# Directed propaganda in the majority-rule model

F. L. Forgerini  Universidade Federal do Sul da Bahia, Itabuna/BA - Brazil
<EMAIL_ADDRESS>N. Crokidakis  Instituto de Física,
Universidade Federal Fluminense, Niterói/RJ - Brazil
<EMAIL_ADDRESS>M. A. V. Carvalho  Universidade Federal do Sul da
Bahia, Itabuna/BA - Brazil
<EMAIL_ADDRESS>
###### Abstract

Advertisement and propaganda have changed continuously in the past decades,
mainly due to the people’s interactions at online platforms and social
networks, and operate nowadays reaching a highly specific online audience
instead targeting the masses. The impacts of this new media effect, oriented
directly for a specific audience, is investigated on this study, in which we
focus on the opinion evolution of agents in the majority-rule model,
considering the presence of directed propaganda. We introduce $p$ as the
probability of a “positive” external propaganda and $q$ as the probability to
the agents follow the external propaganda. Our results show that the usual
majority-rule model stationary state is reached, with a full consensus, only
for two cases, namely when the external propaganda is absent or when the media
favors only one of the two opinions. However, even for a small influence of
external propaganda, the final state is reached with a majority opinion
dominating the population. For the case in which the propaganda influence is
strong enough among the agents, we show that the consensus can not be reached
at all. In addition, we show through analytical and numerical results that the
system undergoes to an order-disorder transition that occurs at $q_{c}=1/3$
for the case $p=0.5$.

Opinion dynamics; Majority-rule model; Directed propaganda; Collective
phenomena

##  I Introduction

The challenge of understanding the collective behavior emerging from
individuals in society has fascinated physicists for the last few decades and
their contributions gave rise to what have been called sociophysics, an
attempt to explain how a social collective behavior emerges from individual
agents [1, 2, 3]. However, the lineage of the agent-based models can be traced
back to the 1940’s, with the theoretical conception of Von Neumann’s self-
replicating machines [4] and Sakoda’s general discrete dynamical model for
social interactions [5]. It is important to note that both developments
emerged alongside the early computers, and, on several occasions, the initial
simulations were conducted without their assistance.

Von Neumann’s devices would follow instructions in order to create identical
copies of themselves. Stanislaw Ulam, a mathematician and close associate of
von Neumann, further refined this idea by suggesting that the machine could be
represented on paper as a grid of cells, giving rise to what would later be
termed cellular automata. Intrigued by Ulam’s proposition, von Neumann
proceeded to develop the initial design, thus pioneering the concept of
cellular automata.

The foundational Sakoda model encompasses the dynamics of social interactions
within a network involving two distinct groups of individuals. The model
incorporates social specific attitudes such as attraction, repulsion, and
neutrality, which guide the evolution of social bondings. Each individual in
the network evaluates its social expectations across all feasible locations,
displaying a preference for proximity to individuals associated with positive
attitudes of attraction while avoiding locations near those linked to negative
attitudes of repulsion. This evaluative process is randomly iterated across
all individuals, leading to the recursive implementation of Sakoda’s
algorithm. When certain conditions are met, this iterative process drives the
system towards the emergence of a well-organized spatial pattern.

Departing from the paradigm of von Neumann’s machine, John Conway’s Game of
Life [6] functioned based on elementary rules operating within a simulated
realm portrayed as a two-dimensional field. Furthermore, the emergence of the
agent-based model as a framework for studying social systems can be largely
attributed to the efforts of computer scientist Craig Reynolds [7], who sought
to model the dynamics of living biological agents, commonly referred to as
artificial life. But it was the groundbreaking model of Sugarscape, developed
by Joshua M. Epstein and Robert Axtell [8], which served as a platform for
conducting large-scale simulations and investigations. It enabled the
exploration of intricate dynamics encompassing a wide range of social
phenomena, such as seasonal migrations, pollution, sexual reproduction,
combat, disease transmission, and cultural diffusion.

In 1999, Gilbert Nigel, in collaboration with Klaus G. Troitzsch, made a
significant scholarly contribution by publishing the inaugural textbook on
social simulation, entitled “Simulation for the Social Scientist” [9]. This
pioneering work provided a comprehensive foundation for understanding and
applying simulation techniques in the context of social science research.

Kathleen M. Carley made a crucial contribution in promoting the utilization of
simulation methodologies in the context of organizational systems. Her
scholarly work, “Computational Organizational Science and Organizational
Engineering” [10]. Following these initial advancements, a proliferation of
models, methodologies, and scholarly publications in the field of social
simulation ensued. The topics have also diversified, encompassing knowledge
transmission, institutions, reputation, social norms, elections, economics,
among many others. Of particular relevance to this study is the theme of
advertisement and propaganda, or belief transmission. A variety of
sociological relevant problems have been studied by the means of the
statistical mechanics techniques jointly with the massive use of computational
modeling and simulations [11]. Different realms from social sciences such as
opinion [12, 13, 14, 15, 16] and language dynamics [17, 18], community
formation and detection [19, 20], gossip and rumor spreading [21, 22, 23], to
name a few, have been extensively studied by physicists in collaboration with
scientist from different areas.

The basic starting point of thermodynamics and statistical mechanics focus on
the study of an isolated system with interacting particles, in which the
macrostates of the system are described by a set of parameters of its
microscopic constituents. By direct analogy from the statistical physics one
can apply the methods used in the framework of the statistical mechanics to
the study of social problems, such as opinion dynamics, in which the
individuals can interact with other individuals as the particles do in the
standard thermodynamics systems [24, 25].

Traditional advertisement and propaganda have enormously changed in the last
decades, from the usual mass media that reach a broad audience with same type
of information, to the directed advertisement, mediated by complex algorithms
and artificial intelligence operating in social media and online platforms on
the web [26, 27]. This new form of how the information is reaching people is
poorly studied in the context of opinion dynamics and yet it is not well
understood how it can affect the way consensus (or majority) is formed in a
population [28, 29, 30].

We are interested here in the majority-vote model. It was originally proposed
by Serge Galam to describe how a fully-connected population composed by $N$
individuals reach consensus, in which the agents can decide in favor or
against a particular subject [31]. A random number of agents, called
“discussion group”, is selected to debate, interacting among each other and,
after the discussion, all of them follow the most popular opinion, the
majority opinion of the group. The results for the original model show that
the initial majority opinion win the debate after some time, i.e., the steady
states of the model are formed by full consensus situations, where all agents
share the same opinion, represented by variables $+1$ or $-1$. The model was
after extended by several researchers [28, 32, 33, 34, 35, 36].

In this study we investigate the evolution of agents’ opinion in the majority-
rule model. We introduce a directed propaganda, oriented directly for the
discussion group, that we argue can mimic the complex algorithms and
artificial intelligence used in advertisements on social media and online
platforms. Our results suggest that even a small influence of directed
propaganda is enough to avoid consensus in the population. In addition, one
can observe an order-disorder phase transition for a specific value of the
parameters related to the directed propaganda.

This paper is organized as follows: in Section II we present our model as well
their dynamic rules. The numerical and analytical results are discussed in
Section III and our conclusions and final remarks are presented in Section IV.
In Appendix one can see the details for the analytical calculations.

##  II The Majority-Rule Model with directed propaganda

The dynamics of agreement and disagreement is treated in terms of the
variation of the number of different opinion states in population, where each
individual can have one of two possible opinions ($\pm 1$). The average
opinion in the system, represented by $M$, for a population with $N$
individuals, is the order parameter of the system (or the magnetization),
defined as

$M=\frac{1}{N}\left|\sum_{i=1}^{N}S_{i}\right|~{},$ (1)

in which $S_{i}$ is the individual opinion for each agent ($i=1,2,...,N$),
considered at each time step. We define one time step as an attempt to $M$
agents try to change their opinions by interacting in discussions groups. In
our study we considered a discussion group with a constant size of three
agents. Differently from the original model, we are considering an external
directed propaganda, acting like an external field, $H=\pm 1$. However, this
directed propaganda has no effect on the entire system, but affects only
agents on discussion group instead. We argue that the introduction of the
directed propaganda as an external field addressed to a specific group of
agents can influence the discussion group beyond the majority, leading the
entire system to not reach consensus. Furthermore, this introduction of an
external directed field can mimic, in a simple way, the actual new forms of
online propaganda.

One can see in Fig. 1 a schematic representation of our model, in which three
agents are selected at random to form a discussion group 111We are considering
a fully-connected population. However, just as an illustration, Fig. 1
exhibits the population in a grid, but there are no restrictions for the
choose of agents to form the discussion groups.. After the action of the
external influence or the discussion process, the agents’ states are updated
and the system carries on with the opinion dynamics. The dynamic rules for our
model are very simple and they are shown in the following. At each time step,
we:

  1. 1.

Randomly choose the three agents to form the discussion group;

  2. 2.

Set $H=+1$ with probability $p$ and $H=-1$ with probability $1-p$;

  3. 3.

With probability $q$ agents in discussion group follow the propaganda $H$. On
the other hand, with probability $1-q$ the agents follow the majority opinion
inside the group.

Figure 1: The schematic representation of our model. From the lattice (a) we
chose at random (b) the discussion group (c) and, with probability $p$ we set
the external field $H=+1$, for instance (d). With probability $q$, all agents
follow the external field (e) and the return to the lattice to continue the
system dynamics (f). Figure 2: Average opinion as function of time for $p=0$,
$p=0.2$, $p=0.5$, $p=0.8$, $p=1$ and for $q=0.1$. Full consensus is reached
only for $p=0$ or $p=1$.

We introduce two parameters, namely $p$ and $q$. The first one, $p$, is the
probability of a “positive” external propaganda. The parameter $q$ is the
probability to the agents in discussion group change their opinions and follow
the external propaganda direction. In other words, $p$ is a measure of the
propaganda “volume” for one specific direction and $q$ is a measure of how
strong or how effective the propaganda is in order to influence the agents.
Our results in the next section will show that, even for a small value of the
parameter $q$, the system does not show the full consensus at stationary state
that is observed in the original majority-rule model.

##  III Numerical and Analytical Results

The introduction of the propaganda in majority-rule model, acting like an
external field and directly influencing the discussion group, changing the
stationary state of the model and thus the full consensus can no longer be
reached. In fig. 2 we exhibit the time evolution of the average opinion $M$,
obtained from numerical simulations of the model considering Eq. (1), for
$q=0.1$ and typical values of $p$. As one can see in Fig. 2, for a small value
of $q$ the system does not reach a stationary state with full consensus,
except for the cases where $p=0$ or $p=1$. These situations means that there
is no competition between the propaganda, since there is only one possible
direction of $H$ and, therefore, is possible for the system to reach full
consensus. Any other case, due to the competition between the two directions
of propaganda, a majority opinion (but not totality) is obtained, mimicking a
democratic situation. Taking into account an entire population, a very common
situation nowadays, in any real situation and for most democratic states, is
that there is always a fraction of agents with opinion contrary to the
dominant opinion.

In our simulations, the system’s dynamics initiate with a random distribution
of opinions among the agents, i.e., with the same probability each agent can
have $+1$ or $-1$ opinion. In Fig. 3 we show three different instants of time
for the agents’ opinion. For sufficient long times, the systems reaches the
steady state and, even though the individual opinions may change, the average
opinion is constant in time.

Figure 3: Evolution of the agents’ opinions in our model. Three different
instants of time for the same initial configuration is shown. The $-1$ opinion
is represented by a black pixel, while the white one represents the $+1$
opinion. We plot a grid $L\times L$ with $L=500$ agents, and we considered the
parameters $p=0.5$ and $q=0.3$. As initial configuration, we considered equal
fractions for the two opinions, randomly distributed in the grid. From left
panel to right, one can see the beginning of the dynamics, an intermediate
instant of time and the steady state configuration.

