# Magnetic dipole ${\gamma}$-ray strength functions in the crossover from
spherical to deformed neodymium isotopes

A. Mercenne  Center for Theoretical Physics, Sloane Physics Laboratory, Yale
University, New Haven, Connecticut 06520, USA  Department of Physics and
Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA  P.
Fanto  Center for Theoretical Physics, Sloane Physics Laboratory, Yale
University, New Haven, Connecticut 06520, USA  W. Ryssens  Institut
d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, Campus de la
Plaine CP 226, 1050 Brussels, Belgium  Y. Alhassid  Center for Theoretical
Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut
06520, USA

###### Abstract

We calculate the magnetic dipole $\gamma$-ray strength functions in a chain of
even-mass neodymium isotopes 144-152Nd in the framework of the configuration-
interaction (CI) shell model. We infer the strength function by applying the
maximum entropy method (MEM) to the exact imaginary-time response function
calculated with the shell-model Monte Carlo (SMMC) method. The success of the
MEM depends on the choice of a good strength function as a prior distribution.
We investigate two choices for the prior strength function: the static path
approximation (SPA) and the quasiparticle random-phase approximation (QRPA).
We find that the QRPA is a better approximation at low temperatures (i.e.,
near the ground state), while the SPA is a better choice at finite
temperatures. We identify a low-energy enhancement (LEE) in the MEM
deexcitation $M1$ strength functions of the even-mass neodymium isotopes and
compare with recent experimental results for the total deexcitation
$\gamma$-ray strength functions. The LEE is already seen in the SPA strength
function but not in the QRPA strength function, indicating the importance of
large-amplitude static fluctuations around the mean field in reproducing the
LEE. Our method is currently the only one which can reproduce LEE in heavy
open-shell nuclei where conventional CI shell model calculations are
prohibited. With the onset of deformation as number of neutrons increases
along the chain of neodymium isotopes, we observe that some of the LEE
strength transfers to a low-energy excitation, which we interpret as a finite-
temperature “scissors” mode. We also observe a finite-temperature spin-flip
mode.

###### pacs:

Valid PACS appear here

##  I Introduction

The process of creating new elements through the rapid capture of neutrons,
known as r-process nucleosynthesis, can be studied by modeling neutron capture
rates in compound nucleus reactions. These neutron capture rates are
calculated within the Hauser-Feshbach theory Hauser and Feshbach (1952);
Koning and Rochman (2012), which requires as input the $\gamma$-ray strength
function (${\gamma}$SF) Bartholomew  _et al._ . By accurately determining the
${\gamma}$SF, we can improve our ability to model r-process nucleosynthesis
Mumpower  _et al._ (2016) and gain a better understanding of the origins and
abundances of the elements in the universe.

In heavy nuclei, the magnetic dipole ($M1$) ${\gamma}$SF exhibits several
excitation modes that are typically observed in the low-energy tail of the
giant dipole resonance. These modes include a “scissors” mode and a spin-flip
resonance Bohle  _et al._ (1984); Heyde  _et al._ (2010). In strongly deformed
nuclei the orbital scissors mode at lower energies is clearly separated from a
resonance-like structure composed of proton and neutron spin-flip excitations
at higher energies Heyde  _et al._ (2010). The inclusion of the scissors mode
in the $M1$ $\gamma$SF improves the accuracy of neutron radiative capture
rates in Hauser-Feshbach calculations Mumpower  _et al._ (2017).

A low-energy enhancement (LEE), characterized by an increase in strength at
low ${\gamma}$-ray energies, has been observed in the ${\gamma}$SFs of several
mid-mass nuclei Voinov  _et al._ (2004); Guttormsen  _et al._ (2005);
Wiedeking  _et al._ (2012); Larsen  _et al._ (2013, 2018) and several heavier
lanthanide nuclei Kheswa  _et al._ (2015); Naqvi  _et al._ (2019); Simon  _et
al._ (2016); Guttormsen  _et al._ (2022). The LEE was shown to be dipole in
nature Jones  _et al._ (2018). If the LEE persists in heavy, neutron-rich
nuclei, it would significantly increase the radiative neutron capture cross
sections of nuclei near the neutron drip line Larsen and Goriely (2010).

Configuration-interaction (CI) shell-model calculations Loens  _et al._
(2012); Schwengner  _et al._ (2013); Brown and Larsen (2014); Schwengner  _et
al._ (2017); Sieja (2017); Karampagia  _et al._ (2017); Sieja (2018) suggested
that the LEE originates in the $M1$ $\gamma$SF, although relativistic QRPA
calculations Litvinova and Belov (2013) proposed the LEE is of $E1$ origin. CI
shell model calculations were carried out for nuclei in different major shells
and concluded that the LEE is more pronounced near shell closure Mitdbø  _et
al._ (2018).

Conventional CI shell-model methods in heavy open-shell nuclei are prohibited
because of the combinatorial increase of the dimensionality of the many-
particle model space with number of valence nucleons and/or number of valence
orbitals. The shell-model Monte Carlo (SMMC) method Johnson  _et al._ (1992);
Alhassid  _et al._ (1994); Nakada and Alhassid (1997, 1998); Ormand (1997);
Langanke (1998); Alhassid  _et al._ (1999, 2007, 2008); Alhassid  uses
auxiliary-field Monte Carlo techniques within the CI shell-model framework to
enable exact (within statistical errors) calculations of thermal observables
in heavy nuclei. However, in SMMC we cannot calculate directly the finite-
temperature $\gamma$SF, but only its Laplace transform, the imaginary-time
response function. The calculation of the $\gamma$SF requires an analytic
continuation which is carried out in the maximum entropy method (MEM)
Gubernatis  _et al._ (1991); Jarrell and Gubernatis (1996); Gubernatis and
Werner (2016).

The success of the MEM depends on a good choice for a prior strength function
at finite temperature. In Ref. Fanto and Alhassid (2024), the static path
approximation plus random-phase approximation (SPA+RPA) was used to calculate
a prior for the $M1$ $\gamma$SF for a pairing plus quadrupole interaction, and
a LEE was identified in a chain of samarium isotopes. However, for more
general interactions such as the interactions used in SMMC applications to
lanthanides Özen  _et al._ (2013), the SPA+RPA becomes time consuming. Here we
investigate two other choices for the prior: the static path approximation
(SPA) Mühlschlegel  _et al._ (1972); Alhassid and Zingman (1984); Lauritzen
_et al._ (1988); Arve  _et al._ (1988); Attias and Alhassid (1997) and the
finite-temperature quasiparticle random phase approximation (QRPA). In these
approximations we can calculate a strength function directly without the need
for analytic continuation.

We apply our methods to calculate the $M1$ $\gamma$SFs in a chain of even-mass
neodymium isotopes 144-152Nd. We convert these strength functions to
deexcitation $M1$ strength functions and compare our results with recent
experimental results for the deexcitation $\gamma$SFs in 144-150Nd Guttormsen
_et al._ (2022). We also provide a prediction of the deexcitation $M1$
$\gamma$SF in 152Nd, for which no measurements were made.

Comparing the imaginary-time response functions of the SPA and QRPA to exact
SMMC response functions, we find that the QRPA is a better approximation at
low temperatures close to the ground state, but that the SPA is a
significantly better approximation at finite temperature. Consequently, at
finite temperature we use the SPA strength as a prior for the MEM and identify
a LEE in all of the even-mass neodymium isotopes in the chain under study.
With the onset of deformation with increasing number of neutrons along the
chain of neodymium isotopes, we observe that some of the LEE strength
transfers to a low-energy excitation, which we interpret as a finite-
temperature “scissors” mode.

We also find that at finite temperature the LEE is already observed in the SPA
strength function but not in the QRPA strength function. Thus the QRPA, which
is the standard method for calculating finite-temperature strength functions
in nuclear astrophysics, cannot reproduce the LEE Fanto and Alhassid (2024). A
similar conclusion was reached in Ref. Frosini  _et al._ (2024) by comparing
the finite-temperature QRPA $M1$ deexcitation strength function with exact
diagonalization results in the CI shell model. We conclude that large-
amplitude static fluctuations around the mean-field state, which are included
in the SPA but not in the QRPA, play an important role in reproducing the LEE.

##  II Theoretical framework

###  II.1 SMMC and imaginary-time response functions

The SMMC method is based on the representation of the Gibbs operator
${{e}^{-\beta\hat{H}}}$ (where ${\hat{H}}$ is the CI shell model Hamiltonian
and ${\beta=1/T}$ is the inverse temperature) as a functional integral of one-
body propagators $\hat{U}_{\sigma}\equiv\hat{U}_{\sigma}(\beta,0)$. These
propagators describe non-interacting nucleons moving in external auxiliary
fields $\sigma=\sigma(\tau)$, which depend on imaginary time $\tau$. This is
formally expressed in terms of the Hubbard-Stratonovich (HS) transformation
Hubbard (1959); *stratonovich_1957; *stratonovich_1958

$e^{-\beta\hat{H}}=\int D[\sigma]{G}_{\sigma}{\hat{U}}_{\sigma}\;,$ (1)

where ${G}_{\sigma}$ is a Gaussian factor, and $D[\sigma]$ is a measure. The
thermal expectation value of an observable
${\langle\hat{O}\rangle}=\text{Tr}(\hat{O}e^{-\beta\hat{H}})/\text{Tr}e^{-\beta\hat{H}}$
can then be written as

$\langle\hat{O}\rangle=\frac{\int D[\sigma]{W}_{\sigma}{\Phi}_{\sigma}{\langle
O\rangle}_{\sigma}}{\int D[\sigma]{W}_{\sigma}{\Phi}_{\sigma}}\;,$ (2)

where ${{W}_{\sigma}={G}_{\sigma}|\text{Tr}{\hat{U}}_{\sigma}|}$ is a
positive-definite weight,
${{\Phi}_{\sigma}=\text{Tr\,}{\hat{U}}_{\sigma}/|\text{Tr\,}{\hat{U}}_{\sigma}|}$
is the Monte Carlo sign function, and ${{\langle
O\rangle}_{\sigma}=\text{Tr}(\hat{O}{\hat{U}}_{\sigma})/\text{Tr}{\hat{U}}_{\sigma}}$.

