.12 |  |
|  |  | 5323 | 5202 |  | 7.275 | 4.88 |  | 0.0003 | 0.02 |  |
56Ni($0^{+}$) |  56Co($1^{+}$) |  | 1299 | 1720 |  | 4.308 | 4.40 |  | 100 | 100 |  | 56ni
57Ni($3/2^{-}$) |  57Co($3/2^{-}$) |  | 1815 | 1377 |  | 5.583 | 5.64 |  | 75.90 | 64.5 |  | ENSDF
|  |  | 2043 | 1757 |  | 6.372 | 6.22 |  | 3.42 | 5.66 |  |
|  57Co($1/2^{-}$) |  | 1967 | 1504 |  | 5.991 | 6.05 |  | 19.34 | 17.04 |  |
|  57Co($5/2^{-}$) |  | 1879 | 1919 |  | 6.576 | 5.74 |  | 3.42 | 12.3 |  |
|  |  | 2356 | 2133 |  | 6.415 | 8.13 |  | 0.001 | 0.03 |  |
57Cu($3/2^{-}$) |  57Ni($3/2^{-}$) |  | 0 | 0 |  | 3.655 | 3.67 |  | 83.36 | 89.9 |  | 57cu
|  57Ni($5/2^{-}$) |  | 1119 | 768 |  | 8.877 | 5.44 |  | 0.0002 | 0.94 |  |
|  57Ni($1/2^{-}$) |  | 1989 | 1112 |  | 3.752 | 4.37 |  | 16.63 | 8.6 |  |
58Cu($1^{+}$) |  58Ni($0^{+}$) |  | 0 | 0 |  | 3.735 | 4.87 |  | 96.39 | 81.2 |  | 58cu
|  |  | 3004 | 2943 |  | 4.391 | 4.77 |  | 1.99 | 10.1 |  |
Table 2: Continuation. |  |  | Ex. energy (keV) |  | log$ft$ value |  | Branching ($\%$) |  |
---|---|---|---|---|---|---|---|---|---
AZ${}_{i}(J^{\pi})$ |  AZ${}_{f}(J^{\pi})$ |  | Theo. | Expt. |  | Theo. | Expt. |  | Theo. | Expt. |  | Ref.
|  |  | 4296 | 3532 |  | 4.926 | 6.64 |  | 0.004 | 0.07 |  |
|  58Ni($2^{+}$) |  | 1298 | 1454 |  | 5.728 | 6.20 |  | 0.41 | 1.4 |  |
|  |  | 2969 | 2776 |  | 4.837 |  $>$6.4 |  | 0.74 |  $<$0.28 |  |
|  |  | 3856 | 3038 |  | 4.888 | 6.23 |  | 0.24 | 0.32 |  |
|  58Ni($1^{+}$) |  | 3309 | 2902 |  | 7.877 | 5.13 |  | 0.001 | 4.6 |  |
59Cu($3/2^{-}$) |  59Ni($3/2^{-}$) |  | 0 | 0 |  | 5.168 | 5.0 |  | 64.04 | 58.05 |  | 59cu
|  |  | 1169 | 878 |  | 5.154 | 5.3 |  | 12.69 | 8.78 |  |
|  |  | 2603 | 1734 |  | 5.963 | 5.3 |  | 0.08 | 1.99 |  |
|  |  | 2950 | 2414 |  | 5.171 | 5.5 |  | 0.17 | 0.20 |  |
|  59Ni($5/2^{-}$) |  | 542 | 339 |  | 6.169 | 5.8 |  | 3.17 | 5.89 |  |
|  |  | 1644 | 1188 |  | 6.057 | 7.0 |  | 0.66 | 0.11 |  |
|  |  | 2312 | 1679 |  | 6.099 | 5.4 |  | 0.13 | 1.61 |  |
|  |  | 3373 | 2681 |  | 5.731 | 6.0 |  | 0.02 | 0.03 |  |
|  59Ni($1/2^{-}$) |  | 672 | 464 |  | 7.032 | 6.0 |  | 0.36 | 3.39 |  |
|  |  | 1867 | 1301 |  | 4.406 | 4.7 |  | 18.67 | 19.5 |  |
60Cu($2^{+}$) |  60Ni($2^{+}$) |  | 1635 | 1332 |  | 6.129 | 7.3 |  | 10.27 | 5.0 |  | 60cu
|  |  | 2141 | 2158 |  | 5.442 | 6.4 |  | 30.90 | 15.3 |  |
|  |  | 3078 | 3124 |  | 4.925 | 5.1 |  | 36.04 | 52.34 |  |
|  |  | 3456 | 3269 |  | 5.813 | 6.0 |  | 2.85 | 4.99 |  |
|  60Ni($3^{+}$) |  | 2505 | 2626 |  | 5.567 | 6.8 |  | 15.88 | 2.8 |  |
61Cu($3/2^{-}$) |  61Ni($3/2^{-}$) |  | 80 | 0 |  | 5.207 | 5.07 |  | 45.10 | 67 |  | 61cu
|  |  | 1290 | 1099 |  | 5.612 | 5.86 |  | 0.71 | 0.68 |  |
|  |  | 1840 | 1185 |  | 6.114 | 5.00 |  | 0.19 | 4.2 |  |
|  61Ni($5/2^{-}$) |  | 0 | 67 |  | 6.129 | 6.35 |  | 4.46 | 2.9 |  |
|  |  | 1277 | 908 |  | 6.555 | 5.72 |  | 0.12 | 1.30 |  |
|  |  | 1287 | 1132 |  | 8.877 | 6.49 |  | 0.003 | 0.15 |  |
|  61Ni($1/2^{-}$) |  | 592 | 283 |  | 4.860 | 5.53 |  | 43.05 | 8.1 |  |
|  |  | 1523 | 656 |  | 4.725 | 4.95 |  | 5.33 | 13.3 |  |
62Cu($1^{+}$) |  62Ni($0^{+}$) |  | 0 | 0 |  | 4.958 | 5.158 |  | 99.30 | 99.59 |  | 62cu
|  |  | 2192 | 2048 |  | 6.731 | 6.00 |  | 0.01 | 0.08 |  |
|  62Ni($2^{+}$) |  | 1820 | 1172 |  | 5.459 | 7.03 |  | 0.62 | 0.14 |  |
|  |  | 2445 | 2301 |  | 5.974 | 5.98 |  | 0.03 | 0.02 |  |
64Cu($1^{+}$) |  64Ni($0^{+}$) |  | 0 | 0 |  | 5.004 | 4.969 |  | 93.46 | 99.23 |  | 64cu
|  64Ni($2^{+}$) |  | 1637 | 1345 |  | 5.805 | 5.504 |  | 0.003 | 0.76 |  |
58Zn($0^{+}$) |  58Cu($1^{+}$) |  | 0 | 0 |  | 3.258 | 4.1 |  | 80.36 | 18 |  | 58zn
|  |  | 1476 | 1051 |  | 4.424 | 4.1 |  | 2.19 | 10 |  |
59Zn($3/2^{-}$) |  59Cu($3/2^{-}$) |  | 0 | 0 |  | 3.774 | 3.698 |  | 61.71 | 94.1 |  | 59zn
|  |  | 3336 | 4773 |  | 5.657 | 6.5 |  | 0.07 | 0.002 |  |
|  59Cu($1/2^{-}$) |  | 422 | 491 |  | 3.887 | 4.86 |  | 36.95 | 4.8 |  |
|  |  | 3416 | 4347 |  | 5.285 | 6.09 |  | 0.15 | 0.01 |  |
|  59Cu($5/2^{-}$) |  | 1655 | 914 |  | 5.899 | 5.39 |  | 0.16 | 1.1 |  |
60Zn($0^{+}$) |  60Cu($1^{+}$) |  | 476 | 62 |  | 4.901 | 5.3 |  | 55.53 | 20.4 |  | 60zn
|  |  | 549 | 364 |  | 5.023 | 5.9 |  | 25.76 | 3.1 |  |
|  |  | 971 | 670 |  | 4.461 | 4.4 |  | 18.64 | 72.8 |  |
61Zn($3/2^{-}$) |  61Cu($3/2^{-}$) |  | 0 | 0 |  | 5.207 | 5.40 |  | 77.57 | 66.7 |  | 61zn
|  |  | 1650 | 1660 |  | 5.647 | 5.30 |  | 3.82 | 10.80 |  |
|  |  | 2024 | 1932 |  | 6.071 | 6.25 |  | 0.79 | 0.79 |  |
|  |  | 2379 | 2358 |  | 5.701 | 5.70 |  | 0.98 | 1.29 |  |
|  |  | 2605 | 2472 |  | 6.224 | 5.60 |  | 0.19 | 1.29 |  |
|  61Cu($5/2^{-}$) |  | 1397 | 970 |  | 5.923 | 7.10 |  | 2.91 | 0.46 |  |
|  |  | 1669 | 1394 |  | 6.038 | 6.69 |  | 1.51 | 0.66 |  |
|  |  | 1952 | 1904 |  | 5.832 | 6.80 |  | 1.55 | 0.23 |  |
62Zn($0^{+}$) |  62Cu($1^{+}$) |  | 159 | 0 |  | 5.212 | 4.99 |  | 61.0 | 40.2 |  | 62zn
63Zn($3/2^{-}$) |  63Cu($3/2^{-}$) |  | 0 | 0 |  | 5.446 | 5.40 |  | 96.13 | 84.04 |  | ENSDF
|  |  | 1511 | 1547 |  | 5.918 | 6.70 |  | 0.77 | 0.04 |  |
|  63Cu($1/2^{-}$) |  | 824 | 669 |  | 6.647 | 5.82 |  | 1.02 | 7.92 |  |
|  63Cu($5/2^{-}$) |  | 1635 | 962 |  | 5.877 | 5.61 |  | 0.60 | 6.08 |  |
|  |  | 1875 | 1412 |  | 6.297 | 5.87 |  | 0.12 | 0.7 |  |
65Zn($5/2^{-}$) |  65Cu($3/2^{-}$) |  | 0 | 0 |  | 5.895 | 7.450 |  | 28.04 | 49.96 |  | ENSDF
|  65Cu($5/2^{-}$) |  | 1569 | 1115 |  | 5.777 | 5.893 |  | 71.95 | 50.04 |  |

