Bohr Hamiltonian with Hulthen plus ring-shaped potential for triaxial nuclei with deformation-dependent mass term
A. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne
EEPJ manuscript No. (will be inserted by the editor)
Bohr Hamiltonian with Hulth´en plus ring-shaped potential for triaxial nuclei withdeformation-dependent mass term
A. Adahchour , S. Ait El Korchi , A. El Batoul , A. Lahbas and M. Oulne High Energy Physics and Astrophysics Laboratory, Department of Physics, Faculty of Sciences Semlalia, Cadi Ayyad Univer-sity, P.O.B. 2390, Marrakesh 40000, Morocco. ESMaR, Department of Physics, Faculty of Sciences, Mohammed V University in Rabat, Morocco.Received: date / Revised version: date
Abstract.
In this work, we present a new version of the Bohr collective Hamiltonian for triaxial nucleiwithin Deformation-Dependent Mass formalism (
DDM ) using the Hulth´en potential. We shall call thedeveloped model Z(5)-HD. Analytical expressions for energy spectra are derived by means of the recentversion of the Asymptotic Iteration Method. The calculated numerical results of energies and B ( E DDM formalism as well as theoretical predictionsof Z(5)-DD model with Davidson potential using
DDM formalism. The obtained results show an overallagreement with experimental data and an important improvement in respect to the other models.
Key words.
Bohr Hamiltonian, triaxial nuclei, Deformation-Dependent Mass formalism, Hulth´en poten-tial, Davidson potential
PACS.
Since its introduction for the first time in semi-conductorphysics [1], the position dependent mass formalism hasbeen applied in several works in many different fields ofphysics [2,3,4,5,6,7,8,9,10,11,12,13,14]. Such a formal-ism has been adopted in nuclear physics for concrete rea-sons, namely: from the comparison of theoretical calcu-lations with the experimental data, it has been pointedout that the mass tensor should be taken as a function ofthe collective coordinates [10,15,16,17,18,19,20,21]. So,based on these considerations, the Bohr Hamiltonian witha mass depending on collective variable can be elaboratedfor studying collective excited states in nuclei. These lat-ter play an important role in the so-called shape phasetransitions in nuclei which are of quantum type. Thus,several models considered as critical points of symmetries,namely: E(5) [22], X(5) [23], X(3) [24], Z(5) [25], Z(4) [26]have been introduced.In the present work, we will focus on the critical pointsymmetry Z(5) which represents the transition from pro-late axially symmetric SU(3) nuclei to oblate shape. Wewill consider a Bohr Hamiltonian with Hulth´en potentialincluding a mass parameter depending on the collectivecoordinate β . We have chosen this potential for its flat-ness. Indeed, it has been proved that as the considered a corresponding author: [email protected] potential is flat when β increases as the calculated transi-tion probabilities are more precise [27].We will study the same isotopes that have been alreadytreated in [10,28,29] using respectively Davidson poten-tial within the Deformation Dependent Mass formalism(here called Z(5)-DD model), the Hulth´en potential with-out DDM (here called Z(5)-H model) and Davidson po-tential without DDM (called Z(5)-D model). Hence, ouraim is to study:1) The effect of DDM on the energy spectra and B ( E , , , , Xeand , , Pt using the same potential2) The effect of the potential on the energy spectra and B ( E
2) transition probabilities of the same isotopes takinginto account the Deformation Dependent Mass term.The structure of the present work is as follows: In Sec-tions 2, 3 and 4, the theoretical background of the elabo-rated model Z(5)-HD is presented, namely: the β part andthe γ part of the spectrum, the obtained analytical expres-sions for the energy levels by means of Asymptotic Iter-ation Method and the the total wave function. Section 5contains the B ( E
2) transition probabilities, while numer-ical results for energy spectra and B ( E
2) are presented,discussed and compared with other results in section 6.Finally, section 7 contains a conclusion. a r X i v : . [ nu c l - t h ] A ug A. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length
In the framework of Z(5), the original Bohr Hamiltonian[30] is H B = − ¯ h B (cid:20) β ∂∂β β ∂∂β + 1 β sin 3 γ ∂∂γ sin 3 γ ∂∂γ − β (cid:88) k =1 Q k sin ( γ − π k ) (cid:21) + V ( β, γ ) , (1)where B is the mass parameter, which is usually consid-ered constant, β and γ are the usual collective coordinates( β being a deformation coordinate measuring departurefrom spherical shape, and γ being an angle measuring de-parture from axial symmetry), while Q k (k = 1, 2, 3)are the components of angular momentum in the intrinsicframe.By using a mass depending on the radial deformation co-ordinate β , B ( β ) = B ( f ( β )) , (2)where B is the constant mass and f ( β ) the deformationfunction, the Schr¨odinger equation corresponding to theHamiltonian (1) is given by [10] HΨ ( β, γ, θ i ) = (cid:20) − √ fβ ∂∂β β f ∂∂β (cid:112) f − f β sin 3 γ∂∂γ sin 3 γ ∂∂γ + f β (cid:88) k =1 , , Q k sin ( γ − πk )+ V eff (cid:21) Ψ ( β, γ, θ i ) = εΨ ( β, γ, θ i ) , (3)where θ i are the Euler angles and the reduced energies ε , reduced potential v ( β, γ ), effective potential V eff ( β, γ )are respectively ε = B ¯ h E , v ( β, γ ) = B ¯ h V ( β, γ ) V eff ( β, γ ) = v ( β, γ ) + (1 − δ − λ ) f ∇ f + ( − δ )( − λ )( ∇ f ) ,where δ and λ are free parameters, as it was proved in [1]that the most general form of such a Hermitian Hamil-tonian contains two free parameters (denoted by δ and λ in the present work). These parameters came from theconstruction procedure of the kinetic energy term. In Refs[10,31], it has been shown that