In SMMC, the right-hand side of Eq. (2) is calculated using stochastic
sampling of uncorrelated auxiliary-field configurations ${\sigma_{k}}$
according to the positive-definite distribution ${{W}_{\sigma}}$. The thermal
expectation of the observable $\hat{O}$ is then estimated by

$\langle\hat{O}\rangle\approx{\sum_{k}\langle
O\rangle_{\sigma_{k}}\Phi_{\sigma_{k}}\over\sum_{k}\Phi_{\sigma_{k}}}\;.$ (3)

Thermal averages at fixed number of protons and neutrons are calculated by
particle-number projection Alhassid .

The imaginary-time response function of a spherical tensor
${\mathcal{O}}_{\lambda}$ of rank $\lambda$ at temperature $T$ is defined by

$\displaystyle R_{\mathcal{O}_{\lambda}}(T;\tau)$
$\displaystyle=\langle{\mathcal{O}}_{\lambda}(\tau)\cdot{\mathcal{O}}_{\lambda}(0)\rangle=\sum_{\begin{subarray}{c}\alpha_{i}J_{i}\\\
\alpha_{f}J_{f}\end{subarray}}\frac{e^{-\beta E_{\alpha_{i}J_{i}}}}{Z}$
$\displaystyle\times$
$\displaystyle|(\alpha_{f}J_{f}||\hat{\mathcal{O}}_{\lambda}||\alpha_{i}J_{i})|^{2}e^{-\tau(E_{\alpha_{f}J_{f}}-E_{\alpha_{i}J_{i}})}\;,$
(4)

where $\langle\ldots\rangle$ denotes a thermal average,
${\mathcal{O}}_{\lambda}(\tau)\cdot{\mathcal{O}}_{\lambda}(0)=\sum_{\mu}(-)^{\mu}\mathcal{O}_{\lambda\mu}(\tau)\mathcal{O}_{\lambda-\mu}$,
and
$\mathcal{O}_{\lambda\mu}(\tau)=e^{\tau\hat{H}}\mathcal{O}_{\lambda\mu}e^{-\tau\hat{H}}$.
Here $\alpha J$ are eigenstates of the Hamiltonian $\hat{H}$ with spin $J$ and
energy $E_{\alpha J}$.

Using the HS transformation, we can express this response function in the form
Koonin  _et al._ (1997); Alhassid

${R}_{{\mathcal{O}}_{\lambda}}(T;\tau)=\frac{\int
D[\sigma]W_{\sigma}\Phi_{\sigma}{\langle{\mathcal{O}}_{\lambda}(\tau).{\mathcal{O}}_{\lambda}\rangle}_{\sigma}}{\int
D[\sigma]W_{\sigma}\Phi_{\sigma}}$ (5)

where
${{\mathcal{O}}_{\lambda}(\tau)={\hat{U}}_{\sigma}^{-1}(\tau,0){\mathcal{O}}_{\lambda}{\hat{U}}_{\sigma}(\tau,0)}$,
with $\hat{U}_{\sigma}(\tau,0)$ is the propagator between times $0$ and $\tau$
for a given configuration of the auxiliary fields $\sigma$, and the
expectation value ${{\langle\dots\rangle}_{\sigma}}$ is taken with respect to
the propagator ${{\hat{U}}_{\sigma}\equiv{\hat{U}}_{\sigma}(\beta,0)}$. As
mentioned above, in SMMC we sample auxiliary-field configurations according to
the weight function ${{W}_{\sigma}}$ and average over these samples to
determine the imaginary-time response function.

###  II.2 Strength function and maximum entropy method

The finite-temperature strength function
${S}_{{\mathcal{O}}_{\lambda}}(T;\omega)$ at temperature $T$ and transition
energy $\omega$ for a tensor operator of rank $\lambda$ is defined by

$\displaystyle
S_{\mathcal{O}_{\lambda}}(T;\omega)=\sum_{\begin{subarray}{c}\alpha_{i}J_{i}\\\
\alpha_{f}J_{f}\end{subarray}}$ $\displaystyle\frac{e^{-\beta
E_{\alpha_{i}J_{i}}}}{Z}|(\alpha_{f}J_{f}||\hat{\mathcal{O}}_{\lambda}||\alpha_{i}J_{i})|^{2}$
(6) $\displaystyle\times\delta(\omega-E_{f}+E_{i})\;.$

We note that the strength function relevant to experiments is described by the
microcanonical ensemble $\delta(E_{i}-\hat{H})$, where $E_{i}$ is the initial
energy of the nucleus Alhassid and Bush (1990). In the finite-temperature
approach we approximate the microcanonical average by a canonical average as
in Eq. (6). The microcanonical ensemble is related to the canonical ensemble
by an inverse Laplace transform. In the saddle-point approximation, the
temperature $T$ of the canonical ensemble is determined by the condition
$E(T)=E_{i}$, where $E(T)=\langle\hat{H}\rangle$ is the average thermal energy
and $E_{i}$ is the energy of the microcanonical ensemble. In the following we
measure $E_{i}$ with respect to the ground-state energy, i.e., $E_{i}$ will
denote the initial excitation energy of the nucleus.

The response function (II.1) is a Laplace transform of the strength function
(6)

$R_{\mathcal{O}_{\lambda}}(T;\tau)=\int_{-\infty}^{\infty}d\omega\,e^{-\tau\omega}S_{\mathcal{O}_{\lambda}}(T;\omega)\;.$
(7)

Using the symmetry property of the strength function

$S_{\mathcal{O}_{\lambda}}(T;-\omega)=e^{-\beta\omega}S_{\mathcal{O}_{\lambda}}(T;\omega)\;,$
(8)

we can rewrite (7) as an integral over non-negative values of $\omega$

${R}_{{\mathcal{O}}_{\lambda}}(T;\tau)=\int_{0}^{\infty}d\omega
K(\tau,\omega){S}_{{\mathcal{O}}_{\lambda}}(T;\omega)\;,$ (9)

where the kernel is
${K(\tau;\omega)={e}^{-\tau\omega}+{e}^{-(\beta-\tau)\omega}}$.

To determine the strength function from the response function, we have to
invert (9) numerically which is an ill-defined problem with no unique
solution. In particular, the response function is not sensitive to the
strength function at large values of $\omega$ due to the presence of an
exponentially small tail in the kernel for large $\omega$. To select an
optimal strength function, we implement the maximum-entropy method (MEM)
Jarrell and Gubernatis (1996), in which the objective function

$\mathcal{Q}({S}_{{\mathcal{O}}_{\lambda}};\alpha)=\alpha\mathcal{S}-\frac{1}{2}{\chi}^{2}\;$
(10)

is maximized. In Eq. (10)

${\chi}^{2}={\left({\bar{R}}_{{\mathcal{O}}_{\lambda}}-{R}_{{\mathcal{O}}_{\lambda}}\right)}^{T}{\mathcal{C}}^{-1}\left({\bar{R}}_{{\mathcal{O}}_{\lambda}}-{R}_{{\mathcal{O}}_{\lambda}}\right)\;,$
(11)

where ${\bar{R}}$ and ${\mathcal{C}}$ are the SMMC response function and its
covariance matrix, respectively. ${\mathcal{S}}$ is the entropy function

$\displaystyle\mathcal{S}$ $\displaystyle=-\int
d\omega\left({S}_{{\mathcal{O}}_{\lambda}}(T;\omega)-{S}_{{\mathcal{O}}_{\lambda}}^{\text{prior}}(T;\omega)\right.$
$\displaystyle\left.-{S}_{{\mathcal{O}}_{\lambda}}(T;\omega)\ln{\left[{S}_{{\mathcal{O}}_{\lambda}}(T;\omega)/{S}_{{\mathcal{O}}_{\lambda}}^{\text{prior}}(T;\omega)\right]}\right)\;,$
(12)

where ${{S}_{{\mathcal{O}}_{\lambda}}^{\text{prior}}}$ is a suitably chosen
prior for the strength function. The MEM provides a best fit to the known SMMC
response function (minimizing $\chi^{2}$) while at the same time keeping as
close as possible to the prior strength function (maximizing $\mathcal{S}$).
The coefficient $\alpha$ in the objective function (10) controls the relative
importance between the entropy and the $\chi^{2}$ terms.

Here we use Bryan’s method Bryan (1990), in which the final expression for the
MEM strength function is given by

${S}_{{\mathcal{O}}_{\lambda}}^{\text{MEM}}=\int
d\alpha{S}_{{\mathcal{O}}_{\lambda}}^{\alpha}\mathcal{P}\left(\alpha|{\bar{G}}_{{\mathcal{O}}_{\lambda}},\mathcal{C},{S}_{{\mathcal{O}}_{\lambda}}^{\text{prior}}\right),$
(13)

where ${{S}_{{\mathcal{O}}_{\lambda}}^{\alpha}}$ maximizes the objective
function (10) for a given ${\alpha}$, and
${\mathcal{P}\left(\alpha|{\bar{G}}_{{\mathcal{O}}_{\lambda}},\mathcal{C},{S}_{{\mathcal{O}}_{\lambda}}^{\text{prior}}\right)}$
is a probability distribution for $\alpha$ given in Ref. Jarrell and
Gubernatis (1996).

