Microscopic interacting boson model calculations for even–even 128−138Ce nuclei

—journal of May 2007

physics pp. 769–778

Microscopic interacting boson model calculations for even–even

Ce nuclei

NURETTIN TURKAN^1,2 and ISMAIL MARAS³

1Faculty of Arts and Science, Bozok University, 66100 Yozgat, Turkey

2Institute of Science, Erciyes University, 38039 Kayseri, Turkey

3Faculty of Arts and Science, Celal Bayar University, Manisa, Turkey E-mail: nurettin turkan@yahoo.com

MS received 13 September 2006; revised 10 January 2007; accepted 8 March 2007 Abstract. In this study, we determined the most appropriate Hamiltonian that is needed for the present calculations of energy levels andB(E2) values of^128−138Ce nuclei which have a mass aroundA∼= 130 using the interacting boson model (IBM). Using the best- fitted values of parameters in the Hamiltonian of the IBM-2, we have calculated energy levels andB(E2) values for a number of transitions in128,130,132,134,136,138Ce. The results were compared with the previous experimental and theoretical (PTSM model) data and it was observed that they are in good agreement. Also some predictions of this model have better accuracy than those of PTSM model. It has turned out that the interact- ing boson approximation (IBA) is fairly reliable for calculating spectra in the entire set of 128,130,132,134,136,138Ce isotopes and the quality of the fits presented in this paper is acceptable.

Keywords. Interacting boson model; even–even Ce; pair-truncated shell model; electric quadrupole transition probability.

PACS Nos 23.20.-g; 23.20.Js; 23.20.Lv

1. Introduction

The ground-state charge and transition charge densities of the nucleus provide meeting ground between the theory and experiment. Nuclei in the neutron number ofN = 82 near the partial proton shell closure ofZ = 58 provide interesting cases for the investigation of certain aspects of nuclear structure by electron scattering.

Very little is known about the multipolarities of interband transitions in cerium nuclei. In earlier studies on130,132,134Ce, Husaret al [1] and Nolanet al [2] obtained both energy level spacing and life-time results which indicated increased collective behavior with decreasing neutron number. Wells et al [3] obtained evidence for collective behavior in¹²⁸Ce from lifetime measurements, Saladin et al [4] studied evidence for continuum E0 transitions following the decay of high-spin states in

130Ce. No detailed work has been done on the structure of cerium nuclei and it is

necessary to carry out calculations that are comparable to experimental results. So, one of the goals of the present study is to test interacting boson model calculations in the mass region ofA∼= 130 by comparing them with some previous experimental and theoretical results.

The interacting boson model offers a simple Hamiltonian, capable of describing collective nuclear properties across a wide range of nuclei, based on general algebraic group theoretical techniques which have also recently found application in problems in atomic, molecular, and high-energy physics [5,6].

It has been widely used for describing the quadrupole collective states of the medium heavy nuclei and no distinction is made between proton and neutron vari- ables when the first version of the model (IBM-1) is applied. So, triaxiality can be described explicitly through the introduction of cubic terms in the boson operators.

However, the microscopic foundations state certainly that it is very important to describe the proton and neutron variables explicitly. This is also generalized defini- tion of the second version of the IBA-model (IBM-2 model). The building blocks of IBM-2 aresρ anddρ bosons (ρ=π, ν) which are considered to be approximations to the proton (neutron) pairs with spin-parity 0⁺ and 2⁺. The boson images of the fermion operators are given in terms of the OAI mapping [7]. The changes in the structure of the nuclei have been proposed to be related to the exceptionally strong neutron–proton interaction. It is also suggested that the neutron–proton effective interactions have a deformation producing tendency, while the neutron–

neutron and proton–proton interactions are of spheriphying nature [8,9]. Within the region of medium-heavy and heavy nuclei, a large number of nuclei exhibit properties that are neither close to anharmonic quadrupole vibrational spectra nor to deformed rotors. Thus, the standard description of these phenomena has been given in terms of nuclear triaxiality going from rigid triaxial shapes to more soft po- tential energy surfaces when describing such nuclei in a geometric description. IBM Hamiltonian takes different forms depending on the regions (SU(5), SU(3), SO(6)) of the traditional IBA triangle.

