IBFFM LOW-LYING STATES AND THEIR SPIN-DEPENDENT LEVEL DENSITIES FOR EVEN-A NUCLEI ASSOCIATED WITH SU(6) APPROXIMATION AND IT’S O(6) DYNAMICAL SYMMETRY OF IBM

Objective : in this paper spin dependent level density dependence from dynamical, exchange particle-quasiparticle interactions and residual proton-neutron interactions are investigated. Theoretical framework: in the interacting boson-fermion-fermion model(IBFFM) for odd-odd nuclei 196 Au the low-lying states and their densities are calculated for π d 3/2 , π d 5/2 , π s ½ , π h11/2 proton quasiparticles and ν p1/2 , ν f5/2 , ν p3/2 , ν f7/2 , ν i13/2 neutron quasiparticles and ν p1/2, ν f5/2, ν p3/2, ν f7/2, ν i13/2 neutron quasiparticles states, coupled to the IBM core in Su(6) approximation and it's O(6) dynamical symetry of interacting boson model (IBM). Method: we also applied cutting bosons and fermions space and investigated their impacts on spin dependent level density. The results are shown graphically and compared to the previous combinatorial, thermodynamic and spectral distribution approaches for 132Pr, 244 am and 114 Cd. Results and discussion: the IBFFM low-lying states up to 2 MeV are exactely in acordance with experimentala data. The IBFFM spin–dependent level densities can be well accounted by bethe and modified spin distribution formula. For small changes of dynamical, exchange and residual interactions we have a small changes in the spin-dependent level density of states. Research implications: the complex collective features and interplay between collective and quasiparticle degrees of freedom do not influence sizeable the IBFFM spin-dependent level density distribution. Originality/value: this study contributes to the literature of IBFFM low-lying states and their spin-dependent level densities for even-A nuclei


INTRODUCTION
For the last few decades, for describing odd-odd nuclei has been successfully used a model which in literature is known as IBFFM (interacting boson-fermion-fermion model) model.The model was developed from the IBM (interacting boson model) (Arima & Iachello, 1975, 1979, 1987) model for even-even nuclei and the IBFM (interacting boson-fermion model)( ( Iachello & Scholten, 1979;Paar et all., 1982) for the even-odd nuclei.IBFFM (Paar et all., 1985(Paar et all., , 1987(Paar et all., , 1989) is an algebraic model based on the collective nuclear states as representations of a SU(6) group of symmetry associated with proton and neutron quasiparticles coupled to the O(6) limit of the IBM core interacting with each other by residual interaction.In the collective excitations of the core we do not distinguish proton and neutron bosons while the Hamiltonian contains the most two-particles fermion interactions.The dominant interaction between unpaired nucleons is one quadrupole phonon exchange through the polarization of the core.A exchange term of the boson-fermion interaction describes the antisymetrysation of the unpaired(quasi) particles and the internal fermions structure of the phonon.In the present work, firstly the IBFFM Calculations for odd-odd nuclei 196 Au the low lying energetic spectre is investigate for   3/2 ,   5/2 ,   1/2 , π h 11/2 and   1/2 ,   5/2 ,   3/2 ,   7/2 ,   13/2 proton-neutron configurations coupled to the IBM boson core, than we calculated the changes of spin-dependent level density on dynamical, exchange particlequasiparticle interactions, and residual proton-neutron interactions.