Figure 2:  The theoretical and experimental $\beta^{+}$/EC- decay half-life
versus mass number A of the concerned nuclei for $fp$ space.

Figure 3:  The theoretical and experimental $\beta^{+}$/EC- decay half-life versus mass number A of the concerned nuclei for $f_{5/2}pg_{9/2}$ space. Table 3: List of superallowed transitions with $0^{+}$ $\rightarrow$ $0^{+}$. The theoretical log$ft$ values and branching ratios of $\beta^{+}$/EC decay of the concerned nuclei are compared with the experimental values. The experimental ground state energy and parity $J^{\pi}$ of the parent and daughter nucleus are listed along with $Q$ values. In the last column the references of the experimental data are given. $J_{P}^{\pi}$ ( $J_{D}^{\pi}$) and $E_{P}$ ($E_{D}$) are spin-parity and excitation energy of parent (daughter) nuclei, respectively.  |  |  |  |  |  |  | log$ft$ |  | Branch (%) |
---|---|---|---|---|---|---|---|---|---|---
Nuclide | Decay | Q(keV) |  $E_{P}$(keV) | $J_{P}^{\pi}$ |  $E_{D}$(keV) | $J_{D}^{\pi}$ | Theo. | Expt. |  | Theo. | Expt. | Ref.
42Sc |  $\beta^{+}$/EC | 6426.1 | 0.0 | $0^{+}$ | 0.0 | $0^{+}$ | 3.493 | 3.485 |  | 58.02 | 99.98 | 42sc
42Ti |  $\beta^{+}$/EC | 7016.48 | 0.0 | $0^{+}$ | 0.0 | $0^{+}$ | 3.497 | 3.495 |  | 19.73 | 47.74 | 42ti
46V |  $\beta^{+}$/EC | 7051.4 | 0.0 | $0^{+}$ | 0.0 | $0^{+}$ | 3.497 | 3.484 |  | 87.8 | 99.97 | 46v
50Mn |  $\beta^{+}$/EC | 7634.48 | 0.0 | $0^{+}$ | 0.0 | $0^{+}$ | 3.497 | 3.485 |  | 93.44 | 99.93 | 46v
48Fe |  $\beta^{+}$/EC | 11290 | 0.0 | $0^{+}$ | 3036.8 | $0^{+}$ | 3.196 | 3.300 |  | 26.94 | 35.03 | 48fe
50Fe |  $\beta^{+}$/EC | 8151 | 0.0 | $0^{+}$ | 0.0 | $0^{+}$ | 3.497 | 3.490 |  | 52.7 | 74.17 | 50fe
54Ni |  $\beta^{+}$/EC | 8790 | 0.0 | $0^{+}$ | 0.0 | $0^{+}$ | 3.493 | 3.501 |  | 51.13 | 79.17 | 54ni
58Zn |  $\beta^{+}$/EC | 9364 | 0.0 | $0^{+}$ | 203 | $0^{+}$ | 3.497 | 3.486 |  | 17.44 | 72.08 | 58zn
Table 4:  The theoretical (SM) results of $\beta$+-decay half-lives for the concerned nuclei are compared with the experimental data, experimental $Q$ values, $I_{\beta^{+}}$ \+ $I_{\epsilon}$ \- decay probabilities and theoretical quenched $\sum$ $B(GT)$ values are reported in this table. |  |  $Q$ value |  | Sum | Half-life | $I_{\beta^{+}}+I_{\epsilon}$
---|---|---|---|---|---|---
AZ${}_{i}(J^{\pi})$ |  AZf | (keV) | $B(GT)$ | Theo. | Expt. |  | $\%$
41Sc(${7/2}^{-}$) |  41Ca | 6495.48$\pm$16 | 2.436(2.110) | 1213.8(1402.04) ms | 596.3$\pm$17 ms41sc  |  | 100
42Sc($0^{+}$) |  42Ca | 6426.10$\pm$10 | 0.485(0.420) | 822.2(950.4) ms | 680.79$\pm$28 ms42sc  |  | 100
43Sc($7/2^{-}$) |  43Ca | 2220.7$\pm$19 | 0.064(0.055) | 4.4(5.1) h | 3.89$\pm$12 h43sc  |  | 100
42Ti($0^{+}$) |  42Sc | 7016.48$\pm$22 | 2.528(2.188) | 181.8(210.1) ms | 208.65$\pm$80 ms42ti  |  | 100
43Ti($7/2^{-}$) |  43Sc | 6867$\pm$7 | 1.892(1.638) | 1054.7(1218.2) ms | 509$\pm$5 ms43ti  |  | 100
45Ti($7/2^{-}$) |  45Sc | 2062.1$\pm$5 | 0.517(0.448) | 132.4(153.0) min | 184.8$\pm$5 min45ti  |  | 100
44V($2^{+}$) |  44Ti | 1.343$\times 10^{4}$$\pm$12 | 0.804(0.696) | 98(113.2) ms | 111$\pm$7 ms44v  |  | 100
45V($7/2^{-}$) |  45Ti | 7126$\pm$17 | 0.256(0.221) | 2335(2697.1) ms | 547$\pm$6 ms45v  |  | 100
46V($0^{+}$) |  46Ti | 7051.4$\pm$10 | 0.183(0.159) | 810.1(935.7) ms | 422.50$\pm$11 ms46v  |  | 100
47V($3/2^{-}$) |  47Ti | 2930.34$\pm$30 | 0.401(0.347) | 23.9(27.6) min | 32.6$\pm$3 min47v  |  | 100
45Cr($7/2^{-}$) |  45V | 1.291$\times 10^{4}$$\pm$50 | 0.401(0.347) | 110.4(127.5) ms | 60.9$\pm$4 ms45cr  |  | 100
47Cr($3/2^{-}$) |  47V | 7444$\pm$14 | 0.160(0.138) | 2688.7(3105.6) ms | 500$\pm$15 ms47cr  |  | 100
48Mn($4^{+}$) |  48Cr | 13525$\pm$10 | 0.320(0.277) | 176.5(238.9) ms | 157.7$\pm$22 ms48mn  |  | 100
49Mn($5/2^{-}$) |  49Cr | 7715$\pm$24 | 0.192(0.166) | 1848.1(2134.7) ms | 382$\pm$7 msENSDF  |  | 100
50Mn($0^{+}$) |  50Cr | 7634.48$\pm$7 | 0.041(0.035) | 592.5(684.3) ms | 283.19$\pm$10 ms46v  |  | 100