Since we are considering a fully-connected population, we can consider a mean-
field approach to obtain some analytical results for the model. Defining
$f_{1}$ and $f_{-1}$ as the stationary densities of each possible state ($+1$
or $-1$, respectively) and considering the normalization condition where
$f_{1}+f_{-1}=1$, analytical expressions can be derived for those densities.
By considering the probabilities that a given agent change it’s opinion
($+1\rightarrow-1$ or $-1\rightarrow+1$), we calculate the variations of the
average opinion and calculate $M$ at the stationary state (see the Appendix
for a detailed description of our calculations).

As discussed in the Appendix, we can obtain a third-order polynomial for the
fraction of agents with $+1$ opinion, given by

$2\,(q-1)\,f_{1}^{3}-3\,(q-1)\,f_{1}^{2}-f_{1}+p\,q=0~{},$ (2)

which depends on the two parameters $p$ and $q$.

Figure 4: Steady state average opinion (or the magnetization $M$) versus $q$,
for distinct values of p. As one can see on the left panel, when $p=0.5$, for
large enough values of $q$, the system does not present a majority opinion,
representing a paramagnetic state where $M=0$. For the other values of $p$ we
have $M>0$ for all values of $q$. On the right panel, at the critical point
$q_{c}=1/3$, an order-disorder nonequilibrium phase transition occurs for the
case $p=0.5$. The solid line is the analytical solution, Eq. (3), and the red
circles are numerical results.

For the specific case $p=0.5$, we obtain a first solution given by
$f_{1}=f_{-1}=1/2$. The magnetization can be obtained from $M=|f_{1}-f_{-1}|$
which gives us $M=0$, representing a paramagnetic state. The other two
solutions of the third-order polynomial give us $f_{1}\neq f_{-1}$ (see the
Appendix), which indicates a ferromagnetic phase, where one of the two
opinions, $+1$ or $-1$, is the dominant opinion in the population. Considering
again the relation $M=|f_{1}-f_{-1}|$ for the mentioned two solutions, we
obtain

$\displaystyle M=\frac{\sqrt{3q^{2}-4q+1}}{q-1}~{},$ (3)

which can be rewritten in the critical phenomena language as
$M\sim(q_{c}-q)^{\beta}$, where the critical point is $q_{c}=1/3$ and the
order parameter critical exponent is $\beta=1/2$, as shown in Appendix. The
order-disorder nonequilibrium phase transition at $q_{c}$ seems to be in the
Ising mean-field universality class, as it is usual in opinion models with two
or three states [40]. Our model only presents a phase transition for the case
$p=0.5$, as it is discussed in the Appendix.

In fig. 4 we exhibit the numerical results for the steady-state magnetization
for different values of $q$. The critical point $q_{c}$ obtained analytically
for $p=0.5$ are in agreement with our numerical results, as one can see in the
right panel of fig. 4. For the other values of $p$, we observe no phase
transition, as discussed in the Appendix. For such case, we observe that one
of the two opinions, $+1$ or $-1$, is the majority opinion in the population,
for all values of $p$ and $q$.

Figure 5: Steady state average opinion (or the magnetization $M$) versus $p$
for the case $q=1$. The blue circles were obtained from the numerical
simulations of the model, whereas the solid red line is the analytical result
$M=|2p-1|$. Except for $p=0.5$, one opinion is always the dominant opinion in
the population. Full consensus are observed for $p=0$ and $p=1$.

We also studied the steady-state magnetization for the case $q=1$ and
different values of $p$. Considering $q=1$ in Eq. (2), one can obtain
$f_{1}=p$, which by the normalization condition leads to $f_{-1}=1-p$. Thus,
the stationary average opinion is $M=|f_{1}-f_{-1}|=|2p-1|$. Our numerical
simulations are in agreement with the analytical calculation. As one can see
in fig. 5 where, except for the case when $p=0.5$, one of the two possible
opinions is the dominant position in the population. Exceptions are observed
for the extreme cases $p=0$ and $p=1$. Considering the analytical results, we
see that for $p=0$ the stationary state of the system exhibits a population
with all agents sharing opinion $-1$, whereas for $p=1$ the final state of the
population is a full consensus with all agents sharing the same opinion $+1$.
These consensus situations are easily to be understood. Indeed, for $q=1$ the
majority rule will not be applied, and the agents in the discussion group will
always follow the media effect. For the case $p=0$, the directed propaganda
will always favors opinion $-1$, leading all agents to follow the media
opinion after a long time. Analogously, for $p=1$ the directed propaganda will
always favors opinion $+1$, leading to another situation of full consensus at
the steady states.

##  IV Final Remarks

We studied the majority-rule model with the inclusion of an external
propaganda (external field) acting directly to the discussion group instead
the entire population. The local group propaganda can favors opinion $+1$ with
probability $p$ and opinion $-1$ with probability $1-p$. After that, all
agents in the discussion group follow the media opinion with probability $q$,
or with the complementary probability $1-q$ the agents follow the standard
majority rule. We argue that the directed propaganda can mimic in a very
simple way the complex algorithms and artificial intelligence used in
advertisements on social media and online platforms. The inclusion of the
directed propaganda modify the original majority-rule model, introducing an
order-disorder phase transition. We show through analytical and numerical
results that this transition occurs at $q_{c}=1/3$ for the case $p=0.5$. For
$p\neq 0.5$ this transition is absent. Our results show that the usual
majority-rule model stationary state is reached, with a full consensus, for
the case when the external propaganda is absent ($q=0$). In the other limiting
case $q=1$, where the agents will always follow the directed propaganda, the
agents can reach full consensus for the extreme cases $p=0$ and $p=1$.
However, even for a small influence of the external propaganda, the final
state is reached with majority but not full consensus. For the case which the
propaganda influence is strong enough among the agents, we show that the
consensus can not be reached at all.

*

##  Appendix A Analytical calculations

Following the approach of References [37, 38, 32], we computed the stationary
order parameter $M$. Let us first define $f_{1}$ and $f_{-1}$ as the
stationary probabilities of each possible state ($+1$ or $-1$, respectively),
in a way that we have the normalization condition

$f_{1}+f_{-1}=1~{}.$ (4)

We have to calculate the probability that a given agent suffers the change
$+1\to-1$ or $-1\to+1$. We are considering groups of 3 agents, so one can have
distinct variations of the magnetization, depending on the states of the 3
agents and on the probabilities $p$ and $q$. For example, the probability to
choose at random 3 agents with opinions $o=+1$, i.e, a configuration
$(+,+,+)$, is $f_{1}^{3}$. With probability $q$ the agents will follow the
local media effect. In addition, with probability $p$ the configuration
remains $(+,+,+)$, which does not affect the magnetization of the system,
since the local magnetic field in this case is given by $H=+1$. However, with
probability $1-p$ the local field will be given by $H=-1$, so the group will
change opinion to $(-,-,-)$, which cause a variation of $-6$ in the
magnetization. In other words, the magnetization decreases $6$ units with
probability $q\,(1-p)\,f_{1}^{3}$. One can denote this probability as $r(-6)$,
i.e., the probability that the magnetization variation is equal to $-6$.
Generalizing, one can define $r(k)$, with $-6\leq k\leq+6$ in this case, as
the probability that the magnetization variation is $k$ after the application
of the models’ rules. As the order parameter (magnetization) stabilizes in the
steady states, we have that its average variation must vanish in those steady
states, namely,

$6\,[r(+6)-r(-6)]+4\,[r(+4)-r(-4)]+2\,[r(+2)-r(-2)]=0\,.$ (5)

In this case, we have

$\displaystyle r(+6)$ $\displaystyle=$ $\displaystyle p\,q\,f_{-1}^{3}$
$\displaystyle r(-6)$ $\displaystyle=$ $\displaystyle(1-p)\,q\,f_{1}^{3}$
$\displaystyle r(+4)$ $\displaystyle=$ $\displaystyle
3\,p\,q\,f_{1}\,f_{-1}^{2}$ $\displaystyle r(-4)$ $\displaystyle=$
$\displaystyle 3\,(1-p)\,q\,f_{1}{2}\,f_{-1}$ $\displaystyle r(+2)$
$\displaystyle=$ $\displaystyle
3\,p\,q\,f_{1}^{2}\,f_{-1}+3\,(1-q)\,f_{1}^{2}\,f_{-1}$ $\displaystyle r(-2)$
$\displaystyle=$ $\displaystyle
3\,(1-p)\,q\,f_{1}\,f_{-1}^{2}+3\,(1-q)\,f_{1}\,f_{-1}^{2}~{}.$

Thus, the null average variation condition Eq. (5), together with the
normalization condition Eq. (4) written in the form $f_{-1}=1-f_{-1}$, gives
us a third-order polynomial for $f_{1}$, namely

$2\,(q-1)\,f_{1}^{3}-3\,(q-1)\,f_{1}^{2}-f_{1}+p\,q=0~{}.$ (6)

The polynomial gives us three distinct solutions.

Let us first consider the specific case $p=0.5$. The mentioned three solutions
are given by

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{2}$ (7)
$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\frac{(q-1)\pm\sqrt{3q^{2}-4q+1}}{2(q-1)}~{}.$ (8)

The first solution, $f_{1}=1/2$, leads to $f_{-1}=1/2$ due to the
normalization condition, Eq. (4). Thus, $f_{1}=f_{-1}$ indicates a
paramagnetic phase. The other 2 solutions lead to $f_{1}\neq f_{-1}$, which
indicates a ferromagnetic phase, where one of the two opinions, $+1$ or $-1$,
is the majority opinion in the population. As Eq. (8) predicts two solutions
(see the $\pm$ signals), one has two curves for each value of $q$ since
$q<q_{c}$, where $q_{c}$ is the critical point, that will be obtained in the
following. When $f_{1}$ assumes one of the values in Eq. (8), consequently
$f_{-1}$ takes the other one [39].

The order parameter can be obtained from $M=|f_{1}-f_{-1}|$, which give us

$\displaystyle
M=\frac{\sqrt{3q^{2}-4q+1}}{q-1}=\frac{\sqrt{3(1-q)(1/3-q)}}{q-1}\sim(q_{c}-q)^{\beta}~{},$
(9)

where $q_{c}(p=1/2)=1/3$ and $\beta=1/2$. Eq. (9) is valid for $q<q_{c}$ and
it shows an order-disorder nonequilibrium phase transition at $q_{c}=1/3$, and
the mentioned transition appears to be in the Ising mean-field universality
class, as is usual in opinion models with two or three states [40].

For $p\neq 0.5$, the general solution of Eq. (6) leads to 2 complex conjugate
solutions, and the third solution is a real function for $f_{1}$. In such a
case, the absence of a solution $f_{1}=1/2$ indicates the absence of a
paramagnetic phase, and thus there is no phase transition since we always will
have $f_{1}\neq f_{-1}$, leading to a ferromagnetic state where one of the two
opinions is the majority opinion in the population ($M>0$). Thus, the model
only presents a phase transition for the case $p=1/2$.

## Acknowledgments

NC acknowledges financial support from the Brazilian funding agencies Conselho
Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Grant
310893/2020-8) and Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado
do Rio de Janeiro (FAPERJ, Grant 203.217/2017).

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#  Isoscalar, isovector and orbital contributions in $M1$ transitions from
analogous $M1$ and Gamow-Teller transitions in $T=\frac{1}{2}$ mirror nuclei

Subhrajit<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Roorkee, Roorkee 247667, India  P. C. Srivastava222
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Roorkee, Roorkee 247667, India

###### Abstract

The isoscalar and isovector components and their contributions to $M1$
transitions are discussed in the odd-$A$, $T=1/2$ mirror nuclei with mass
ranging from $A=23$ to 37. The orbital contributions in various $M1$
transitions and ground state magnetic moments are calculated by comparing
analogous $M1$ and Gamow-Teller transitions between mirror pairs. The orbital
contributions in different $M1$ transitions are explained on the basis of
configurations of the initial and final states involved. In magnetic moments,
the orbital contributions are found to be dependent on the deformation and
single-particle nature of the states. All the $T=1/2$ mirror pairs are studied
using isospin non-conserving interaction. The results are also compared with
predictions from ab initio effective interaction derived from realistic
nuclear forces.