The structures of ^128−138Ce was undertaken in this study to provide more de- tail on the neutron-rich isotopes. So, the aim is to carry out B(E2) transition probabilities of Ce nuclei around the mass regionA∼= 130 by employing the most appropriate Hamiltonian of IBM-2 in a valence space and to give a clear description about their structure in the dynamic symmetry limits. The results of the present IBM-2 calculations are compared with the previous IBM-2 results of Pudduet al [10] and also those of the pair-truncated shell model (PTSM) [11]. The outline of the this paper is as follows: starting from an approximate IBM-2 formulation for the Hamiltonian in §2, we give a review of the theoretical background of the study. In §3, the previous experimental and theoretical data are compared with calculated values and the general features of Ce isotopes in the rangeA= 128–138 are reviewed. The last section contains some concluding remarks.

2. Theoretical framework

In this study, IBM-2 is carried out to project out the IBM-1 part of Hamiltonian by starting with IBM-2 Hamiltonian and then use PHINT code for IBM-1 calculations.

As is known, IBM-2 formalism is another alternative specifying the parameters of the Hamiltonian used in the calculations and according to the procedure of this formalism, in IBM-2 Hamiltonian the neutrons’ and protons’ degrees of freedom are taken into account explicitly. It has the advantage of being closer to the microscopic theory. However, the matrices that have to be diagonalized are much larger. Also, one can regard the IBM-1 model space, in which neutron and proton degrees of freedom are distinguished as a subspace of the IBM-2 model space, namely that of the fully symmetric states. From the IBA-2 Hamiltonian one can thus project out its IBA-1 piece [12]. In the projection from IBA-2 onto IBA-1 the number of neutron (Nν) and proton (Nπ) bosons play an important role. The program code PHINT [13] is used within the option of specifying the parameters in the neutron–

proton formalism where it takes care of projecting maximum symmetry states. If the lowest states are indeed fully symmetric, the calculation with PHINT will give exactly the same excitation energies. Thus the Hamiltonian can be written as

H=ε(ndν+ndπ) +κ(Qπ.^.Qν) +Vππ+Vνν+Mπν, (1) wherendρ is the neutron (proton)d-boson number operator and

ndρ =d^+.d,˜ ρ=π, ν

d˜ρm= (−1)^mdρ,−m (2)

Qρ = (s⁺_ρd˜ρ+d⁺_ρsρ)⁽²⁾+χρ(d⁺_ρd˜ρ)⁽²⁾ Vρρ =

L=0,2,4

CLρ((d⁺_ρd⁺_ρ)^(L)(d⁺_ρd˜ρ)^(L))⁽⁰⁾; ρ=π, ν (3) and

Mνπ=1

2ξ2[(s⁺_νd⁺_π −d⁺_νs⁺_π)⁽²⁾·(sνd˜π−d˜νsπ)⁽²⁾]

−

L=1,3

ξL[(d⁺_νd⁺_π)^(L)·( ˜dνd˜π)^(L)], (4) wheres⁺_ρ,d⁺_ρm andsρ, dρmrepresent the s- andd-boson creation and annihilation operators. The parameters ε, κ, χρ and CLρ are the free parameters that have been determined so as to reproduce as closely as possible the excitation energy of all positive parity levels for which a clear indication of the spin value exists, following the same procedure described in [14]. The value ofχπhas been kept fixed along the isotopic chain as suggested by microscopic considerations which predict that this parameter depends only on the proton number. Due to admixtures of non-fully symmetric states in the IBA-2 the projection gives different results and parameters, mostly ε (ED) and κ (RKAP), have to be normalized. Mπν affects only the position of the non-fully symmetric states relative to the symmetric ones.

For this reason Mνπ is often referred to as the Majorana force and we kept the coefficientsCLρ= 0 and ξL= 0 constant for all calculations.