###  II.3 The SPA strength function

The success of the MEM depends on a good choice for the prior distribution. In
this work we explore two choices for the prior strength function: the SPA and
the QRPA. In both methods we can calculate directly the strength function
without the need for analytic continuation. The SPA strength function is
discussed below (see also in Ref. Fanto and Alhassid (2024)), while the
calculation of the finite-temperature QRPA strength function in the CI shell
model framework is discussed in Ref. Ryssens and Alhassid  and is based on the
HF-SHELL code developed in Ref. Ryssens and Alhassid (2021).

The SPA takes into account large-amplitude static fluctuations of the
auxiliary fields in the HS transformation and ignore all time-dependent
fluctuations. We apply the SPA in the framework of the CI shell model
Hamiltonian in which the two-body interaction is written in a separable form
Fanto and Alhassid (2021)
$\hat{H}_{2}=-(1/2)\sum_{\alpha}v_{\alpha}\hat{Q}_{\alpha}^{2}$. Using the
adiabatic approximation of Ref. Rossignoli and Ring (1998), the SPA strength
function of a transition operator $\mathcal{O}_{\lambda}$ of rank $\lambda$ is
given by

$S_{\mathcal{O}_{\lambda}}(T;\omega)\approx\frac{\int d\sigma
M(\sigma)Z(\sigma)S_{\mathcal{O}_{\lambda}}(T,\sigma;\omega)}{\int d\sigma
M(\sigma)Z(\sigma)}\,,$ (14)

where $\sigma$ are static auxiliary fields, and $M(\sigma)$ is a measure
function given in Ref. Fanto and Alhassid (2021). $Z(\sigma)$ is a
$\sigma$-dependent partition function,
$Z(\sigma)=\text{Tr}\,e^{-\beta\left(\hat{h}_{\sigma}-\sum_{\lambda=p,n}\mu_{\lambda}\hat{N}_{\lambda}\right)}$,
where

$\hat{h}_{\sigma}=\hat{H}_{1}-\sum_{\alpha}v_{\alpha}\sigma_{\alpha}\hat{Q}_{\alpha}$
(15)

is a one-body Hamiltonian, and ${\mu}_{p}$ and ${\mu}_{n}$ are the proton and
neutron chemical potentials, respectively.

$S_{\mathcal{O}_{\lambda}}(T,\sigma;\omega)$ is the thermal strength function
for static fields $\sigma$ given by

$\begin{split}&S_{\mathcal{O}_{\lambda}}(T,\sigma;\omega)=S_{\mathcal{O_{\lambda}}}^{(0)}(T,\sigma)\delta(\omega)\\\
&+\frac{1}{1-e^{-\beta\omega}}\sum_{\mu,kl}\frac{1}{2}|\mathcal{O}_{\lambda\mu,kl}|^{2}(\tilde{f}_{l}-\tilde{f}_{k})\delta(\omega-(\tilde{E}_{k}-\tilde{E}_{l}))\;,\end{split}$
(16)

where $\mathcal{O}_{\lambda\mu,kl}=\langle
k|\mathcal{O}_{\lambda\mu}|l\rangle$ are the matrix elements of the one-body
operator $\mathcal{O}_{\lambda\mu}$, $\tilde{E}_{k}$ are the generalized
quasi-particle energies Fanto and Alhassid (2021) and
$\tilde{f}_{k}=(1+e^{\beta\tilde{E}_{k}})^{-1}$ are their corresponding Fermi-
Dirac occupation numbers.

The first term on the right-hand-side of Eq. (16) is the SPA strength function
in the $\omega\to 0$ limit Rossignoli and Ring (1998)

$S_{\mathcal{O_{\lambda}}}^{(0)}(T,\sigma)=\frac{1}{2}\sum_{\mu}\sum_{\begin{subarray}{c}kl,k^{\prime}l^{\prime}\\\
E_{k}=E_{l}\end{subarray}}\mathcal{O}_{\lambda\mu,kl}^{*}\mathcal{O}_{\lambda\mu,k^{\prime}l^{\prime}}\langle\alpha_{k}^{\dagger}\alpha_{l}\alpha_{k^{\prime}}^{\dagger}\alpha_{l^{\prime}}\rangle_{\sigma}\;,$
(17)

where $\alpha_{k},\alpha_{k}^{\dagger}$ are, respectively, the annihilation
and creation quasiparticles operators in the eigenbasis of $\hat{h}_{\sigma}$
with positive energies $E_{k}$. The expectation values in Eq. (17) for a given
auxiliary field configuration $\sigma$ are evaluated using Wick’s theorem. We
note that the sum in Eq. (17) include both states with $k=l$ as well as
distinct quasiparticle states $k\neq l$ that have the same energies
$E_{k}=E_{l}$.

The above formulation assumed the grand-canonical ensemble, while the
appropriate ensemble to use is the canonical ensemble of fixed number of
protons and neutrons. Rather than implementing a full particle-number
projection, we carry out the canonical projection approximately in the saddle-
point approximation together with an exact number-parity projection. The
latter describes well odd-even effects due to pairing correlations and is
carried out using the number-parity projection operator
$\hat{P}_{\eta}=\Pi_{\nu=p,n}(1+\eta_{\nu}e^{i\pi\hat{N}_{\nu}})/2$, where
$\eta_{\nu}=1$ ($\eta_{\nu}=-1$) for even (odd) number parity.

In this formulation, the SPA expression for the canonical thermal strength
function is

${S}_{{\mathcal{O}}_{\lambda}}(T;\omega)\simeq\frac{\int d\sigma
M(\sigma)\mathcal{Z}_{\eta}(\sigma){S}_{{\mathcal{O}}_{\lambda},\eta}(T,\sigma;\omega)}{\int
d\sigma M(\sigma)\mathcal{Z}_{\eta}(\sigma)}\;,$ (18)

where
$\mathcal{Z}_{\eta}=Z_{\eta}e^{\sum_{\nu=p,n}(\ln\zeta_{\nu}-\beta\mu_{\nu}N_{\nu})}$
with
${{Z}_{\eta}(\sigma)=\text{Tr}\left[{\hat{P}}_{\eta}{e}^{-\beta\left({\hat{h}}_{\sigma}-\sum_{\nu=\text{p,n}}{\mu}_{\nu}{\hat{N}}_{\nu}\right)}\right]}$
being the number-parity projected one-body partition function and
$\zeta_{\nu}$ saddle-point factors given by Eq. (25) of Ref. Fanto and
Alhassid (2021). The chemical potentials $\mu_{p}$ and $\mu_{n}$ are
determined by the saddle-point conditions for number of protons $N_{p}$ and
number of neutrons $N_{n}$, respectively.

$S_{{\mathcal{O}}_{\lambda},\eta}(T,\sigma;\omega)$, the number-parity
projected strength function for a given configuration of the static auxiliary
fields $\sigma$, is given by Eqs. (16) and (17) but with the quasi-particle
occupation numbers replaced with the number-parity projected occupation
numbers $f_{k}^{\eta}$ (see Eqs. (22) and (21) in Ref. Fanto and Alhassid
(2021)), and $\langle\ldots\rangle_{\sigma}$ replaced by the projected
expectation value $\langle\ldots\rangle_{\sigma,\eta}$.

##  III Application to a chain of neodymium isotopes

###  III.1 CI shell model space and Hamiltonian

We use the theoretical framework of Sec. II to calculate finite-temperature
$M1$ strength functions in a chain of even-mass neodymium isotopes 144-152Nd,
which describes the crossover from spherical to well-deformed nuclei. We apply
the MEM to extract finite-temperature strength functions from imaginary-time
response functions calculated with SMMC, with the aid of prior functions
calculated from either SPA or QRPA.

The CI shell-model single-particle space consists of a set of spherical
orbitals ${{a}_{\nu}=({n}_{{a}_{\nu}},{l}_{{a}_{\nu}},{j}_{{a}_{\nu}})}$,
where ${\nu=\\{{\rm p,n}\\}}$. The single-particle orbitals and their energies
${{\epsilon}_{{a}_{\nu}}}$ are taken from a central Woods-Saxon potential plus
spin-orbit interaction Bohr and Mottelson . The interaction includes a
monopole pairing term and multipole-multipole interaction terms. The
Hamiltonian is then given by

$\displaystyle\hat{H}$
$\displaystyle=\sum_{{a}_{\nu}}{\epsilon}_{{a}_{\nu}}{\hat{n}}_{{a}_{\nu}}-\sum_{\nu=p,n}{g}_{\nu}{\hat{P}}_{\nu}^{\dagger}{\hat{P}}_{\nu}$
$\displaystyle-\sum_{\lambda}{\chi}_{\lambda}:({\hat{O}}_{\lambda;p}+{\hat{O}}_{\lambda;n}).({\hat{O}}_{\lambda;p}+{\hat{O}}_{\lambda;n}):$
(19)

with
${{\hat{P}}_{\nu}^{\dagger}=\sum_{({a}_{\nu}m)>0}{a}_{{a}_{\nu}m}^{\dagger}{a}_{\overline{{a}_{\nu}m}}^{\dagger}}$
the ${J=0}$ pair creation operator, where ${\ket{{a}_{\nu}m}}$ is a shell
model single-particle state with magnetic quantum number ${m}$ for particle
species ${\lambda}$, ${({a}_{\nu}m)>0}$ describes half the single-particle
states, and
${\ket{\overline{{a}_{\nu}m}}={(-)}^{{j}_{{a}_{\nu}}+{l}_{{a}_{\nu}}+m}\ket{{a}_{\nu}-m}}$
is the time-reversed state of ${\ket{{a}_{\nu}m}}$. Here, ${::}$ denotes
normal ordering and
${{\hat{O}}_{\lambda;\nu}=\frac{1}{\sqrt{2\lambda+1}}\sum_{ab}\bra{{j}_{{a}_{\nu}}}|\frac{d{V}_{\text{WS}}}{dr}{Y}_{\lambda}|\ket{{j}_{{b}_{\nu}}}{\left[{a}_{{a}_{\nu}}^{\dagger}\times{\tilde{a}}_{{b}_{\nu}}\right]}^{\lambda}}$
is a surface-peaked multipole operator, with ${{V}_{\text{WS}}}$ being the
central Woods-Saxon potential. We include quadrupole, octupole, and
hexadecupole terms (i.e. ${\lambda=2,3,4}$ ) with corresponding strengths
${\chi}_{\lambda}=\chi k_{\lambda}$. The parameter ${\chi}$ is determined
self-consistently Alhassid  _et al._ (1996) and ${{k}_{\lambda}}$ are
renormalization factors accounting for core polarization effects.

