Spectroscopy of odd-odd nuclei within the interacting boson-fermion-fermion model based on the Gogny energy density functional
K. Nomura R. Rodríguez-Guzmán L. M. Robledo





Physics Department, Faculty of Science, University of
Zagreb, HR-10000 Zagreb, Croatia
Advanced Science Research Center, Japan Atomic Energy
Agency, Tokai, 319-1195 Ibaraki, Japan


Physics Department, Kuwait University, 13060 Kuwait, Kuwait


Departamento de Física Teórica, Universidad
Autónoma de Madrid, E-28049 Madrid, Spain

Center for Computational Simulation,
Universidad Politécnica de Madrid,
Campus de Montegancedo, Boadilla del Monte, 28660-Madrid, Spain





We present a method to calculate spectroscopic properties of odd-odd 
nuclei within the framework of the Interacting Boson-Fermion-Fermion 
Model based on the Gogny energy density functional. The 
(β,γ)-deformation energy surface of the even-even 
(boson-)core nucleus, spherical single-particle energies and occupation 
probabilities of the odd neutron and odd proton, are provided by the 
constrained self-consistent mean-field calculation within the 
Hartree-Fock-Bogoliubov method with the Gogny-D1M functional. These 
quantities are used as a microscopic input to fix most of the 
parameters of the IBFFM Hamiltonian. Only a few coupling constants for 
the boson-fermion Hamiltonian and the residual neutron-proton 
interaction are specifically adjusted to reproduce experimental 
low-energy spectra in odd-mass and odd-odd nuclei, respectively. In 
this way,  the number of free parameters involved in the IBFFM 
framework is reduced significantly. The method is successfully applied 
to the description of the low-energy spectra and  electromagnetic 
transition rates in the odd-odd ^194,196,198Au nuclei. 




Spectroscopy of odd-odd nuclei within the interacting boson-fermion-fermion model based on the Gogny energy density functional
    L. M. Robledo
    December 30, 2023
==============================================================================================================================







§ INTRODUCTION





The unified theoretical description of low-lying states in even-even, 
odd-mass, and odd-odd nuclei is one of the major goals of nuclear 
structure. In even-even systems at low energy, nucleons are coupled 
pairwise and the type of couplings determines the low-lying collective 
structure of vibrational and rotational states. The microscopic 
description of low-lying collective states in even-even systems has 
been extensively pursued with numerous theoretical methods 
<cit.>. However, the 
description of odd-mass and odd-odd nuclei is more cumbersome, due to 
the fact that in those systems both collective and single-particle 
motions have to be treated on the same footing <cit.>. 


The Interacting Boson Model (IBM) <cit.> has been remarkably successful in
the phenomenological study of low-lying structures in medium-mass and heavy even-even nuclei. 
In its simplest version, the building blocks of the IBM are the monopole
s and quadrupole d bosons, which represent the collective pairs of
valence nucleons coupled to spin and parity 0^+ and 2^+, respectively <cit.>.
The microscopic foundation of the IBM starting from the nucleonic degrees
of freedom has been extensively pursued in the literature
<cit.>. In
particular, a systematic
method of deriving the IBM Hamiltonian from microscopic input has been developed in
<cit.>. 
In this approach, the deformation energy surface that is obtained from
the self-consistent mean-field (SCMF) calculation based on a given energy
density functional (EDF) is mapped onto the expectation value of the IBM
Hamiltonian in the boson coherent state <cit.>. This
procedure completely determines the strength parameters of the IBM Hamiltonian. 
Since the EDF framework allows for a global mean-field description of
intrinsic properties of nuclei over the entire  Segré's
chart, it has become possible to determine in a unified way the 
parameters of the IBM Hamiltonian basically
for any arbitrary nucleus. 



The method mentioned above has been recently 
extended to odd-mass systems <cit.> by considering the coupling between bosonic
(collective) degrees of freedom and an unpaired nucleon within the
framework of the Interacting Boson-Fermion Model (IBFM) <cit.>. 
In this extension, the even-even core (IBM) Hamiltonian, the single-particle energies and occupation
probabilities of the odd particle, which are building blocks of the IBFM
Hamiltonian, have been completely determined based on the output of a SCMF calculation. 
Even though a few strength parameters for the particle-boson coupling
are treated as free
parameters, the method allows for
an accurate, systematic, and computationally feasible description of 
various low-energy properties of odd-mass medium-mass and heavy nuclei: 
e.g., signatures of shape phase transitions 
<cit.>, octupole
correlations in neutron-rich odd-mass Ba 
isotopes <cit.>, and the structure of neutron-rich
odd-mass Kr isotopes <cit.>. 