these parameters had noeffect on the obtained results. The predictions for theo-retical spectra turn out to be independent of the choicemade for these two free parameters. Also, in the presentwork (section 6), it will be seen that these parameters playpractically no role.The function f ( β ) depends only on the radial coordinate β . So, only the β part of the above equation is affected. In order to achieve a separation of variables, we assumethat the reduced potential v ( β, γ ) depends on the vari-ables β and γ and has the form [32,33,34,35,36] v ( β, γ ) = u ( β ) + f β w ( γ ) , (4)with w ( γ ) having a deep minimum at γ = π and the wavefunctions have the form Ψ ( β, γ, θ i ) = ξ ( β ) Φ ( γ, θ i ) . (5)The separation of variables gives (cid:20) − √ fβ ∂∂β β f ∂∂β (cid:112) f + f β Λ + 14 (1 − δ − λ ) f ∇ f + 12 ( 12 − δ )( 12 − λ )( ∇ f ) + u ( β ) (cid:21) ξ ( β ) = ε ξ ( β ) , (6)and (cid:20) − γ ∂∂γ sin 3 γ ∂∂γ + 14 (cid:88) k =1 , , Q k sin ( γ − πk )+ w ( γ ) (cid:21) Φ ( γ, θ i ) = Λ Φ ( γ, θ i ) , (7)where Λ is the separation constant and equation (6) canbe simplified by performing the derivatives12 f ξ (cid:48)(cid:48) + (cid:18) f f (cid:48) + 2 f β (cid:19) ξ (cid:48) + (cid:20) ( f (cid:48) ) f f (cid:48)(cid:48) f f (cid:48) β − f β Λ + ε − v eff (cid:21) ξ = 0 , (8)with v eff = u ( β )+ 14 (1 − δ − λ ) f ( 4 f (cid:48) β + f (cid:48)(cid:48) )+ 12 ( 12 − δ )( 12 − λ )( f (cid:48) ) . (9)In the present work, we use the Hulth´en potential [37,38]with a unit depth as in [39,40] u ( β ) = − e τβ − , (10)where τ = b is a screening parameter and b is the range ofthe potential. This potential has some properties, namely:it behaves as a short-range potential for small values of β and decreases exponentially for very large values of β .By inserting the function F ( β ) = β ξ ( β ) in the radialequation (8), one obtains f F (cid:48)(cid:48) + 2 f f (cid:48) F (cid:48) + (cid:18) ε − v eff + f + βf f (cid:48) β + f Λ β − ( f (cid:48) ) − f f (cid:48)(cid:48) (cid:19) F = 0 . (11)In order to make connection between our results and thoseobtained in Ref.[28], we have replaced 2 ε by (cid:15) and divided u ( β ) by 2 in the above equation. So, one obtains f F (cid:48)(cid:48) + 2 f f (cid:48) F (cid:48) + ( (cid:15) − u eff ) F = 0 , (12) . Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length 3 where u eff = v eff + f + βf f (cid:48) β + f Λ β − ( f (cid:48) ) − f f (cid:48)(cid:48) . (13)The special form for the deformation function is f ( β ) = 1 + aβ , a << . (14)By inserting these forms for the potential and the defor-mation function in Eq. (13), one gets2 u eff = k β + k + k − β − e τβ − , (15)where k = a (cid:18) − δ − λ ) + (1 − δ )(1 − λ ) + 4 + Λ (cid:19) ,k = a (cid:18) − δ − λ ) + 7 + 2 Λ (cid:19) , (16) k − = 2 + Λ. Equation (12) becomes f F (cid:48)(cid:48) ( β )+2 ff (cid:48) F (cid:48) ( β )+ (cid:18) (cid:15) − k β − k − k − β + 1 e τβ − (cid:19) F ( β ) = 0 . (17) To simplify equation (17), we will proceed to a change ofthe function R( β ) by F ( β ) = R ( β )1 + aβ . (18)Thus, equation (17) becomes R (cid:48)(cid:48) ( β ) + (cid:18) (cid:15) − k β − k (1 + aβ ) − a aβ +1(1 + aβ ) ( e τβ − − k − (1 + aβ ) β (cid:19) R ( β ) = 0 . (19) From this equation, if we set the deformation parameter a = 0, we recover the equation (7) of Ref. [28]. Becauseof the centrifugal potential and the form of the Hulth´enone, the Schr¨odinger equation (19) cannot be solved ana-lytically. So we will proceed to a rigorous approximationthat allows to tackle this problem. For a small β deforma-tion, the centrifugal potential could be approximated bythe following expression as in Refs. [41,42,43]1 β ≈ τ e − τβ ( e − τβ − . (20)This approximation is also valid for small values of thescreening parameter τ . By using the new variable y = e − τβ , we obtain1 e τβ − y − y , β = 1 − yτ √ y , aβ = a (1 − y ) + τ yτ y . (21) Rewriting equation (19) by using the new variable y , weget R (cid:48)(cid:48) ( y ) + 1 y R (cid:48) ( y ) + (cid:20) ( (cid:15) − k ) τ + (2 − y ) k ( ay + ( τ − a ) y + a ) − k + 2 ay ( ay + ( τ − a ) y + a ) + τ y (1 − y )( ay + ( τ − a ) y + a ) − τ k − y (1 − y ) ( ay + ( τ − a ) y + a ) (cid:21) R ( y ) = 0 . (22) If a = 0, the dependence of the mass on the deformationis canceled, then we easily check that we recover the equa-tion (9) of Ref. [28].The Schr¨odinger equation (22) cannot yet be solved an-alytically because of some terms. Hence, in the absenceof a rigorous solution to this equation, we can use a fur-ther approximation. For a small deformation parameter a ( a << k is proportional to a param-eter, as a first approximation, equation (22) becomes R (cid:48)(cid:48) ( y ) + 1 y R (cid:48) ( y ) + (cid:20) ( (cid:15) − k ) τ + 2 k τ y + 1 τ y (1 − y ) − k − y (1 − y ) (cid:21) R ( y ) = 0 . (23)In order to transform the above differential equation to amore compact one, we use the following variables µ = − ( (cid:15) − k ) τ + 2 k τ , ν = 12 (1 + (cid:112) k − ) . (24)So, the differential equation (23) becomes R (cid:48)(cid:48) ( y ) + 1 y R (cid:48) ( y ) − (cid:20) µ y − τ y (1 − y ) + ν − νy (1 − y ) (cid:21) R ( y ) = 0 . (25)To apply the asymptotic iteration method of Refs. [44,45],the reasonable physical wave function that we propose isas follows R ( y ) = y µ (1 − y ) ν χ ( y ) . (26)For this form of the radial wave function, Eq. (25) reads χ (cid:48)(cid:48) ( y ) = − ω ( y ) σ ( y ) χ (cid:48) ( y ) − κ n σ ( y ) χ ( y ) , (27)with ω ( y ) = (2 µ + 1) − (2 µ + 2 ν + 1) y, (28a) σ ( y ) = y (1 − y ) , (28b) κ n = 1 τ − ν (2 µ + ν ) . (28c)Equation (27) leads us directly to the energy eigenvaluesusing the new generalized formula [46] which replaced theiterative calculations in the original AIM formulation [47]. κ n = − n ω (cid:48) ( y ) − n ( n − σ (cid:48)(cid:48) ( y ) . (29) A. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length