V. Paar et al. used calculations of 198 Au (Paar et all., 1989(Paar et all., , 1996(Paar et all., , 1993) ) for the predictions of 196 Au.In those calculations, the nuclei 200  Hg, 199  Hg and 199 Au were used to obtain a description of the boson(phonon) core and to determine the parameters of the boson- Arima, and F. Iachello proposed the description of collective quadrupole states for eveneven nuclei in terms of SU( 6) boson group approach.They have shown that vibrational and rotational states can be obtained within this model associated with O(6) and SU(3) limits.In the ref.(Arima & Iachello, 1975) they point out that the group SU(6) of six dimensional special unitary transformations might provide the appropriate framework for a unified description of collective nuclear states.They shows that six-dimensional special unitary transformation group SU( 6) can be a suitable framework for a unified description of collective nuclear states.A number of positive -parity states in the even-even nuclei can be generated as the state of a system of N-bosons having no intrinsic spin but have the ability to occupy two levels, a groundstate level with angular momentum L=0, and an excited state with angular momentum L=2.In the case in which the two levels are degenerate and there is no interaction between bosons, the five components of the excited L=2 state, called d-boson for convenience, and the single component of the ground L=0, called s-boson, span a six-dimensional vector space which provides the basic for the representations of the unitary group U(6).
The calculation of 198 Hg is provided in respect to the particle vibrational model(PVM) Hamiltonian developed by Janssen et al. (Janssen et all.,1974)which describes collective quadropole phonon states in even-even nuclei in terms of quadrupole phonon creation(annihilation) operators   + (  ) with the z-component μ of the angular momentum.The particle vibrational model(PVM) Hamiltonian has the form: Here, N ˆ denoting the phonon number operator,  b ~ denote the spehric tensor, . h.c.denotes hermite conjugation operator.This Hamiltonian is identical to the IBM Hamiltonian in ref (Arima & Iachello, 1975

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as the phenomenological treatment of the parameters is concerned; among other features, all limiting symmetries (SU(3), SU(5), O(5), etc.) and the results obtained by fitting the parameters are exactly the same in both models.The parameters for IBM and PVM Hamiltonian are correlated via a parameter transformation (Arima & Iachello, 1975;(Janssen et all.,1974)The boson number in HIBM is equivalent to the phonon number in the HPVM.In PVM hamiltonian occurs an integer parameter N , which limits the maximum number of phonons.For this reason, the model is known as Truncated Quadrupole Phonon Model (HQPM).The immediate link between N and collective amplitudes does not exist, but the fact that the phonon is linked with the creation and deletation of two quasiparticles means that N is equal to one half of the total number of valent particles.For  117 79 196 the closest shell is with 82 protons and 126-82=44neutrons, thats mean the number of valent particles are 82-79=3 protons or 1 boson(phonon) +1 kvaziproton nonpairing and 126-117=9 neutrons or 4 bosons(phonons) and 1 kvazineutron nonpairing that is 5-N bosons(phonons) coupled in L=2 with the maximal total angular momentum is I=10.Space cutting is also embedded in the IBM model, and the way of cutting the space(boson space and fermion space) will have an impact on the IBFFM energy spectrum and total and spin-dependent level densities ( Kabashi et all., 2016) The impact of the strength of monopol, dynamic and exchange interactions A0, 0  and 0  respectively on the spin-dependent level densities will be discussed in this paper.
V. Paar et al. included spin-spin and tensor interactions into the calculations.
The residual interactions are defined such as.(Paar et all., 1994) Where: .2A 1/3 fm.The parameters for the spin-spin interactions is Vσσ =0.7, for the tensor interaction VT = -0.025for the positive parity and VT = -0.01 for the negative parity.
The IBFFM wave functions are expressed in the form: Jr= Jr (jj)j,ndI (jj)j,ndI;J>, where Jr> denotes the r-th state of the angular momentum J.The parameters for IBFFM calculation are taken from first, second, third , and fourth step from IBM, IBFM(), IBFM() and HRES.(Chumnei, 1995;Paar et all.,1998)TheIBFFM diagonalization is performed by a numerical procedure in a given space of the state.The space dimension is determined by the physical properties, intuition and capabilities of the computer.The calculations was performed using the computer code IBFFM (Paar et all., 1985).In this paper we investigate the IBFFM low-lying energy spectrum for 196 Au , spin dependent nuclear level density dependence from residual , dynamical and exchange interactions.The resulting IBFFM low-lying energy spectrum of 196 Au is presented in Fig. 1.For both parities and in compared with experimental data in Fig. 2. see Nucl. Data Sheets, 83: 145, 1998(Chumnei, 1998)   Below we will investigate how spin-dependent level density, depend on dynamical, and residual interactions.In previous calculations for spin-dependent level densities (Kabashi, et all ., 2009), is taken into account the residual proton-neutron interactions Eq.( 7).