In this work, we extend these studies to odd-odd nuclei by using
the Interacting Boson-Fermion-Fermion Model (IBFFM)
<cit.>. 
The IBFFM is an extension of the IBFM that considers odd-odd nuclei as a 
system composed of an IBM core plus an unpaired neutron and an unpaired proton.
The IBM-core and particle-boson coupling Hamiltonians are determined in
way similar to that employed for odd-mass nuclei <cit.>. 
The only additional parameters are the coefficients of the residual
neutron-proton interaction. They are determined to reasonably reproduce
the experimental data for the low-lying spectra of the considered odd-odd
nuclei. 
The microscopic input used to determine part of the IBFFM Hamiltonian is obtained by
constrained SCMF calculations within the Hartree-Fock-Bogoliubov (HFB) method
based on the Gogny D1M EDF <cit.>.  
The two most relevant parametrizations of the finite range Gogny force, namely
D1S <cit.> and D1M <cit.> have proven along the years to
provide a reliable description of many collective phenomena all over
the periodic table (see <cit.> for some examples).
Our choice of D1M is based solely on its better performance to describe
binding energies.

As an application of the proposed methodology, we specifically study the
properties of the odd-odd ^194,196,198Au nuclei.  
Their low-lying structures are described by unpaired neutron and proton
holes coupled with the even-even core nuclei ^196,198,200Hg. 
The IBM parameters for the even-even cores were already obtained in Ref. <cit.>
as part of a comprehensive study of  shape coexistence
and low-lying structures in the entire Hg isotopic chain within the
configuration-mixing IBM method based on the Gogny-D1M EDF. The results obtained
suggest that the nuclei ^196,198,200Hg have weakly oblate deformed to
nearly spherical ground-state shapes. 
For the neighbouring odd-N nuclei ^195,197,199Hg and odd-Z nuclei
^195,197,199Au, there are plenty of experimental data to determine
the boson-fermion strength parameters. 
In addition, the odd-odd Au nuclei in this mass region have previously
been extensively studied within the IBFFM framework: e.g., by means of 
numerical studies <cit.>, or by 
pure-algebraic approaches <cit.> in the
context of nuclear supersymmetry <cit.>. Those
results will be a good reference to compare with our less phenomenological
results.



On the other hand, it is worth to mention that microscopic nuclear
structure models are also  applied in the
spectroscopic studies of odd-mass and/or odd-odd nuclei with the Gogny force. 
As an example, let us mention the studies of 
various low-energy properties of
odd-mass systems at the mean-field level using full blocking 
<cit.> or the equal filling approximation <cit.>.
To our knowledge there is only one study <cit.> of odd-odd 
nuclei focused on the ability to reproduce the empirical Gallagher-Moszkowski (GM) rule.
As shown in this reference, the GM rule is not fulfilled by the Gogny force
and the failure is traced back to the lack of additional proton-neutron
interaction terms in the interaction. This difficulty and the inability
of any effective interaction to reach spectroscopic accuracy for the 
spectra of odd nuclei <cit.> point to the necessity to add extra terms with
extra parameters that can be fitted locally to improve the quality of 
the description of odd and odd-odd nuclei. This is achieved in our model
through the set of extra terms added with parameters not fixed by the EDF input.
Another source of difficulties hampering to reach  spectroscopic accuracy in the description of odd nuclei 
with EDFs is the impact of dynamical 
correlations as those coming from symmetry restoration <cit.>.  In the
last few years it has been possible to include time-reversal symmetry and blocking effects
along with angular momentum and particle number projection <cit.>
but the complexity of the problem prevents its use beyond very light 
systems like ^24Mg <cit.>. 


This paper is organized as follows: In Sec. <ref> we describe
the procedure to construct the IBFFM Hamiltonian based on the SCMF
calculation. 
In Sec. <ref>, the spectroscopic properties of 
the even-even Hg nuclei are briefly reviewed. In the same section, the
results for low-energy spectra in the odd-N Hg and the odd-Z Au isotopes
are discussed, followed by the results of the spectroscopic calculations
for the odd-odd Au nuclei. 
Finally, a short summary and concluding remarks are given in
Sec. <ref>. 