The above formulation gives the energy spectrum of the β equation (cid:15) n = − τ ( n + + (cid:113) + k − ) − τ ( n + + (cid:113) + k − ) − k τ + k , (30)where n is the principal quantum number and k − , k and k are defined previously as a function of Λ , which repre-sents the eigenvalues of the γ -vibrational plus rotationalpart of the Hamiltonian for triaxial nuclei. If we set thedeformation parameter a = 0, our energy formula Eq. (30)matches with that obtained in previous works [28,48,49,50].For Eq.(7), which represents the γ variable, we use a newgeneralized potential proposed in [51] which is inspired bya ring-shaped potential w ( γ ) = c + s cos (3 γ )sin (3 γ ) , (31)where c and s are free parameters. Inserting this form ofthe potential in equation (7), we get (cid:20) − γ ∂∂γ sin 3 γ ∂∂γ + 14 (cid:88) k =1 , , Q k sin ( γ − πk )+ c + s cos (3 γ )sin (3 γ ) (cid:21) Φ ( γ, θ i ) = Λ Φ ( γ, θ i ) . (32)Since the potential is minimal at γ = π , then the angularmomentum term can be written as [52,53]14 (cid:88) k =1 , , Q k sin ( γ − πk ) ≈ Q − Q , (33)with Q = Q + Q + Q and the wave functions are givenin the form Φ ( γ, θ i ) = Γ ( γ ) D LM,α ( θ i ) . (34)Thus, the separation of variables leads to the following setof differential equations (cid:20) − γ ∂∂γ sin 3 γ ∂∂γ + c + s cos (3 γ )sin (3 γ ) (cid:21) Γ ( γ ) = Λ (cid:48) Γ ( γ ) , (35)[ Q − Q ] D LM,α ( θ i ) = ¯ Λ D
LM,α ( θ i ) . (36)The resolution of the above equation is carried out byMeyer-ter-Vehn [54,55] with the results:¯ Λ = L ( L + 1) − α , (37) D LM,α ( θ i ) = (cid:115) L + 116 π (1 + δ α, ) (cid:20) D ( L ) M,α ( θ i ) + ( − L D ( L ) M, − α ( θ i ) (cid:21) , (38) where D ( θ i ) denotes Wigner functions of the Euler angles θ i ( i = 1 , , L is the total angular momentum quantumnumber, while M and α are the quantum numbers of theprojections of angular momentum on the laboratory fixed z -axis and the body-fixed x (cid:48) -axis, respectively.In the study of triaxial nuclei, instead of the projec-tion α of the angular momentum on the x (cid:48) -axis, we usethe wobbling quantum number n w = L − α [54,55]. Byreplacing α by L − n w in Eq. (37), one obtains¯ Λ = L ( L + 4) + 3 n w (2 L − n w )4 . (39)For the sake of solving equation (35) through the AIM,we introduce a new variable z = cos(3 γ ) and we proposethe following ansatz for the eigenvectors Γ ( γ ) Γ ( z ) = (1 − z ) √ c + s η ( z ) , (40)leading to and η (cid:48)(cid:48) ( z ) = − √ c + s ) zz − η (cid:48) ( z ) − (3 √ c + s + c − Λ (cid:48) )9( z − η ( z ) . (41) By using the generalized formula of AIM given in Eq. (29),we derive the eigenvalues: Λ (cid:48) = 9 n γ ( n γ + 1) + 3 √ c + s (2 n γ + 1) + c, (42)where n γ is the quantum number related to γ -excitations.Finally, the analytical expression of Λ , which representsthe eigenvalues of the γ -vibrational plus rotational part ofthe Hamiltonian for triaxial nuclei is Λ = 9 n γ ( n γ + 1) + 3 √ c + s (2 n γ + 1) + c + L ( L + 4) + 3 n w (2 L − n w )4 . (43)The solution of equation (41) gives the eigenfunctions whichare obtained in terms of Legendre polynomials. η ( z ) = N n γ (1 − z ) − √ c + s P √ c + sn γ + √ c + s ( z ) , (44)where N n γ is a normalisation constant.From equation (40), the γ angular wave function for tri-axial nuclei is given by Γ ( γ ) = N n γ P √ c + sn γ + √ c + s (cos(3 γ )) , (45)and the normalisation constant is obtained by using thenormalisation condition (cid:90) π Γ ( γ ) | sin(3 γ ) | dγ = 1 . (46)By using the orthogonality relation of Legendre polyno-mials (See Ref. [56], Eq.(7.112), page 769), we obtain N n γ = (cid:20) (3( n γ + √ c + s/
3) + ) ( n γ !)(2 √ c + s/ n γ )! (cid:21) . (47) . Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length 5 The total wave function has the form Ψ ( β, γ, θ i ) = ξ ( β ) Φ ( γ, θ i )= β − R ( β ) Γ ( γ ) D LM,α i ( θ i ) , (48)where R ( β ) is the radial function corresponding to theeigenvectors of Eq. (23), Γ ( γ ) is the angular wave functionof the γ -part given by Eq.(35) and D LM,α i ( θ i ) are the eigen-functions of the angular momentum given by Eq. (38). Byusing the parametrization given in Eqs. (24)-(26) and thegeneral solution of Asymptotic Iteration Method (AIM),the solution of equation (27) is obtained as χ ( y ) = N n F ([ − n, µ + 2 ν + n ] , [2 µ + 1] , y ) , (49)where F are hyper-geometrical functions and N n is anormalization constant. So, by using the connection be-tween the hyper-geometrical functions and Jacobi polyno-mials, we derive χ ( y ) = N n Γ (2 µ + 1) Γ ( n + 1) Γ (2 µ + n + 1) P n (2 µ, ν − y ) . (50)By proceeding to a change of variable y in the function R ( y )(Eq. (26)) by t = 1 − y , we obtain finally the follow-ing wave function R ( t ) = N n (1 − t ) µ (1 + t ) ν − µ + ν ) Γ (2 µ + 1) Γ ( n + 1) Γ (2 µ + n + 1) P n (2 µ, ν − t ) . (51)With regard to obtaining the normalisation constant N n ,we use the orthogonality relation of Jacobi polynomials(See Ref. [56], Eq. 7.391, page 806). N n = (cid:18) τ µ ( µ + ν + n ) ν + n (cid:19) (cid:20) n ! Γ (2 µ + n + 1) Γ (2 ν + 2 µ + n )( Γ (2 µ + 1) Γ ( n + 1)) Γ (2 ν + n ) (cid:21) . (52) In general, the quadrupole operator is given by T ( E M = tβ (cid:20) D (2) M, ( θ i ) cos( γ − π √ (cid:18) D (2) M, ( θ i )+ D (2) M, − ( θ i ) (cid:19) sin( γ − π (cid:21) , (53)where t is a scaler factor and D (2) M,α ( θ i )( α = 0 , , −
2) de-notes the Wigner functions of Euler angles. Around γ = π (triaxial nuclei), this expression is simplified into T ( E M = − √ tβ (cid:18) D (2) M, ( θ i ) + D (2) M, − ( θ i ) (cid:19) . (54) The B ( E