###### pacs:

21.60.Cs, 23.20.-g, 21.10.Ky, 27.30.+t

##  I Introduction

The magnetic dipole ($M1$) and weak Gamow-Teller ($GT$) transitions provide
crucial information for understanding the structure and configurations of
atomic nuclei. The $GT$ transition is induced by the spin $\sigma\tau$ charge
exchange operator, while the $M1$ transition involves both orbital ($l$) and
spin ($\sigma$) angular momentum operators caused by the electromagnetic
interaction BohrMottelsonBook . Considering the charge-independent nature of
strong interaction or isospin symmetry, the $M1$ operator can be decomposed
into isoscalar ($l$ and $\sigma$) and isovector ($l\tau$ and $\sigma\tau$)
components where the isovector (IV) or particularly $\sigma\tau$ component is
dominant 27AlSi . So, the $M1$ and $GT$ operators have similar
characteristics, although they have different meson-exchange current
contributions Brown1987 ; Etchegoyen1988 ; Sagawa2018 ; Matsubara2021 . The
studies of similarities/differences between the $M1$ and $GT$ transitions
between analogous states and understanding their origins are essential for
gaining deeper insights into the microscopic nature of $M1$ transitions.

The mirror nuclei offer unique opportunities to investigate and analyze
different components of $M1$ transition. In the low-energy regime, the
structural properties of mirror nuclei are equivalent, and they exhibit
analogous states at similar excitation energies Wendt2014 ; AIPConf2005 ;
Lenzi2001 . Extensive investigations into $M1$ and $GT$ transitions between
analogous states have been conducted in various mirror nuclei or isospin
multiplets through different experimental methods, such as charge exchange
reactions FujitaReview ; Fujita_A27 ; 23NaMg ; 25MgAl ; Fujita_A26 ;
Fujita_A25 ; Shimbara_A37 ; Adachi2012 . The contributions of the IV orbital
term ($l\tau$) to $M1$ transitions, which can be obtained by comparing
analogous $M1$ and $GT$ transitions, have been in focus. The $l\tau$ term can
interfere constructively or destructively with the $\sigma\tau$ term and is
responsible for enhancement or reduction in $B(M1)$ strength compared to
$B(GT)$ strength. The experimental and theoretical analysis performed on some
$T=1/2$ mirror nuclei in Refs. 27AlSi ; 23NaMg ; 25MgAl  reveal that the
orbital contributions are largely dependent on the configurations involved in
the transitions and the structure of nuclei. Large orbital contributions are
observed in deformed nuclei.

In this article, we aim to study the analogous $M1$ and $GT$ transitions
between low-energy states in the $T_{z}=\pm 1/2$ odd-mass mirror pairs with
$A=23$ to 37 within $sd$\- model space. The IS and IV components are
calculated for various $M1$ transitions between low-lying states, and the
corresponding orbital contributions to the $M1$ transitions have been analyzed
in detail, followed by a similar discussion on ground state magnetic moments.
They are compared with available measured data and predictions are made where
experimental data are not available yet. The transitions involving large
orbital contributions and their configurations have been investigated in all
the $T=1/2$ mirror pairs.

The interacting shell model, which serves as one of the most powerful tools to
provide a unified description of both the structure and decays of atomic
nuclei, has been used for this purpose. However, it’s worth noting that most
shell model interactions are founded on isospin symmetry. But, the Coulomb
interaction between protons and other isospin-breaking effects of strong
interaction can change the energy of analogous states in mirror nuclei and
cause asymmetry to the $M1$ decays Lam2013 ; Kaneko2017 . Hence, we have
addressed the mirror pairs with the recently developed phenomenological $sd$\-
shell interaction USDC USDCint  that accounts for all such effects. Further,
the recent advances and development in the ab initio methods have enabled the
decoupling of effective interactions for valance space starting from the
underlying $NN$ and $NNN$ forces. The valance space effective interactions
derived using ab initio approaches like No Core Shell Model (NCSM)
ncsm_effint_sdshell , In-medium Similarity Renormalization Group (IM-SRG)
imsrg_ensembleReference ; imsrg_Miyagi , and Coupled Cluster theory (CC)
CCEIint  have shown promising results in explaining ground state, low-lying
excitation energies, electromagnetic, and other collective properties of open-
shell nuclei Priyanka2023 ; Na_work_NPA ; Archana2017 . These effective
interactions preserve the fundamental characteristics of strong nuclear forces
like charge symmetry, Coulomb interaction, and charge-dependent components of
nuclear forces. In this work, we have also discussed the $M1$, $GT$
transitions and their respective components from valance space effective
interaction derived using the Coupled Cluster approach and have provided the
ab initio predictions for the $T=1/2$ mirror nuclei.

The paper is organized in the following way. The formalism and important
expressions related to the evaluation of different components of $M1$
transitions are briefly described in Sec. II. The results for IS, IV
components, and orbital contributions in various $M1$ transitions and ground
state magnetic moments of $T=1/2$ mirror pairs are presented in Sec. III. The
summary and conclusions are given in Sec. IV.

##  II Formalism

To obtain the isoscalar (IS), isovector (IV) components, and other parameters
of $M1$ transition and magnetic moments, we have explicitly followed the
formalism as those described by Fujita et al. in Ref. 27AlSi  and in Refs.
23NaMg ; 25MgAl . In this section, we have briefly outlined the important
expressions for these quantities.

The $M1$ transition or magnetic moment operator $\vec{\mu}$ consisting of an
orbital part $g_{l}\vec{l}$ and spin part $(1/2)g_{s}\vec{\sigma}$ can be
written in terms of IS and IV components as

$\vec{\mu}=\sum_{i=1}^{A}\left[(g_{l}^{IS}\vec{l}_{i}+(1/2)g_{s}^{IS}\vec{\sigma}_{i})-(g_{l}^{IV}\vec{l}_{i}+(1/2)g_{s}^{IV}\vec{\sigma}_{i})\tau_{zi}\right]$
(1)

in the units of nuclear magneton ($\mu_{N}$). Here $\tau_{zi}=-1$ for protons
and +1 for neutrons. The IS and IV $g$\- factor coefficients are give by
$g_{l}^{IS}=\frac{1}{2}(g_{l}^{\pi}+g_{l}^{\nu})$,
$g_{s}^{IS}=\frac{1}{2}(g_{s}^{\pi}+g_{s}^{\nu})$,
$g_{l}^{IV}=\frac{1}{2}(g_{l}^{\pi}-g_{l}^{\nu})$, and
$g_{s}^{IV}=\frac{1}{2}(g_{s}^{\pi}-g_{s}^{\nu})$, where $g_{l}$ and $g_{s}$
corresponds to the orbital and spin $g$\- factors of bare protons ($\pi$) and
neutrons ($\nu$), respectively.

The $M1$ transition strength ($B(M1)$) between initial state with spin $J_{i}$
and isospin $T_{i}$ and final state with corresponding quantum number $J_{f}$
and $T_{f}$, respectively is given by

$B(M1)=\frac{1}{2J_{i}+1}\frac{3}{4\pi}{|\langle
J_{f}T_{f}T_{zf}\|\vec{\mu}\|J_{i}T_{i}T_{zi}\rangle|}^{2},$ (2)

and the magnetic moment of a state with spin $J$ and isospin $T$ is given by

$\mu=\sqrt{\frac{J}{(J+1)(2J+1)}}\langle JTT_{z}\|\vec{\mu}\|JTT_{z}\rangle.$
(3)

By applying the Wigner-Eckart theorem in the isospin space, the $B(M1)$
operator can be written as larry

$B(M1)=\frac{1}{2J_{i}+1}\frac{3}{4\pi}{\left[M^{IS}_{M1}-\frac{CG_{M1}}{\sqrt{2T_{f}+1}}M_{M1}^{IV}\right]}^{2},$
(4)

where $CG_{M1}=(T_{i}T_{zi}10|T_{f}T_{zf})$ is the Clebsch-Gordan (or CG)
coefficient. The IS and IV $M1$ matrix elements of Eq. (4) are given by

$M^{IS}_{M1}=g_{l}^{IS}M_{M1}(l)+\frac{1}{2}g_{s}^{IS}M_{M1}(\sigma)$ (5)

and

$M^{IV}_{M1}=g_{l}^{IV}M_{M1}(l\tau)+\frac{1}{2}g_{s}^{IV}M_{M1}(\sigma\tau).$
(6)

Here $M_{M1}(x)$ represents the reduced matrix elements defined by $\langle
J_{f}T_{f}\|\sum_{i=1}^{A}\vec{x}_{i}\|J_{i}T_{i}\rangle$ where
$\vec{x}=\\{{l,\sigma,l\tau,\sigma\tau\\}}$. The IV spin term and hence the
$M^{IV}_{M1}$ matrix element of $M1$ transition is usually larger than
$M^{IS}_{M1}$ due to the coefficient $g_{s}^{IV}$. Therefore, in the $M1$
transition, the IS term may interfere constructively or destructively with the
IV term. Additionally, when the IS term is very small, the orbital IV term (
$g_{l}^{IV}M_{M1}(l\tau)$ ) interferes constructively or destructively with
the IV spin term ( $\frac{1}{2}g_{s}^{IV}M_{M1}(\sigma\tau)$ ). The
contributions of the IS term or the orbital IV term to a $M1$ transition are
discussed later in this section.

A similar application of the Wigner-Eckart theorem to the magnetic moment
operator results

$\mu=\sqrt{\frac{J}{(J+1)(2J+1)}}\left[M^{IS}_{M1}-\frac{CG_{M1}}{\sqrt{2T+1}}M_{M1}^{IV}\right],$
(7)

where $CG_{M1}=(TT_{z}10|TT_{z})$.

For analogous $M1$ transitions between $T_{z}=\pm T$ mirror nuclei, the IS and
IV part of the $M1$ transition strengths are extracted from Eq. (4) as (see
Ref. 27AlSi )

$B_{IS}(M1)=\frac{1}{4}{\left[\sqrt{B(M1)}_{T_{z}=+T}-\sqrt{B(M1)}_{T_{z}=-T}\right]}^{2},$
(8)

and

$B_{IV}(M1)=\frac{1}{4}{\left[\sqrt{B(M1)}_{T_{z}=+T}+\sqrt{B(M1)}_{T_{z}=-T}\right]}^{2}.$
(9)

Similarly, using Eq. (7), the IS and IV components of the magnetic moment can
be written as $\mu_{IS}=\frac{1}{2}(\mu_{T_{z}=T}+\mu_{T_{z}=-T})$ and
$\mu_{IV}=\frac{1}{2}(\mu_{T_{z}=T}-\mu_{T_{z}=-T})$.