For the lanthanides we use a model space consisting of the following orbitals:
${0{g}_{7/2}}$, ${1{d}_{5/2}}$, ${1{d}_{3/2}}$, ${2{s}_{1/2}}$,
${0{h}_{11/2}}$, and ${1{f}_{7/2}}$ for protons; ${0{h}_{11/2}}$,
${0{h}_{9/2}}$, ${1{f}_{7/2}}$, ${1{f}_{5/2}}$, ${2{p}_{3/2}}$,
${2{p}_{1/2}}$, ${0{i}_{13/2}}$, and ${1{g}_{9/2}}$ for neutrons. The values
of the pairing strengths $g_{\nu}$ and renormalization factors $k_{\lambda}$
are given in Ref. Alhassid  _et al._ (2008).

We consider the $M1$ transition operator

${\hat{\mathcal{O}}}_{M1}=\sqrt{\frac{3}{4\pi}}\frac{{\mu}_{N}}{\hbar
c}({g}_{l}\mathbf{l}+{g}_{s}\mathbf{s})\;,$ (20)

where ${\mathbf{l}}$ and ${\mathbf{s}}$ are the orbital and spin angular
momentum operators, respectively. In our calculations, we used the free-
nucleon $g$ factors ${{g}_{l,p}=1}$, ${{g}_{l,n}=0}$, ${{g}_{s,p}=5.5857}$,
and ${g=-3.8263}$.

###  III.2 Results

Figure 1: The $M1$ imaginary-time response function ${R}_{M1}(\tau)$ as a
function of imaginary time $\tau$ for a chain of even-mass neodymium isotopes
144,148,152Nd. The left column shows results for near-zero temperatures $T$
(i.e., close to the ground state) while the right column corresponds to the
neutron separation energy. The QRPA (orange solid line) and SPA (green solid
line) response functions are compared with the SMMC response functions (blue
diamonds). The SMMC response functions include statistical errors but these
errors are smaller than the size of the symbols.

We calculated the finite-temperature $M1$ imaginary-time response function
$R_{M1}(T;\tau)$ in SMMC using Eq. (5). We discretize the imaginary time
interval $[0,\beta]$ into slices of length $\Delta\beta$. We carried out the
calculations for time slices $\Delta\beta=1/32,1/64$ MeV-1 and extrapolated to
$\Delta\beta=0$. The MEM is then applied to determine the $M1$ strength
function $S_{M1}(T;\tau)$ using for the prior either the SPA or QRPA strength
functions. For both priors, we replace the $\delta$ functions [see, e.g., the
$\sigma$-dependent SPA strength function in Eq. (16)] by a Lorentzian with a
fixed width parameter of $0.2$ MeV. This width is comparable to the bin width
used in conventional CI shell-model studies Schwengner  _et al._ (2013);
Karampagia  _et al._ (2017); Schwengner  _et al._ (2017); Sieja (2017); Mitdbø
_et al._ (2018). In the following, we refer to the MEM with the QRPA prior as
MEM-QRPA, and the MEM with the SPA prior as MEM-SPA.

####  III.2.1 $M1$ response and strength functions using the QRPA and SPA
priors

The reliability of MEM depends upon a good choice of the prior strength
function. In Fig. 1 we compare the SMMC $M1$ imaginary-time response function
$R_{M1}$ (blue solid diamonds) with the SPA (green lines) and QRPA (orange
lines) response functions at several temperatures for spherical (144Nd),
transitional (148Nd) and deformed (152Nd) neodymium isotopes. $E_{i}$ in the
panels denotes the initial excitation energy of the nucleus which in the
finite-temperature formalism is given by the average thermal energy $E(T)$
(measured with respect to the ground-state energy). We observe that at very
low temperatures close to the ground state (left column in Fig. 1 both the
QRPA and SPA response functions are quite close to the exact SMMC results,
with the QRPA response function being essentially on top of the SMMC response
function. However, with increasing temperature, the SPA response function is
in closer agreement with the SMMC results, when compared to the QRPA response
functions. In particular at temperatures that correspond to the neutron
separation energy of $E_{i}\sim 7-8$ MeV (right column in Fig. 1), the SPA is
a superior approximation to the QRPA. We conclude that large-amplitude static
fluctuations around the mean field that are accounted for in the SPA but are
missing in the QRPA play an important role at finite temperature.

Figure 2: Strength function $S_{M1}(\omega)$ for 144Nd at different
temperatures, from close to the ground state ($T=0.05$ MeV) up to a
temperature of $T=0.73$ MeV, which corresponds to the neutron separation
energy. The MEM results (blue solid lines) are compared with the prior results
(orange dashed lines). The top row shows results with the QRPA strength as
prior, while the bottom row shows results with the SPA prior.

In Fig. 2 we show the temperature dependence of the $M1$ strength function
$S_{M1}$ versus transition energy $\omega$ for the spherical isotope 144Nd.
The MEM strength functions are shown in blue solid lines while the prior
strength functions are the orange dashed lines. In the top row the prior is
the QRPA strength function while in the bottom row the prior is the SPA
strength function. The various temperatures correspond to average initial
excitation energies $E_{i}$ shown in the panels. The near ground-state
strength functions ($T=0.05$ MeV) obtained with QRPA and SPA are similar. For
the QRPA, the MEM strength function essentially coincides with the QRPA prior,
while for the SPA we observe a small difference. As the temperature increases
towards the neutron separation energy $\sim 7.94$ MeV, we observe larger
differences between the MEM and prior strength functions in the QRPA case as
compared with the SPA case. This is consistent with the comparison between the
various response functions shown in Fig. 1. We conclude that the strength
function at very low temperatures near the ground state is well described by
the QRPA while the SPA is a better prior to use at the neutron resonance
energy.

As the initial excitation energy increases, we notice the appearance of an
$\omega=0$ peak that becomes larger as we approach the neutron resonance
energy. This will be interpreted as the LEE.

In general the response function is less sensitive to the behavior of the
strength function at large values of $\omega$. This is attributed to the
exponential suppression of the large-$\omega$ contribution in the Laplace
transform. On the other hand, we expect the low-$\omega$ results and in
particular the LEE to be highly reliable. We note that the MEM strength
function reproduces the LEE even for the QRPA prior that do not exhibits a
LEE.

####  III.2.2 Ground-state $M1$ strength functions

The ground-state $M1$ strength functions for the chain of even-mass neodymium
isotopes 144-152Nd are shown in Fig. 3 using very low temperatures. The top
row corresponds to the QRPA prior and the bottom row corresponds to the SPA
prior. For each prior we show the corresponding strength functions by orange
dashed lines, while the MEM strength functions are described by the blue solid
lines. The overall structure of the MEM strength functions exhibits similar
behavior when using either the QRPA or the SPA strength functions as prior.
The deviation of the MEM from the prior is smaller for the QRPA, confirming
our earlier conclusion that at very low temperatures the QRPA is a better
approximation than the SPA.

In general the strength function $S_{M1}$ describes the absorption of
$\gamma$-rays for $\omega>0$ and their emission for $\omega<0$. For
transitions from the ground state the strength is non-zero for only
$\omega>0$.

For the spherical isotopes 144Nd and 146Nd we observe in Fig. 3 a pronounced
peak at $\sim 6$ MeV, which we interpret as the spin-flip mode. This mode
involves a transitions of the type $(j=l+\frac{1}{2})\to(j=l-\frac{1}{2})$ and
is characterized by the flip of one or more nuclear spins, which results in a
change in the total angular momentum of the nucleus. The strength of the spin-
flip mode becomes fragmented in the transitional 148Nd isotope and in the
deformed 150-152Nd isotopes.

We also observe in the ground-state $M1$ strength function a structure around
$\omega\simeq 2-3$ MeV, which we interpret as the scissors mode. This mode
describes the collective oscillations of the proton and neutron clouds against
each other like scissors blades Heyde  _et al._ (2010).

Figure 3: Near ground-state strength functions $S_{\rm M1}(\omega)$ versus
$\omega$ for the even-mass 144-152Nd isotopes. The MEM results (blue solid
lines) are compared with the prior results (orange dashed lines). The top row
shows results with the QRPA prior, while the bottom row shows results with the
SPA prior.