§ THEORETICAL FRAMEWORK








 §.§ Hamiltonian



In this work we use the version of the IBFFM that distinguishes between
neutron and proton degrees of freedom (denoted hereafter as IBFFM-2). 
The IBFFM-2 Hamiltonian is expressed as: 

    Ĥ_ = Ĥ_B + Ĥ_F^ν + Ĥ_F^π + Ĥ_BF^ν + H_BF^π + V̂_res.

The first term in Eq. (<ref>) is the neutron-proton IBM (IBM-2)
Hamiltonian <cit.>  that describes the even-even core nuclei
^196,198,200Hg. The second and third terms represent the
Hamiltonian for an odd neutron and an odd proton, respectively. 
The fourth and fifth terms correspond to the interaction Hamiltonians describing the couplings
of the odd
neutron and of the odd proton to the IBM-2 core, respectively.
The last term in Eq. (<ref>) is the residual interaction between
the odd neutron and odd proton. 

For the boson-core Hamiltonian Ĥ_B the standard IBM-2 Hamiltonian is adopted: 

    Ĥ_B = ϵ(n̂_d_ν + n̂_d_π)+κQ̂_ν·Q̂_π + κ'L̂·L̂,

where n̂_d_ρ=d^†_ρ·d̃_ρ (ρ=ν,π) is the
d-boson number operator, Q̂_ρ=d_ρ^† s_ρ +
s_ρ^†d̃_ρ^† + χ_ρ(d^†_ρ×d̃_ρ)^(2) is the quadrupole operator, and L̂=L̂_ν + L̂_π is the angular momentum operator with L̂_ρ=√(10)(d_ρ^†×d̃_ρ)^(1). The different parameters
of the Hamiltonian are denoted by 
ϵ, κ, χ_ν, χ_π, and κ'. 
The doubly-magic nucleus ^208Pb is taken as the inert 
core for the boson space. The numbers of neutron N_ν and proton
N_π bosons equal the number of 
neutron-hole and proton-hole pairs, respectively. As a consequence,  N_π=1 and N_ν=5, 4, and
3 for the ^196,198,200Hg nuclei, respectively. 

The Hamiltonian for the odd nucleon reads: 

    Ĥ_F^ρ = -∑_j_ρϵ_j_ρ√(2j_ρ+1)
      (a_j_ρ^†×ã_j_ρ)^(0)

with ϵ_j_ρ being the single-particle energy of the odd
nucleon. 
j_ν (j_π) stands for the angular momentum of the odd neutron
(proton).
a_j_ρ^(†) represents the fermion annihilation (creation)
operator, and ã_j_ρ is defined as ã_jm=(-1)^j-ma_j-m.
For the fermion valence space, we consider the full neutron major
shell N=82-126, i.e., 3p_1/2, 3p_3/2, 2f_5/2, 2f_7/2, 
1h_9/2, and 1i_13/2 orbitals, and the full proton major shell
Z=50-82, i.e., 3s_1/2, 2d_3/2, 2d_5/2, 1g_7/2, and
1h_11/2 orbitals. 



For the boson-fermion interaction term Ĥ_ BF^ρ in
Eq. (<ref>), we use the following form: 

    Ĥ_BF^ρ = Γ_ρQ̂_ρ'·q̂_ρ 
    +
      Λ_ρV̂_ρ'ρ + A_ρn̂_d_ρn̂_ρ

where ρ'≠ρ, and the first, second, and third terms are the
quadrupole dynamical, exchange, and monopole terms, respectively. The parameters
of the interaction Hamiltonian are denoted by  
Γ_ρ, Λ_ρ, and A_ρ. 
As in the previous studies
<cit.>, we assume that both the dynamical and exchange terms 
are dominated by the interaction between unlike particles (i.e., between
the odd neutron and proton bosons and between the odd proton and neutron
bosons). We also assume that for the monopole term the interaction between
like-particles (i.e., between the odd neutron and neutron bosons and between
the odd proton and proton bosons) plays a dominant role. 
In Eq. (<ref>) Q̂_ρ is the same bosonic quadrupole operator as 
in the IBM-2 Hamiltonian in Eq. (<ref>). The fermionic quadrupole 
operator q̂_ρ reads: 

    q̂_ρ=∑_j_ρ j'_ργ_j_ρ j'_ρ(a^_j_ρ×ã_j'_ρ)^(2),
 
where γ_j_ρ
j'_ρ=(u_j_ρu_j'_ρ-v_j_ρv_j'_ρ)Q_j_ρ
j'_ρ and  Q_j_ρ j'_ρ=⟨
l1/2j_ρ||Y^(2)||l'1/2j'_ρ⟩ represents
the matrix element of the fermionic 
quadrupole operator in the considered single-particle basis.
The exchange term V̂_ρ'ρ in Eq. (<ref>) reads: 