The IS contribution to the $M1$ transitions can be analyzed through a
parameter $R_{IS}$ as given below;

$R_{IS}=\frac{B(M1)}{B_{IV}(M1)}.$ (10)

The value of $R_{IS}$ larger than unity indicates that the IS term
constructively interferes with the IV term and vice-versa. For the magnetic
moment, the $R_{IS}$ parameter can be written as

$R_{IS}={\left(\frac{\mu}{\mu_{IV}}\right)}^{2}.$ (11)

The Gamow-Teller ($GT$) transition strength between initial (i) and final (f)
states with $J$ and $T$ as good quantum numbers is given by

$B(GT)=\frac{1}{2}\frac{1}{2J_{i}+1}\frac{CG_{GT}^{2}}{2T_{f}+1}{\left[M_{GT}(\sigma\tau)\right]}^{2}$
(12)

where $M_{GT}(\sigma\tau)=\langle
J_{f}T_{f}\|\sum_{i=1}^{A}\vec{\sigma}_{i}\vec{\tau}_{i}\|J_{i}T_{i}\rangle$
and $CG_{GT}=(T_{i}T_{zi}1\pm 1|T_{f}T_{zf})$ is the CG coefficient. For
analogous states, the IV spin operator ($\sigma\tau$) is similar in both $M1$
and $GT$ transition. However, the contributions of meson exchange currents (or
MEC) are different for $M1$ and $GT$ operators. So, for $M1$ and $GT$
transitions between analogous states, a relation can be established between
the IV spin type $GT$ matrix ($M_{GT}(\sigma\tau)$) and the IV spin term of
$M1$ transition ($M_{M1}(\tau\sigma)$) through a parameter $R_{MEC}$ defined
as

$R_{MEC}=\frac{{[M_{M1}(\sigma\tau)]}^{2}}{{[M_{GT}(\sigma\tau)]}^{2}}.$ (13)

The MEC contributions to $M1$ and $GT$ operators, including core polarization
effects, are taken care of by the parameter $R_{MEC}$ (for details, see 27AlSi
). This implies, when $M_{M1}(\sigma\tau)$ is dominant in $M1$ transition, the
$B(M1)$ value in one of the mirror pair should be
$B(M1)\approx\frac{3}{8\pi}{(g_{s}^{IV})}^{2}\frac{CG_{M1}^{2}}{CG_{GT}^{2}}R_{MEC}B(GT)$
23NaMg . Any significant deviation from this value will imply constructive or
destructive interference of the orbital term with the spin term in
$M^{IV}_{M1}$ element. The orbital contribution to the IV term of $M1$
transition can be readily identified by the following ratio 27AlSi ;

$R_{oc}=\frac{8\pi}{3{(g_{s}^{IV})}^{2}}\frac{1}{R_{MEC}}\frac{CG^{2}_{GT}}{CG^{2}_{M1}}\frac{B_{IV}(M1)}{B(GT)},$
(14)

and for magnetic momentum $R_{oc}$ becomes

$R_{oc}=\frac{2(J+1)}{J{(g_{s}^{IV})}^{2}}\frac{CG^{2}_{GT}}{CG^{2}_{M1}}\frac{1}{R_{MEC}}\frac{\mu^{2}_{IV}}{B(GT)},$
(15)

where the squared of CG co-efficient is 2 for $T=1/2$ nuclei. Here, $R_{oc}>1$
indicates that the orbital part makes constructive interference with the spin
term in $M_{M1}^{IV}$ to enhance the $M1$ transition and vice versa unless the
transition strength is very small where the spin term is hindered and much
smaller than the orbital one, i.e.,
$|g_{l}^{IV}M_{M1}(l\tau)|>>|\frac{1}{2}g_{s}^{IV}M_{M1}(\tau\sigma)|$.

To find the orbital contribution to the $M1$ operator or $R_{oc}$, one needs
to calculate the IV component of $B(M1)$ transition. However, in many cases,
for analogous $M1$ transitions in mirror pairs, the $B(M1)$ strengths are not
available or not yet measured in one of the nuclei within mirror pairs. In
such cases, similar $B(M1)$ strengths are expected in mirror pairs due to
isospin symmetry, and without necessarily determining $B_{IV}(M1)$, the
combined contribution of IS and orbital term to the IV spin term can be
inferred through a parameter $R_{ISO}$ as follows 23NaMg ; 25MgAl

$R_{ISO}=\frac{1}{2.644}\frac{CG^{2}_{GT}}{CG^{2}_{M1}}\frac{1}{R_{MEC}}\frac{B(M1)}{B(GT)}.$
(16)

Since the IS contributions are usually very small, the $R_{ISO}$ values
closely resemble with $R_{oc}$.

In the experimental works 23NaMg ; 27AlSi ; 25MgAl , the average value of
$R_{MEC}$ was taken as 1.25 for $sd$ shell nuclei and the IS, IV components of
$M1$ transition or magnetic moment, $R_{oc}$ and $R_{ISO}$ parameters have
been deduced. In order to facilitate the comparison between our calculated
results and these experimental findings, we have adopted the same value for
$R_{MEC}$ and $g_{s}^{IV}=4.706$ in our calculations. In particular, Eq. (8)
to (11) and Eq. (14) to (16) are used to calculate various components of $M1$
transitions and different parameters, as described above.

In the framework of shell-model, the $M1$ and $GT$ operators for the mirror
pairs are calculated from universal $sd$\- shell variant or USD-based USDC
interaction. Starting from a renormalized G-matrix approach Gmatrix , the two-
body matrix elements (TBMEs) of USD-based interactions are determined and
adjusted to fit several experimental datasets USDA_USDB . The USDC interaction
incorporates Coulomb interaction between protons and other isospin-breaking
parts of the strong interaction USDCint . The g.s., low-lying excited states,
and other isospin-symmetry breaking effects are well captured by the USDC
interaction in most of the $sd$\- shell nuclei within experimental uncertainty
compared to other existing interactions. The $M1$ and $GT$ components are then
analyzed from an ab initio coupled-cluster effective interaction (CCEI)
derived from realistic chiral $NN$ and $NNN$ forces using the coupled-cluster
approach CCEIint . The CCEI is decoupled nonperturbatively from the chiral
potentials N3LO ; N2LO  for open-shell nuclei with valance protons and
neutrons and retains the properties of the original nucleon-nucleon
interaction. The CCEI interaction provides a good description of the ground
state binding energies as well as collective behaviors of open-shell nuclei
from the first principles. The shell model Hamiltonians are diagonalized using
NushellX nushellx  and KSHELL kshell .

##  III Results and Discussion

In the shell-model approach, various factors, such as the effects of core
polarization, model space truncation, MEC contributions etc, are taken into
account by introducing effective model space operators Brown1987 ;
Etchegoyen1988 . Such studies had been performed with USD-type Hamiltonians
(USD, USDA, and USDB) in Ref. effE2M1GT  and effective $E2$, $M1$, and $GT$
operators or effective charges, $g$\- factors ($g^{eff}$), and quenching
factor ($q$), respectively were proposed for these Hamiltonians. It was found
that the operators are less sensitive to the USD Hamiltonians, with their
values remaining approximately the same in all USD-type Hamiltonians.
Following this observation, we have used the same effective $g$\- factors
(${(g_{l}^{\pi})}^{eff}=1.159,{(g_{l}^{\nu})}^{eff}=-0.09$ and
${(g_{s}^{\pi})}^{eff}=5.15,{(g_{l}^{\nu})}^{eff}=-3.55$) and $q=0.76$, as
determined for the USDB Hamiltonian, to calculate the $M1$ and $GT$ transition
matrix elements from USD-based USDC interaction. The analysis of $GT$
strengths performed in Ref. qfactorCCEI  using CCEI yields $q=0.78$,
approximately the same as that obtained for USDB. Since the major components
of $M1$ and $GT$ transitions are similar ( $\sigma\tau$ ), identical effective
$g$\- factors can be expected for CCEI as those found in the USDB Hamiltonian.
Hence, to ensure consistency, we have employed the same effective $g$\- and
quenching factors for CCEI as those employed for USDC.

The mirror nuclei exhibit analogous states ($J^{\pi}$) with similar excitation
energies ($E_{x}$). The $M1$ transition strengths between the ground state
(g.s.) and lower excited states within the mirror nuclei and the $GT$
transition strengths for the analogous transitions between the mirror pair are
calculated for $T=1/2$ nuclei from $A=23$ to $37$ in Table 1. The IS and IV
components of the $M1$ transitions ($B_{IS}(M1)$ and $B_{IV}(M1)$), IS
contributions in each mirror nuclei ($R_{IS}$) and orbital contribution to the
IV term of $M1$ transitions ($R_{oc}$ or $R_{ISO}$) are listed in different
columns of Table 1. The first row ( or the transition from g.s. to g.s. ) in
each mirror pair represents the magnetic moment ($\mu$) instead of
$B(M1)\uparrow$, thus indicating the IS ($\mu_{IS}$) and IV components
($\mu_{IV}$), along with IS and orbital contributions to the g.s. magnetic
moments. The phenomenological interactions have supremacy in describing the
experimental behaviors and are always used as a benchmark. So, we have made a
one-to-one comparison of the experimental data wherever available, with the
results obtained from the USDC interaction in Table 1. Then, the predictions
of ab initio CCEI are compared with the USDC results through various figures.
First, we have discussed the $M1$ components in each mirror pair, followed by
an analysis of the g.s. magnetic moments and their corresponding components
for all mirror pairs within the $sd$-shell.

###  III.1 $\mathbf{{}^{23}Na\rightarrow^{23}Mg}$

The $M1$ transition strengths in the 23Na and 23Mg nuclei and the
corresponding $GT$ strengths for analogous transitions between the mirror pair
are well reproduced from the USDC interaction. As discussed earlier, the IS
components are found to be much smaller than the IV components. For stronger
$M1$ transitions i.e., from $3/2_{1}^{+}$ to $5/2_{1}^{+}$, $1/2_{2}^{+}$ and
$5/2_{3}^{+}$, the $GT$ strengths are much smaller than the $B(M1)$ values.
For these cases, the $R_{IS}$ values remain approximately 1, suggesting that
the IS contribution is small. This implies that, it is the constructive
interference between the orbital and spin IV terms which enhances the $M1$
transition strengths. The $R_{oc}$ or $R_{ISO}$ values of these transitions
also reflect the same. The calculated $R_{ISO}$ value is smaller than the
measured value in $3/2_{1}^{+}\rightarrow 5/2_{3}^{+}$, which is associated
with large experimental uncertainty. Due to the lack of experimental data, the
measured $R_{IS}$ values are available only for a few transitions. Further,
except $3/2_{1}^{+}\rightarrow 5/2_{1}^{+}$ transition, only the $R_{ISO}$
values have been deduced for all transitions from the experimental data 23NaMg
due to the unavailability of $B(M1)$ strengths in 23Mg. We have calculated
$R_{IS}$, $R_{oc}$, and $R_{ISO}$ parameters for all transitions corresponding
to the experimental data. In the stronger transitions, the IS term has a small
effect around $8-10\%$, while it modifies the $M1$ strengths by $\sim 20\%$ in
other cases. In the transitions $3/2_{1}^{+}\rightarrow 3/2_{2}^{+}$ and
$3/2_{1}^{+}\rightarrow 5/2_{2}^{+}$, the $R_{oc}$ values suggest that the
orbital parts do not interfere constructively with the spin parts in the IV
$M1$ term. The 23Na - 23Mg nuclei are largely deformed, and the orbital
contributions to different transitions in these nuclei were discussed using
deformed Nilsson’s model and particle rotor model in Ref. 23NaMg . The
deformed structure and collective properties of these isotopes can be well
explained from the shell model theory E2A23 . In the framework of the shell
model, the energy eigenstates arise from the mixing of various configurations
involving $sd$ -shell orbitals in deformed nuclei. In 23Na, the ground state
emerges from a configuration $\pi(d_{5/2}^{3})\otimes\nu(d_{5/2}^{4})$. The
major components of the wave functions of the states $5/2_{1}^{+}$,
$1/2_{2}^{+}$, and $5/2_{3}^{+}$ are also the same. The transitions from the
g.s. to these states are associated with large positive orbital contributions.
Due to isospin symmetry, identical structures are found in the mirror partner
23Mg with proton and neutron configurations interchanged. Similar
interpretations can be made for 23Mg.

Figure 1: Upper panel: $R_{oc}$ (filled triangles) or $R_{ISO}$ (unfilled
triangles) of different transitions. Lower panel: The black, magenta, and
green diamonds correspond to the Expt., USDC, and CCEI data. The filled
(unfilled) symbols refer to the $T_{z}=1/2$ ($T_{z}=-1/2$) nucleus in the
$A=23$ pair.

The ab initio results are compared with the USDC results in Fig. 1. They lie
very close to the experimental data and results obtained from the USDC
interaction. Large deviation arises only in the case of
$3/2_{1}^{+}\rightarrow 1/2_{2}^{+}$. For this transition, a large $R_{ISO}$
value or a relatively much smaller $GT$ transition strength is observed
compared to the $B(M1)$ values in the CCEI interaction. The orbital
contribution is solely responsible for such enhancement in the $M1$ transition
strength since the IS contribution from CCEI interaction remains the same as
that of USDC. From the experiment and USDC interaction, a very weak $M1$
strength is observed for the transition $3/2_{1}^{+}\rightarrow 1/2_{1}^{+}$.
The deduced $R_{oc}$ and $R_{IS}$ values of such weak transitions are not
reliable 23NaMg  and are not shown in Fig. 1. But in CCEI, the $M1$ strengths
for this transition are not negligible, and its prediction is depicted in Fig.
1.