2) transition rates are given by [57] B ( E L i α i → L f α f ) = 516 π | < (cid:32)L f α f || T ( E || L i α i > | (2 L i + 1) , (55)where the reduced matrix element is obtained through theWigner-Eckart theorem [57] | < (cid:32)L f α f | T ( E M | L i α i > | = ( L i L f | α i M α f ) (cid:112) L f + 1 × | < (cid:32)L f α f || T ( E || L i α i > | . (56)In equation (56), the integral over γ leads to unity (be-cause of the normalisation of Γ ( γ ) ), the integral over theEuler Angles is performed by using the standard integralsof three Wigner functions and the integral over β takesthe form I β ( n i , L i , α i , n f , L f , α f ) = (57) (cid:90) ∞ β ξ n i ,L i ,α i ( β ) ξ n f ,L f ,α f ( β ) β dβ, where the factor β comes from the volume element [30]and the factor β comes from Eq.(54). The final result givesthe general expression of E B ( E L i α i → L f α f ) = 516 π t δ α i , )(1 + δ α f , ) (cid:20) ( L i L f | α i α f ) + ( L i L f | α i − α f )+ ( − L i ( L i L f | − α i α f ) (cid:21) I β ( n i , L i , α i , n f , L f , α f ) . (58)The Clebsch-Gordan coefficients (CGCs) appearing in equa-tion (58) impose a ∆α = ± γ band,c) within the odd levels of the γ band,d) between the even levels of γ band and the g.s band,e) between the odd levels of γ band and the g.s band,f) between the odd levels and the even levels of γ band. In this section, we present all results we have obtainedwith Z(5)-HD model. This model was applied for calcu-lating energy ratios of excited collective states and reduced E , , , , Xeand , , Pt. Such nuclei have been previously chosento be studied within other models [10,28,29] because ofundergoing the signature of the triaxial rigid rotor [58,59] ∆E = | E + g + E + γ − E + γ | ≈ . (59) A. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length
This equation is used in an approximate way, because theexperimental data for the eight nuclei , , , , Xeand , , Pt, respectively lead to the values ∆E ( keV ) = 49 , , , , , , , . (60)By referring to the values of equation (60), the isotopes , Xe and , , Pt are good candidates for a tri-axial rotor model and hence, presumptively could presenttriaxial deformation in their structure. Such a feature willbe checked afterwards through another important signa-ture. So, the above formula (59) served us in the first stepas a guide in choosing the candidate nuclei and therefore,we have added the isotopes , , Xe in our presentstudy.The allowed bands (i.e. ground state (g.s), β and γ ) arelabelled by the quantum numbers, n , n w , n γ and L . As de-scribed in the framework of the rotation-vibration model[60], the lowest bands for Z (5) are as follows1. The g.s band is characterized by n = 0, n γ = 0, n w = 02. The β band is characterized by n = 1, n γ = 0, n w = 0.3. The γ band composed by the even L levels with n = 0, n γ = 0, n w = 2 and the odd L levels with n = 0, n γ = 0, n w = 1.Discussion on nature of such bands can be found in therecent review article [61].The energy spectrum is given by equation (30) and de-pends on four parameters, namely: the screening parame-ter τ in the β potential, the ring-shape parameters c and s of the γ potential and the mass deformation parameter a . Our task is to fit these parameters to reproduce theexperimental data by applying a least-squares fitting pro-cedure for each considered isotope. We evaluate the rootmean square (r.m.s) deviation between the theoretical val-ues and the experimental data by σ = (cid:115) (cid:80) mi =1 ( E i ( exp ) − E i ( th )) ( m − E (2 +1 ) , (61)where E i ( exp ) and E i ( th ) represent the experimental andtheoretical energies of the i th level, respectively, while m denotes the number of states. E (2 +1 ) is the energy of thefirst excited level of the g.s band. The corresponding freeparameters ( τ , c , s ) and the mass deformation parameter a are listed in table 1. In this table, we give the fitted pa-rameters allowing to reproduce the experimental data [62]and Z (5) model [25]. The results presented here have beenobtained for δ = λ = 0. Different choices for δ and λ leadto a renormalization of the parameters values τ , c , s and a , so that the predicted energy levels remain unchanged.In Ref [10] (Z(5)-DD model), the authors have presentedthe analytical results for triaxial nuclei with γ = π byusing Davidson potential, but they did not present theirnumerical results. So, in order to compare our results ob-tained with Hulth´en potential with those obtained withDavidson potential, we have used the analytical formulasof Ref [10] and all obtained numerical results are presented nuclei τ c s a L g L β L γ m Xe 0.071 8 192 0.0025 12 4 9 16
Xe 0.050 2 140 0.0000 10 2 7 12
Xe 0.010 0 140 0.0000 14 0 5 11
Xe 0.080 72 226 0.0000 6 0 5 7
Xe 0.080 78 187 0.0000 6 0 5 7
Pt 0.050 19 73 0.0010 10 4 8 14
Pt 0.059 6 195 0.0030 10 4 8 13
Pt 0.086 7 120 0.0059 10 4 8 13 Z (5) 0.039 11 406 - 14 4 9 17 Table 1.
The Hulth´en potential and deformation parametersvalues fitted to the experimental data [62] as well as the resultsof Z (5) model [25]. L g , L β and L γ characterize the angularmomenta of the highest levels of the ground state, β and γ bands respectively, included in the fit, while m is the totalnumber of experimental states involved in the r.m.s fit. in table 2. Here, the parameters β , c and a are respec-tively the Davidson potential parameter, the γ -potentialparameter and the mass deformation parameter. nuclei β c a L g L β L γ m Xe 1.19 1.38 0.0060 12 4 9 16
Xe 0.94 4.52 0.0000 10 2 7 12
Xe 0.11 4.53 0.0000 14 0 5 11
Xe 0.00 0.00 0.0000 6 0 5 7
Xe 0.00 0.00 0.0000 6 0 5 7
Pt 1.05 8.47 0.0035 10 4 8 14
Pt 1.04 6.88 0.0055 10 4 8 13
Pt 0.84 3.05 0.0093 10 4 8 13 Z (5) 1.37 7.52 - 14 4 9 17 Table 2.