    V̂_ρ'ρ =    -(s_ρ'^d̃_ρ')^(2)·{∑_j_ρj'_ρj”_ρ√(10/N_ρ(2j_ρ+1))β_j_ρj'_ρβ_j”_ρj_ρ
       :((d_ρ^×ã_j”_ρ)^(j_ρ)×
    (a_j'_ρ^×s̃_ρ)^(j'_ρ))^(2):
    } + (H.c.),

with β_j_ρj'_ρ=(u_j_ρv_j'_ρ+v_j_ρu_j'_ρ)Q_j_ρj'_ρ.
In the second line of the above equation the notation :(⋯): indicates normal ordering. 
In the monopole interactions, the number operator for the odd fermion is
expressed as n̂_ρ=∑_j_ρ(-√(2j_ρ+1))(a^_j_ρ×ã_j_ρ)^(0). 



In previous IBFFM
calculations <cit.>, the residual interaction V̂_res in Eq. (<ref>) contained a quadrupole-quadrupole,
delta, spin-spin-delta, spin-spin, and tensor interaction. 
However, we find that only the delta and spin-spin-delta terms are 
enough to provide a good description of the low-lying states in the odd-odd nuclei considered
here. Therefore, the residual interaction used here reads: 

    V̂_res=4πδ( r_ν-
      r_π)(u_0+u_1σ_ν·σ_π),

with u_0 and u_1 the parameters. 
Furthermore, the matrix element of the residual interaction 
V̂_res, denoted by V_res', can be expressed as
<cit.>: 

    V_res'
       = (u_j_ν' u_j_π' u_j_ν u_j_ν + v_j_ν' v_j_π' v_j_ν v_j_ν)
    V^J_j_ν' j_π' j_ν j_π
        - (u_j_ν'v_j_π'u_j_νv_j_π +
     v_j_ν'u_j_π'v_j_νu_j_π) 
       ×∑_J' (2J'+1)
    
    {[ j_ν'  j_π   J';  j_ν j_π'    J ]} 
    V^J'_j_ν'j_π j_ν j_π',

where

    V^J_j_ν'j_π'j_ν j_π = ⟨ j_ν'j_π';J|V̂_res|j_ν j_π;J⟩

is the matrix element between the neutron-proton pairs, and J stands
for the total angular momentum of the neutron-proton pair. 
The bracket in Eq. (<ref>) stands for the Racah coefficient. 
Also in Eq. (<ref>) the terms resulting from contractions are ignored as in Ref. <cit.>.
A similar residual neutron-proton interaction is used in the
two-quasiparticle-rotor-model calculation in Ref. <cit.>. 





 §.§ Procedure to build the IBFFM-2 Hamiltonian



The ingredients of the  IBFFM-2 Hamiltonian Ĥ in
Eq. (<ref>) are determined with the following procedure. 

 
  * Firstly, the IBM-2 Hamiltonian is determined by using the methods of
       Refs. <cit.>: the (β,γ)-deformation
       energy surface obtained from the constrained Gogny-D1M HFB calculation is mapped
       onto the expectation value of the IBM-2 Hamiltonian in the boson
       coherent state <cit.>. This procedure
       completely determines the parameters ϵ, κ,
       χ_ν, and χ_π in the IBM-2 Hamiltonian. Only the
       strength parameter κ' for the L̂·L̂ term is
       determined separately from the other parameters, by adjusting the cranking moment of inertia in
       the boson intrinsic state to the corresponding Thouless-Valatin <cit.>
       moment of inertia obtained by the Gogny-HFB SCMF calculation at
       the equilibrium mean-field minimum <cit.>. 


  * Second, the strength parameters for the boson-fermion coupling
      Hamiltonians Ĥ_BF^ν and Ĥ_BF^π for
      the odd-N Hg and odd-Z Au nuclei, respectively, is
      determined by using the procedure of
      <cit.>: Single-particle energies and 
      occupation probabilities of the odd nucleon are provided by the Gogny-HFB
      calculation constrained to zero deformation (see,
      Ref. <cit.>, for details); Optimal values of the
      parameters Γ_ν, Λ_ν, and A_ν 
      (Γ_π, Λ_π, and A_π), are chosen,
      separately for positive and negative parity, so as to reproduce
      the experimental low-energy levels of each of the considered odd-N Hg (odd-Z Au) nuclei. 