Where: η, is spin corrections parameter, this parameter appears in addition to the spin cutoff parameter σ.In the low-spin limit of Eq.( 12), the spin correction parameter η cancels in the expression; then, the spin distribution is determined solely by the spin cutoff parameter σ as in Eq.( 10).On the other hand, the reduction of spin distribution for high spins is due the effect of spin correction parameter η.As we can see from Fig. 3. at low excitation energies (0-1MeV) and (1-2MeV) for low spins (J = 0,1,2), IBFFM results show systemic deviation from the spin-dependent Bethe formula Eq.( 10).In the high-spin tail of the spin distribution when the spin increase to the maximum , the IBFFM spin distribution falls monotonically below the predictions of the Bethe formula.We see that already in the interval 2-3 MeV, we have almost perfect agreement with Bethe formula for spin J=2.The fact that it's only a statistical fluctuation is confirmed by the fact that in interval 0-1 MeV and for hight spins there are slightly higher point fluctuations than in other energy intervals.This means that simple analytical forms are expected in areas where the spectrum is dense.The monotonic high-spin reduction has been previously investigated in the framework of combinatorial calculations (Paar et all., 1993).
The nature of this reduction was discussed in the context of polynomial expansion (Gilbert & Cameron ,1965;Paar et all.,1996).
For spins higher than about half of spin of the yrast state, the Bethe formula with spin dependence Eq.( 10) predicts too many states and is off by several orders of magnitude at the maximum possible spin (Lang ,1965).
To some extent this can be accounted for in thermodynamic approach by assuming that an energy equivalent to the rotational energy is not available for excitation.Therefore, the state density decreases as the angular momentum increases.However, the fits of thermodynamic formula to the combinatorial spin-dependent level density showed that deviation remain large ( Paar et all.,1993).A similar result was obtained in this work for the IBFFM spin distribution.
Oscillatory behavior at low energy for a small spins is understandable because in that area the energetic spectrum is relatively rare and for a small number of states we have greater statistical deviations.In Fig. 3. we present the fit of the modified formula Eq.( 12) to the obtained IBFFM spin dependence results (the three lowest spins (J=0,1,2) have been omitted from the fit.We can see that the dependence of yH (J) can be quite nicely described by the expression Eq.( 12).
This leads to the conclusion that two essentially different models, the IBFFM and the schematic model obtained by the GPM(Gaussian Polynomial Method), have some common features.IBFFM spin-dependent level density values for parameters  , , D in the modified spin distribution formula Eq.( 12) and the values of  , D in Bethe formula Eq.( 10) for 196 Au.The values are given for four energy intervals in the low-lying section of the spectrum for pozitive parity and with the changes in strength of dynamical interactions interactions to =0.513 and =0.513(for Fig. 3. left) and =0.13 and =0.13(forFig. 3. right) with exluted residual interactions Hres=0 in both cases.In the Table 1.a, b. are given IBFFM spin-dependent level density values for parameters  , , D, in the modified spin distribution formula Eq.( 12) and the values of  , D in Bethe formula Eq.( 10) for 196 Au, for the positive and negative parity.From Table .1a, b we see that spin cutoff parameter σ, and spin correction parameter η, calculated in IBFFM increase with increasing energy.The first four spins J=0, 1,2,3 are not taken in fitting.The energy dependence of η is slightly faster than the energy dependence of σ.Similar behaviour was obtained previously in the combinatorial calculations for 114 Cd and 244 Am and 132 Pr (Kabashi et all., 2009;Paar et all., 1993;Gilbert & Cameron ,1965).