51Mn($5/2^{-}$) |  51Cr | 3207.5$\pm$3 | 0.033(0.028) | 47.3(54.6) min | 46.2$\pm$1 min51mn  |  | 100
48Fe($0^{+}$) |  48Mn | 11290$\pm$90 | 1.397(1.209) | 11.1(12.7) ms | 45.5$\pm$8 ms48fe  |  | 100
49Fe($7/2^{-}$) |  49Mn | 16895$\pm$73 | 0.204(0.177) | 267.8(309.4) ms | 64.7$\pm$3 ms49fe  |  | 100
50Fe($0^{+}$) |  50Mn | 8151$\pm$8 | 1.220(1.056) | 241.7(279.2) ms | 152.0$\pm$6 ms50fe  |  | 100
51Fe($5/2^{-}$) |  51Mn | 8041$\pm$9 | 0.332(0.287) | 1221.1(1410.4) ms | 305$\pm$2 ms51fe  |  | 100
53Fe($7/2^{-}$) |  53Mn | 3742.6$\pm$17 | 0.058(0.050) | 9.3(10.7) min | 8.51$\pm$2 min53fe  |  | 100
50Co($6^{+}$) |  50Fe | 16895$\pm$73 | 0.051(0.040) | 255.3(294.4) ms | 38.8$\pm$2 ms50co  |  | 100
52Co($6^{+}$) |  52Fe | 14340 | 0.354(0.306) | 163.1(188.4) ms | 104$\pm$7 ms52co  |  | 100
53Co($7/2^{-}$) |  53Fe | 8300$\pm$18 | 0.298(0.258) | 977.6(1129.2) ms | 240$\pm$20 ms53co  |  | 100
55Co($7/2^{-}$) |  55Fe | 3451.8$\pm$4 | 0.021(0.018) | 9.9(11.5) h | 17.53$\pm$3 hENSDF  |  | 100
57Co($7/2^{-}$) |  57Fe | 836.0$\pm$4 | 0.002(0.001) | 198.5(229.3) d | 271.74$\pm$6 dENSDF  |  | 100
58Co($2^{+}$) |  58Fe | 2307.6$\pm$12 | 0.002(0.001) | 44.4(51.3) d | 70.86$\pm$6 dENSDF  |  | 100
54Ni($0^{+}$) |  54Co | 8.79$\times 10^{3}$$\pm$5 | 1.128(0.976) | 164.1(189.5) ms | 114.2$\pm$3 ms54ni  |  | 100
56Ni($0^{+}$) |  56Co | 2136$\pm$12 | 0.192(0.165) | 0.05(0.06) d | 6.07 $\pm$10 dENSDF  |  | 100
57Ni($3/2^{-}$) |  57Co | 3264.2$\pm$26 | 0.029(0.025) | 32.8(37.9) h | 35.60$\pm$6 hENSDF  |  | 100
57Cu($3/2^{-}$) |  57Ni | 8770$\pm$16 | 1.656(1.769) | 173.2(151.8) ms | 196.3$\pm$7 ms57cu  |  | 100
58Cu($1^{+}$) |  58Ni | 8565.6$\pm$14 | 1.216(1.298) | 0.3(0.2) s | 3.204$\pm$7 s58cu  |  | 100
59Cu($3/2^{-}$) |  59Ni | 4798.4$\pm$4 | 0.271(0.289) | 121.2(106.2) s | 81.5$\pm$5 s59cu  |  | 100
60Cu($2^{+}$) |  60Ni | 6128.0$\pm$16 | 0.095(0.101) | 0.75(0.65) min | 23.7$\pm$4 min60cu  |  | 100
61Cu($3/2^{-}$) |  61Ni | 2237.8$\pm$10 | 0.194(0.207) | 2.8(2.4) h | 3.336$\pm$10 h61cu  |  | 100
62Cu($1^{+}$) |  62Ni | 3958.90$\pm$48 | 0.094(0.101) | 5.98(5.24) min | 9.67$\pm$3 min62cu  |  | 100
64Cu($1^{+}$) |  64Ni | 1674.62$\pm$21 | 0.051(0.055) | 21.0(18.4) h | 12.7006$\pm$20 h64cu  |  | 100
58Zn($0^{+}$) |  58Cu | 9364$\pm$50 | 2.451(2.620) | 44.8(42.0) ms | 86$\pm$8 ms58zn  |  | 100
59Zn($3/2^{-}$) |  59Cu | 9142.8$\pm$6 | 1.425(1.522) | 137.3(120.3) ms | 178.6$\pm$18 ms59zn  |  | 100
60Zn($0^{+}$) |  60Cu | 4170.8$\pm$16 | 0.235(0.252) | 4.2(3.7) min | 2.38$\pm$5 min60zn  |  | 100
61Zn($3/2^{-}$) |  61Cu | 5635$\pm$16 | 0.145(0.155) | 65.5(57.37) s | 89.1$\pm$2 s61zn  |  | 100
62Zn($0^{+}$) |  62Cu | 1619.5$\pm$7 | 0.152(0.163) | 15.1(13.3) h | 9.193$\pm$15 h62zn  |  | 100
63Zn($3/2^{-}$) |  63Cu | 3366.5$\pm$16 | 0.093(0.100) | 48.2(42.2) min | 38.47$\pm$5 minENSDF  |  | 100
65Zn($5/2^{-}$) |  65Cu | 1352.1$\pm$3 | 0.012(0.013) | 4.9(4.4) d | 243.93$\pm$9 dENSDF  |  | 100

##  4 Results and discussions

The computed Phase Space Factors (PSF) for $\beta^{+}$/EC decay along with
log$(f^{(+)}_{0}+f^{(EC)}_{0})$ values are given in table 1. For higher $Q$
value, the relation $f^{(+)}_{0}$$\gg$$f^{(EC)}_{0}$ and the half-life of a
$\beta^{+}$/EC transition is determined by the $\beta^{+}$ decay. For small
decay energies the electron-capture dominates as reflected from the table. We
have calculated $f^{(+)}_{0}$ and $f^{(EC)}_{0}$ for small decay energies and
compared them in table 1. We considered the calculated
log$(f^{(+)}_{0}+f^{(EC)}_{0})$ values using Eq. 7 in our shell model
calculations for 13 nuclei (43Sc, 45Ti, 53Fe, 55Co, 57Co, 58Co, 56Ni, 57Ni,
61Cu, 64Cu, 60Zn, 62Zn and 65Zn) where $f^{(EC)}_{0}$ were significant. The
shell model results of half-lives and branching ratios are improved for these
nuclei after including the dominant EC phase space factor in our calculations.