  * By following previous IBFFM calculations
      <cit.>, the same strength parameters Γ_ν, Λ_ν, and A_ν
      (Γ_π, Λ_π, and A_π) as those obtained for
      the odd-N Hg (odd-Z Au) nuclei in
      the previous step, are used for the odd-odd nuclei. The
      single-particle energies and occupation probabilities are, 
      however, newly calculated for the odd-odd systems. 


  * Finally, the parameters in the residual 
      interaction V̂_res, i.e., u_0 and u_1, are
      determined so as to reasonably reproduce the low-lying
      spectra in the studied odd-odd nuclei. The fixed values u_0=-0.3 MeV
      and u_1=-0.033 MeV for positive parity, 
      and u_0=-0.3 MeV and u_1=0.0 MeV for negative-parity
      states, are adopted. The ratio, u_0/u_1≈ 9, was also
      considered in <cit.>. 



The values of the IBM-2 parameters employed in the present work are
shown in Table <ref>. They are exactly the same as those
used in Ref. <cit.>. 
The fitted strength parameters for the Hamiltonian Ĥ_BF^ν
(Ĥ_BF^π), i.e., Γ_ν, Λ_ν, and
A_ν (Γ_π, Λ_π, and A_π) are shown in Table <ref>
(Table <ref>). 
The fixed value Γ_ρ=0.8 MeV is used for the strength
parameter for the quadrupole dynamical term for
all the odd-mass and odd-odd nuclei and for both parities. 
Other parameters do not differ too much between neighbouring isotopes. 
Tables <ref>, <ref>, and <ref> summarize the
single-particle energies and occupation probabilities obtained from the
Gogny-HFB SCMF calculations for the studied odd-N Hg, odd-Z Au, and
odd-odd Au isotopes, respectively. 
We note that the single-particle energies and occupation probabilities for the odd-N
Hg (Table <ref>) and odd-Z Au (Table <ref>)
nuclei are almost identical to those computed for the odd-odd Au nuclei (see,
Table <ref>). 






































Once all the parameters of the IBFFM-2 Hamiltonian are obtained, it
is diagonalised numerically in the basis |L_ν L_π(L);j_ν
j_π(J):I⟩, using the computer program TWBOS <cit.>. 
L_ν (L_π) and L are the angular momentum for neutron (proton)
bosons and the total angular momentum for the even-even boson core, 
respectively. Finally, I stands for the total angular momentum of the coupled system. 





 §.§ Transition operators


Using the eigenstates of the IBFFM-2 Hamiltonian, we can determine
the electric quadrupole (E2) and magnetic dipole (M1) properties of the odd-odd nuclei. 
In the present framework, the E2 operator T̂^(E2) takes the following form: 

    T̂^(E2)   = e_ν^BQ̂_ν + e_π^BQ̂_π
    -1/√(5)∑_ρ=ν,π∑_j_ρj'_ρ
       ×(u_j_ρu_j'_ρ-v_j_ρv_j'_ρ)⟨
    j'_ρ||e^F_ρr^2Y^(2)||j_ρ⟩(a_j_ρ^†×ã_j'_ρ)^(2),

where e^B_ρ and e^F_ρ
stand for the effective charges for the boson and fermion systems,
respectively. 
The fixed values e^B_ν=e^B_π=0.15 eb, which are taken from
Ref. <cit.>, and e^F_ν=0.5 eb and e^F_π=1.5 eb
are used. 


The M1 transition operator T̂^(M1) reads: 

    T̂^(M1)   =√(3/4π){
    g_ν^BL̂^B_ν + g_π^BL̂^B_π
    -1/√(3)∑_ρ=ν,π∑_jj'
       × (u_j_ρu_j'_ρ+v_j_ρv_j'_ρ)⟨
    j'_ρ||g_l^ρ l+g_s^ρ s||j_ρ⟩(a_j_ρ^†×ã_j'_ρ)^(1)}.

In this expression, g_ν^B and g_π^B are the g-factors for the neutron and
proton bosons, respectively. The fixed values g_ν^B=0 μ_N and
g_π^B=1.0 μ_N  <cit.> are used in this work. 
For the neutron (proton) g-factors, the usual Schmidt values 
g_l^ν=0 μ_N and g_s^ν=-3.82 μ_N
(g_l^π=1.0 μ_N and g_s^π=5.58 μ_N) are used. 
The g_s value for both the proton and neutron are quenched by 30
%. 

We note that the forms of the operators T̂^(E2)
(Eq. (<ref>)) and T̂^(M1) (Eq. (<ref>)) have been
used in previous IBFFM-2 calculations
<cit.>. 