Table 1:  Experimental and calculated data for $B(M1)\uparrow$, $B(GT)$, and their corresponding components in $T=\frac{1}{2}$ mirror nuclei. For more details, see text.  |  | $E_{x}$ |  | $B(M1)\uparrow$ |  |  |  | $R_{IS}$ |
---|---|---|---|---|---|---|---|---|---
$\mathbf{{}^{23}Na\rightarrow^{23}Mg}$ |  |  |  |  |  |  |  |  |  |  |  |
Expt. 23NaMg  |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  23Na |  23Mg |  |  23Na |  23Mg |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  23Na |  23Mg |  $R_{oc}$($R_{ISO}$)
23 | $3/2^{+}_{1}$ | 0.0 | 0.0 |  | 2.218 111Represents $\mu$ | -0.536 111Represents $\mu$ | 0.190(4) | 0.841 222Represents $\mu_{IS}$ | 1.337 333Represents $\mu_{IV}$ | 2.59 | 0.15 | 2.3(1)
| $5/2^{+}_{1}$ | 0.440 | 0.459 |  | 0.554(34) | 0.591(64) | 0.146(6) | $1\times 10^{-4}$ | 0.57(4) | 0.97(7) | 1.03(13) | 2.4(2)
| $1/2^{+}_{1}$ | 2.391 | 2.359 |  | 0.0017(3) | 0.0017(4) | 0.055(4) | $1\times 10^{-8}$ | 0.0017(3) | 1.0(2) | 1.0(2) | 0.019(3)
| $3/2^{+}_{2}$ | 2.982 | 2.906 |  | 0.292(41) |  | 0.193(11) |  |  |  |  | 0.92(14) 444Represents $R_{ISO}$
| $5/2^{+}_{2}$ | 3.914 | 3.860 |  | 0.090(15) |  | 0.055(4) |  |  |  |  | 0.99(18) 444Represents $R_{ISO}$
| $1/2^{+}_{2}$ | 4.430 | 4.357 |  | 1.02(7) |  | 0.250(13) |  |  |  |  | 2.48(22) 444Represents $R_{ISO}$
| $5/2^{+}_{3}$ | 5.379 | 5.291 |  | 0.33(12) |  | 0.066(5) |  |  |  |  | 3.0(11) 444Represents $R_{ISO}$
USDC |  |  |  |  |  |  |  |  |  |  |  |
23 | $3/2^{+}_{1}$ | 0.0 | 0.0 |  | 2.162 111Represents $\mu$ | -0.396 111Represents $\mu$ | 0.164 | 0.883 222Represents $\mu_{IS}$ | 1.279 333Represents $\mu_{IV}$ | 2.85 | 0.09 | 2.40
| $5/2^{+}_{1}$ | 0.408 | 0.397 |  | 0.537 | 0.457 | 0.134 | $8\times 10^{-4}$ | 0.496 | 1.08 | 0.92 | 2.24
| $1/2^{+}_{1}$ | 2.190 | 2.162 |  | 0.020 | 0.002 | 0.071 | $2\times 10^{-3}$ | 0.008 | 2.31 | 0.23 | 0.07
| $3/2^{+}_{2}$ | 2.751 | 2.746 |  | 0.333 | 0.230 | 0.207 | $2\times 10^{-3}$ | 0.280 | 1.19 | 0.82 | 0.82(0.97 444Represents $R_{ISO}$)
| $5/2^{+}_{2}$ | 3.748 | 3.739 |  | 0.058 | 0.038 | 0.046 | $5\times 10^{-4}$ | 0.047 | 1.22 | 0.80 | 0.62(0.76 444Represents $R_{ISO}$)
| $1/2^{+}_{2}$ | 4.414 | 4.425 |  | 1.047 | 0.863 | 0.241 | $2\times 10^{-3}$ | 0.953 | 1.09 | 0.91 | 2.40(2.63 444Represents $R_{ISO}$)
| $5/2^{+}_{3}$ | 5.363 | 5.367 |  | 0.345 | 0.297 | 0.125 | $4\times 10^{-4}$ | 0.321 | 1.07 | 0.93 | 1.55(1.67 444Represents $R_{ISO}$)
$\mathbf{{}^{25}Mg\rightarrow^{25}Al}$ |  |  |  |  |  |  |  |  |  |  |  |
Expt. 25MgAl  |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  25Mg |  25Al |  |  25Mg |  25Al |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  25Mg |  25Al |  $R_{oc}$
25 | $5/2^{+}_{1}$ | 0.0 | 0.0 |  | -0.855 111Represents $\mu$ | 3.645 111Represents $\mu$ | 0.408(2) | 1.395 222Represents $\mu_{IS}$ | -2.250 333Represents $\mu_{IV}$ | 0.14 | 2.62 | 2.51
| $3/2^{+}_{1}$ | 0.975 | 0.945 |  | 0.0011(1) |  | 0.003(1) |  |  |  |  |
| $7/2^{+}_{1}$ | 1.612 | 1.613 |  | 0.83(12) |  | 0.165(7) |  |  |  |  | 3.1(8) 444Represents $R_{ISO}$
| $5/2^{+}_{2}$ | 1.790 | 1.965 |  | 0.0014(2) |  | 0.019(2) |  |  |  |  |
| $3/2^{+}_{2}$ | 2.674 | 2.801 |  | 0.007(2) |  | 0.017(2) |  |  |  |  |
USDC |  |  |  |  |  |  |  |  |  |  |  |
25 | $5/2^{+}_{1}$ | 0.0 | 0.0 |  | -0.861 111Represents $\mu$ | 3.757 111Represents $\mu$ | 0.360 | 1.448 222Represents $\mu_{IS}$ | -2.309 333Represents $\mu_{IV}$ | 0.14 | 2.65 | 3.0
| $3/2^{+}_{1}$ | 1.072 | 1.015 |  | 0.003 | 0.005 | 0.003 | $6\times 10^{-5}$ | 0.004 | 0.76 | 1.27 | 0.80
| $7/2^{+}_{1}$ | 1.708 | 1.723 |  | 0.810 | 0.881 | 0.131 | $4\times 10^{-4}$ | 0.845 | 0.95 | 1.04 | 3.90(3.70 444Represents $R_{ISO}$)
| $5/2^{+}_{2}$ | 2.012 | 1.882 |  | 0.003 | 0.002 | $7\times 10^{-5}$ | $2\times 10^{-5}$ | 0.002 | 1.21 | 0.81 | 21.40
| $3/2^{+}_{2}$ | 2.834 | 2.739 |  | 0.014 | 0.019 | 0.010 | $1\times 10^{-4}$ | 0.016 | 0.85 | 1.15 | 0.99
$\mathbf{{}^{27}Al\rightarrow^{27}Si}$ |  |  |  |  |  |  |  |  |  |  |  |
Expt. 27AlSi  |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  27Al |  27Si |  |  27Al |  27Si |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  27Al |  27Si |  $R_{oc}$
27 | $5/2^{+}_{1}$ | 0.0 | 0.0 |  | 3.642 111Represents $\mu$ | -0.865 111Represents $\mu$ | 0.307(44) | 1.388 222Represents $\mu_{IS}$ | 2.254 333Represents $\mu_{IV}$ | 2.6 | 0.15 | 3.3(5)
| $3/2^{+}_{1}$ | 1.014 | 0.957 |  | 0.015(1) | 0.019(1) | 2.2(3) $\times 10^{-4}$ | 6(5)$\times 10^{-5}$ | 0.017(1) | 0.89(7) | 1.12(9) | 51.3(80)
| $7/2^{+}_{1}$ | 2.211 | 2.164 |  | 0.150(4) | 0.102(10) | 0.079(6) | 1.2(5)$\times 10^{-3}$ | 0.125(6) | 1.20(6) | 0.82(8) | 0.96(12)
| $5/2^{+}_{2}$ | 2.735 | 2.648 |  | 0.046(7) | 0.022(5) | 0.039(4) | 1.1(8)$\times 10^{-3}$ | 0.033(4) | 1.40(28) | 0.67(18) | 0.51(10)
| $3/2^{+}_{2}$ | 2.982 | 2.866 |  | 0.245(13) |  | 0.173(12) |  |  |  |  | 0.85(3) 444Represents $R_{ISO}$
| $3/2^{+}_{3}$ | 3.957 | 3.804 |  | 0.145(12) |  | 0.079(7) |  |  |  |  | 1.11(1) 444Represents $R_{ISO}$
| $5/2^{+}_{3}$ | 4.410 | 4.289 |  | 0.226(28) | 0.075(37) | 0.097(9) | 0.010(7) | 0.140(28) | 1.61(37) | 0.53(28) | 0.88(19)
USDC |  |  |  |  |  |  |  |  |  |  |  |
27 | $5/2^{+}_{1}$ | 0.0 | 0.0 |  | 3.576 111Represents $\mu$ | -0.710 111Represents $\mu$ | 0.275 | 1.433 222Represents $\mu_{IS}$ | 2.143 333Represents $\mu_{IV}$ | 2.78 | 0.11 | 3.38
| $3/2^{+}_{1}$ | 1.063 | 1.117 |  | 0.001 | 0.007 | 0.003 | $7\times 10^{-4}$ | 0.003 | 0.30 | 2.10 | 0.67
| $7/2^{+}_{1}$ | 2.289 | 2.346 |  | 0.128 | 0.100 | 0.075 | $4\times 10^{-4}$ | 0.113 | 1.13 | 0.88 | 0.92
| $5/2^{+}_{2}$ | 2.692 | 2.733 |  | 0.074 | 0.050 | 0.058 | $6\times 10^{-4}$ | 0.061 | 1.20 | 0.82 | 0.64
| $3/2^{+}_{2}$ | 2.843 | 2.835 |  | 0.347 | 0.242 | 0.206 | $2\times 10^{-3}$ | 0.292 | 1.19 | 0.83 | 0.86(1.02 444Represents $R_{ISO}$)
| $3/2^{+}_{3}$ | 3.983 | 3.951 |  | 0.204 | 0.110 | 0.067 | $3\times 10^{-3}$ | 0.153 | 1.33 | 0.72 | 1.38(1.84 444Represents $R_{ISO}$ )
| $5/2^{+}_{3}$ | 4.367 | 4.432 |  | 0.300 | 0.220 | 0.090 | $1\times 10^{-3}$ | 0.258 | 1.16 | 0.85 | 1.74
| $7/2^{+}_{2}$ | 4.675 | 4.689 |  | 0.064 | 0.037 | 0.019 | $9\times 10^{-4}$ | 0.050 | 1.30 | 0.74 | 1.58
Table 1:  Continued. |  | $E_{x}$ |  | $B(M1)\uparrow$ |  |  |  | $R_{IS}$ |
---|---|---|---|---|---|---|---|---|---
$\mathbf{{}^{29}Si\rightarrow^{29}P}$ |  |  |  |  |  |  |  |  |  |  |  |
Expt. NNDC  |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  29Si |  29P |  |  29Si |  29P |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  29Si |  29P |  $R_{oc}$
29 | $1/2^{+}_{1}$ | 0.0 | 0.0 |  | -0.556 111Represents $\mu$ | 1.235 111Represents $\mu$ |  | 0.340 222Represents $\mu_{IS}$ | -0.895 333Represents $\mu_{IV}$ | 0.38 | 1.90 |
| $3/2^{+}_{1}$ | 1.273 | 1.383 |  | 0.127(9) | 0.193(50) |  | $2\times 10^{-3}$ | 0.158(50) | 0.80(19) | 1.22(21) |
| $3/2^{+}_{2}$ | 2.425 | 2.422 |  | 0.233(28) | 0.233(78) |  |  |  |  |  |
| $1/2^{+}_{2}$ | 4.840 | 4.759 |  |  $>$0.086 | NA |  |  |  |  |  |
USDC |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  29Si |  29P |  |  29Si |  29P |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  29Si |  29P |  $R_{oc}$
29 | $1/2^{+}_{1}$ | 0.0 | 0.0 |  | -0.500 111Represents $\mu$ | 1.144 111Represents $\mu$ | 0.141 | 0.322 222Represents $\mu_{IS}$ | -0.822 333Represents $\mu_{IV}$ | 0.37 | 1.94 | 2.07
| $3/2^{+}_{1}$ | 1.263 | 1.337 |  | 0.031 | 0.051 | 0.026 | $6\times 10^{-4}$ | 0.040 | 0.77 | 1.26 | 0.94
| $3/2^{+}_{2}$ | 2.505 | 2.579 |  | 0.422 | 0.585 | 0.322 | $3\times 10^{-3}$ | 0.500 | 0.84 | 1.17 | 0.94
| $1/2^{+}_{2}$ | 4.760 | 4.756 |  | 0.268 | 0.437 | 0.242 | $5\times 10^{-3}$ | 0.347 | 0.77 | 1.25 | 0.87
| $3/2^{+}_{3}$ | 6.075 | 6.129 |  | 0.004 | $1\times 10^{-4}$ | 0.003 | $7\times 10^{-4}$ | 0.001 | 2.98 | 0.07 | 0.27
| $1/2^{+}_{3}$ | 6.656 | 6.717 |  | 0.154 | 0.283 | 0.112 | $5\times 10^{-3}$ | 0.213 | 0.72 | 1.32 | 1.15
$\mathbf{{}^{31}P\rightarrow^{31}S}$ |  |  |  |  |  |  |  |  |  |  |  |
Expt. NNDC  |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  31P |  31S |  |  31P |  31S |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  31P |  31S |  $R_{oc}$
31 | $1/2^{+}_{1}$ | 0.0 | 0.0 |  | 1.131 111Represents $\mu$ | (-)0.488 111Represents $\mu$ |  | 0.325 222Represents $\mu_{IS}$ | 0.810 333Represents $\mu_{IV}$ | 1.95 | 0.36 |
| $3/2^{+}_{1}$ | 1.266 | 1.249 |  | 0.070(4) | 0.072(18) |  | $3\times 10^{-6}$ | 0.071(12) | 0.98(20) | 1.01(20) |
| $1/2^{+}_{2}$ | 3.134 | 3.079 |  | 0.175(78) | NA |  |  |  |  |  |
| $3/2^{+}_{2}$ | 3.506 | 3.436 |  | 0.104(16) | NA |  |  |  |  |  |
USDC |  |  |  |  |  |  |  |  |  |  |  |
31 | $1/2^{+}_{1}$ | 0.0 | 0.0 |  | 1.047 111Represents $\mu$ | -0.427 111Represents $\mu$ | 0.181 | 0.310 222Represents $\mu_{IS}$ | 0.737 333Represents $\mu_{IV}$ | 2.02 | 0.33 | 1.30
| $3/2^{+}_{1}$ | 1.179 | 1.186 |  | 0.019 | 0.007 | 0.037 | $7\times 10^{-4}$ | 0.012 | 1.55 | 0.57 | 0.20
| $1/2^{+}_{2}$ | 3.268 | 3.272 |  | 0.216 | 0.164 | 0.041 | $9\times 10^{-4}$ | 0.190 | 1.14 | 0.87 | 2.80
| $3/2^{+}_{2}$ | 3.605 | 3.644 |  | 0.195 | 0.113 | 0.153 | $3\times 10^{-3}$ | 0.151 | 1.29 | 0.75 | 0.60
| $3/2^{+}_{3}$ | 4.392 | 4.412 |  | 0.017 | 0.014 | 0.015 | $3\times 10^{-5}$ | 0.015 | 1.10 | 0.90 | 0.62
| $1/2^{+}_{3}$ | 5.013 | 5.020 |  | 0.112 | 0.128 | 0.046 | $1\times 10^{-4}$ | 0.120 | 0.93 | 1.06 | 1.57
$\mathbf{{}^{33}S\rightarrow^{33}Cl}$ |  |  |  |  |  |  |  |  |  |  |  |
Expt. NNDC  |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  33S |  33Cl |  |  33S |  33Cl |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  33S |  33Cl |  $R_{oc}$
33 | $3/2^{+}_{1}$ | 0.0 | 0.0 |  | 0.644 111Represents $\mu$ | 0.755 111Represents $\mu$ |  | 0.700 222Represents $\mu_{IS}$ | -0.055 333Represents $\mu_{IV}$ | 134.64 | 185.05 |
| $1/2^{+}_{1}$ | 0.841 | 0.810 |  | 0.027(4) | 0.015(5) |  | $4\times 10^{-4}$ | 0.021(9) | 1.31(22) | 0.73(16) |
| $5/2^{+}_{1}$ | 1.967 | 1.986 |  | 0.057(21) | 0.108(48) |  | 0.002 | 0.080(66) | 0.71(8) | 1.34(10) |
| $3/2^{+}_{2}$ | 2.314 | 2.352 |  | 0.0077(14) | 0.004(2) |  |  |  |  |  |
| $5/2^{+}_{2}$ | 2.867 | 2.839 |  | 0.214(162) | 0.465(120) |  | 0.012 | 0.327(308) | 0.65(48) | 1.42(84) |
USDC |  |  |  |  |  |  |  |  |  |  |  |
33 | $3/2^{+}_{1}$ | 0.0 | 0.0 |  | 0.664 111Represents $\mu$ | 0.841 111Represents $\mu$ | 0.099 | 0.752 222Represents $\mu_{IS}$ | -0.088 333Represents $\mu_{IV}$ | 56.30 | 90.30 | 0.02
| $1/2^{+}_{1}$ | 0.893 | 0.899 |  | 0.011 | 0.021 | 0.007 | $4\times 10^{-4}$ | 0.015 | 0.71 | 1.35 | 1.35
| $5/2^{+}_{1}$ | 1.956 | 1.965 |  | 0.016 | 0.031 | 0.021 | $6\times 10^{-4}$ | 0.023 | 0.70 | 1.35 | 0.66
| $3/2^{+}_{2}$ | 2.326 | 2.345 |  | 0.001 | $1\times 10^{-4}$ | 0.007 | $1\times 10^{-4}$ | $4\times 10^{-4}$ | 2.31 | 0.23 | 0.04
| $5/2^{+}_{2}$ | 2.952 | 2.948 |  | 0.408 | 0.566 | 0.207 | $3\times 10^{-3}$ | 0.484 | 0.84 | 1.17 | 1.41
| $1/2^{+}_{2}$ | 3.928 | 3.926 |  | 0.036 | 0.034 | 0.017 | $7\times 10^{-6}$ | 0.035 | 1.03 | 0.97 | 1.24
$\mathbf{{}^{35}Cl\rightarrow^{35}Ar}$ |  |  |  |  |  |  |  |  |  |  |  |
Expt. NNDC  |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  35Cl |  35Ar |  |  35Cl |  35Ar |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  35Cl |  35Ar |  $R_{oc}$
35 | $3/2^{+}_{1}$ | 0.0 | 0.0 |  | 0.822 111Represents $\mu$ | NA |  |  |  |  |  |
| $1/2^{+}_{1}$ | 1.220 | 1.184 |  | 0.085(2) | NA |  |  |  |  |  |
| $5/2^{+}_{1}$ | 1.763 | 1.450 |  | 0.003(3) | NA |  |  |  |  |  |
Table 1:  Continued. |  | $E_{x}$ |  | $B(M1)\uparrow$ |  |  |  | $R_{IS}$ |
---|---|---|---|---|---|---|---|---|---
USDC |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  35Cl |  35Ar |  |  35Cl |  35Ar |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  37Ar |  37K |  $R_{oc}$
35 | $3/2^{+}_{1}$ | 0.0 | 0.0 |  | 0.878 111Represents $\mu$ | 0.636 111Represents $\mu$ | 0.077 | 0.757 222Represents $\mu_{IS}$ | 0.121 333Represents $\mu_{IV}$ | 52.65 | 27.63 | 0.04
| $1/2^{+}_{1}$ | 1.244 | 1.150 |  | 0.027 | 0.017 | 0.026 | $3\times 10^{-4}$ | 0.022 | 1.24 | 0.78 | 0.50
| $5/2^{+}_{1}$ | 1.725 | 1.706 |  | 0.005 | 0.008 | 0.027 | $9\times 10^{-5}$ | 0.006 | 0.78 | 1.25 | 0.14
| $3/2^{+}_{2}$ | 2.688 | 2.601 |  | 0.050 | 0.033 | 0.033 | $4\times 10^{-4}$ | 0.041 | 1.22 | 0.80 | 0.75
| $5/2^{+}_{2}$ | 3.091 | 3.059 |  | 0.059 | 0.036 | 0.014 | $7\times 10^{-4}$ | 0.047 | 1.26 | 0.77 | 2.02
$\mathbf{{}^{37}Ar\rightarrow^{37}K}$ |  |  |  |  |  |  |  |  |  |  |  |
Expt. NNDC  |  |  |  |  |  |  |  |  |  |  |  |
$A$ | $J^{\pi}$ |  37Ar |  37K |  |  37Ar |  37K |  $B(GT)$ |  $B_{IS}(M1)$ |  $B_{IV}(M1)$ |  37Ar |  37K |  $R_{oc}$
37 | $3/2^{+}_{1}$ | 0.0 | 0.0 |  | 1.145(5) 111Represents $\mu$ | 0.203 111Represents $\mu$ |  | 0.674 222Represents $\mu_{IS}$ | 0.471 333Represents $\mu_{IV}$ | 5.91 | 0.18 |
| $1/2^{+}_{1}$ | 1.409 | 1.370 |  | NA | NA |  |  |  |  |  |
| $5/2^{+}_{1}$ | 2.796 | 2.750 |  | 0.185(107) | 1.96(86) |  |  |  |  |  |
| $5/2^{+}_{2}$ | 3.170 | 3.239 |  | 0.021(6) | NA |  |  |  |  |  |
USDC |  |  |  |  |  |  |  |  |  |  |  |
37 | $3/2^{+}_{1}$ | 0.0 | 0.0 |  | 1.107 111Represents $\mu$ | 0.330 111Represents $\mu$ | 0.232 | 0.718 222Represents $\mu_{IS}$ | 0.388 333Represents $\mu_{IV}$ | 8.12 | 0.72 | 0.15
| $1/2^{+}_{1}$ | 1.439 | 1.437 |  | 0.016 | 0.014 | $4\times 10^{-4}$ | $1\times 10^{-5}$ | 0.015 | 1.06 | 0.93 | 22.67
| $5/2^{+}_{1}$ | 2.729 | 2.766 |  | 0.273 | 0.447 | 0.504 | 0.005 | 0.354 | 0.77 | 1.26 | 0.42
| $5/2^{+}_{2}$ | 3.140 | 3.165 |  | 0.465 | 0.550 | 0.148 | $9\times 10^{-4}$ | 0.506 | 0.92 | 1.08 | 2.07
| $3/2^{+}_{2}$ | 3.641 | 3.711 |  | 0.018 | 0.013 | 0.036 | $1\times 10^{-4}$ | 0.015 | 1.17 | 0.84 | 0.26
| $1/2^{+}_{2}$ | 4.162 | 4.214 |  | $3\times 10^{-4}$ | $5\times 10^{-4}$ | 0.031 | $6\times 10^{-6}$ | $4\times 10^{-4}$ | 0.76 | 1.27 | 0.01