The theoretical shell-model (SM) results of excitation energies as well as
$\beta$-decay properties such as log$ft$ values, and branching percentages of
the concerned nuclei are compared with the experimental data in table 2.
Columns 3 and 4 represent the theoretical and experimental excitation
energies, respectively of each state associated with the $\beta^{+}$ \-
decays. Columns 5 and 6 represent quenched theoretical and experimental
log$ft$ values, respectively. The theoretical log$ft$ values are quenched by
quenching factor q = 0.719 for $fp$ space and q = 0.743 for $f_{5/2}pg_{9/2}$
space, respectively. The theoretical and experimental branching fractions are
listed in columns 7 and 8, respectively. The theoretical results of excitation
energies, log$ft$ values, and the branching ratios are in reasonable agreement
with the experimental data.

Table 3 represents the comparison of theoretical and experimental
$\beta$-decay half-lives of concerned nuclei. The experimental $\beta$-decay
$Q$ values, $I_{\beta^{+}}$+$I_{\epsilon}$ decay probabilities and theoretical
quenched $\sum B(GT)$ values are also reported. The first and second columns
represent parent and daughter nuclei, respectively. The experimental $Q$
values are taken from ENSDF  and listed in column 3. The theoretical quenched
$\sum{B(GT)}$ values are presented in column 4. In column 4, the quenched
$\sum{B(GT)}$ values are reported by quenching factor q = 0.719 and 0.743 for
$fp$ and $f_{5/2}pg_{9/2}$ space, respectively, while in the small bracket,
the theoretical results are quenched by quenching factor q = 0.669 and 0.768
for $fp$ and $f_{5/2}pg_{9/2}$ space, respectively. The quenched $\sum{B(GT)}$
values are smaller with q = 0.669 than 0.719 thus the quenched half-life with
q = 0.669 are a little bit larger than q = 0.719 for $fp$ space. In the case
of $f_{5/2}pg_{9/2}$ space, the quenched $\sum{B(GT)}$ values with quenching
factor q = 0.768 are larger than q = 0.743 thus the quenched half-lives with q
= 0.768 are a little bit smaller than q = 0.743 and are in general closer to
the experiment.

Columns 5 and 6 present theoretical and experimental $\beta$-decays half-
lives, respectively. In column 5, the quenched half-lives are reported by
quenching factor q = 0.719 and 0.743 for $fp$ and $f_{5/2}pg_{9/2}$ space,
respectively, while in the small bracket, the theoretical results are quenched
by quenching factor q = 0.669 and 0.768 for $fp$ and $f_{5/2}pg_{9/2}$ space,
respectively. Overall the quenched half-life results with q = 0.719 are closer
to the experimental data for $fp$ space while in case of $f_{5/2}pg_{9/2}$
space, the results with q = 0.768 are closer to the experimental data. The
experimental $I_{\beta^{+}}$+$I_{\epsilon}$ decay probabilities are presented
in the last column. In our present work the ground state spin parity $J^{\pi}$
of parent nuclei is determined using shell model for nuclei where the
experiment ground state $J^{\pi}$ is not confirmed, it is indicated by * in
table 2.

For 41Sc, 43Sc, 42Ti nuclei experimental data are available in refs. 41sc ;
43sc ; 42ti  and the experimental $\beta^{+}$-decay half-lives for these
nuclei are found to be 596.3 $\pm$ 17 ms, 3.891 $\pm$ 12 h, and 208.65 $\pm$
80 ms, respectively, whereas the SM results with q = 0.719 are 1213.8 ms, 4.4
h and 181.8 ms, respectively. The quenched SM results are closer to the
observed values. In the case of 43Sc, the $f^{(EC)}_{0}$ values for the
transitions at $7/2^{-}$ (0 MeV) and $5/2^{-}$ (0.373 MeV) are significant
compared to $f^{(+)}_{0}$. The quenched half life is 4.4 h with q = 0.719
using computed log$(f^{(+)}_{0}+f^{(EC)}_{0})$ values, where as without
$f^{(EC)}_{0}$ the half-life is 4.6 h, also the branching ratios shifted
towards experimental value after including $f^{(EC)}_{0}$ in calculation.

The half-life of 43Ti was found to be 509$\pm$5 ms in ref. 43ti  which is
determined from the high-energy $\gamma$-rays of mass-separated samples, while
the calculated shell model result is 1054.7 ms with q = 0.719. For 45Ti, the
$f^{(EC)}_{0}$ is significant at the transitions $7/2^{-}$ (0 MeV), $5/2^{-}$
(0.543 MeV), $5/2^{-}$ (0.720 MeV) and $7/2^{-}$ (1.408 MeV) with
log$(f^{(+)}_{0}+f^{(EC)}_{0})$ values 0.652, -0.287, -0.614, and -0.932,
respectively. The predicted half-life is 132.4 min with quenching value 0.719.
The branching ratios are also improved.

The 48,49Fe and 50Co nuclei are proton-rich nuclei 44v  which are produced by
fragmentation of a 58Ni beam at 650 MeV/u with the GSI Projectile-Fragment
Separator FRS. The theoretical quenched $\beta^{+}$-decay half-lives, quenched
log$ft$ values and branching ratios of these nuclei are also calculated and
compared with the experimental data. There are five transitions in 53Fe at
$7/2^{-}$ (0 MeV), $5/2^{-}$ (0.378 MeV), $9/2^{-}$ (1.619 MeV), and $5/2^{-}$
(2.273 MeV) where the EC decay is significant. The SM result of half-life is
9.3 min with q = 0.719 which is close to the experimental value 8.51 min.

In the case of 42Sc $\rightarrow$ 42Ca, 42Ti $\rightarrow$ 42Sc, 46V
$\rightarrow$ 46Ti, 50Mn $\rightarrow$ 50Cr, 48Fe $\rightarrow$ 48Mn, 50Fe
$\rightarrow$ 50Mn, 54Ni $\rightarrow$ 54Co, and 58Zn $\rightarrow$ 58Cu
transitions, the Fermi matrix elements are non-zero. So, we reported the
superallowed Fermi decay for $0^{+}$ $\rightarrow$ $0^{+}$ transition for
these nuclei and compared the calculated log$ft$ values and branching ratios
with experimental data in table 3. The calculated half-lives and branching
ratios of these nuclei improved after including the Fermi transition. There
are notable differences observed between calculated and experimental log$ft$
values for 53Fe, 55Co and 60Zn decay, probably the results may be improved if
we take extended model space such as $fpg_{9/2}$. There are six transitions
observed in 55Co at $5/2^{-}$ (0.931 MeV), $7/2^{-}$ (1.316 MeV), $7/2^{-}$
(1.408 MeV), $5/2^{-}$ (2.144 MeV), $9/2^{-}$ (2.212 MeV), and $9/2^{-}$
(2.301 MeV) in which the EC branching ratios are significant. Our calculations
in table 1 also support that the $f^{(EC)}_{0}$ is significant for these
transitions. After using the calculated log$(f^{(+)}_{0}+f^{(EC)}_{0})$ values
the SM result of half-life is 9.9 h with q = 0.719, where as without the
$f^{(EC)}_{0}$, the SM result is 1.36 h. In case of 57Co, 99.8$\%$ EC
transition is observed at $5/2^{-}_{1}$ (0.136 MeV) with half-life 271.7 d.
The calculated $f^{(EC)}_{0}$ $>$ $f^{(+)}_{0}$ and the calculated SM results
with quenching factor 0.719 is 275.1 d which is close to the experimental
value 271.74 d. In case of 58Co the dominant EC transitions are observed, our
calculations also show that $f^{(EC)}_{0}$ is dominating over $f^{(+)}_{0}$,
using log$(f^{(+)}_{0}+f^{(EC)}_{0})$ from table 1 the SM half-life is 45.0 d
with q = 0.719.