As we show later, we have computed the B(E2) and B(M1) transition rates, the
spectroscopic quadrupole moment Q(I), and the magnetic moment μ(I)
for the odd-odd nuclei ^194,196,198Au, using the computer code
TWBTRN <cit.>. 




§ RESULTS AND DISCUSSION 











 §.§ Even-even Hg isotopes










In this section, we briefly discuss relevant results for the even-even
nuclei ^196,198,200Hg, which were already presented in Ref. <cit.>. 
We plot in Fig. <ref> the Gogny-D1M and mapped IBM-2 energy
surfaces for the ^196,198,200Hg nuclei. 
In Fig. <ref>, the Gogny-D1M energy surface for the ^196Hg
nucleus exhibits a single oblate minimum located at β≈ 0.13. 
The oblate minimum becomes less pronounced in ^198Hg and, finally,
the ^200Hg nucleus exhibits a near spherical shape with a very shallow
oblate minimum at β≈ 0.08. 
The mapped IBM-2 energy surfaces on the right-hand side in
Fig. <ref> reproduce the basic features of the original Gogny-D1M
ones around the global minimum, but look rather flat in the region away from the minimum. This is due to
the restricted boson model space <cit.>, which only comprises a finite number of
bosons. 


The calculated and experimental <cit.> low-lying spectra are shown
in Fig. <ref>. The yrast levels for all the
considered even-even nuclei are described reasonably well. 
However, the theoretical energy levels, in particular for ^196,198Hg,
look more stretched than the experimental ones. 
For instance, our calculation is not able to account for the
excitation energy of the low-lying 0^+_3 level of ^196,198Hg. 
This discrepancy could be remedied by including in the IBM-2 model space
the intruder configurations that are associated with coexisting mean-field
minima. 
These configurations are, however, not considered for the nuclei studied 
here <cit.>, since their corresponding Gogny-D1M energy surfaces only exhibit a
single mean-field minimum (see, Fig. <ref>). 





 §.§ Odd-mass Hg and Au isotopes


 







Next, we discuss the spectroscopic properties of the odd-mass nuclei,
obtained within the neutron-proton IBFM (IBFM-2). 
For the diagonalization of the IBFM-2 Hamiltonian, the computer code PBOS is used. 
The theoretical and experimental low-energy spectra for the odd-N nuclei
^195,197,199Hg are compared in Fig. <ref>. 
Especially for the positive-parity states, which are based on the unique-parity
ν i_13/2 configuration, the present calculation
provides an excellent description of the experimental spectra for the
considered odd-N nuclei, although only three parameters are involved (see, Table <ref>). 
The calculation reproduces nicely the ground-state band built on the
13/2^+_1 state, which follows the Δ I=2 systematic of the
weak-coupling limit. 
For the ^195,197Hg nuclei, the calculation suggests that the
negative-parity yrast states near the ground state are based mainly on the odd neutron in the
3p_1/2 single-particle orbital coupled to the IBM-2 core. 
In the case of the nucleus ^199Hg, however, in most of the yrast
states in the vicinity of the ground state three configurations
3p_1/2, 3p_3/2, and 2f_5/2 are more strongly mixed than in ^195,197Hg. 
Such a change in the structure of the low-lying state from ^195,197Hg
to ^199Hg, reflects the evolution of shapes in the
corresponding even-even systems from ^198Hg (weakly oblate
deformed) to ^200Hg (nearly spherical). 


In Fig. <ref> we show similar plots for the odd-Z isotopes
^195,197,199Au. 
In general, our calculation is in a very good agreement with the
experimental data. 
Our calculation suggests that  the 
IBFM-2 wave functions of the lowest positive-parity states for
the considered odd-Z Au nuclei are composed, with a probability of more than 80 %, of the 
3s_1/2 and 2d_3/2 single-particle configurations, which are
substantially mixed with each other. 
On the other hand, the 2d_5/2 and 1g_7/2 configurations turn
out to play minor roles in describing the lowest-lying states. 


We confirm that both the E2 and M1 properties of the
considered odd-mass nuclei are reasonably described with the present approach. 




 §.§ Odd-odd Au isotopes











Let us now focus on the discussion of the results for the odd-odd nuclei. 
The low-lying spectra computed with the IBFFM-2 for the odd-odd
^194,196,198Au nuclei are depicted in Fig. <ref>, and
compared with the experimental data <cit.>. 