In particular, in this work we focus on the E2 transition that is one of the important factors within the collective nuclear structure. So the electromagnetic

transitions can also be analyzed in the framework of IBM-2 and the most general E2 transition operator can be written as

T(E2) =eπxQ⁽²⁾π +eνxQ⁽²⁾ν , (5) where

Q⁽²⁾_ρ = (s⁺_ρd˜ρ+d⁺_ρsρ)⁽²⁾+χρ(d⁺_ρd˜ρ)⁽²⁾, ρ=π, ν. (6) Here,χρis a non-dimensional coefficient andeρ is the effective quadrupole charge.

After mapping the IBM-2 quadrupole operator to IBM-1 space (puttingeπ =eρ), theT(E2) operator takes the same form as in eq. (6) but withoutρindex. In the PHINT code, the effective charges for the sd and dd parts are denoted by E2SD and E2DD respectively.

3. Results and discussion

The^128−138Ce isotopes have Nπ = 4, andNν varies from 6 to 1, while the para- metersκ, χρ andε, as well asCLρ, withL= 0,2,4 were treated as free parameters and their values were estimated by fitting to the measured level energies. This procedure was made by selecting the ‘traditional’ values of parameters and then allowing one parameter to vary keeping the others constant until a best fit was obtained. This was carried out iteratively until an overall fit was achieved. Having obtained wave functions for the states in^128−138Ce after fitting the experimental energy levels in IBM-2, we can calculate the electromagnetic transition rates be- tween states using computer code FBEM (a subroutine of PHINT). As pointed out by Bijker et al [15], nuclei withχπ+χν = 0 have properties close to those of the SO(6) limit. The Hamiltonian sets of parameters which have been varied along the isotopic chain are shown as a function of the neutron number for Ce isotopes in figure 1a. As seen, the large variation between Ce and the neighboring Te–Xe–Ba [16–18] isotopes points in the same direction.

In this study, we takeχπ =−1.2 in the fit for all Ce isotopes. In particular, the spectrum of theSU(5) nuclei is dominated byε, which is large in comparison with the other parameters, whereas O(6) nuclei are characterized byκ, which is large compared toε[19]. The energy level fit with these parameters is shown in figures 1b and 1c along with experimental levels. As can be seen, the agreement between the experiment and the presented results are quite good and the general features are reproduced well. Figures 1b and 1c show the experimental [20] positive-parity spectra of the ground-state band, quasi-beta band and quasi-gamma band for Ce isotopes and compare them with the presented IBM-2 results. The excitation en- ergies for all members of bands are well-reproduced by the calculation. We observe discrepancy between theory and experiment forJ^π = 4⁺,6⁺,8⁺ and 10⁺ in these isotopes. But one must be careful in comparing theory with experiment, since all calculated states have a collective nature, whereas some of the experimental states may have a particle-like structure. As can be seen from the figures the calculated values are generally in good agreement with experiment (taken from ref. [20]).

(a)(b)(c) Figure1.(a)TheparametersusedintheHamiltonianforIBM-2calculationsforCe.(b)Resultsofthecalculated energiesofgroundstatebandandtheenergyratioE(J^{π i})/E(2^{+ 1})fortheJ^{π i}=4

+ 1,6

+ 1,8

+ 1

and10

+ 1

levelsin128−138Ce isotopeswithvibrationalandrotationallimitsshownontheextremeleftandextremerightofthefigure,respectively[21]. Filledspacesdenotethepresentedresultswhileemptyspacesshowtheexperimentalones.(c)Resultsofthecalculated energiesofquasi-betaandquasi-gammabandsareshownalongwiththeexperimentalonesfor128−138 Ceisotopes.