####  III.2.3 Finite-temperature $M1$ strength functions

As the temperature $T$ increases (which translates into increasing initial
excitation energy $E_{i}$ of the nucleus), a peak centered at $\omega=0$
develops in the strength function. This is clearly seen in Fig. 2) for 144Nd.
We see such a peak already $T=0.32$ MeV (corresponding to $E_{i}\simeq 0.96$
MeV) and it becomes more pronounced with increasing temperature. We will
identify this peak with the LEE observed in the deexcitation strength function
at low $\gamma$-ray energies $E_{\gamma}$. We note that the SPA strength
function already reproduces the LEE and the MEM-SPA makes it more enhanced. On
the other hand, the LEE is missing in the QRPA strength function but appears
in the MEM-QRPA. This suggests that the LEE is a very robust feature of the
finite-temperature $M1$ strength function.

In Fig. 4, we show the $M1$ strength functions for the even-mass neodymium
isotopes for initial energies that correspond to the neutron separation
energy. The orange dashed lines are the SPA strength functions, while the blue
solid lines are the MEM-SPA. We limit our calculations to the SPA prior since
at the neutron resonance energy, the SPA response function is closer to the
SMMC response function when compared with the QRPA response function (see Fig
1). We observe a LEE structure for all the neodymium isotopes with the LEE
peak decreasing with increased neutron number, i.e., in the crossover from
spherical to deformed nuclei. A slight shallow structure, which we interpret
as the scissors mode built on top of excited states, seems to develop close to
$\sim 2$ MeV in the deformed isotopes. We conclude that some of the LEE
strength is transferred to the scissors mode as the nucleus becomes more
deformed. This scissors mode is more noticeable in the finite-temperature QRPA
calculations as is seen in the QRPA results for the spherical nucleus 144Nd in
Fig. 2). A scissors mode built on top of excited states has been observed
experimentally in lanthanide nuclei Krtička and Bečvàř (2011). The increase in
the scissors mode in deformed nuclei has been observed experimentally at the
neutron separation energy in neodymium isotopes Guttormsen  _et al._ (2022),

A spin-flip mode at $\sim 6$ MeV is observed in all of the neodymium isotopes
also at the neutron separation energy (see Fig. 4) and its height decreases in
the crossover from spherical to deformed neodymium isotopes.

Figure 4: Strength functions $S_{\rm M1}(\omega)$ versus $\omega$ for the
even-mass 144-152Nd isotopes at temperatures that correspond to the neutron
separation energies. The orange dashed lines are the SPA strength functions
and the blue solid lines are the MEM strength functions using the SPA as
prior. Figure 5: Deexcitation ${\gamma}$-ray strength functions $f_{\rm M1}$
as a function of the $\gamma$-ray energy $E_{\gamma}$ at the neutron
separation energy for the even-mass neodymium isotopes 144-152Nd. The MEM
results (blue solid lines) are compared with the SPA prior results (orange
dashed lines) and with the experimental data (black symbols) of Ref.
Guttormsen  _et al._ (2022).

####  III.2.4 Deexcitation $M1$ strength function $f_{\rm M1}$ and the LEE

In experiments, one usually measures the deexcitation $M1$ strength function,
$f_{\rm M1}$. This function describes the probability for the compound nucleus
to decay by $M1$ $\gamma$-rays with energy $E_{\gamma}$. The deexcitation
strength $f_{\rm M1}$ for initial energy $E_{i}$ and $\gamma$-ray energy
$E_{\gamma}$ is related to the strength function $S_{\rm M1}$ by Fanto and
Alhassid (2024)

$f_{M1}(E_{i},E_{\gamma})\approx\frac{1}{3}a\frac{\tilde{\rho}(E_{i})}{\tilde{\rho}(E_{i}-E_{\gamma})}S_{M1}(T;\omega=-E_{\gamma})\,,$
(21)

where $a=\frac{16\pi}{9(\hbar c)^{3}}$, $T$ is the temperature that
corresponds to an initial excitation energy $E_{i}$, and $\tilde{\rho}(E_{x})$
is the level density at excitation energy $E_{x}$. The factor of $1/3$ takes
into account the 3 possible value of the final spin for given initial value of
the spin dal .

Using SMMC level densities Guttormsen  _et al._ (2021) and the MEM strength
functions $S_{M1}(T;\omega)$, we calculated the deexcitation strength function
$f_{\rm M1}$ using Eq. (21) for the even mass 144-152Nd at an initial energy
that corresponds to the neutron separation energy. In Fig. 5 we show $f_{\rm
M1}$ versus $E_{\gamma}$. The dashed orange lines are the SPA $f_{\rm M1}$ and
the blue solid lines are the MEM-SPA results. We clearly observed the LEE at
low $\gamma$-ray energies (below $\sim 2$ MeV) for all the even-mass neodymium
isotopes.

We compare our results with recent experimental data for the deexcitation
$\gamma$SF Guttormsen  _et al._ (2022). We note, however, that the
experimental results include contributions from both $E1$ and $M1$ and a
detailed comparison with experiment requires the calculation of the $E1$
$\gamma$SF. In the spherical isotopes 144,146Nd, a clear LEE is observed
experimentally, in overall agreement with our theoretical results. In the
transitional isotope 148Nd, a LEE seems to develop below $\sim 1.5$ MeV. No
LEE is observed experimentally in the deformed isotope 150Nd while our
calculated $f_{M1}$ clearly indicates the existence of a LEE. We note,
however, the absence of experimental data at low $\gamma$-ray energies. Thus
the missing LEE in the experimental results for 150Nd does not preclude the
possibility of a residual LEE in neutron-rich neodymium isotopes. In fact, our
calculations suggest its persistence in 152Nd.

Figure 6: The LEE slope $\kappa$ (on a logarithmic scale) as a function of
temperature $T$ for the even-mass neodymium isotopes 144-152Nd in the MEM-SPA
calculations.

As suggested in Refs. Schwengner  _et al._ (2013); Karampagia  _et al._
(2017), we find that the LEE is well described by an exponential form
Schwengner  _et al._ (2013)

$f_{\rm M1}(E_{\gamma})\approx C_{0}{e}^{-\kappa{E}_{\gamma}}\;,$ (22)

where $C_{0}$ is the value of the LEE at $E_{\gamma}=0$ and $\kappa$ describes
the slope of the LEE on a logarithmic scale. In Fig. 6, the values of $\kappa$
for 144-152Nd are plotted as a function of temperature $T$. The results show
that $\kappa$ remains roughly constant across a wide range of temperatures for
each the neodymium isotopes. This finding, which is consistent with
conventional CI shell-model results in smaller model space Karampagia  _et
al._ (2017), indicates that the slope of the LEE depends only weakly on the
initial energy. On average, the $\kappa$ values for the different isotopes are
similar.

Strictly speaking, in the micorcanonical ensemble for which the initial energy
is fixed, the temperature fluctuates and its variance is given by
$\langle(\Delta T)^{2}\rangle=T^{2}/C$, where $C$ is the canonical heat
capacity. For the neodymium isotopes the r.m.s. $\sqrt{\langle(\Delta
T)^{2}\rangle}$ of temperature fluctuations varies between $0$ at $T=0$ and
$\sim 0.15-0.18$ MeV at the neutron separation energies. Since the LEE depends
only weakly on temperature, we expect our results for the LEE to be robust
despite these temperature fluctuations.

In Table 1 we list the values of $\kappa$ at the neutron separation energy. We
find that the slope $\kappa$ increases while the LEE maximal value $C_{0}$
decreases with the onset of deformation along the chain of neodymium isotopes.
In particular, we find close agreement between our value of $\kappa=2.1$ MeV-1
in the spherical isotopes 144,146Nd and the experimental value of $\kappa=1.9$
MeV-1 Guttormsen  _et al._ (2022).

|  $\kappa$ (${\text{MeV}}^{-1}$) |  $C_{0}$ (${\text{MeV}}^{-3}$)
---|---|---
144Nd | 2.1 | $4.1\times{10}^{-7}$
146Nd | 2.1 | $3.9\times{10}^{-7}$
148Nd | 2.6 | $3.3\times{10}^{-7}$
150Nd | 2.6 | $3.1\times{10}^{-7}$
152Nd | 2.7 | $2.9\times{10}^{-7}$
Table 1: LEE slopes $\kappa$ and LEE maximal values $C_{0}$ at $E_{\gamma}=0$
obtained from fits to the deexcitation strength function $f_{\rm M1}$ in Fig.
5 at low $\gamma$-ray energies.

##  IV Conclusions

In this study, we employed the MEM to deduce $M1$ $\gamma$SF in the even-mass
144-152Nd isotopes from the exact SMMC imaginary-time $M1$ response functions,
using the SPA and QRPA strength functions as prior. We find that at low
temperatures (close to the ground state) the QRPA response function is
somewhat closer to the SMMC response function than the corresponding SPA, but
that at higher temperatures and, in particular close to the neutron resonance
energy, the SPA is a better starting point to use as prior. We identify a
spin-flip mode in both the ground state and finite-temperature $M1$
$\gamma$SF. A scissors mode, clearly visible in the ground-state $\gamma$SF,
is suppressed at finite temperature but is still visible in the deformed
nuclei.