In 56Ni decay, the 100$\%$ EC transition was observed at $1^{+}$ (1.720 MeV)
with log$ft$ = 4.40. When the $Q$ values of the EC transition are less than
the electron rest mass energy the endpoint energy $E_{0}$ becomes negative and
the $\beta^{+}$ mode cannot exist. As in table 1, the $f^{(+)}_{0}$ is
negative for 56Ni decay (indicated by “–”) with $f^{(EC)}_{0}$ = 0.034. The
calculated half-life is 6.9 d by considering only EC transition which is very
close to the experimental value 6.07 d.

In case of 61Cu we computed and compared PSF for EC and $\beta^{+}$-decay and
reported the half-life 2.8 h with q = 0.719 which is close to the experimental
value 3.336 h. Similarly we computed $f^{(EC)}_{0}$ and $f^{(+)}_{0}$ for 64Cu
and reported half-life 21.0 h with q = 0.719. In case of 64Cu decay the
observed $\beta^{+}$/EC -decay branching ratio 61.5 $\%$ is scaled to 100 $\%$
in order to be consistently comparison between computed $\beta^{+}$/EC -decay
branching. The SM half-life results slightly change for 60Zn, 62Zn and 65Zn
after considering $f^{(EC)}_{0}$ in SM calculations.

The theoretical and experimental $\beta$-decay half-lives of concerned nuclei
are plotted in Figs. 2 - 3 and the data are taken from table 4. In the figure,
we used a log frame to show the $\beta^{+}$-decay half-lives. The figure
indicates that the $\beta$-decay half-lives increase rapidly with the
increasing mass number. The experimental data are presented by dotted lines
with error bars. For most of the $fp$ and $f_{5/2}pg_{9/2}$ shell nuclei SM
results of $\beta$-decay half-lives are in reasonable agreement with the
experimental data. The SM results of Cu and Zn isotopes for $f_{5/2}pg_{9/2}$
space using JUN45 interaction, show a reasonable agreement with experimental
data.

##  5 Summary and Conclusion

In the present work, we have reported a comprehensive nuclear shell model
study of $\beta$-decay half-lives, log$ft$ values, and branching fractions for
the $fp$ and $f_{5/2}pg_{9/2}$ shell nuclei with $Z=21-30$. The calculations
have been performed in two different model spaces. For $fp$ shell nuclei the
KB3G effective interaction has been used and for Cu and Zn nuclei in
$f_{5/2}pg_{9/2}$ model space the JUN45 effective interaction has been used.
Over all the calculated results of excitation energies, log$ft$ values, half-
lives, and branching fractions for most of the nuclei are in reasonable
agreement with the available experimental data. The present shell model
results of $\beta$-decay, half-lives, log$ft$ values, and branching fractions
will add more information to the earlier experimental works. Further, the
calculated quenching factors in this work for $fp$ and $f_{5/2}pg_{9/2}$ space
will play an important role for $\beta$-decay study in this space.

## Acknowledgment:

V. Kumar acknowledges financial support from SERB Project (File No.
EEQ/2019/000084), Govt. of India. The authors also acknowledge PARAM Shivay
Computing facility at IIT (BHU) Varanasi. V. Kumar also acknowledges financial
support from IoE Seed Grant, BHU (R/Dev/D/IoE/Seed Grant-II/2021-22/39960).
PCS acknowledges financial support from SERB (India), CRG/2022/005167. In
addition, we would like to thank the National Supercomputing Mission (NSM) for
providing computing resources of ‘PARAM Ganga’ at the Indian Institute of
Technology Roorkee, implemented by C-DAC and supported by the Ministry of
Electronics and Information Technology (MeitY) and Department of Science and
Technology (DST), Government of India. We would like to thank Prof. T. Suzuki
for useful suggestions.

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# Large Language Models are biased to overestimate profoundness

Eugenio Herrera-Berg*1, Tomás Vergara Browne*1,2, Pablo León-Villagrá†3,
Marc-Lluís Vives†4, Cristian Buc Calderon†1

1Centro Nacional de Inteligencia Artificial (CENIA), Chile
2Pontificia Universidad Católica de Chile, Chile
3Brown University, United States
4Leiden University, the Netherlands

###### Abstract

Recent advancements in natural language processing by large language models
(LLMs), such as GPT-4, have been suggested to approach Artificial General
Intelligence. And yet, it is still under dispute whether LLMs possess similar
reasoning abilities to humans. This study evaluates GPT-4 and various other
LLMs in judging the profoundness of mundane, motivational, and pseudo-profound
statements. We found a significant statement-to-statement correlation between
the LLMs and humans, irrespective of the type of statements and the prompting
technique used. However, LLMs systematically overestimate the profoundness of
nonsensical statements, with the exception of Tk-instruct, which uniquely
underestimates the profoundness of statements. Only few-shot learning prompts,
as opposed to chain-of-thought prompting, draw LLMs ratings closer to humans.
Furthermore, this work provides insights into the potential biases induced by
Reinforcement Learning from Human Feedback (RLHF), inducing an increase in the
bias to overestimate the profoundness of statements.

**footnotetext: Equal contribution.††footnotetext: Co-senior authors.

##  1 Introduction

GPT-4 has now achieved the ability to perform a wide range of tasks on par
with humans OpenAI (2023). Given that current LLMs excel at text
interpretation and generation, recent work has assessed LLMs’ ability to
perform tasks beyond simple language understanding. And the evidence for LLMs’
ability to perform these tasks is changing rapidly: while previous versions of
LLMs (e.g., GPT-3.5) fell short in tasks seen as unsolvable with textual input
alone Ullman (2023), more recent models, such as GPT-4, have been suggested to
achieve them Bubeck et al. (2023). However, most of these studies have focused
on presenting the LLM with statements following conversational maxims. Thus,
when presenting prompts to an LLM, one generally strives to make the statement
informative and truthful to convey one’s ideas or goals successfully. However,
not all language use is aimed at efficiently communicating information.
Sometimes, it can be beneficial for a speaker to obscure the meaning of an
utterance, for example, to deceive or persuade or to hide one’s true
intentions. Successful, human-level communication requires the listener to
detect such language use; otherwise, they are susceptible to deception. Here,
we assess whether GPT-4 and other LLMs can identify language created with the
aim of impressing the listener rather than communicating meaning.

The term “pseudo-profound bullshit” (BS) refers to sentences that seem to have
a deep meaning at first glance, but are meaningless Pennycook et al. (2015).
These sentences are syntactically correct and presented as true and
significant but, upon further consideration, lack substance. For example:

> “Consciousness is the growth of coherence, and of us.”