In the upper part of figure 1b, E(2⁺₁) of ^128−138Ce nuclei rises smoothly till N = 80 and so it can be said that the collectivity decreases smoothly until this neutron number which is the minimal neutron open-shell nucleus; two neutron holes. Figure 1b also contains the systematic of basic observables in Ce isotopes showingR4/2, R6/2, R8/2, R10/2 as a function of neutron numbers changing from 66 to 80. The energy ratioE(J_i^π)/E(2⁺₁) for theJ_i^π= 4⁺₁,6⁺₁, 8⁺₁ and 10⁺₁ levels for the doubly even Ce isotopes with both the vibrational and rotational limits for this ratio are given and those limits are shown on the extreme left and extreme right of the figure respectively [21]. The behavior of the ratio of the energies of the first 4⁺ and 2⁺ states are good criterion for the shape transition. The value of R_4/2 ratio has the limiting value 2 for a quadrupole vibrator, 2.5 for a non-axial gamma-soft rotor and 3.33 for an ideally symmetric rotor. As can be seen in the lower part of figure 1b it decreases gradually from about 2.3 to about 2.0. The agreement between the experimental values and the calculated ones is especially very good and the calculated results show that R_4/2 > 2 for all Ce isotopes and it means that their structures seem to be varying from gamma soft rotor to near-harmonic vibrator (HV). So, the energy spectrum of the ^128−138Ce nuclei can be situated between the pure vibrational and rotational limit.

The stable even–even nuclei in Te, Xe, Ba and Ce isotopic chains represent excel- lent opportunities for studying the behavior of the total low-lyingE2 strengths in the transitional region from deformed to spherical nuclei. Calculations of electro- magnetic transitions give a good test of nuclear model wave functions. To determine the boson effective charges E2DD and E2SD, we perform a fit to the experimental B(E2) values in the Ce isotopes. The matrix elements of theE2 operator of eq.

(6) have been calculated using the effective charges of E2SD = 0.139, E2DD = 0.08 for N= 70; E2SD = 0.140, E2DD = 0.08 for N= 72; E2SD = 0.140, E2DD = 0.100 forN= 74; E2SD = 0.123, E2DD = 0.100 for N= 76; E2SD = 0.120, E2DD = 0.300 forN= 78 and E2SD = 0.127, E2DD = 0.100 for N= 80. Here, the units are in eb and the effective charges appear as new parameters.

Even–even Ba [22–25] and Te [26–31] are also the member of the chain existing along the mass region of A ∼ 120–130 and they are neighboring isotopes to Ce nuclei. Yoshinaga and Higashiyama [32] have stated that the experimental data of ^130−134Ba simulate the O(6) limit prediction of IBM. In figure 2a, we show that B(E2; 2⁺₁ → 0⁺₁), B(E2; 4⁺₁ →2⁺₁), B(E2; 2⁺₂ →2⁺₁), B(E2; 2⁺₂ → 0⁺₁) and B(E2; 2⁺₁ → 0⁺₂) are values changing with neutron numbers for Ce, Ba and Te nuclei. In the figure the calculated values which are presented by full circle were compared with that of some previous experimental and theoretical results [3] and it is seen that they are in good agreement. The B(E2) values of Ce as well as Ba [33] decrease smoothly as neutron numberN approachesN=80. Among these valuesB(E2; 2⁺₁ →0⁺₁) andB(E2; 4⁺₁ →2⁺₁) are of the same order of magnitude and display a typical decrease towards the end of the shell. In figure 3, some of the presented B(E2) results are compared with the values obtained in PTSM model [11] and it is seen that the presented ones have better agreement with experiment than the predictions of the PTSM model. The largeB(E2) value in all ^128−138Ce nuclei is the main indicator of the vibrational behavior of these nuclei. They are nicely reproduced by the experiment and the fits of them are satisfactory. No experimental values exist for B(E2; 2⁺₂ →2⁺₁), B(E2; 2⁺₂ →0⁺₁) and B(E2; 2⁺₁ →

Figure 2. (a) The calculated B(E2; 2⁺₁ → 0⁺₁), B(E2; 4⁺₁ → 2⁺₁), B(E2; 2⁺₂ → 2⁺₁), B(E2; 2⁺₂ → 0⁺₁) and B(E2; 2⁺₁ → 0⁺₂) values chang- ing as a function of neutron number for Ce, Ba and Te nuclei along with some theoretical and experimental values. (b) Comparison of systematic of basic observables in Ce isotopes showing R1(=B4/2) = ^B(E2;4_B(E2;2^1→21)

1→01), R2 = ^B(E2;2_B(E2;2^2→21)

1→0₁),R3 = ^B(E2;0_B(E2;2^2→21)

1→0₁),R4 = ^B(E2;2_B(E2;2^2→01)

2→2₁),R5 = ^B(E2;3_B(E2;3^1→21)

1→4₁), R6 = ^B(E2;4_B(E2;4^2→41)

2→22) and R7 = ^B(E2;4_B(E2;2^1→21)

2→21) ratios of ^128−138Ce isotopes with those ofSU(5), SU(3), O(6) dynamical symmetry limits [19].