The Davidson potential and deformation parametersvalues fitted to the experimental data [62] as well as the resultsof Z (5) model [25]. L g , L β and L γ characterize the angularmomenta of the highest levels of the ground state, β and γ bands respectively, included in the fit, while m is the totalnumber of experimental states involved in the rms fit. From table 1, containing the results obtained withHulth´en potential, the following remarks are applying:1. For the isotope
Xe, the term of the deformation be-comes necessary, leading to nonzero value from the fit.2. For the isotopes , , , Xe, however, the fittingleads to a mass deformation parameter a = 0. So, theyare vibrational isotopes.3. For the isotopes , , Pt, the fitting leads to nonzero values of the mass deformation parameter whichincrease with mass number A .From table 2, the results are obtained with Davidson po-tential and the following remarks are applying:1. For Xe, the β and a terms become necessary, lead-ing to non zero values of both them. . Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length 7
2. For , Xe, the fitting leads to non zero β , howeverthe mass deformation parameter a = 0.3. For , Xe, both β and a parameters are equal tozero. Therefore, there is no need for deformation de-pendence in the potential.4. For , , Pt, the fitting leads to non zero valuesof β and a , and the mass deformation parameter in-creases with mass number A .From results displayed in tables 1 and 2, we concludethat , Xe are vibrational isotopes and , Xe arepure vibrators (both parameters β and a are null). Thus,our results confirm those obtained in Refs [10,29] concern-ing the vibrational nature of these isotopes. Such a resultis corroborated by the obtained values below for the ratio R / which are close to the vibrator’s characteristic value:2.In Fig.1, we compare the values of the mass deformationparameter given by our model Z(5)-HD and those ob-tained with Z(5)-DD [10] model for Xe and , , Pt.We note that there is a strong correlation between them( ρ = 0 . a is not a simplefitting parameter to be adjusted, but a structural param-eter of the model. It is not influenced by the presence ofthe parameters of the used potential. Moreover, this pa-rameter plays an important role for the moment of inertiainasmuch as it moderates the variation of the latter whenthe nuclear deformation β increases as can be seen fromFig.2 with arbitrary values of a and Fig.3 for a concretecase.In table 3, we compare the quality measure σ of our ● ● ● ● - Hulthèn a - D a v i d s on Correlation coefficient ρ = Fig. 1.
The comparison of deformation parameter a given byour model Z(5)-HD and Z(5)-DD model [10] for isotopes Xeand , , Pt. results (Z(5)-HD) with that of Z(5)-H model [28], Z(5)-DD model [10] and Z(5) model [25]. From this table, onecan see that our model is generally more efficient thanthe all others. Moreover, one can observe that the r.m.sfor the two isotopes
Xe and
Xe are equal for bothmodels Z(5)-HD and Z(5)-H and so smaller than that forZ(5)-DD. Hence, we can conclude that Hulth´en potential a = = = = β β / ( + a . β ) Fig. 2.
The function β /f ( β ) = β / (1 + aβ ) plotted as afunction of the nuclear deformation β for different arbitraryvalues of parameter a a = a H = a D = β β / ( + a . β ) Fig. 3.
The function β /f ( β ) = β / (1 + aβ ) plotted as afunction of the nuclear deformation β for values of the param-eter a obtained for Pt isotope with Z(5)-HD and Z(5)-DDmodels is more suitable for describing pure vibrators than theDavidson one.In figures 4-11, we have plotted the energy spectra of theisotopes , , , , Xe and , , Pt. From Fig.4,one can see that in the g.s band of
Xe, our model Z(5)-HD reproduces well the experimental levels in comparisonwith Z(5)-H and Z(5)-DD. Also, in the β band, Z(5)-HDis more precise than the others, while in the γ band thedifference between all models’ calculations is not signifi-cant.Fig.5 shows the spectrum of Xe, where our model isstill more efficient for mostly all levels in the g.s band ex-cept the last one, while in β and γ bands there is generallyno significant difference between all models. However, onecan observe that the β band head is better reproducedwith both Z(5)-HD and Z(5)-H models.The spectrum of Xe presented in Fig.6 shows that thelevels from L = 2 to L = 4, in the g.s band, are well re-produced by all models. Nevertheless, for the levels above L = 4 except the one with L = 12, Z(5)-DD is more pre- A. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Lengthnuclei Z(5)-HD Z(5)-H Z(5)-DD Z(5)
Xe 0.716 0.835 0.791 1.082
Xe 0.508 0.508 0.495 0.802
Xe 0.443 0.443 0.297 1.564
Xe 0.181 0.181 0.422 1.013
Xe 0.123 0.123 0.790 1.524
Pt 0.517 0.521 0.528 0.886
Pt 0.544 0.553 0.566 0.973
Pt 0.602 0.718 0.746 1.448
Table 3.
The root mean square (rms) deviation between ex-perimental data [62] and the theoretical results correspondingto Z(5)-HD, Z(5)-H [28], Z(5)-DD [10] and Z(5) [25] of givenisotopes. cise than the others. However, the β band head is betterreproduced with Z(5)-HD and Z(5)-H than Z(5)-DD. Asto the γ band, all models are almost equal in the repro-duction of all levels except the levels L = 4 and L = 5where Z(5)-DD is slightly more precise.From Fig.7 and Fig.8, representing the energy spectra for Xe and
Xe, one can see that the calculations of Z(5)-HD and Z(5)-H are identical because in this case, the massdeformation parameter is null. Besides, these results arefairly better than those obtained within Z(5)-DD. So, asit was already mentioned above, the Hulth´en potential ismore appropriate for nuclei possessing a vibrational na-ture than the Davidson potential.Fig.9 presents the spectrum of
Pt. Here, one can seethat, in the g.s band, all levels are well reproduced by allmodels, but with some prevalence of Z(5)-HD followed byZ(5)-H except the last level. In the β band, the levels 0 + and 4 + are better calculated with Z(5)-DD, but the 2 + iswell reproduced with Z(5)-HD. In the γ band, our modelZ(5)-HD followed by Z(5)-H show some performance inrespect to Z(5)-DD except for levels 4 + and 6 + .From the spectra of Pt and
Pt given respectively inFig.10 and Fig.11, we can make the same observation inthe g.s band like for the isotope
Pt. However, in the β band of Pt, all levels are well described with Z(5)-HDin respect to the others, while for
Pt, the calculationsof Z(5)-HD are the most precise followed by those of Z(5)-DD. As to the γ band, the situation for both isotopesis similar to that of Pt. But, here, we have to noticethat the common feature of all presented spectra is theobserved inversion of the levels 6 + and 7 + . The origin ofthis effect has been already explained in Ref. [28] where ithas been also mentioned that such a feature appears justin spectra of triaxial nuclei and hence could be consideredas a signature of triaxiality in nuclei.By using the potential and deformation parametersvalues ( τ , c , s , a ) given in table 1 for Hulth´en poten-tial and ( β , c , a ) given in table 2 for Davidson potential,which are obtained by fitting the energy ratios, we havecalculated the intra-band and inter-band B ( E