Bethe formula
The energy dependence of  calculated in IBFFM is slightly stronger than prediced by Eq.( 13), with the backshift E1 taken from Table 1a.The parameter  increasese with energy somewhat faster than .This IBFFM result is in accordance with the results of previous IBFFM calculations for 132 Pr (Paar et all.,1996),combinatorial calculations for 114 Cd and 244 Am.In the present calculation the average ratio of  /  ̅̅̅̅̅̅̅ 1.67; which is approximately as in ref (Kabashi et all., 2009), 1.74; for 196 Au and 1.8, for 132 Pr (Paar et all.,1993;Gilbert & Cameron,1965) and which is somewhat lower than the ratios 2.5, obtained in previous combinatorial calculations for 114 Cd and 2.4,for 244 Am (Paar et all.,1993).Let us compare the IBFFM results for spin cutoff parameter , to the systematics based on available data and to previus calculations in thermodynamic and spectral distribution approaches.Using the values 12 of a and E1 from systematics (Paar et all., 1993(Paar et all., , 1995(Paar et all., ,1996;;Gilbert & Cameron ,1965;Egidy et all.,1988;Thomas, 1969).The rigid-body estimate for the spin cutoff parameter at the energy E2MeV is rig = 5.1, which is close to the calculated IBFFM value.Such a large value of calculated  is in qualitative accordance with previous calculations for some nuclei in thermodynamic and spectral distribution approach (Lang ,1965,1969&Tomas, 1969).In all these calculations the spin cutoff parameter  is systematically larger than expected.As shown in this work, this feature is not affected by accounting for collectivity and for interaction between collective and quasiparticle degrees of freedom in IBFFM.

DEPENDENCE FROM EXCHANGE AND RESIDUAL INTERACTIONS
Below we will investigate how spin-dependent level density, depend on exchange, and residual interactions.In previous calculations for spin-dependent level densities (Kabashi et al. 2009), we took into account the residual proton-neutron interactions Eq.( 7).In the Fig. 4. We present the plot of y H (J) versus (J+0.5) 2 , for our IBFFM spin-dependent level density distribution for 196 Au, with the IBFFM parameters taken from fifth steps of IBFFM calculations similar as in ref. (Kabashi et al. 2009), for positive parity but with the changes in strength of exchange interactions to Λ 0  =0.5 and Λ 0  =0.05 (Fig. 4. left) and Λ 0  =1.0 and Λ 0  =0.1 (Fig. 4. right) with incluted residual interactions in both cases HRES≠0( In ref. (Kabashi et al. 2009) the strength of exchange interactions were Λ 0  =0.85 and Λ 0  =0.05 HRES≠0) SU( 6 with Fig. 3, we see that the IBFFM spin-depedent distributions almost coincide.This again confirms the previus conclusions in ref. (Kabashi et all., 2009;Paar et all.,1993;Gilbert & Cameron,1965) 1) For small spins (up to Jmax / 2) the IBFFM spin -depedent distribution of level densities we can describe by the Bethe expression Eq.( 10).
2) Spin values greater than Jmax / 2 up to Jmax, the distribution goes below the predictions of Bethe's formula and can be described by the spin-modified expression Eq.( 12).
In the Table 1  14 the average ratio of  /  ̅̅̅̅̅̅̅ is 1.695 aproximately the same value as in Table 1a, b and from the ref.(Kabashi et all., 2009;Paar et all.,1993;Gilbert & Cameron,1965) .Now it can be said that the above conclusions can apply generally that, for small changes of Hamiltonian parameters (which includes the dynamical, exchange and residual interactions) we will have little changes in the IBFFM spin-dependent level density of the states.On the basis of this, it can be said that similar behaviour we have for other odd-odd nuclei (Paar et all.,1993).

CONCLUSION
The IBFFM levels capture most of the states in low-lying energy regions.However, with increasing of energy excitation an increasing fractions of states is expected to lie outside the frame of the IBFFM model space.The IBFFM low-lying states up to 2 MeV are exactely in acordance with experimental data.On basis of IBFFM calculation, the predicted most probable ground-state spin-parity assignments might be J  =2 -or 5 + .