Figure 3. Comparison of someB(E2) values derived in the present IBM-2 calculation with those in PTSM model [11].

0⁺₂) transitions for ^128−138Ce. Contrary to the results of Puddu et al [10], the presentB(E2; 2⁺₂ →2⁺₁) values decrease as a function of neutron number,N = 70, 72, 74, 76, and 78. Also,B(E2; 2⁺₂ →0⁺₁) results derived in this paper are different from those calculated in ref. [10].

SomeB(E2) transition ratios of ^128−138Ce isotopes are discussed as R1= B(E2; 4₁→2₁)

B(E2; 2₁→0₁), R2=B(E2; 2₂→2₁) B(E2; 2₁→0₁), R3= B(E2; 0₂→2₁)

B(E2; 2₁→0₁), R4=B(E2; 2₂→0₁) B(E2; 2₂→2₁), R5= B(E2; 3₁→2₁)

B(E2; 3₁→4₁), R6=B(E2; 4₂→4₁) B(E2; 4₂→2₂) and

R7=B(E2; 4₁→2₁) B(E2; 2₂→2₁)

and the calculated ratios are compared with that ofSU(5), O(6), SU(3) ratio lim- its [19] in figure 2b. Moreover, R1 also represents the change of the rate of B4/2 = B(E2; 4⁺₁ → 2⁺₁)/B(E2; 2⁺₁ → 0⁺₁) with neutron number. B4/2 is 2.0 in a pure geometric vibrator and about 1.5 in the finite particle interacting boson approximation (IBA) model. It is 1.43 in a pure rotor. As can be seen in the figure, the use of complete Hamiltonian shows that gamma-soft rotor features exist in Ce, but with a dominancy of vibrational character. As can be seen in the figure,B_4/2 increases from about 1.40 forN= 70 to 1.50 forN= 80 and it is showing a case of vibrational structure. There is only one experimental value (R1 ofN= 70) and it is not in good agreement with the calculated value. As stated by Ramanet al[34], in the 50≤(N, Z)≤82 (‘tin’) region, empirical models apparently fare poorly.

4. Conclusion

It is well-known that some of the even–even Ce isotopes exhibit back-bending, superdeformed bands etc. No attempt is made in this paper to study these and the focus is only on the structure of low-lying states. It turns out that the interacting boson approximation (IBA) is fairly reliable for the calculation of spectra in the entire set of^128−138Ce isotopes. Also, it can be said that the observed systematic of low-lying spectra andB(E2) transition probabilities are found to get reproduced with a reasonable accuracy. Moreover, many predictions of this model are closer to experiment than those of PTSM model. The calculated results of the excitation energies shown in figures 1b and 1c and those of the electric quadrupole transition probabilities shown in figures 2a and 2b, are in good agreement with experimental values. So, it indicates that the set of parameters used in the calculation of Ce isotopes is a good approximation and the quality of the fits presented in this paper is acceptable. The Ce isotopes are close to both the proton and the neutron closed shell and these nuclei are not expected to be deformed. Finally, it can also be stated that gamma-soft rotor features exist in Ce, but with a dominancy of vibrational character.

Acknowledgements

We would like to thank Dr R F Casten for fruitful discussions during the Symposium of Physics (BPU-6) in Turkey.

References

[1] D Husar, S J Mills, H Graf, U Neumann, D Pelte and G Seiler-Clark,Nucl. Phys.