2) transi-tion rates, normalized to B ( E
2; 2 +0 , → +0 , ). Let us sim-ply note that the reduced E B ( E
2) transition rates for the iso-topes , , , Xe, in the case of Davidson potential,exceptionally we have used the analytical outcome givenin Ref [29]. All our results are presented in tables 4 and 5alongside with those obtained with Z(5)-H, Z(5)-DD, esMand Z(5) models as well as the experimental data. Fromthese tables, one can make the following observations:1) For transitions between the lower levels, our theo-retical results obtained with Z(5)-HD model are slightlyhigher than the experimental data, but generally remaincloser to them in comparison particularly with Z(5)-DD.2) In respect to Z(5)-H model, our results for the iso-topes , , , Xe are the same because the mass de-formation parameter a = 0. For Xe and , , Pt, allour results are slightly higher. Thus, the precision gainedin terms of energy ratios, thanks to the introduction of amass deformation parameter, is somewhat weakly lost intransition rates’ outcomes. However, the precision of ourmodel’s calculations persists versus Z(5)-DD. This latterobservation is due to the flatness of the Hulth´en poten-tial. Indeed, as it was mentioned in the introduction, theprecision of transition rates calculations depends on thisflatness insofar as it increases as the potential is flatter,when the β coordinate increases too. From Fig.12, onecan see that Hulth´en potential is flatter than Davidsonpotential.Another important signature for triaxiality in nuclei isthe odd-even staggering of energy levels which happens in γ band and which is described by the following relation[64]: S ( J ) = E ( J + γ ) + E (( J − + γ ) − E (( J − + γ ) E (2 +1 ) . (62)This relation gives the relative displacement of the ( J − + γ to the average of its neighbors, J + γ and ( J − + γ , normal-ized to the energy of the first excited state of the g.s band, E (2 +1 ). It was shown [65] that γ -soft shapes exhibit stag-gering with negative S ( J ) values at even-J and positive S ( J ) values at odd-J spins. However, for triaxial nucleithe opposite signs are seen, i.e. positive S ( J ) at even-Jand negative S ( J ) at odd-J. This is a sensitive probe oftriaxiality, as shown, for example, in Ref. [65]. It shouldalso be pointed out that for , Xe the data show be-havior opposite to all models (see Fig. 13). The same holdsmore or less for , Xe. In these two nuclei the Z(5)-HDmodel does get correctly the minimum at J=6, but its val-ues show no staggering (jumping up and down). Actuallyoccasional disagreements between theory and experimentcan sometimes lead later to interesting physical insights,therefore they should be pointed out. Moreover, from Fig.13, one can see that generally all studied nuclei exhibita stronger odd-even staggering than that observed exper-imentally. However, one can remark that the amplitudeof such an effect was attenuated within Z(5)-HD calcula-tions tending to the experimental behavior. Besides, suchan effect for
Pt and
Pt is well reproduced by ourmodel which concords with the previous studies in Refs.[65,28]. So, the two isotopes
Pt and
Pt are consid-ered as good candidates for triaxial deformation because . Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length 9 g.s band
Exp0 + + + + + + + ( )- HD0 + + + + + + + ( )- H0 + + + + + + + ( )- DD0 + + + + + + + ( ) + + + + + + + β band Exp0 + + + ( )- HD0 + + + ( )- H0 + + + ( )- DD0 + + + ( ) + + + γ band Exp2 + + + + + + + + ( )- HD2 + + + + + + + + ( )- H2 + + + + + + + + ( )- DD2 + + + + + + + + ( ) + + + + + + + + Xe Fig. 4.
The comparison between our theoretical energy spectra, given by Eq. (30) using the parameters in Table 1 for
Xeisotope with the experimental data [62], those obtained in Ref. [28] and in Ref. [10] using parameters in table 2 and those fromfree parameters model Z(5) [25]. g.s band
Exp0 + + + + + + ( )- HD0 + + + + + + ( )- H0 + + + + + + ( )- DD0 + + + + + + ( ) + + + + + + β band Exp0 + + ( )- HD0 + + ( )- H0 + + ( )- DD0 + + ( ) + + γ band Exp2 + + + + + + ( )- HD2 + + + + + + ( )- H2 + + + + + + ( )- DD2 + + + + + + ( ) + + + + + + Xe Fig. 5.
The comparison between our theoretical energy spectra, given by Eq. (30) using the parameters in Table 1 for
Xeisotope with the experimental data [62], those obtained in Ref. [28] and in Ref. [10] using parameters in table 2 and those fromfree parameters model Z(5) [25].0 A. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length g.s band
Exp0 + + + + + + + + ( )- HD0 + + + + + + + + ( )- H0 + + + + + + + + ( )- DD0 + + + + + + + + ( ) + + + + + + + + β band Exp0 + ( )- HD0 + ( )- H0 + ( )- DD0 + ( ) + γ band Exp2 + + + + ( )- HD2 + + + + ( )- H2 + + + + ( )- DD2 + + + + ( ) + + + + Xe Fig. 6.
The comparison between our theoretical energy spectra, given by Eq. (30) using the parameters in Table 1 for
Xeisotope with the experimental data [62], those obtained in Ref. [28] and in Ref. [10] using parameters in table 2 and those fromfree parameters model Z(5) [25]. g.s band
Exp0 + + + + ( )- HD0 + + + + ( )- H0 + + + + ( )- DD0 + + + + ( ) + + + + β band Exp + ( )- HD0 + ( )- H0 + ( )- DD0 + ( ) + γ band Exp2 + + + + ( )- HD2 + + + + ( )- H2 + + + + ( )- DD2 + + + + ( ) + + + + Xe Fig. 7.
The comparison between our theoretical energy spectra, given by Eq. (30) using the parameters in Table 1 for
Xeisotope with the experimental data [62], those obtained in Ref. [28] and in Ref. [10] using parameters in table 2 and those fromfree parameters model Z(5) [25].. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length 11 g.s band
Exp0 + + + + ( )- HD0 + + + + ( )- H0 + + + + ( )- DD0 + + + + ( ) + + + + β band Exp0 + ( )- HD0 + ( )- H0 + ( )- DD0 + ( ) + γ band Exp2 + + + + ( )- HD2 + + + + ( )- H2 + + + + ( )- DD2 + + + + ( ) + + + + Xe Fig. 8.
The comparison between our theoretical energy spectra, given by Eq. (30) using the parameters in Table 1 for
Xeisotope with the experimental data [62], those obtained in Ref. [28] and in Ref. [10] using parameters in table 2 and those fromfree parameters model Z(5) [25]. g.s band
Exp0 + + + + + + ( )- HD0 + + + + + + ( )- H0 + + + + + + ( )- DD0 + + + + + + ( ) + + + + + + β band Exp0 + + + ( )- HD0 + + + ( )- H0 + + + ( )- DD0 + + + ( ) + + + γ band Exp2 + + + + + + + ( )- HD2 + + + + + + + ( )- H2 + + + + + + + ( )- DD2 + + + + + + + ( ) + + + + + + + Pt Fig. 9.