At finite temperature, and, in particular, at the neutron separation energy, a
LEE structure develops in the $\gamma$SF of all the even-mass neodymium
isotopes 144-152Nd. The peak of the LEE is strongest in the spherical isotopes
and decreases in the deformed nuclei with some of its strength transferring to
the scissors mode at $\omega\approx 2-3$ MeV built on top of excited states.
The LEE structure is not reproduced in the finite-temperature QRPA strength
function but is observed in the SPA strength function. This LEE structure is
very robust and emerges also in the MEM with the QRPA prior despite its
absence in the QRPA strength function. In addition to the LEE and the scissors
mode, we observe a spin-flip mode around $\omega\approx 6$ MeV, with its
strength decreasing with deformation.

We also calculated the deexcitation strength function $f_{M1}$ as a function
of the $\gamma$-ray energy $E_{\gamma}$ at the neutron separation energy, and
identified a LEE structure in all the neodymium isotopes under study. Our
results are overall consistent with the experimental findings of Ref.
Guttormsen  _et al._ (2022). The latter did not observe a LEE in the deformed
isotope 150Nd but this might be due to the lack of data at the very low
$\gamma$-ray energies. We identify a LEE also in the deformed 152Nd for which
there are no experimental data. We also find that the slope of the LEE (on a
logarithmic scale) is approximately independent of the initial temperature (or
excitation energy).

Currently our method is the only one capable of reproducing a LEE in heavy
open-shell nuclei for which conventional CI shell-model calculations are
prohibited. If the LEE persists in heavy neutron-rich nuclei, it will have
significant effects on $r$-process nucleosynthesis by enhancing the neutron-
capture rates.

###### Acknowledgements.

This work was supported in part by the U.S. DOE grant No. DE-SC0019521. The
calculations used resources of the National Energy Research Scientific
Computing Center (NERSC), a U.S. Department of Energy Office of Science User
Facility operated under Contract No. DE-AC02-05CH11231. We thank the Yale
Center for Research Computing for guidance and use of the research computing
infrastructure. W. R. is a Research Associate of the F.R.S.-FNRS and a member
of BLU-ULB (Brussels laboratory of the Universe).

## References

  * Hauser and Feshbach (1952) W. Hauser and H. Feshbach, Phys. Rev. 87, 366 (1952).
  * Koning and Rochman (2012) A. J. Koning and D. Rochman, Nucl. Data Sheets 113, 2841 (2012).
  * (3) G. A. Bartholomew, E. D. Earle, A. J. Ferguson, J. W. Knowles, and M. A. Lone,  _Advances in Nuclear Physics_ , edited by M. Baranger and E. Vogt (Springer Boston 1973) pp. 267–298.
  * Mumpower  _et al._ (2016) M. Mumpower, R. Surman, G. C. McLaughlin, and A. Aprahamian, Prog. Part. Nucl. Phys. 86, 86 (2016).
  * Bohle  _et al._ (1984) D. Bohle, A. Richter, W. Steffen, A. E. L. Dieperink, N. L. Iudice, F. Palumbo, and O. Scholten, Phys. Lett. B 137, 27 (1984).
  * Heyde  _et al._ (2010) K. Heyde, P. von Neumann-Cosel, and A. Richter, Rev. Mod. Phys. 82, 2365 (2010).
  * Mumpower  _et al._ (2017) M. R. Mumpower, T. Kawano, J. L. Ullmann, M. Krtička, and T. M. Sprouse, Phys. Rev. C 96, 024612 (2017).
  * Voinov  _et al._ (2004) A. Voinov, E. Algin, U. Agvaanluvsan, T. Belgya, R. Chankova, M. Guttormsen, G. E. Mitchell, J. Rekstad, A. Schiller, and S. Siem, Phys. Rev. Lett. 93, 142504 (2004).
  * Guttormsen  _et al._ (2005) M. Guttormsen, R. Chankova, U. Agvaanluvsan, E. Algin, L. A. Bernstein, F. Ingebretsen, T. Lönnroth, S. Messelt, G. E. Mitchell, J. Rekstad, A. Schiller, S. Siem, A. C. Sunde, A. Voinov, and S. Ødegård, Phys. Rev. C 71, 044307 (2005).
  * Wiedeking  _et al._ (2012) M. Wiedeking, L. A. Bernstein, M. Krtička, D. L. Bleuel, J. M. Allmond, M. S. Basunia, J. T. Harke, P. Fallon, R. B. Firestone, B. L. Goldblum, R. Hatarik, P. T. Lake, I.-Y. Lee, S. R. Lesher, S. Paschalis, M. Petri, L. Phair, , and N. D. Scielzo, Phys. Rev. Lett. 108, 162503 (2012).
  * Larsen  _et al._ (2013) A. C. Larsen, N. Blasi, A. Bracco, F. Camera, T. K. Eriksen, A. Görgen, M. Guttormsen, T. W. Hagen, S. Leoni, B. Million, H. T. Nyhus, T. Renstrøm, S. J. Rose, I. E. Ruud, S. Siem, T. Tornyi, G. M. Tveten, A. V. Voinov, , and M. Wiedeking, Phys. Rev. Lett. 111, 242504 (2013).
  * Larsen  _et al._ (2018) A. C. Larsen, J. E. Midtbø, M. Guttormsen, T. Renstrøm, S. N. Liddick, A. Spyrou, S. Karampagia, B. A. Brown, O. Achakovskiy, S. Kamerdzhiev, D. L. Bleuel, A. Couture, L. C. Campo, B. P. Crider, A. C. Dombos, R. Lewis, S. Mosby, F. Naqvi, G. Perdikakis, C. J. Prokop, S. J. Quinn, and S. Siem, Phys. Rev. C 97, 054329 (2018).
  * Kheswa  _et al._ (2015) B. V. Kheswa, M. Wiedeking, F. Giacoppo, S. Goriely, M. Guttormsen, A. C. Larsen, F. L. B. Garrote, T. K. Eriksen, A. Görgen, T. W. Hagen, P. E. Koehler, M. Klintefjord, H. T. Nyhus, P. Papka, T. Renstrøm, S. Rose, E. Sahin, S. Siem, and T. Tornyi, Phys. Lett. B 744, 268 (2015).
  * Naqvi  _et al._ (2019) F. Naqvi, A. Simon, M. Guttormsen, R. Schwengner, S. Frauendorf, C. S. Reingold, J. T. Harke, N. Cooper, R. O. Hughes, S. Ota, and A. Saastamoinen, Phys. Rev. C 99, 054331 (2019).
  * Simon  _et al._ (2016) A. Simon, M. Guttormsen, A. C. Larsen, C. W. Beausang, P. Humby, J. T. Harke, R. J. Casperson, R. O. Hughes, T. J. Ross, J. M. Allmond, R. Chyzh, M. Dag, J. Koglin, E. McCleskey, M. McCleskey, S. Ota, and A. Saastamoinen, Phys. Rev. C 93, 034303 (2016).
  * Guttormsen  _et al._ (2022) M. Guttormsen, K. O. Ay, M. Ozgur, E. Algin, A. C. Larsen, F. L. Bello Garrote, H. C. Berg, L. Crespo Campo, T. Dahl-Jacobsen, F. W. Furmyr, D. Gjestvang, A. Görgen, T. W. Hagen, V. W. Ingeberg, B. V. Kheswa, I. K. B. Kullmann, M. Klintefjord, M. Markova, J. E. Midtbø, V. Modamio, W. Paulsen, L. G. Pedersen, T. Renstrøm, E. Sahin, S. Siem, G. M. Tveten, and M. Wiedeking, Phys. Rev. C 106, 034314 (2022).
  * Jones  _et al._ (2018) M. D. Jones, A. O. Macchiavelli, M. Wiedeking, L. A. Bernstein, H. L. Crawford, C. M. Campbell, R. M. Clark, M. Cromaz, P. Fallon, I. Y. Lee, M. Salathe, A. Wiens, A. D. Ayangeakaa, D. L. Bleuel, S. Bottoni, M. P. Carpenter, H. M. Davids, J. Elson, A. Görgen, M. Guttormsen, R. V. F. Janssens, J. E. Kinnison, L. Kirsch, A. C. Larsen, T. Lauritsen, W. Reviol, D. G. Sarantites, S. Siem, A. V. Voinov, and S. Zhu, Phys. Rev. C 97, 024327 (2018).
  * Larsen and Goriely (2010) A. C. Larsen and S. Goriely, Phys. Rev. C 82, 014318 (2010).
  * Loens  _et al._ (2012) H. Loens, K. Langanke, G. Martínez-Pinedo, and K. Sieja, Eur. Phys. J. A 48, 34 (2012).
  * Schwengner  _et al._ (2013) R. Schwengner, S. Frauendorf, and A. C. Larsen, Phys. Rev. Lett. 111, 232504 (2013).
  * Brown and Larsen (2014) B. A. Brown and A. C. Larsen, Phys. Rev. Lett. 113, 252502 (2014).
  * Schwengner  _et al._ (2017) R. Schwengner, S. Frauendorf, and B. A. Brown, Phys. Rev. Lett. 118, 092502 (2017).
  * Sieja (2017) K. Sieja, Phys. Rev. Lett. 119, 052502 (2017).
  * Karampagia  _et al._ (2017) S. Karampagia, B. A. Brown, and V. Zelevinsky, Phys. Rev. C 95, 024322 (2017).
  * Sieja (2018) K. Sieja, Phys. Rev. C 98, 064312 (2018).
  * Litvinova and Belov (2013) E. Litvinova and N. Belov, Phys. Rev. C 88, 031302(R) (2013).
  * Mitdbø  _et al._ (2018) J. E. Mitdbø, A. C. Larsen, T. Renstrøm, F. L. B. Garrote, and E. Lime, Phys. Rev. C 98, 064321 (2018).
  * Johnson  _et al._ (1992) C. W. Johnson, S. E. Koonin, G. H. Lang, and W. E. Ormand, Phys. Rev. Lett. 69, 3157 (1992).
  * Alhassid  _et al._ (1994) Y. Alhassid, D. J. Dean, S. E. Koonin, G. Lang, and W. E. Ormand, Phys. Rev. Lett. 72, 613 (1994).
  * Nakada and Alhassid (1997) H. Nakada and Y. Alhassid, Phys. Rev. Lett. 79, 2939 (1997).
  * Nakada and Alhassid (1998) H. Nakada and Y. Alhassid, Physics Letters B 436, 231 (1998).
  * Ormand (1997) W. E. Ormand, Phys. Rev. C 56, R1678 (1997).
  * Langanke (1998) K. Langanke, Phys. Lett. B 438, 235 (1998).
  * Alhassid  _et al._ (1999) Y. Alhassid, S. Liu, and H. Nakada, Phys. Rev. Lett. 83, 4265 (1999).
  * Alhassid  _et al._ (2007) Y. Alhassid, S. Liu, and H. Nakada, Phys. Rev. Lett. 99, 162504 (2007).
  * Alhassid  _et al._ (2008) Y. Alhassid, L. Fang, and H. Nakada, Phys. Rev. Lett. 101, 082501 (2008).
  * (37) Y. Alhassid,  _Emergent Phenomena in Atomic Nuclei from Large-Scale Modeling: a Symmetry-Guided Perspective_ , edited by K. D. Launey (World Scientific, Singapore, 2017) pp. 267–298.
  * Gubernatis  _et al._ (1991) J. E. Gubernatis, M. Jarrell, R. N. Silver, and D. S. Sivia, Phys. Rev. B 44, 6011 (1991).
  * Jarrell and Gubernatis (1996) M. Jarrell and J. E. Gubernatis, Phys. Rep. 269, 133 (1996).
  * Gubernatis and Werner (2016) J. E. Gubernatis and P. Werner,  _Quantum Monte Carlo methods: algorithms for lattice models_ (Cambridge University Press, Cambridge, 2016).
  * Fanto and Alhassid (2024) P. Fanto and Y. Alhassid, Phys. Rev. C 109, L031302 (2024).
  * Özen  _et al._ (2013) C. Özen, Y. Alhassid, and H. Nakada, Phys. Rev. Lett. 110, 042502 (2013).
  * Mühlschlegel  _et al._ (1972) B. Mühlschlegel, D. J. Scalapino, and R. Denton, Phys. Rev. B 6, 1767 (1972).
  * Alhassid and Zingman (1984) Y. Alhassid and J. Zingman, Phys. Rev. C 30, 684 (1984).
  * Lauritzen  _et al._ (1988) B. Lauritzen, P. Arve, and G. F. Bertsch, Phys. Rev. Lett. 61, 2835 (1988).
  * Arve  _et al._ (1988) P. Arve, G. Bertsch, B. Lauritzen, and G. Puddu, Annals of Physics 183, 309 (1988).
  * Attias and Alhassid (1997) H. Attias and Y. Alhassid, Nucl. Phys. A 652, 565 (1997).
  * Frosini  _et al._ (2024) M. Frosini, W. Ryssens, and K. Sieja, Phys. Rev. C 110, 014307 (2024).
  * Hubbard (1959) J. Hubbard, Phys. Rev. Lett. 3, 77 (1959).
  * Stratonovich (1957) R. L. Stratonovich, Dokl. Akad. Nauk SSSR 115, 1097 (1957).
  * Stratonovich (1958) R. L. Stratonovich, Sov. Phys. Dokl. 2, 416 (1958).
  * Koonin  _et al._ (1997) S. E. Koonin, D. J. Dean, and K. Langanke, Phys. Rep. 278, 1 (1997).
  * Alhassid and Bush (1990) Y. Alhassid and B. Bush, Nuclear Physics A 509, 461 (1990).
  * Bryan (1990) R. K. Bryan, Eur. Biophys. J. 18, 165 (1990).
  * (55) W. Ryssens and Y. Alhassid, Solution of the finite-temperature (q)rpa equations for shell-model Hamiltonians: HF-SHELL v2, in preparation .
  * Ryssens and Alhassid (2021) W. Ryssens and Y. Alhassid, Eur. Phys. J. A 57, 76 (2021).
  * Fanto and Alhassid (2021) P. Fanto and Y. Alhassid, Phys. Rev. C 103, 064310 (2021).
  * Rossignoli and Ring (1998) R. Rossignoli and P. Ring, Nuclear Physics A 633, 613 (1998).
  * (59) A. Bohr and B. R. Mottelson,  _Nuclear Structure_ , Vol. 1 (Benjamin, New York, 1969).
  * Alhassid  _et al._ (1996) Y. Alhassid, G. F. Bertsch, D. J. Dean, and S. E. Koonin, Phys. Rev. Lett. 77, 1444 (1996).
  * Krtička and Bečvàř (2011) M. Krtička and F. Bečvàř, Int. J. Mod. Phys. E 20, 488 (2011).
  * (62) D. DeMartini, private communication.
  * Guttormsen  _et al._ (2021) M. Guttormsen, Y. Alhassid, W. Ryssens, K. Ay, M. Ozgur, E. Algin, A. Larsen, F. Bello Garrote, L. Crespo Campo, T. Dahl-Jacobsen, A. Görgen, T. Hagen, V. Ingeberg, B. Kheswa, M. Klintefjord, J. Midtbø, V. Modamio, T. Renstrøm, E. Sahin, S. Siem, G. Tveten, and F. Zeiser, Physics Letters B 816, 136206 (2021).