A292, 267 (1977)

[2] P J Nolan, D M Todd, P J Smith, D J G Love, P J Twin, O Andersen, J D Garrett, G B Hagemann and B Herskind,Phys. Lett.B108, 269 (1982)

[3] J C Wells, N R Johnson, J Hattula, M P Fewell, D R Haenni, I Y Lee, F K McGowan, J W Johnson and L L Riedinger,Phys. Rev.C30, 1532 (1984)

[4] J X Saladin, M P Metlay, D F Winchell, M S Kaplan, I Y Lee, C Baktash, M L Halbert, N R Johnson and O Dietzsch,Phys. Rev.C53, 652 (1996)

[5] O S Van Roosmalen, A E L Dieperink and F Iachello,Chem. Phys. Lett.85, 32 (1982) [6] M E Kellman and P R Herrick,Phys. Rev.A22, 1536 (1980)

[7] T Otsuka, A Arima and F Iachello,Nucl. Phys.A309, 1 (1979) [8] P Federman and S Pittel,Phys. Lett.B69, 385 (1977)

[9] C K Nair, A Ansari and L Satpathy,Phys. Lett.B71, 257 (1977) [10] G Puddu, O Scholten and T Otsuka,Nucl. Phys.A348, 109 (1980)

[11] T Takahashi, N Yoshinaga and K Higashiyama,Phys. Rev.C71, 014305 (2005) [12] O Scholten, The interacting boson fermion model and its applications, PhD Thesis

(University of Groningen, 1980)

[13] O Scholten,The program package PHINT (KVI Reports, University of Groningen, 1990)

[14] A Giannatiempo, A Nannini, P Sona and D Cutoiu,Phys. Rev.C52, 2969 (1995) [15] R Bijker, A E L Dieperink and O Scholten,Nucl. Phys.A344, 207 (1980) [16] S F Hickset al,Phys. Rev.C71, 34307 (2005)

[17] N Yoshida, A Gelberg, T Otsuka, I Wiedenh¨over, H Sagawa and P von Brentano, Nucl. Phys.A619, 65 (1997)

[18] T Otsuka,Nucl. Phys.A557, 531c (1993)

[19] J Stachel, P van Isacker and K Heyde,Phys. Rev.C25, 650 (1982)

[20] R B Firestone,Table of isotopes, USAedited by V S Shirley (J. Wiley-Interscience Press, New York, 1996)

[21] D De Frenne, H Thierens, E Jacobs, P De Gelder, P D’hondt, A DeClercq, K Heyde and A J Deruytter,Phys. Rev.C18, 486 (1978)

[22] Yan-An Luo, Jin-Quan Chen and J P Draayer,Nucl. Phys.A669, 101 (2000) [23] N Idrissi, A Gizon, J Genevey, P Paris, V Barci, D Barneoud, J Blachot, D Bucurescu,

R Duffait, J Gizon, C F Liang and B Weiss,Zeit. Phys.A341, 427 (1992) [24] J Yan, O Vogel, P von Brentano and A Gelberg,Phys. Rev.C48, 1046 (1993) [25] E S Paul, D B Fossan, Y Liang, R Ma and N Xu,Phys. Rev.C40, 1255 (1989) [26] C J Barton, M A Caprio, D Shapira, N V Zamfir, D S Brenner, R L Gill, T A Lewis,

J R Cooper, R F Casten, C W Beausanget al,Phys. Lett.B551, 269 (2003) [27] M J Bechara, O Dietzsch, M Samuel and U Smilansky,Phys. Rev.C7, 628 (1978) [28] Yu K Shubnyi,Sov. Phys. JETP 18, 316 (1964)

[29] J Barrette, M Barrette, R Haroutunian, G Lamoureux and S Monaro, Phys. Rev.

C10, 1166 (1974)

[30] A M Kleinfeld, G M¨aggi and D Werdecker,Nucl. Phys.A248, 342 (1975)

[31] A Christy, I Hall, R P Harper, I M Naqib and B Wakefield,Nucl. Phys.A142, 591 (1970)

[32] N Yoshinaga and K Higashiyama,Phys. Rev.C69, 054309 (2004) [33] K Uchiyamaet al,Eur. Phys. J.A2, 13 (1998)

[34] S Raman, J A Sheikh and K H Bhatt,Phys. Rev. C52, 1380 (1995) [35] D Huser, S J Mills, H Grafet al,Nucl. Phys.A292, 267 (1977) [36] W Dehnhardtet al,Nucl. Phys.A255, 1 (1974)