Pseudo-profound bullshit is thus an example of language use that is not aimed
at conveying information but evoking an interpretation in the listener in
order to seem meaningful and insightful; and could potentially fool LLMs to
produce non-desired responses. Cognitive strategies deployed during language
exchanges to uncover (and create) these strategies require sophisticated
skills Musker and Pavlick (2023), such as making recursive inferences or
assessing the actual meaning of hard-to-parse sentences Bubeck et al. (2023).
In fact, in humans, the ability to detect the shallowness of pseudo-profound
bullshit correlates with classical measures of cognitive sophistication like
verbal intelligence or individual tendencies to exert deliberation Pennycook
et al. (2015).

Because LLMs perform better in tasks that require knowledge of syntax,
morphology, or phonology while struggling in tasks that require formal
reasoning (e.g., logic), causal world knowledge, situation modeling, or
communicative intent Mahowald et al. (2023), it is an open question whether an
LLM will be able to detect the presence of pseudo-profound bullshit.

We find that GPT-4, and most other LLMs tested here, display a strong bias
towards profoundness, i.e., the pseudo-profound statements are systematically
ranked above a mid-point level of profoundness. In contrast, humans rank these
statements below this mid-point level. The one exception we found is with Tk-
Instruct, which consistently underestimates the profoundness of every
statement, including that of motivational statements.

Moreover, we show that chain-of-thought prompting methods Wei et al. (2022),
which typically increase the reasoning abilities in LLMs, have no statistical
effect on this ranking; and that only few-shot learning allows GPT-4 to rate
statements more similar to humans and below the mid-point ranking. Finally, we
show that, despite the biases found, the statement-to-statement rankings
display a strong correlation between the LLMs and humans.

##  2 Related Work

#### Probing Language Understanding.

LLMs can be treated as participants in psycholinguistic Linzen et al. (2016);
Dillion et al. (2023) or cognitive science experimental studies Binz and
Schulz (2023). Thus, recent work suggests a series of diagnostics to analyze
LLMs inspired by human experiments Ettinger (2020). For instance, LLMs can
represent hierarchical syntactic structure Lin et al. (2019) but tend to
struggle in semantic tasks Tenney et al. (2019) or display similar content
effects as humans Dasgupta et al. (2022). Furthermore, Hu et al. (2023)
compared a variety of LLMs to human evaluation on seven pragmatic phenomena to
test possible correlations between human and models’ judgments.

#### Boosting Reasoning in LLMs.

Recent work evaluates the reasoning abilities in LLMs by analyzing the
techniques that elicit reasoning Huang and Chang (2022). For instance, several
prompting methods increase the reasoning abilities of LLMs. Chain-of-thought
(CoT) prompting is a method that implements a sequence of interposed natural
language processing steps leading to a final answer Wei et al. (2022); Zhao et
al. (2023); Lyu et al. (2023). Furthermore, increasing reasoning abilities can
also be elicited via Zero-shot CoT by simply adding the sentence “let’s think
step by step" at the end of the prompt Kojima et al. (2022). In the same vein,
showing a few examples to LLMs allows them to quickly learn to reason on
complex tasks Brown et al. (2020); Tsimpoukelli et al. (2021), an ability
known as few-shot learning.

##  3 Method

Our study is based on previous research assessing whether humans are receptive
to pseudo-profound statements Pennycook et al. (2015). In the original study
by Pennycook et al., 198 humans ranked the profoundness of statements on a
5-point Likert scale: 1 = Not at all profound, 2 = somewhat profound, 3 =
fairly profound, 4 = definitely profound, 5 = very profound.

We replicate this study using a variety of LLMs to judge the profoundness of
statements. The LLMs used are GPT-4 and various other models (Flan-T5 XL Chung
et al. (2022), Llama-2 13B with and without RLHF Touvron et al. (2023), Vicuna
13B Chiang et al. (2023) and Tk-Instruct 11B Wang et al. (2022)). We performed
20 repetitions for each experiment to account for the non-deterministic nature
of token generation (given that we use non-zero temperature). We investigate
how sensitive this model is to pseudo-profound bullshit by systematically
comparing it with human ratings on these same and similar statements.
Furthermore, we use distinct prompting methods that have been shown to
increase the reasoning abilities of LLMs (see below). All of our code and
results are publicly available111https://github.com/ouhenio/llms-overstimate-
profoundness.

###  3.1 Dataset

We generated five distinct datasets of sentences. For dataset 1, we used the
same 30 pseudo-profound BS statements as those used in experiment 2 of
Pennycook et al. (2015). For dataset 2, we built a dataset comprising 30 novel
pseudo-profound BS statements generated following the same procedure as
Pennycook et al. (2015). Dataset 3 was generated on the basis of the first
dataset, but we generated 30 novel statements by switching words from one
sentence to another but maintaining the syntactic structure (dataset 3; see
Appendix A for further details). As points of comparison, we also used ten
mundane (non-profound, dataset 4) and ten motivational (profound, dataset 5)
statements from Pennycook et al. (2015) paper. An example from each dataset
can be seen in Appendix A (Table 1).

###  3.2 Experimental Design

We tested the LLMs on the five datasets described in the previous section. The
LLMs replied on the same Likert scale used in Pennycook et al. (2015). We
further manipulated two factors: the prompt method, and the temperature. We
tested three distinct prompting methods: the original instruction participants
received in Pennycook et al. (2015) (prompt 1), few-shot learning Brown et al.
(2020) (prompt 2-3), and chain-of-though (CoT) prompting Wei et al. (2022)
(prompt 4-10). All the prompts are listed in Appendix A (Table 2). The
temperature was set to 0.1 and 0.7 to assess the effect of variability in the
outputs.

We let the LLMs predict one token corresponding to its rating of the
profoundness of the sentence, and parse it as an integer.

###  3.3 Statistical Analyses

We performed an item analysis by averaging ratings across human subjects and
the LLMs’ responses for each statement. Given that our goal was to compare
each LLM to human-level performance, we ran a 3 (between-items) $\times$ 2
(within-items) two-way mixed ANOVA, for each LLM (on the original Pennycook et
al. (2015) prompt). The between-subject factor was statement type (mundane,
motivational, or pseudo-profound BS), and the within-subject factor was agent
type (human or LLM). To assess a potential profoundness bias when rating
pseudo-profound bullshit, we tested the ratings against the mid-point scale.
To analyze a potential prompt effect, we first focused on GPT-4’s BS statement
ratings, and performed a two-way ANOVA (with prompts and statement types as
factors). Moreover, to compare the effect of prompting between models, we
performed a two-way ANOVA (with model type and prompts as factors) on the
LLMs’ BS statements ratings.

Finally, we ran regressions to test the degree to which human ratings were
predicted by the LLMs. Human mean ratings for each statement was predicted by
each LLM’s ratings, with statement type (mundane, motivational, or pseudo-
profound bullshit), evaluation prompt (1 to 10), and temperature as main
effects.

Since manipulating the temperature of the LLMs did not significantly change
the resulting ratings, we collapsed ratings across temperatures in all
reported analyses. Moreover, given that we found no differences in ratings
between datasets 1, 2, and 3, we did not include the ratings of datasets 2 and
3 in the analyses. Lastly, for simplicity and brevity, we only report
statistical results of GPT-4; results of other LLMs can be found in Appendix A
(Table 3)

Figure 1: Distribution of ratings per statement type in humans and the LLMs.