The IBFFM spin-dependent level density depedence, on the selected value for the strength of dynamical, exchange interactions, and with included and excluded residual interaction, which are presented with their respectively IBFFM spin-depedent level densities distributions almost coincide.This again confirms the previus conclusions in ref. (Kabashi et , 2009;Paar et all.,1993;Gilbert & Cameron,1965).At low excitation energies (0-1MeV) and (1-2MeV) for low spins (J = 0,1,2), IBFFM results show systemic deviation from the spindependent Bethe formula .Oscillatory behavior at low energy for a small spins is understandable, because in that area the energetic spectrum is relatively rare and for a small number of states we have greater statistical deviations.
For spins greater than J=2 up to Jmax / 2, the IBFFM spin -depedent level densities distribution we can describe by the Bethe's formula.
In the high-spin tail of the spin distribution, when the spin values increase from Jmax / 2 to the maximum spin values Jmax, the distribution goes below the predictions of Bethe's formula and is similar as in the combinatorial calculations (Paar et all.,1993),and can be well fitted by the modified spin-dependent level density formula, which was introduced to fit the results of combinatorial calculations.The values of spin cutoff parameter σ calculated in IBFFM are slightly higher than rigid body values.
The values of the new spin truncation parameter η calculated in IBFFM are by factor ≈1.68 higher than the value of σ, and the introduction of η causes only a minor change of the value of spin cutoff parameter σ with respect to spin dependent Bethe formula.
Thus for small changes of dynamical, exchange and residual interactions we have a small changes in the spin-dependent level density of states, which means that the complex collective features and interplay between collective and quasiparticle degrees of freedom, do not influence sizeable the IBFFM spin-dependent level density distribution.
IBFFM Low-Lying States and Their Spin-Dependent Level Densities For Even-A Nuclei Associated with SU(6) Approximation and It's O(6) Dynamical Symmetry of IBM ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.9 | p.1-17 | e08270 | 2024.4 fermion(particle-phonon) interactions in 198 Au.Thus, following this scheme, the even-even nucleus 198 Hg and the odd-even nuclei 197 Hg and 197 Au are included in the calculations of 196 Au.

Figure 4
Figure 4IBFFM spin-dependent level density distribution for196 Au, with the IBFFM parameters taken from fifth steps of IBFFM calculations similar as in( Kabashi et al. 2009), for positive parity, but with the changes in strength of exchange interactions to  0  =0.5 and  0  =0.05 (Fig. 4. left) and  0  =1.0 and  0  =0.1 (Fig. 4.right) with incluted residual interactions in both cases.The calculated values yH=ln[N(J)/(J+0.5)]are presented for the energy intervals 0-1 MeV(closed squares), 1-2 MeV(closed circles), 2-3 MeV (closed triangles) and 0-5 MeV(closed diamonds).N(J) denotes the number of levels of spin J obtained by IBFFM calculation in the corresponding energy intervals.Dashed lines present the fits of spin-dependent Bethe formula (10) to the states of spinss J=4-8, and solid lines the fits of modified spin-dependent formula (12) . a, b. are presented, IBFFM spin-dependent level density values for parameters , , D, in the modified spin distribution formula Eq.(12), and the values of  and D in Bethe formula Eq.(10) for 196 Au.The values are given for four energy intervals in the lowlying section of the spectrum for pozitive parity, and with the changes in strength of exchange interactions interactions to Λ 0  =0.5 and Λ 0  =0.05 (Figure4.left) and Λ 0  =1.0 and Λ 0  =0.1 (Fig. 4. right) with incluted residual interactions in both cases.From the results taken from IBFFM Low-Lying States and Their Spin-Dependent Level Densities For Even-A Nuclei Associated with SU(6) Approximation and It's O(6) Dynamical Symmetry of IBM ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.9 | p.1-17 | e08270 | 2024.15 all.