  §.§.§ ^194Au


Firstly, we observe that the present IBFFM-2 result for the ^194Au nucleus
is in a very good agreement with the experimental spectra, especially
for the positive-parity states. 
The main component (≈ 72 %) in the wave functions of the lowest
three positive-parity states, i.e., 5^+_1, 7^+_1, and 6^+_1 states, is the (ν p_1/2⊗π
h_11/2) neutron-proton pair coupled to the boson core. 
For the negative-parity, the energy levels near the 1^- ground state
are reasonably reproduced in the present calculation. The main
components of IBFFM-2 wave function of the 1^- state are the 
(ν p_1/2⊗π s_1/2)_J=1^- (17 %), and 
(ν f_5/2⊗π s_1/2)_J=3^- neutron-proton pairs (13 %). 
However, the calculation is not able to reproduce the experimental 11^-, 12^-, 
and 13^- levels, which are below 1 MeV excitation. The excitation energies for
these states are predicted to be much larger (> 3 MeV).
Empirically, these higher-spin negative parity states are mainly made of
the pair composed of the unique-parity orbitals, i.e., (ν
i_13/2⊗π h_11/2) <cit.>.
The corresponding IBFFM-2 wave functions obtained in the present work
are, however, made of the coupling between the odd neutron and 
proton in the normal-parity orbitals: for instance, the main components
of the predicted 11^-_1 states are the 
(ν p_1/2⊗π s_1/2)_J=1^- (35 %), 
(ν f_5/2⊗π s_1/2)_J=3^- (19 %), and (ν
p_1/2⊗π d_5/2)_J=3^- (13 %) neutron-protons pairs. 




The experimental information about the electromagnetic properties is
rather scarce for ^194Au. Nevertheless, we show in
Table <ref> the calculated B(E2) and B(M1) transition
rates, and quadrupole Q(I) and magnetic μ(I) moments  in
comparison with the available data. 
The predicted B(E2) values seem to be qualitatively in a good
agreement with the data. 
The calculated B(M1) values are, however, too large as compared
to the experimental values. 
The sign of the predicted Q(1^-)  moment is opposite
to that given by the experiment. 





  §.§.§ ^196Au


From the comparison of energy levels of the odd-odd nucleus ^196Au, shown in
Fig. <ref>, one concludes that the calculation is able to reproduce
both the experimental positive- and negative-parity levels reasonably well. 
The calculated low-lying positive-parity states for ^196Au are similar in
structure to those for ^194Au: 67 % and 66 % of the predicted
5^+_1 and 7^+_1 states are dominated by the 
(ν p_1/2⊗ h_11/2)_J=5^+ pair component, respectively. 
The spin of the calculated lowest negative parity state is
I=1^-. This is at variance with the experiment, although the
experimental 1^-_1 level is only 6 keV above the 2^-_1 ground
state. 
Furthermore, the present calculation considerably overestimates the 2^-_1 energy level.
The non-negligible components (> 10 %) of the corresponding IBFFM-2 wave functions for the
2^-_1 and 1^-_1 states are
the following: 
(ν p_1/2⊗π d_3/2)_J=2^- (24 %),  
(ν p_1/2⊗π s_1/2)_J=1^- (11 %), and 
(ν f_5/2⊗π s_1/2)_J=2^- (11 %) for the 2^-_1 state, and 
(ν p_1/2⊗π s_1/2)_J=1^- (38 %) and 
(ν p_1/2⊗π d_3/2)_J=1^- (10 %) for the 1^-_1 state. 



Table <ref> exhibits the calculated and experimental
electromagnetic properties. Regarding the B(E2) rates, our results are
in a reasonable agreement with the experiment. 
However, similarly to ^196Au the calculated B(M1) values are
generally much larger than the experimental values.


A number of experimental B(E2) and B(M1) transition rates from the 1^-
state at the excitation energy of E_x=298.5 keV are available <cit.>. 
However, there are also too many experimental 1^- states below 298.5 keV, and it 
is not clear which theoretical 1^- state corresponds to the
experimental one observed at E_x=298.5 keV. 
For this reason, we do not compare our results with the 
experimental B(E2) and B(M1) transitions rates from 
the 1^-(298.5 keV) state.