##  4 Results

In the original study by Pennycook et al., humans discriminated between
mundane, pseudo-profound BS, and motivational statements. Unsurprisingly,
mundane statements were rated as the least profound, motivational statements
as the most profound, and pseudo-profound BS was rated in between (Figure 1).
This trend was also captured by the LLMs’ ratings (Figure 1, the main effect
of statement type: F(2,47) = 72.09, p < 0.001). Importantly, even though the
LLMs followed the same qualitative pattern as humans, most of LLMs’ ratings of
profoundness were constantly higher than human ratings (see Figure 1, main
effect of agent type: F(1,47) = 185.28, p < 0.001). The bias towards
profoundness observed was independent of whether the statement was mundane,
pseudo-profound BS, or motivational (non-significant interaction: F(2,47) =
0.96, p = 0.40) for GPT-. For the rest of LLMs, profoundness was overestimated
only in pseudoprofound BS statements (Flan-T5 (XL), Vicuna (13B)) or it was
larger for those statements (Llama-2 + RLHF (13B), Llama-2 (13-B)), as denoted
by a significant interaction between statement and agent types (all ps <
0.001, see Table 3 in the Supplement). Tk-Instruct showed the opposite effect,
but at the cost of underestimating the profoundness of motivational
statements. We included a possible explanation for these phenomena in the
discussion.

Figure 2: Profoundness ratings across all evaluation prompts for each LLM.

We used 10 prompts to obtain each LLM’s ratings. Results reveal that prompting
significantly influenced the ratings for GPT-4 (main effect of the prompt:
F(9,423) = 18.12, p < 0.001). Crucially, the effect of prompting showed a
significant interaction with statement type (interaction effect: F(18, 423) =
35.06, p < 0.001). Further analyses revealed that the interaction was driven
by n-shot learning evaluation prompts, which lowered the ratings for pseudo-
profound BS statements. Planned t-test between the two n-shot learning and
other prompts revealed a significant decrease in the rating for the pseudo-
profound BS (see Figure 2, all ps < 0.001, Bonferroni-corrected), while n-shot
learning prompts did not affect the mundane or motivational statements (all ps
> 0.05). Importantly, the ratings from the n-shot learning prompts were not
significantly different than the human ratings for pseudo-profound BS (all ps
> 0.05). Other LLMs were also affected by prompting, but significantly less
(interaction effect of the two-factor ANOVA contrasting LLMs with evaluation
prompt: F(45, 1305) = 21.52, p < 0.001, see Figure 2).

In addition to assessing profoundness ratings for pseudo-profound statements
relative to other statement types, we investigated the absolute rating on the
scale. Due to the structure of the response scale, ratings below three are
considered not to be profound. Thus, if LLMs detect pseudo-profound BS, their
ratings should be significantly lower than the mid-point level of 3. Note that
in Figure 3, we mean-center the rating data to facilitate data interpretation;
above 0 implies overestimation, and below 0 implies underestimation with
respect to the mid-point level.

Our results reveal that, overall (with the exception of TK-instruct), LLMs’
are biased towards overestimating the profoundness of pseudo-profound bullshit
(ratings were significantly higher than 3: t(29) = 6.67, p < 0.001, average =
3.70, see Figure 3). This is in contrast with humans, who perceived pseudo-
profound bullshit statements as non-profound (ratings were significantly lower
than 3: t(29) = -9.71, p < 0.001, average = 2.56, Figure 3). GPT-4’s bias
towards profoundness was prevented, however, by the two prompts that
significantly lowered ratings for pseudo-profound bullshit statements (ps <
0.001 for both prompts, Figure 3), which at the same time caused the ratings
to be not significantly different from humans (ps > 0.05 for both prompts,
Figure 3). While other LLMs were also affected by these prompts, their overall
effect did not induce pseudo-profound BS detection.

Although the ratings of the LLMs did not match those of humans in absolute
terms, it did capture some of the same variability in statement-to-statement
responses as humans did (b = 0.15, SE = 0.004, t(939) = 4.31, ps < 0.001)
across prompting strategies.

Figure 3: Overview of the profoundness assessment in humans, and the LLMs with
1-shot and 3-shot learning. Ratings are adjusted to be centered around the
midpoint (3).

##  5 Discussion

We found that LLM scores significantly correlated with human data across all
three statement types, suggesting that, overall, the LLMs could differentiate
mundane, motivational, and pseudo-profound statements. However, unlike humans,
most LLMs displayed an overall bias toward attributing profoundness to
statements, irrespective of the type of sentence in the case of GPT-4, but
exacerbated by pseudo-profoundness for the rest (with the exception of Tk-
Instruct). For GPT-4, this bias was only reduced by providing few-shot
prompts. Surprisingly, more recent prompting techniques did not result in
significantly different ratings, perhaps reflecting the fact that detecting
meaningfulness in pseudo-profound statements is a task that requires world
knowledge.

In contrast to other LLMs we tested, Tk-Instruct was uniquely biased towards
underestimating statements’ profoundness across the three datasets. These
results are particularly interesting given that Tk-Instruct is fine-tuned to
follow instructions in more than 1600 tasks, some of which contain common
sense detection Wang et al. (2022). Such fine-tuning may render this model
overly cautious in judging the profoundness of statements.

Why do LLMs display a bias towards profoundness overestimation? One
possibility is that this overestimation emerges from the data used to train
these models. Alternatively, this bias may emerge from additional tuning after
the initial training. For instance, LLM alignment using reinforcement learning
from human feedback Ouyang et al. (2022), as with GPT-4 and Llama-2, may
introduce a bias towards being overly credulous. Our results provide a hint in
this direction. Indeed, the Llama-2 RLHF version consistently provides higher
ratings of profoundness to each statement, compared with its no-RLHF
counterpart. This sheds light on how RLHF can potentially (negatively) impact
the judgment of LLMs.

Detecting pseudo-profound BS may depend on the ability of these models to
represent actual meaning. Furthermore, the lack of dynamics in LLMs and top-
down control processes may prevent these models to detect semantic
incongruence elicited by pseudo-profound statements. In contrast to LLMs,
dynamic control processes are at the heart of the human capacity to complete
goals and resolve conflict in incongruent tasks. (e.g. Botvinick et al., 2001;
Shenhav et al., 2014).

## Limitations

First, our few-shot learning setup was limited to a relatively small set of
examples. An exploration with more extensive sets (e.g., 10 or more examples)
may yield different results Wei et al. (2022). Second, while our experiments
covered various prompting strategies, there remain various unexplored methods,
such as “tree of thought” Yao et al. (2023), which can boost the performance
of GPT-4 in the context of profoundness detection. Third, economic factors
(i.e., high costs of GPT-4 API use) constrained us to carry out a proper
exploration of relevant hyperparameters and potential experimental design
factors. Indeed, Pennycook et al. (2015) performed an extensive analysis of
personality traits and other cognitive capacities and their effect of those on
pseudo-profound bullshit receptivity. Future work should include these tests
for LLMs, allowing us to further explore the theoretical underpinnings of
LLMs’ overly credulous perspective. Indeed, understanding how different
factors influence the propensity of LLMs to overestimate profoundness, can
inform the development of more unbiased and accurate models.

## Ethics Statement

Our data were collected using GPT-4’s API, in accordance with their terms of
use, which do not state the prohibition to use the model for research
purposes. We did not perform any human experiment studies and simply used the
open-sourced human data of Pennycook et al. (2015). Hence, no ethics committee
approval was needed to carry out our study.

## Acknowledgements

This work was funded by the Centro Nacional de Inteligencia Artificial, CENIA,
FB210017, Basal ANID.