###  III.2 $\mathbf{{}^{25}Mg\rightarrow^{25}Al}$

In contrast to the $A=23$ pair, weak $M1$ and $GT$ strengths are measured for
transitions between analogous states in the $A=25$ system except for the
transition $5/2_{1}^{+}\rightarrow 7/2_{1}^{+}$. The results obtained from the
USDC interaction display a similar pattern. Only in the case of
$5/2_{1}^{+}\rightarrow 5/2_{2}^{+}$ the calculated $GT$ strength was
significantly smaller than the experimental data. The $R_{IS}$ remains almost
1 in $5/2_{1}^{+}\rightarrow 7/2_{1}^{+}$, and the large $R_{oc}$ value
exceeding unity implies constructive interference between the orbital and the
spin IV term. The $A=25$ system 25Mg and 25Al are also strongly deformed
DeformationOtsuka  like those $A=23$, $T_{z}=\pm 1/2$ nuclei. The orbital
contributions to $M1$ transitions of the $A=25$ pair have been discussed in
the context of deformed Nilsson’s model and particle rotor model in Ref.
25MgAl . In shell model, the g.s. in 25Mg arises from a configuration
$\pi(d_{5/2}^{4})\otimes\nu(d_{5/2}^{5})$. The major component of the wave
function configurations is $\pi(d_{5/2}^{4})\otimes\nu(d_{5/2}^{5})$ only in
the $7/2_{1}^{+}$ state, and the $B(M1)$ strength from g.s. to this state is
enhanced by the constructive interference of the orbital IV term in the $M1$
transition. Similar interpretations can be made for the mirror partner 25Al,
where the corresponding configuration is
$\pi(d_{5/2}^{5})\otimes\nu(d_{5/2}^{4})$. The results obtained from CCEI are
similar to the USDC results in which the transition strengths remain strong
only for the transition from $5/2_{1}^{+}$ to $7/2_{1}^{+}$. The associated
$R_{oc}$ (or $R_{ISO}$) value for this transition is 3.39 (or 3.04).