  §.§.§ ^198Au


As one sees from the comparison between the theoretical and experimental
low-energy spectra for the odd-odd nucleus ^198Au in
Fig. <ref>, the description of the positive-parity states is
generally good. 
As in the case of ^196Au, however, our calculation fails to
reproduce the spin of 
the lowest negative-parity state. 
The structure of the 2^-_1 and 1^-_1 wave functions for ^198Au
turn out to be rather similar to those of ^196Au, that is, 
(ν p_1/2⊗π d_3/2)_J=2^- (26 %), 
(ν p_1/2⊗π s_1/2)_J=1^- (19 %), and 
(ν p_1/2⊗π s_1/2)_J=0^- (12 %) for the 2^-_1 state, and 
(ν p_1/2⊗π s_1/2)_J=1^- (41 %) and 
(ν p_1/2⊗π d_3/2)_J=1^- (13 %) for the 1^-_1 state. 
The previous IBFFM calculation of <cit.> obtains an excellent
description of both the positive- and negative-parity levels. 
The IBFFM wave functions they obtained are predominantly described by the (ν p_1/2⊗π
d_3/2)_J=2^- component (> 70 %) for the 2^-_1 state, and (ν
p_1/2⊗π d_3/2)_J=1^- (50 %) for the 1^-_1 state. 
The difference between our result and that of <cit.> could be
accounted for by the different single-particle energies used in each study. 
In the present calculation, the 2d_3/2 single-particle orbital is
about 0.9 MeV above the 3s_1/2 (see, Table <ref>). 
On the other hand, in <cit.> the 2d_3/2 orbital 
is below the 3s_1/2 one and, consequently, the π d_3/2
single-particle configuration plays a more dominant role in low-energy
region than in our calculation. 



In Table <ref> the calculated B(E2) values for ^198Au are, in
general, in good agreement with the experiment. 
We also present the calculated B(M1), but for most of the
available data only a lower limit for this quantity is known. 
The calculated magnetic moment of the 5^+_1 state, μ(5^+_1), has the opposite sign  and is a factor of 4
larger in magnitude than the experimental one. 
Similar results have been obtained for the ^194,196Au
nuclei as we obtain μ(5^+_1)≈ 5
μ_N. As already mentioned, the 5^+_1 states obtained in the
present calculation for the considered odd-odd Au nuclei are dominated by the (ν
p_1/2⊗π h_11/2)_J=5^+ neutron-proton pair configuration,
and the predicted μ(5^+_1) moments are mostly accounted for by this
configuration, in particular, by the odd-proton part of the M1 matrix
element, which takes large positive value. 
On the other hand,  empirical studies for the low-lying level structure of
^194Au <cit.> assume the 5^+_1 state and
the band built on it to be based mainly on the (ν i_13/2^-1⊗π
d_3/2^-1)_J=5^+ configuration, leading to the correct sign of the
μ(5^+_1) moment. 







§ SUMMARY AND CONCLUDING REMARKS





In this work, we extend the recently developed method of
Ref. <cit.> for calculating the spectroscopy of odd-mass
nuclei to odd-odd systems. 
The (β,γ)-deformation energy surfaces of the even-even core
nuclei, and spherical single-particle energies and occupation
probabilities of the odd neutron and the odd proton are calculated
by the constrained HFB method based on the Gogny D1M EDF. 
These quantities are then used as microscopic input to build most
of the different terms of the IBFFM-2 Hamiltonian. 
The strength parameters for the boson-fermion interaction terms in the
IBFFM-2 Hamiltonian are taken from those of the neighbouring odd-mass
nuclei. 
Two coefficients in the residual interaction between odd neutron and
proton are the only new parameters, and are determined as to
reproduce the low-energy levels of each odd-odd nucleus. 
In this way, we are able to reduce significantly the number of
free parameters in the IBFFM-2 framework. 
It is shown that the method provides a reasonable description of
low-energy spectra and electromagnetic properties of the odd-odd nuclei
^194,196,198Au. 
Even though a few strength parameters in the boson-fermion and
fermion-fermion interactions are treated as free parameters, the method
developed in this paper, as well as in Ref. <cit.>, in which the
even-even IBM-core Hamiltonian is
determined fully microscopically and only one or two unpaired nucleon
degrees of freedom are added via the particle-boson coupling, allows for
a simultaneous description of a large number of even-even, odd-mass, and
odd-odd medium-mass and heavy nuclei.





This work was supported in part by the QuantiXLie Centre of Excellence, a project
co-financed by the Croatian Government and European Union through the
European Regional Development Fund - the Competitiveness and Cohesion
Operational Programme (Grant KK.01.1.1.01.0004).
The  work of LMR was 
supported by Spanish Ministry of Economy and Competitiveness (MINECO)
Grants No. FPA2015-65929-MINECO and FIS2015-63770-MINECO.