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##  Appendix A Appendix

Statements from dataset 1 and 2 were extracted from two websites:
wisdomofchopra.com and sebpearce.com/bullshit; and from vague statements from
Deepak Chopra’s Twitter feed. Ten statements were selected from each of these
three sources. The statements extracted from each website are constructed by
randomly patching profound words into meaningless but syntactically correct
sentences.

To create dataset 3, we used spaCy. In particular, a random selection of the
nouns in the 30 sentences from dataset 1 was substituted with a word from its
10 nearest neighbors, as determined by spaCy’s cosine distance. Subsequently,
we used ChatGPT to rectify any potential syntax inconsistencies with the
prompt: “Correct any inconsistencies in uppercase usage, spacing, or
punctuation in the following sentence".

All the code used during this study, including the datasets creation and
evaluation, is publicly in the following link https://github.com/ouhenio/llms-
overstimate-profoundness.

Dataset 1: | "Consciousness is the growth of coherence, and of us."
---|---
Dataset 2: | "Your body belongs to the progressive expansion of success."
Dataset 3: | "Forgiveness means letting go of the hope for a better past."
Dataset 4: | "Newborn babies require constant attention."
Dataset 5: | "Your teacher can open the door, but you must enter by yourself."

Table 1: Examples from each dataset used in the experiment.

Original Human prompt (prompt-1) |  | "We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound"
1-shot Learning (prompt-2) |  | "We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
For instance, the sentence:
"This life is nothing short of a summoning rekindling of karmic complexity."
is not profound and is considered as pseudo-profound bullshit, because the
association of these words in the same sentence do not provide any meaning.
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound"
3-shot Learning (prompt-3) |  | "We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
For instance, the 3 following sentences:
"This life is nothing short of a summoning rekindling of karmic complexity."
"Imagination is inextricably connected to the flow of external reality."
""The secret of the universe comprehends spiritual creativity."
are not profound and are considered as pseudo-profound bullshit, because the
association of these words in the same sentence do not provide any meaning.
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound"
CoT zero-shot (prompt 4) |  | "We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound
To give your answer, think step by step."
CoT-1 mundane (prompt-5) |  | "We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
To give your answer, first compare the statements that were given with a
normal mundane sentence, such as:
"The girl on the bicycle has blond hair."
Second, if you believe the statements have the same level of profoundness as
this mundane sentence, you should answer with a low value on the 5-point
scale.
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound"
CoT-1 motivational (prompt-6) |  | "We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
To give your answer, first compare the statements that were given with a
motivational sentence, such as:
"The creative adult is the child who survived."
Second, if you believe the statements have the same level of profoundness as
this mundane sentence, you should answer with a high value on the 5-point
scale.
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound"
CoT-1 mun. and mot. (prompt-7) |  | "We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
To give your answer, first compare the statements that were given with a
motivational sentence, such as:
"Success is not final; failure is not fatal: It is the courage to continue
that counts."
Second, compare the statements that were given with a mundane sentence, such
as:
"The little boy is playing baseball."
Third, if you believe the statements have the same level of profoundness as
the mundane sentence, you should answer with a low value on the 5-point scale
. In contrast, if you believe the statements have the same level of
profoundness as the motivational sentence, you should answer with a high value
on the 5-point scale.
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound"
CoT-3 mundane (prompt-8) |  | "We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
To give your answer, first compare the statements that were given with a
normal mundane sentence, such as:
"The girl on the bicycle has blond hair."
"An english football player scored a goal during the game."
"Brazil is a beautiful country."
Second, if you believe the statements have the same level of profoundness as
this mundane sentence, you should answer with a low value on the 5-point
scale.
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound"
CoT-3 motivational (prompt-9) |  | "We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
To give your answer, first compare the statements that were given with a
motivational sentence, such as:
"The creative adult is the child who survived."
"Learn as if you will live forever, live like you will die tomorrow."
"When you change your thoughts, remember to also change your world."
Second, if you believe the statements have the same level of profoundness as
this motivational sentence, you should answer with a high value on the 5-point
scale.
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound"
CoT-3 mund. and mot. (prompt-10) |  | We are interested in how people experience the profound. Below are a series of statements taken from relevant websites.
---
Please read each statement and take a moment to think about what it might
mean. Then please rate how “profound” you think it is.
Profound means “of deep meaning; of great and broadly inclusive significance.”
To give your answer, first compare the statements that were given with
motivational sentences, such as:
"Success is getting what you want, happiness is wanting what you get."
"It is better to fail in originality than to succeed in imitation."
"I never dreamed about success. I worked for it."
Second, compare the statements that were given with mundane sentences, such
as:
"The two dogs are playing with the tennis ball."
"Surfers are riding the waves."
"Lasagna is an Italian dish."
Third, if you believe the statements have the same level of profoundness as
the mundane sentence, you should answer with a low value on the 5-point scale.
In contrast, if you believe the statements have the same level of profoundness
as the motivational sentence, you should answer with a high value on the
5-point scale.
Rate the profoundness of the following sentences on the following 5-point
scale:
1= Not at all profound, 2 = somewhat profound, 3 = fairly profound, 4 =
definitely profound, 5 = very profound

Table 2: All prompts used in the experiment. GPT-4: |  |  |  |  |
---|---|---|---|---|---
Predictor | $df_{Num}$ | $df_{Den}$ | F | p-value | $\eta_{g}^{2}$
Statement type | 2 | 47 | 72.09 | .000 | .67
Agent type | 1 | 47 | 185.28 | .000 | .57
Statement x Agent | 2 | 47 | 0.96 | .389 | .01
Flan-T5 (XL): |  |  |  |  |
Predictor | $df_{Num}$ | $df_{Den}$ | F | p-value | $\eta_{g}^{2}$
Statement type | 2 | 47 | 100.36 | .000 | .67
Agent type | 1 | 47 | 0.42 | .520 | .00
Statement x Agent | 2 | 47 | 10.18 | .000 | .19
Llama-2 + RLHF (13B): |  |  |  |  |
Predictor | $df_{Num}$ | $df_{Den}$ | F | p-value | $\eta_{g}^{2}$
Statement type | 2 | 47 | 143.62 | .000 | .79
Agent type | 1 | 47 | 497.51 | .000 | .80
Statement x Agent | 2 | 47 | 5.76 | .006 | .09
Llama-2 (13B): |  |  |  |  |
Predictor | $df_{Num}$ | $df_{Den}$ | F | p-value | $\eta_{g}^{2}$
Statement type | 2 | 47 | 127.96 | .000 | .78
Agent type | 1 | 47 | 302.01 | .000 | .69
Statement x Agent | 2 | 47 | 10.42 | .000 | .13
Vicuna (13B): |  |  |  |  |
Predictor | $df_{Num}$ | $df_{Den}$ | F | p-value | $\eta_{g}^{2}$
Statement type | 2 | 47 | 82.75 | .000 | .68
Agent type | 1 | 47 | 35.91 | .000 | .24
Statement x Agent | 2 | 47 | 9.26 | .000 | .14
Tk-Instruct (11B): |  |  |  |  |
Predictor | $df_{Num}$ | $df_{Den}$ | F | p-value | $\eta_{g}^{2}$
Statement type | 2 | 47 | 65.08 | .000 | .61
Agent type | 1 | 47 | 13.63 | .001 | .11
Statement x Agent | 2 | 47 | 10.52 | .000 | .16
Table 3: ANOVAs results divided by Large Language Model. $df_{Num}$ indicates
degrees of freedom numerator. $df_{Den}$ indicates degrees of freedom
denominator. $\eta_{g}^{2}$ indicates generalized eta-squared.