The comparison between our theoretical energy spectra, given by Eq. (30) using the parameters in Table 1 for
Ptisotope with the experimental data [62], those obtained in Ref. [28] and in Ref. [10] using parameters in table 2 and those fromfree parameters model Z(5) [25].2 A. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length g.s band
Exp0 + + + + + + ( )- HD0 + + + + + + ( )- H0 + + + + + + ( )- DD0 + + + + + + ( ) + + + + + + β band Exp0 + + + ( )- HD0 + + + ( )- H0 + + + ( )- DD0 + + + ( ) + + + γ band Exp2 + + + + + + ( )- HD2 + + + + + + + ( )- H2 + + + + + + + ( )- DD2 + + + + + + + ( ) + + + + + + + Pt Fig. 10.
The comparison between our theoretical energy spectra, given by Eq. (30) using the parameters in Table 1 for
Ptisotope with the experimental data [62], those obtained in Ref. [28] and in Ref. [10] using parameters in table 2 and those fromfree parameters model Z(5) [25]. g.s band
Exp0 + + + + + + ( )- HD0 + + + + + + ( )- H0 + + + + + + ( )- DD0 + + + + + + ( ) + + + + + + β band Exp0 + + + ( )- HD0 + + + ( )- H0 + + + ( )- DD0 + + + ( ) + + + γ band Exp2 + + + + + + ( )- HD2 + + + + + + + ( )- H2 + + + + + + + ( )- DD2 + + + + + + + ( ) + + + + + + + Pt Fig. 11.
The comparison between our theoretical energy spectra, given by Eq. (30) using the parameters in Table 1 for
Ptisotope with the experimental data [62], those obtained in Ref. [28] and in Ref. [10] using parameters in table 2 and those fromfree parameters model Z(5) [25].. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length 13 Xe Xe L in,n w L fn,n w exp Z (5) − HD Z (5) − H Z (5) − DD exp Z (5) − HD Z (5) − H Z (5) − DD Z (5)4 , , ... 1.568 1.551 1.600 1.468 1.602 1.604 1.648 1.5906 , , ... 2.325 2.261 2.340 1.941 2.440 2.444 2.464 2.2038 , , ... 3.196 3.030 3.019 2.388 3.441 3.445 3.228 2.63510 , , ... 4.366 3.999 3.688 2.737 4.789 4.785 3.981 2.9672 , , ... 1.590 1.574 1.624 1.194 1.625 1.626 1.673 1.6204 , , ... 0.365 0.365 0.369 ... 0.382 0.383 0.389 0.3486 , , ... 0.227 0.219 0.225 ... 0.243 0.244 0.239 0.1988 , , ... 0.166 0.160 0.158 ... 0.184 0.184 0.170 0.1293 , , ... 1.288 1.254 1.302 ... 1.350 1.352 1.370 1.2435 , , ... 1.176 1.110 1.105 ... 1.266 1.267 1.182 0.9727 , , ... 1.201 1.092 1.000 ... 1.318 1.315 1.080 0.8089 , , ... 1.327 1.144 0.936 ... 1.458 1.447 1.017 0.6964 , , ... 0.736 0.726 0.759 ... 0.759 0.760 0.791 0.7366 , , ... 1.272 1.208 1.202 ... 1.374 1.376 1.286 1.0318 , , ... 2.593 2.342 2.066 ... 2.867 2.862 2.237 1.59010 , , ... 4.552 3.840 2.911 ... 5.016 4.968 3.172 2.0355 , , ... 1.321 1.282 1.326 ... 1.389 1.391 1.400 1.2357 , , ... 2.375 2.233 2.182 ... 2.575 2.578 2.341 1.8519 , , ... 3.753 3.379 2.982 ... 4.139 4.129 3.229 2.30811 , , ... 5.706 4.839 3.751 ... 6.276 6.220 4.084 2.6653 , , ... 2.203 2.157 2.242 ... 2.291 2.294 2.345 2.1715 , , ... 1.629 1.533 1.510 ... 1.765 1.766 1.618 1.3137 , , ... 2.040 1.820 1.603 ... 2.246 2.238 1.737 1.2609 , , ... 2.550 2.118 1.622 ... 2.777 2.745 1.768 1.164 Xe Xe L in,n w L fn,n w exp Z (5) − HD Z (5) − H Z (5) − DD exp Z (5) − HD Z (5) − H Z (5) − DD Z (5)4 , , ... 1.611 1.611 1.711 1.238 1.685 1.685 1.834 1.5906 , , ... 2.469 2.469 2.629 ... 2.630 2.630 2.919 2.2038 , , ... 3.503 3.503 3.496 ... 5.964 5.964 3.955 2.63510 , , , , ... 1.633 1.633 1.739 1.775 1.707 1.707 1.865 1.6204 , , ... 0.386 0.386 0.414 ... 0.203 0.203 0.459 0.3486 , , ... 0.247 0.247 0.259 ... 0.189 0.189 0.292 0.1988 , , ... 0.188 0.188 0.185 ... 0.049 0.049 0.211 0.1293 , , ... 1.366 1.366 1.459 ... 1.677 1.677 1.618 1.2435 , , , , ... 1.347 1.348 1.181 ... 0.992 0.992 1.351 0.8089 , , ... 1.491 1.491 1.119 ... 0.438 0.438 1.287 0.6964 , , ... 0.764 0.746 0.834 ... 0.300 0.300 0.912 0.7366 , , ... 1.400 1.400 1.396 ... 0.455 0.455 1.582 1.0318 , , ... 2.936 2.936 2.454 ... 0.855 0.855 2.816 1.59010 , , ... 5.132 5.132 3.498 ... 0.777 0.777 4.038 2.0355 , , ... 1.406 1.406 1.497 ... 0.045 0.045 1.667 1.2357 , , ... 2.626 2.626 2.544 ... 0.046 0.046 2.891 1.8519 , , ... 4.237 4.237 3.540 ... 1.389 1.389 4.061 2.30811 , , ... 6.421 6.421 4.500 ... 1.124 1.124 5.191 2.6653 , , ... 2.313 2.313 2.483 ... 2.650 2.650 2.731 2.1715 , , ... 1.799 1.799 1.756 ... 1.368 1.368 1.991 1.3137 , , ... 2.298 2.298 1.904 ... 1.326 1.326 2.183 1.2609 , , ... 2.836 2.836 1.948 ... 0.639 0.639 2.246 1.164 Table 4.