Torsion-rotational transitions in methanol… Torsion-rotational transitions in
methanol as a probe of fundamental physical constants – electron and proton
masses J. S. Vorotyntseva, S. A. Levshakov Vorotyntseva, Levshakov *

#  Torsion-rotational transitions in methanol as a probe of fundamental
physical constants – electron and proton masses

J. S. Vorotyntseva<EMAIL_ADDRESS>S. A. Levshakov  Ioffe Institute,
Politechnicheskaya 26, St. Petersburg 194021, Russia

###### Abstract

We report on the using of torsion-rotational transitions in the CH3OH and
13CH3OH molecules to evaluate possible variations of the physical constant
$\mu=m_{e}/m_{p}$ – the electron-to-proton mass ratio – from spectral
observations of emission lines detected in the microwave range towards the
dense molecular cloud Orion-KL. An estimate of the upper limit on the relative
changes in $\mu$ is obtained by two independent ways – with 13CH3OH lines and
with the combination of 13CH3OH and CH3OH lines. The calculated upper limit
$\Delta\mu/\mu<1.1\times 10^{-8}$ (1$\sigma$) is in line with the most
stringent constraints on the variability of fundamental physical constants
established by other astrophysical methods.

???

The existence of Dark Matter in the Universe follows from a number of
observational facts, including flat rotation curves of galaxies at large
galactocentric distances, gravitational lensing of cosmologically distant
objects and the large-scale structure of the spatial distribution of galaxies
[1]. To explain the nature of Dark Matter, various models have been
considered, including phenomena extending the Standard Model of particle
physics (SM). Several of these models assume the existence of hypothetical
scalar fields interacting with the baryonic component of an ordinary substance
[2, 3, 4]. The result of such interactions can lead to the space-time
variations of dimensionless physical constants, such as the fine structure
constant $\alpha$ and the electron-to-proton mass ratio $\mu$
\̃cite[cite]{[\@@bibref{}{missing}{}{}]}bib:Uz-[8]. Therefore, the new
theories can be tested experimentally using the relative measurements of
$\mu=m_{e}/m_{p}$. It should be noted that the mass of the electron $m_{e}$ is
directly related to Higgs-like scalar fields, while the main contribution to
the mass of the proton $m_{p}$ comes from the binding energy of quarks.

Electron-vibro-rotational transitions in molecular spectra have a specific
dependence on $\mu$, individual for each transition [10, 11]. The reaction of
the transition to a change in $\mu$ is characterized by a dimensionless
sensitivity coefficient $Q_{\mu}$, which is defined as

$Q_{\mu}=\frac{df/f}{d\mu/\mu},$ (1)

here $df/f$ is the relative frequency offset, and $d\mu/\mu$ is given by

$\frac{\Delta\mu}{\mu}=\frac{\mu_{obs}-\mu_{lab}}{\mu_{lab}},$ (2)

where $\mu_{obs}$, $\mu_{lab}$ are astronomical and laboratory values of
$\mu$, correspondingly. At the same time, the sensitivity coefficients can
take different signs, which leads to an increase or decrease in the observed
frequency compared to its laboratory value.

The most stringent limits on $\mu$-variations at large redshifts $z$ were
obtained from extragalactic observations towards the quasar J1443+2724 ($z$ =
4.22). From the analysis of the Lyman and Werner absorption lines of molecular
hydrogen H2 the upper limit on $\Delta\mu/\mu<8\times 10^{-6}$ was obtained
[12]. In the Milky Way, the most stringent upper limits were established from
observations of the 23 GHz inversion transition in ammonia NH3, which has a
sensitivity coefficient $Q_{\mu}=4.46$ [13], compared to the pure rotational
transitions in HC3N, HC5N, and HC7N, which have $Q_{\mu}=1$:
$\Delta\mu/\mu<7\times 10^{-9}$ [14]. Independent estimates from observations
of the thermal emission lines of methanol CH3OH in the core of the molecular
cloud L1498 lead to a value of $\Delta\mu/\mu<2\times 10^{-8}$ [15]. Similar
constraints follow from the measurements of the radial velocities of methanol
maser lines: $\Delta\mu/\mu<2\times 10^{-8}$ [16] and $\Delta\mu/\mu<2.7\times
10^{-8}$ [17]. All estimates of $\Delta\mu/\mu$ are given at a 1$\sigma$
confidence level.