As described before, the parameter $R_{oc}$ and its interpretations were based
on the approximation that the IV spin term is not negligible or hindered. In
weak $M1$ or $GT$ transitions, the IV spin terms are found to be very small,
and in such cases, the $R_{oc}$ values are not reliable or could not be
interpreted with the usual meaning 23NaMg ; 27AlSi ; 25MgAl . Further, the
measured $B(GT)$ strengths less than 0.04 units are considered to be less
reliable 25MgAl ; Fujita_A27 ; Fujita2018 . Hence, from here on, the
discussion and depiction through figures of the $M1$ components and $R_{oc}$
values are excluded or avoided where $B(M1)$ and $B(GT)$ transition strengths
are very weak or generally smaller than 0.03 units in the text.

###  III.3 $\mathbf{{}^{27}Al\rightarrow^{27}Si}$

The calculated $M1$ and $GT$ transition strengths in the $A=27$ pair are in
good agreement with the experimental data. The corresponding $R_{oc}$ (or
$R_{ISO}$) values are also well reproduced. A notable deviation in $B(M1)$
strength is observed between the calculated and experimental data for the
transition $5/2_{1}^{+}\rightarrow 5/2_{3}^{+}$ in 27Si. The measured $B(M1)$
strength for $5/2_{1}^{+}\rightarrow 5/2_{3}^{+}$ has a large uncertainty, and
it differs nearly by $\sim$ 3 times from the respective $B(M1)$ strength of
the mirror nucleus 27Al. However, in the calculated results, the difference is
only $\sim$ 1.36 times, and the $R_{oc}$ parameter for this transition is also
larger than unity while the experimental value with errors is close to 1.
Therefore, further measurements with more precision are required to reaffirm
the orbital contributions in this transition.

Figure 2: Same as Fig. 1 but for $A=27$ pair.

The mirror nuclei 27Al - 27Si are less deformed compared to the $A=23$ and 25
pair, and the energy states contain single-particle-like configurations. In
such cases, the wave functions of the energy eigenstates are dominated by a
single particle state or an orbital where the odd nucleon resides. The IV
orbital and spin ( $l\tau$ and $\sigma\tau$ ) terms are hindered for the
transition from $0d$ to $0s$ orbital, while the $l\tau$ terms are facilitated
between single-particle states with the same orbital angular momentum $l$
provided that $l\neq 0$ SuhonenBook . Further, it has been observed that the
$M1$ transition probabilities are enhanced for transitions between spin-orbit
partners M1_32S . Because of isospin symmetry, the structure and
configurations of the $Tz=-1/2$ nucleus closely resemble those of the $Tz=1/2$
nucleus, with only protons and neutrons configurations interchanged. So here
and in the subsequent mirror pairs in the text, our discussion on the
transitions will be based on the $Tz=1/2$ nucleus of the mirror pair.
Analogous arguments apply to the $Tz=-1/2$ nucleus. In 27Al, the g.s.
$5/2_{1}^{+}$ is formed out of a configuration $\pi(d_{5/2}^{-1})$ while the
$3/2_{1}^{+}$ state obtained from $\pi(s_{1/2}^{1})$ type configurations i.e.
when a proton in $s_{1/2}$ orbit coupled to $2_{1}^{+}$ state of 26Mg core.
So, the transition between $5/2_{1}^{+}$ and $3/2_{1}^{+}$ state is strongly
hindered. The $5/2_{2}^{+}$ and $5/2_{3}^{+}$ states arise from the mixing of
a large number of configurations. The $B(E2)$ strengths serve as stringent
probes to understand the structure of nuclear wave functions. The calculated
$E2$ strength between $5/2_{1}^{+}$ to $5/2_{2}^{+}$ is 14.33 $e^{2}fm^{4}$
while it is 0.07 $e^{2}fm^{4}$ for transition from $5/2_{1}^{+}$ to
$5/2_{3}^{+}$. This implies a change in the configuration of the wave function
of $5/2_{3}^{+}$. This state is expected to dominate with $d_{3/2}$ orbital. A
large orbital contribution is observed for the transition between
$5/2_{1}^{+}$ and $5/2_{3}^{+}$ or between the spin-orbit partner. The
$3/2_{3}^{+}$ and $7/2_{2}^{+}$ states are also of such nature, and the $E2$
strengths from $5/2_{1}^{+}$ to these states are 0.04 $e^{2}fm^{4}$ and 1.75
$e^{2}fm^{4}$ respectively. The $R_{oc}$ value for the transition from g.s. to
these states remains larger than one.

Within CCEI, very weak transition strengths are predicted for transitions
other than $5/2_{1}^{+}$ to $5/2_{2}^{+}$, $3/2_{2}^{+}$ and $5/2_{3}^{+}$.
The CCEI results are compared with USDC results in Fig. 2, wherever the
transition strengths are not negligible. While the IS contributions are
approximately the same or exhibit small differences compared to those found in
USDC, the predicted $R_{oc}$ (or $R_{ISO}$) parameters suffer deviations from
those obtained with USDC interactions.

###  III.4 $\mathbf{{}^{29}Si\rightarrow^{29}P}$

The $GT$ transition strengths are not measured yet between the 29Si - 29P
mirror pair. We have calculated the $M1$ transition strengths from g.s. to
several excited states and the $B(GT)$ values for corresponding analogous
states in the $A=29$ pair. The parameter $R_{IS}$ is a signature of IS
contribution to $M1$ transition. Though the $B_{IS}(M1)$ values are small, the
$R_{IS}$ varies in the range 0.7-1.3, implying that the $M1$ transition
strengths can be modified by $\pm 30\%$ in the mirror pair due to constructive
or destructive contributions of the IS term.

Figure 3: Same as Fig. 1 but for $A=29$ pair.

The g.s. $1/2_{1}^{+}$ of 29Si is described by
$\pi(d_{5/2}^{6})\otimes\nu(d_{5/2}^{6}s_{1/2}^{1})$ configuration. The major
configuration of $3/2_{1}^{+}$ and $3/2_{3}^{+}$ states is
$\pi(d_{5/2}^{6})\otimes\nu(d_{5/2}^{6}d_{3/2}^{1})$ or dominated by $d_{3/2}$
orbital. So, weak transition strengths are observed for the transition from
g.s. to $3/2_{1}^{+}$ and $3/2_{3}^{+}$. If we look at the $E2$ transition
probabilities, the $B(E2;1/2_{1}^{+}\rightarrow 3/2_{3}^{+})$=0.2
$e^{2}fm^{4}$ and $B(E2;1/2_{1}^{+}\rightarrow 3/2_{1}^{+})$=58.7
$e^{2}fm^{4}$, respectively. It implies that the $3/2_{3}^{+}$ has a
completely different configuration than $1/2_{1}^{+}$ or characterized by
$d_{3/2}$ orbital, and the transition from $1/2_{1}^{+}$ to $3/2_{3}^{+}$ is
strongly hindered. The major component of the wave function of $1/2_{2}^{+}$
is the same as the $1/2_{1}^{+}$ state, while the $3/2_{2}^{+}$ and
$1/2_{3}^{+}$ appear from large configuration mixing. The $R_{oc}$ parameters
for the transitions from g.s. to these states are close to 1, suggesting that
the orbital term has negligible or no contributions in these transitions. This
is evident since the orbital IV term ($l\tau$) vanishes between $0s$ states.

The CCEI results are compared with those obtained from USDC in Fig. 3. In
addition to $1/2_{1}^{+}\rightarrow 3/2_{3}^{+}$, the CCEI also predicts weak
transition strengths for $1/2_{1}^{+}\rightarrow 1/2_{3}^{+}$ and are not
shown in Fig. 3. The CCEI interaction predicts a large orbital contribution to
the IV term of $M1$ operator for the transition $1/2_{1}^{+}\rightarrow
3/2_{2}^{+}$ while it is $\sim 1$ in USDC interaction. For other transitions,
the calculated $R_{oc}$ values from CCEI are close to USDC results.

###  III.5 $\mathbf{{}^{31}P\rightarrow^{31}S}$

Few experimental data are available on the $M1$ transition strengths in the
$A=31$ mirror pair. We have reported the $B(M1)$ strengths from g.s. to
various excited states as well as the $B(GT)$ values for the analogous
transitions. The low-energy excited states of the mirror nuclei 31P - 31S
exhibit similar patterns like the $A=29$ pair. In 31P, the g.s. is obtained
from a $\pi(s_{1/2})$ type configuration. But the wave functions of
$1/2_{2}^{+}$ and $1/2_{3}^{+}$ states are rather mixed and dominated by the
mixture $\pi(d_{5/2}^{6}s_{1/2}^{1})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2})$ and
$\pi(d_{5/2}^{6}s_{1/2}^{1})\otimes\nu(d_{5/2}^{6}s_{1/2}^{1}d_{3/2}^{1})$
configurations. The $M1$ transition from $1/2_{1}^{+}$ to $1/2_{2}^{+}$ and
$1/2_{3}^{+}$ states remain strong. However, the orbital contributions in
these transitions are different than those of the $A=29$, $T_{z}=\pm 1/2$
pair. Unlike the $A=29$ pair, these transitions are associated with large
orbital contributions, which implies that they may have large deformations.
Again, the major configurations of $3/2_{1}^{+}$ and $3/2_{3}^{+}$ states are
dominated with $d_{3/2}$ orbital while the $3/2_{2}^{+}$ state is obtained
from large configurations mixing. So, the transition strengths from g.s. to
$3/2_{1}^{+}$ and $3/2_{3}^{+}$ are small or negligible. The IS and orbital
contributions to the transitions in the $A=31$ mirror pair calculated from
CCEI are displayed in Fig. 4. The transition strengths from g.s. to
$3/2_{3}^{+}$ are very weak and not shown. In CCEI, the $B(M1)$ strengths are
not as strong as those observed in USDC in $1/2_{1}^{+}\rightarrow
1/2_{2}^{+}$, and the orbital IV term interferes destructively with the spin
IV term. Also, the IS contributions to mirror pair in the transition
$1/2_{1}^{+}\rightarrow 3/2_{2}^{+}$ calculated from CCEI differ from those
obtained with USDC.

Figure 4: Same as Fig. 1 but for $A=31$ pair.