The normalized B ( E
2) transition rates of the Z(5)-HD model, compared to the experimental data [62], Z(5)-H model[28], Z(5)-DD model [10] and the free parameters model Z(5) [25] predictions for , , , Xe isotopes.4 A. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length Xe Pt L in,n w L fn,n w exp Z (5) − HD Z (5) − H Z (5) − DD exp Z (5) − HD Z (5) − H Z (5) − DD esM Z (5)4 , , , , ... 0.275 0.275 2.919 1.224 2.310 2.303 2.360 2.213 2.2038 , , ... 0.192 0.192 3.955 ... 3.146 3.126 3.053 2.735 2.63510 , , ... 1.766 1.766 4.975 ... 4.227 4.178 3.737 3.163 2.9672 , , ... 1.775 1.775 1.865 1.905 1.588 1.586 1.632 1.586 1.6204 , , ... 0.519 0.519 0.459 ... 0.363 0.362 0.372 0.350 0.3486 , , ... 0.005 0.005 0.292 ... 0.226 0.225 0.227 0.1988 , , ... 0.045 0.045 0.211 ... 0.166 0.166 0.160 0.1293 , , ... 1.995 1.995 1.618 0.664 1.281 1.277 1.313 1.236 1.2435 , , ... 0.770 0.770 1.445 ... 1.154 1.146 1.118 0.9727 , , ... 0.677 0.677 1.351 ... 1.157 1.143 1.013 0.8089 , , ... 0.333 0.333 1.287 ... 1.238 1.212 0.950 0.6964 , , ... 0.758 0.758 0.912 ... 0.735 0.734 0.765 0.734 0.7366 , , ... 0.450 0.450 1.583 ... 1.255 1.247 1.216 1.081 1.0318 , , ... 0.679 0.679 2.816 ... 2.495 2.460 2.095 1.715 1.59010 , , ... 0.622 0.622 4.038 ... 4.200 4.086 2.955 2.0355 , , ... 2.408 2.408 1.667 ... 1.312 1.308 1.384 1.250 1.2357 , , ... 1.156 1.156 2.891 ... 2.330 2.312 2.208 1.943 1.8519 , , ... 1.088 1.088 4.061 ... 3.601 3.549 3.023 2.285 2.30811 , , ... 0.893 0.893 5.191 ... 5.278 5.149 3.806 2.907 2.6653 , , ... 2.952 2.952 2.731 1.783 2.194 2.188 2.259 2.147 2.1715 , , ... 4.997 4.997 1.991 ... 1.598 1.585 1.528 1.3137 , , ... 0.924 0.924 2.183 ... 1.945 1.912 1.626 1.2609 , , ... 0.493 0.493 2.246 ... 2.321 2.254 1.647 1.164 Pt Pt L in,n w L fn,n w exp Z (5) − HD Z (5) − H Z (5) − DD esM exp Z (5) − HD Z (5) − H Z (5) − DD esM Z (5)4 , , , , , , , , , , , , , , ... 0.226 0.224 0.229 0.394 0.247 0.243 0.249 0.1988 , , ... 0.166 0.164 0.162 ... 0.177 0.183 0.179 0.1293 , , ... 1.282 1.272 1.323 1.305 ... 1.377 1.348 1.411 1.200 1.2435 , , ... 1.159 1.139 1.129 ... 1.335 1.260 1.228 0.9727 , , ... 1.168 1.132 1.025 ... 1.473 1.306 1.127 0.8089 , , ... 1.261 1.198 0.961 ... 1.797 1.434 1.065 0.6964 , , , , ... 1.259 1.239 1.228 1.112 1.207 1.426 1.369 1.340 1.022 1.0318 , , ... 2.521 2.436 2.121 1.697 ... 3.166 2.839 2.343 1.589 1.59010 , , ... 4.291 4.041 2.995 ... 5.971 4.918 3.329 2.0355 , , ... 1.314 1.303 1.350 1.316 ... 1.417 1.387 1.447 1.205 1.2357 , , ... 2.341 2.296 2.232 1.994 ... 2.720 2.563 2.439 1.834 1.8519 , , ... 3.641 3.515 3.061 2.468 ... 4.642 4.097 3.379 2.306 2.30811 , , ... 5.387 5.087 3.857 2.801 ... 7.687 6.160 4.283 2.633 2.6653 , , ... 2.196 2.182 2.274 2.264 ... 2.326 2.288 2.409 2.094 2.1715 , , ... 1.606 1.575 1.544 ... 1.876 1.756 1.682 1.3137 , , ... 1.969 1.896 1.645 ... 2.598 2.219 1.814 1.2609 , , ... 2.377 2.229 1.668 ... 3.735 2.717 1.851 1.164 Table 5.
The normalized B ( E
2) transition rates of the Z(5)-HD model, compared to the experimental data [62], Z(5)-H model[28], Z(5)-DD model [10], esM model [66] and the free parameters model Z(5) model [25] predictions for
Xe , , , Ptisotopes.. Adahchour, S. Ait El Korchi, A. El Batoul, A. Lahbas, M. Oulne: Title Suppressed Due to Excessive Length 15 τ = τ = τ = β = β = β = - - - β P o t e n ti a l v a l u e s Fig. 12.
The Hulthn and Davidson potentials with parameters given in tables 1 and 2 for , , Pt isotopes of satisfying both signatures for triaxiality, namely: thatof the triaxial rigid rotor and the staggering effect.
In the present work, we have solved the eigenvalues andeigenvectors problem with the Bohr collective Hamilto-nian for triaxial nuclei within Deformation Dependent Massformalism using Hulth´en potential for β -part and a newRing-Shaped potential [28] for the γ one. The obtainedresults with our proposed model dubbed Z(5)-HD werein overall agreement with the experimental data for en-ergy ratios and transition rates of the nuclei , , Xeand , , Pt and comparatively better in general thanother models. Moreover, our model was an improvement ofthe previously proposed one in Ref.[28]. Indeed, the intro-duction of a mass deformation parameter has significantlyincreased the precision of energy ratios calculations par-ticularly. Besides, the flatness of Hulth´en potential versusDavidson one has played an important role in a satisfac-tory reproduction of experimental data of transition rates.Moreover, we have shown that Hulth´en potential is moreappropriate for describing nuclei presenting a vibrational structure. Despite the difference between the parametersof the potential in both cases, namely: Hulth´en and David-son, it was found a strong correlation between the values ofthe mass deformation parameter in these two cases whichcorroborates once again the fact that the mass deforma-tion parameter is not just a simple one to be adjusted forreproducing experimental data, but should be consideredas a model’s structural one. Among the studied nuclei, wehave confirmed that the better candidates for triaxialitywere , Pt. These two isotopes undergo both signa-tures for triaxiality, namely: that of the triaxial rigid rotorand the staggering effect.
References
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