It should be noted that in previous works methanol isotopologues have not been
used extensively. The first estimates of the upper limit on $\mu$-variations
were obtained from the TMRT 65-m telescope [18] observations of the thermal
emission lines of 13CH3OH in the star-forming region NGC 6334I:
$\Delta\mu/\mu<3\times 10^{-8}$. [19].

In the molecules with hindered internal motion, enhanced sensitivity
coefficients $Q_{\mu}$ are inherent for tunneling transitions, since the
tunneling probability depends exponentially on the masses of tunneling
particles [13, 20, 21]. The most promising molecule for these studies is
methanol (CH3OH), where the methyl group CH3 can make torsional vibrations
relative to the hydroxyl group OH. In this case, the hydrogen atom of the
hydroxyl group can be located in three possible positions with equal energies,
and to move from one configuration to another, it must pass through a
potential barrier caused by the three hydrogen atoms of the methyl group. So,
there is an internal hindered motion of the hydrogen atom relative to the
methyl group.

The sensitivity coefficients $Q_{\mu}$ for methanol were calculated by two
independent methods in 2011 [20, 21]. The results show that low-frequency (in
the range of 1–50 GHz) transitions have high values of $Q_{\mu}$ with
different signs: $-17\leq Q_{\mu}\leq+43$, which, compared to the sensitivity
coefficients of the molecular hydrogen H2 lines ($|Q_{\mu}|\sim 10^{-2}$) give
a gain in the limiting estimates of $\Delta\mu/\mu$ by a factor of more than
1000.

In our previous work [19], a list of molecules with the enhanced sensitivity
coefficients was extended due to the methanol isotopologues – 13CH3OH with
$-32\leq Q_{\mu}\leq+78$, and CH318OH with $-109\leq Q_{\mu}\leq+33$.

Turning to the practical measurements of $\Delta\mu/\mu$, we note that to
estimate this value, pairs of molecular lines with different coefficients
$Q_{\mu,1}$ and $Q_{\mu,2}$ are used [20]:

$\frac{\Delta\mu}{\mu}=\frac{V_{1}-V_{2}}{c(Q_{\mu,2}-Q_{\mu,1})},$ (3)

where $V_{1}$ and $V_{2}$ are the measured radial velocities of these lines,
and $c$ is the speed of light. The conversion from the frequency scale $f$ to
the velocite scale $V$ is made by the radio astronomical definition
$V/c=(f_{lab}-f_{obs})/f_{lab}$.

The accuracy of the $\Delta\mu/\mu$ measurements is due to the influence of
various factors. Uncertainties in laboratory frequencies and line centers in
astronomical spectra are the main sources of errors. In addition, there are
systematic errors which can be estimated from observations of different
objects in different molecular transitions. Transitions in methanol
isotopologues have high sensitivity coefficients of both signs, i.e., they are
the most suitable candidates for such studies.

High precision measurements can be carried out from high-resolution spectra,
which were recently obtained for the Orion-KL molecular cloud [22]. The
published spectra contain lines of methanol CH3OH and its two isotopologues –
13CH3OH and CH318OH. The CH318OH lines are found to be quite weak, and have
large errors in radial velocities. However, the 13CH3OH emission spectra
display lines with higher intensities, and their positions are determined
quite accurately (with errors of 100 m s-1, which are acceptable for our
purposes). Thus, it becomes possible to estimate $\Delta\mu/\mu$ independently
– with only the lines of 13CH3OH and in combination with the transition
$J_{K_{u}}\to J_{K_{l}}=15_{2}-15_{1}E$ in CH3OH (see Table 1 below).

Table 1:

Table 1. Selected 13CH3OH and CH3OH transitions in Orion-KL [22].
Given in parentheses are errors in the last digits. Molecule Transition
$f_{lab}$ $\Delta v_{\rm D}$ $V_{\rm LSR}$ ${Q_{\mu}}$ $J_{K_{u}}-J_{K_{l}}$
[MHz] [km s-1] [km s-1] 13CH3OH $6_{2}-5_{3}$A- 27992.990 2.3(3) 6.9(1) 16.3
13CH3OH $9_{2}-9_{1}$E 27581.630 3(1) 6.8(1) $-$14.7 CH3OH $15_{2}-15_{1}$E
28905.812 2.9(1) 6.6(1) $-13.3^{1}$ 1 – The sensitivity coefficient was
calculated in the present work.

We selected from the published spectra the pairs of molecular transitions with
approximately equal values of the Doppler widths $\Delta v_{D}$, such that the
difference in the sensitivity coefficients $\Delta Q_{\mu}$ is maximized. The
selected lines and their parameters are given in Table 1. The first column
lists the transition described by two quantum numbers – the total angular
momentum $J$ and its projection $K$ on the principal axis of the molecule –
for the upper ($u$) and lower ($l$) levels, the second column displays the
transition frequency, the third and fourth columns show the Doppler width and
the Local Standard of Rest radial velocity, $V_{LSR}$, respectively. The fifth
column contains the sensitivity coefficient $Q_{\mu}$, taken from [19]. The
calculation of $Q_{\mu}$ for methanol CH3OH was performed in this work using
our previously developed method [20]. As can be seen from Table 1, the
difference in the sensitivity coefficients is $\Delta Q_{\mu}\approx 30$,
which provides a confident estimate on the $\mu$-variance.

Using Eq. (3) for the transitions in 13CH3OH, we obtain
$\Delta\mu/\mu=(-1.1\pm 1.5)\times 10^{-8}$, which corresponds to the upper
limit on $\Delta\mu/\mu<1.5\times 10^{-8}$. A similar calculation for the
combination of the $6_{2}-5_{3}A^{-}$ line of 13CH3OH and the $15_{2}-15_{1}E$
line of CH3OH gives $\Delta\mu/\mu=(-3.4\pm 1.6)\times 10^{-8}$, and the upper
limit on $\Delta\mu/\mu<1.6\times 10^{-8}$. This yields the average value of
$\langle\Delta\mu/\mu\rangle=(-2.3\pm 1.1)\times 10^{-8}$, and the
corresponding upper limit on the changes in $\mu$ – $\Delta\mu/\mu<1.1\times
10^{-8}$. This upper limit agrees well with previously obtained constraints
from Galactic CH3OH [15, 16] and 13CH3OH [19] observations.

The results of these investigations do not indicate any significant systematic
errors in the estimates of $\Delta\mu/\mu$. It follows that the assumed
effects of Higgs-like scalar fields on the masses of elementary particles do
not exceed the 10-8 level in the disk of the Galaxy. This upper limit, 10-8,
also coincides with a limit on the influence of the hypothetical fifth force
on hadronic interactions [23], so it can be considered as the most robust at
the current time.

FUNDING
This work was supported by ongoing institutional funding and was carried out
within the framework of the topic of the State Assignment of the Ioffe
Institute number FFUG-2024-0002. No additional grants to carry out or direct
this particular research were obtained.

CONFLICT OF INTEREST
The authors declare no conflict of interest.

## References

  * [1] G. Bertone, D. Hooper, Rev. Mod. Phys. 90, 045002 (2018).
  * [2] R. Onofrio, Phys. Rev. D 82, 065008 (2010).
  * [3] F. D. Albareti et al., Phys. Rev. D 95, 044030 (2017).
  * [4] S. Alexander et al., CQD 33, 14LT01 (2016).
  * [5] J.-P. Uzan, Living Rev. Phys. 14, 2 (2011).
  * [6] J. D. Bekenstein, Phys. Rev. D 25 1527 (1982).
  * [7] P. Brax, Phys. Rev. D 90 023505 (2014).
  * [8] K. A. Olive, M. Pospelov, Phys. Rev. D 77, 043524 (2008).
  * [9] R. I. Thompson, Astrophys. Lett. 16, 3 (1975).
  * [10] D. A. Varshalovich, S. A. Levshakov, J. Exp. Theor. Phys. Lett. 58, 237 (1993).
  * [11] M. G. Kozlov, S. A. Levshakov, Ann. Phys. 525, 452 (2013).
  * [12] J. Bagdonaite et al., Phys. Rev. Lett. 114, 071301 (2015).
  * [13] V. V. Flambaum, M. G. Kozlov, Phys. Rev. Lett. 98, 240801 (2007).
  * [14] S. A. Levshakov et al., Mem. S. A. It., 85, 90 (2014).
  * [15] M. Daprà et al., MNRAS 472, 4434 (2017).
  * [16] S. A. Levshakov et al., MNRAS 511, 413 (2022).
  * [17] S. Ellingsen, M. Voronkov, S. Breen, Phys. Rev. Lett 107, 270801 (2011).
  * [18] J. H. Wu et al., ApJS 265, 49 (2023).
  * [19] J. S. Vorotyntseva, M. G. Kozlov, S. A. Levshakov, MNRAS 527, 2750 (2024).
  * [20] S. A. Levshakov, M. G. Kozlov, D. Reimers, ApJ 738, 26 (2011).
  * [21] P. Jansen et al., Phys. Rev. Lett. 106, 100801 (2011).
  * [22] X. Liu et al., ApJS, 106, 19 (2024).
  * [23] E. J. Salumbides, W. Ubachs, V. I. Korobov, J. Mol. Spec. 300, 65 (2014).