###  III.6 $\mathbf{{}^{33}S\rightarrow^{33}Cl}$

The calculated $M1$ transition strengths for $A=33$ pair are compared with the
corresponding experimental data in Table 1. We have reported the $B(GT)$
strengths of the analogous transitions between 33S - 33Cl nuclei for which
experimental data are not available. In 33S, the g.s. $3/2_{1}^{+}$ comes out
from a configuration
$\pi(d_{5/2}^{6}s_{1/2}^{2})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{1})$.
The $1/2_{1}^{+}$ state arises from a configuration
$\pi(d_{5/2}^{6}s_{1/2}^{2})\otimes\nu(d_{5/2}^{6}s_{1/2}^{1}d_{3/2}^{2})$ or
characterized by $\nu s_{1/2}$ orbital. The $3/2_{2}^{+}$ state is also found
to exhibit such characters. So, the transitions from $3/2_{1}^{+}$ to
$1/2_{1}^{+}$ and $3/2_{2}^{+}$ are strongly hindered. The transition
strengths from g.s. to $1/2_{2}^{+}$ are also weak. The dominant wave function
configuration of the $5/2_{2}^{+}$ state
$\pi(d_{5/2}^{6}s_{1/2}^{1}d_{3/2}^{1})\otimes\nu(d_{5/2}^{6}s_{1/2}^{1}d_{3/2}^{2})$
is a mixture of $sd$ orbitals. The g.s. and $5/2_{2}^{+}$ state are connected
with a strong $M1$ transition, and a positive orbital contribution is
observed. But the $M1$ strength is weak in the transition
$3/2_{1}^{+}\rightarrow 5/2_{1}^{+}$. Because the major constituent of the
$5/2_{1}^{+}$ state is
$\pi(d_{5/2}^{6}s_{1/2}^{2})\otimes\nu(d_{5/2}^{6}s_{1/2}^{1}d_{3/2}^{2})$
i.e. this state can be interpreted as a neutron occupying in $s_{1/2}$ orbital
coupled to $2_{1}^{+}$ state of core 32S. Transition strengths from CCEI
interaction for $3/2_{1}^{+}\rightarrow 1/2_{1}^{+}$ and
$3/2_{1}^{+}\rightarrow 3/2_{2}^{+}$ are also observed to be very weak. The
CCEI results are similar to USDC results for the transition
$3/2_{1}^{+}\rightarrow 5/2_{1}^{+}$, but they diverge for the transition
$3/2_{1}^{+}\rightarrow 5/2_{2}^{+}$. It also predicts a very small $B(GT)$
strength of less than 0.02 units for the transition $3/2_{1}^{+}\rightarrow
1/2_{2}^{+}$.

###  III.7 $\mathbf{{}^{35}Cl\rightarrow^{35}Ar}$

The experimental data on $M1$ and $GT$ transition strengths are limited in
35Cl - 35Ar pair, particularly in 35Ar. We have reported several states and
provided estimates of the $B(M1)$ and $B(GT)$ strengths for various
transitions. The calculated transition strengths are found to be either weak
or of moderate magnitude, generally $\sim 0.05$ units. Like other mirror
pairs, transition strengths larger than 0.1 units are not observed. The
orbital IV term interferes constructively with the spin IV term only in the
transition from g.s. to $5/2_{2}^{+}$. The g.s. $3/2_{1}^{+}$ has a
configuration
$\pi(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{1})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{2})$
in 35Cl. The main components of the wave function of $5/2_{1}^{+}$ state are
$\pi(d_{5/2}^{6}s_{1/2}^{1}d_{3/2}^{2})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{2})$
and
$\pi(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{1})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{2})$
while those in $5/2_{2}^{+}$ state are
$\pi(d_{5/2}^{6}s_{1/2}^{1}d_{3/2}^{2})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{2})$
and
$\pi(d_{5/2}^{5}s_{1/2}^{2}d_{3/2}^{2})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{2})$.
This suggests that the $d_{5/2}$ orbital character present in $5/2_{2}^{+}$
state is the possible cause of a large $R_{oc}$ parameter in the transition
from g.s. to this state. Within the CCEI framework, the $M1$ and $GT$
strengths for transitions involving g.s. and $1/2_{1}^{+}$, $3/2_{2}^{+}$ are
very small, typically $\leq 0.01$ units. In the transition from g.s. to
$5/2_{1}^{+}$, the $B(M1)$ strengths are very weak, while the $B(GT)$ strength
is negligible in the transition $3/2_{1}^{+}\rightarrow 5/2_{2}^{+}$.

###  III.8 $\mathbf{{}^{37}Ar\rightarrow^{37}K}$

Figure 5: Same as Fig. 1 but for $A=37$ pair.

For the mirror pair 37Ar - 37K, limited experimental data are available, and
they are associated with large uncertainty. The $M1$ and $GT$ transition
strengths between g.s. and lower excited states are calculated from USDC
interaction. The $M1$ and $GT$ strengths remain strong for the transitions
$3/2_{1}^{+}\rightarrow 5/2_{1}^{+}$ and $3/2_{1}^{+}\rightarrow 5/2_{2}^{+}$.
Both $3/2_{1}^{+}$ and $5/2_{1}^{+}$ arise from a configuration
$\pi(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{2})\otimes\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{3})$
in 37Ar. But the major components of wave function configurations of
$5/2_{2}^{+}$ state are of $\nu(d_{5/2}^{-1})$ type. The transition from g.s.
to $5/2_{2}^{+}$ or between the spin-orbit partner is accompanied by
constructive interference of the orbital term in the $M1$ transition. The USDC
results are compared with the calculated results obtained from CCEI in Fig. 5.
The CCEI couldn’t reproduce the $R_{oc}$ parameter in the transition
$3/2_{1}^{+}\rightarrow 5/2_{2}^{+}$.

### Magnetic moments:

Figure 6: Left panel: The filled (unfilled) black symbols refer to the
experimental data for g.s. magnetic moments of the $Tz=1/2$ ($Tz=-1/2$)
nucleus, while color widths correspond to calculated results. Right panel:
$R_{oc}$ parameter in the g.s. magnetic moments of the mirror pairs.

In this section, the ground state magnetic moments and their respective
components are analyzed for the mirror pairs. The calculated g.s. magnetic
moments of the $T_{z}=\pm 1/2$ pair, along with the experimental data, are
shown in the left panel of Fig. 6. We can see that all available measured data
are well reproduced from USDC interaction. The CCEI results also lie close to
the experimental data or USDC results. The orbital contribution to the ground
state magnetic moments or $R_{oc}$ parameters are shown on the right panel of
Fig. 6. Both USDC and CCEI make similar predictions for $R_{oc}$ values in all
mirror pairs except the $A=27$ pair. The g.s. magnetic moments obtained with
CCEI for the $A=27$ pair exhibit larger differences from the corresponding
experimental data compared to other pairs (see left panel of Fig. 6). The CCEI
also fails to reproduce the $GT$ strength between ground states in 27Al - 27Si
pair, which results in a large $R_{oc}$ value ($\sim 7$) and is out of the
scale of Fig. 6. The parameter $R_{oc}$ has large values ($R_{oc}>1$) in
$A=23$ to $31$ but reduced significantly or lies near zero in the mirror pairs
with mass $A=33$ to $37$.

Large asymmetry is observed in the g.s. magnetic moments of the $A=23$ to $27$
pair, or the magnetic moments of these $T_{z}=\pm 1/2$ pair have huge
differences. This difference is decreased in $A=29$ and 31 and further reduced
significantly in the mirror nuclei $A=33$ to $37$. Accordingly, the $R_{oc}$
parameters are large within the $A=23$ to $27$ mass range, decrease at $A=29$
and $31$, but are still larger than 1. Then, they reduced to nearly zero in
$A=33$ to $37$. Actually, when the magnetic moments of the mirror pairs
exhibit substantial differences, the corresponding IV part or $\mu_{IV}$
components are larger, resulting in an increased $R_{oc}$ parameter (see Eq.
(15)). From Table 1, it can be observed that the corresponding IS components
are smaller than the IV components in these nuclei. However, in $A=33$ to 37
pairs, the magnetic moments of $T_{z}=1/2$ and $T_{z}=-1/2$ nuclei have
similar values, the associated IS terms are large while the IV components are
small, leading to a reduction in the $R_{oc}$ parameter.

As discussed before, the ground state wave functions in the $A=33$ to $37$
nuclei show single-particle structures. The calculated magnetic moments also
do not differ much from the corresponding Schmidt value or single-particle
magnetic moments. The IV orbital term destructively interferes in the $M1$
transition between the ground states of these mirror pairs.

Figure 7:  The filled (unfilled) symbols are g.s. quadrupole deformation in
the $Tz=1/2$ ($Tz=-1/2$) nucleus of the mirror pairs joined by a solid
(dashed) line.

The $1/2_{1}^{+}$ ground states in $A=29$ and $31$ pair have $\pi(s_{1/2})$ or
$\nu(s_{1/2})$ like characters. Since the orbital contributions vanish for
transitions between $s$ orbits, their $R_{oc}$ values should lie close to 1,
which is seen in the case of $A=31$ pair. However, a higher $R_{oc}$ value in
the $A=29$ mirror pair suggests possible ground state deformation. To
visualize the deformation in the ground states and its impact on the
corresponding $R_{oc}$ values, we have plotted the quadrupole deformation
parameter ($\beta_{2}$) against the mass number for the $T=1/2$ pairs in Fig.
7. The deformation parameter $\beta_{2}$ is calculated using the formula Beta2

$\beta_{2}=\frac{4\pi}{3ZR_{0}^{2}}\times\frac{\sqrt{B(E2;J_{f}\rightarrow
J_{i})}}{\langle J_{f}K20|J_{i}K\rangle},$ (17)

where $K$ is the projection of nuclear spin on the symmetry axis,
$R_{0}=1.2A^{1/3}$, $J_{i}$ is the g.s., and $J_{f}$ corresponds to the
excited state of the g.s. band. It can be noted that $J_{f}$ may or may not
necessarily be the same as the first excited state and is identified from the
g.s. band structures of the mirror pairs NNDC ; gsBand29 ; gsBand31 ; gsBand33
; gsBand35 ; gsBand37 . The $\beta_{2}$ parameters are obtained with the
standard assumption that $K=J_{i}$. Both USDC and CCEI make similar
predictions for $\beta_{2}$ in all mirror pairs. From Fig. 7, it can be
observed that the mirror nuclei $A=23$, 25 are largely deformed, followed by
$A=27$, 29 pairs, and are associated with large positive orbital
contributions. The deformation is reduced in $A=31$ where $R_{oc}\sim 1$. The
ground states of the $A=33$ to 37 mirror pairs are characterized by small
deformation and are dominated by single-particle structures. The orbital terms
($l\tau$) have a destructive effect on the g.s. magnetic moments in these
nuclei, and the $R_{oc}$ values lie close to zero.

##  IV Summary

The analogous $M1$ and $GT$ transitions of the odd mass $T=1/2$ mirror pairs
were studied using isospin non-conserving phenomenological and ab initio
effective interactions within $sd$\- model space. The IS, IV components of the
$M1$ operator, the IS, and orbital contributions to $M1$ transitions are
discussed in detail for various $M1$ transitions in each mirror pair. The USDC
interaction well reproduces the available experimental data and is used for
predictions between low-lying states where the experimental data are
unavailable. Along with USDC results, the ab initio CCEI predictions are also
shown. In several cases, the CCEI provides predictions similar to those
obtained using the USDC interaction. In the mirror pairs, the IS contributions
are relatively small, accounting for only about $\sim 10\%$ to $20\%$ change
in the $M1$ transitions. In the strong $M1$ transitions, the IV component, or
particularly the spin IV part, is dominant where the orbital IV term
interferes constructively or destructively depending upon the configurations
of initial and final states. The g.s. magnetic moments of mirror nuclei are
well explained from both USDC and CCEI interactions. Their respective IS, IV
components and orbital contributions are studied and analyzed. The deformed
ground states are found to exhibit large orbital contributions.

##  V Acknowledgment

S.S. would like to thank UGC (University Grant Commission), India, for
providing financial support for his Ph.D. thesis work. P.C.S. acknowledges a
research grant from SERB (India), CRG/2022/005167.

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