Atomic Nuclei 9 Springer-Verlag 1990

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1 Z. Phys. A - Atomic Nuclei 335, (1990) Zeitschrift far Physik A Atomic Nuclei 9 Springer-Verlag 1990 Effective Boson Number Calculations in Mo and Cd Isotopes G. Cata, D. Bucurescn, D. Cutoiu, M. Iva~cu, and N.V. Zamfir Central Institute of Physics, Bucharest, Romania Received October 23, 1989 The effects of the neutron-proton interaction on the low-lying levels of Mo and Cd isotopes have been considered in the frame of the IBA-1 model by taking into account an effective boson number (Neff). Both an empirical procedure based on previous IBA-2 mixing calculations and the Np AT, scheme provide comparable Neff values. Level spectra and electromagnetic transitions are investigated. The results support the idea that IBA-1 calculations with a suitable Neff can largely simulate IBA-2 mixing calculations, taking advantage of simplicity and a smaller number of parameters. PACS: Ev I. Introduction The effects of the neutron-proton (n-p) interaction in the nuclear structure of the A-100 nuclei have been recognized as very important since the pioneering work of Federmann and Pittel [1, 2]. In particular, this interaction accounts for the onset of deformation, subshell structure and intruder states in this nuclear region. More recently, these ideas led to the introduction of the N v N, scheme by Casten, which gave a unifying view of the evolution of the nuclear collectivity across many nuclear regions [3, 4]. A particular role in the nuclear structure is played by the interaction between the valence neutrons and protons which belong to spin-orbit partner (s.o.p.) orbitals [1]. Due to this, the structure of nuclei with proton number close to subshell or shell closure such as Mo (Z = 42) and Cd (Z = 48), is notoriously difficult to be described by geometrical or IBA models [5, 8]. A possible way to do this within the IBA-2 model was suggested by Duval and Barett [6], namely by mixing two bosonic configurations differing by two proton bosons. This procedure has been successfully applied to isotopes of Hg [6], Cd [7, 9, 10] and Mo [8]. The IBA-2 mixing calculations imply a large number of model parameters (for the two IBA-2 Hamiltonians and for the mixing term). The purpose of this work is to present a model description of the Mo and Cd nuclei, with a greatly reduced number of parameters. To achieve this, we consider simple IBA-1 model calculations [-1 lj, based on the assumption that a great deal of the effects of the mixing can be simulated by considering the number of bosons as an effective value, derived either from the previous IBA-2 mixing calculations or from the Np N, scheme. A comparison of these calculations with the experimental data and the existing IBA-2 mixing results justify this approach. 2. The Nuclear Model The interacting boson model (IBA-1) introduced in 1975 by Arima and Iachello [-llj, which we use here, has proved its versatility in many applications over the entire nuclear chart. We present here only the parametrization of the Hamiltonian which has been used in this work. The most commonly used form of the IBA-1 Hamiltonian, which allows the easiest understanding of the role of each term in determining the final structure of the nucleus under consideration, is the so-called multipole expansion. In this parametrization the various boson-boson interactions are grouped in such

2 272 G. Cata et al.: Effective Boson Number Calculations PROTONS NEUTRONS PROTONS NEUTRONS / q/ / r :: :: g7/2 -: :- :: ds/2 r :: d3/2 / _-: h11/2 ~-// ~/ :: :: g7/2 f ~ d5/2 lo2 Mo60 42 g9/2 g912 ~ ill/////~ g9/2 112(--H6~ Fig. 1. Valence configurations of l~ and ~12Cd. The monopole interaction between the spin-orbit partner orbitals is explicitly indicated. Particles and holes are represented by filled and empty circles, respectively a way that the Hamiltonian takes the form: 1 1 H=~ha+ao(P.P)+-2-al(~.Q)+~a2(L.L ) (1) where the operators are: ~d = ~(d + ~)~o) _ 1 r~(o) L = -}fio(d + ~) (1~ (~ =- [(s + ~-F d + s) (2) + ~55 (d + 2)(2)] The E2 electromagnetic transitions between eigenstates of the Hamiltonian (1) are described by the operator: 7"(E 2)= e,[(s + ~+ d + s)(2) + z(d + ~)(2)] (2) with two free parameters, eb and Z. The numerical diagonalization of (1) and the B(E2) value calculations have been performed with the computer codes PHINT and FBEM, respectively [ The Phenomenological Effective Bosun Number As emphasized in the introduction, the main idea of our calculations is to bypass the mixing IBA-2 description of low-lying levels in Mo and Cd nuclei by introducing into the IBA-1 model an effective number of bosons. We evaluate this number by a procedure which is in the spirit of the IBA-2 mixing description. The combined effect of the monopole and quadrupole components of the neutron-proton interaction is a crucial factor in determining the main features of the nuclear structure at low excitation energies [3, 13, 14"1. In the Mo isotopes, when the neutrons start filling the g7/2 orbital, the strong monopole interaction between the s.o.p, orbitals /~g9/2 and vgv/2 depresses the s.p. energy of the/~g9/2 orbital, destroying the Z = 40 subshell gap and thus increasing the probability of protonic excitations across the Z--40 gap, such as to increase the number of n-p valence pairs interacting through the strong quadrupole force. The valence configuration of the isotope ~~ is sche- matically shown in Fig. 1 a. The situation in the Cd chain is illustrated in Fig. 1 b, where the relevant s.o.p. orbitals are gds/2 and vd3/z and the proton excitations take place across the Z = 50 gap. Due to these n--p interaction effects in the Mo and Cd nuclei, it is impossible to define the boson number according to the usual manner of IBA, as a unique value. Indeed, in calculations with the IBA model with an unique boson configuration in both the Mo and Cd nuclei many low-lying states remain outside the model description. A successful description of these unusual ("intruder") states has been achieved in the IBA model frame by mixing of two boson configurations: a normal one with proton boson number N (1) = 1 and an excited one with N} 2) = 3 [7-10] (Fig. 1). For our IBA-1 calculations we define the effective proton boson number by the empirical formula: Neff n ]kt(1) _L n N(2) --F1 ~'rc LF2 (3) where Pl and P2 are the percentages of the normal and excited configuration, respectively, in the structure of the ground state wave function of each isotopes. Then, the total effective boson number used was Neff=Neff+ Nv, where N, is half the number of valence neutrons. For the Mo isotopes, the pl and P2 values were taken from [8]. The obtained N eff values are plotted in Fig. 2; for numerical calculations they were rounded to the nearest integer, for each isotope. In Fig. 2 are also drawn the Nff f values provided by

3 G. Cata et al.: Effective Boson Number Calculations N~r ~.2 MO /!! / 0 I I I I I N Fig. 2. The effective proton boson number for the Mo isotopes, as given by our empirical procedure (3) from IBA-2 mixing calculations [8] (continuous line), and from Casten's - Np N, scheme [13] (dashed line) the Np Nn scheme [14]. One observes that for the 98Mo and l~176 isotopes, which were found to present the strongest mixing of the two boson configurations [8], the Ng ff values provided by the two procedures are in a remarkably good agreement. In the case of the Cd nuclei, where the transition spherical to deformed is more gradual than in the Mo nuclei, we found that choosing p~=pz=0.5 in (3) gives a reasonable description of the data, as will be shown below. 4. Results of Neff IBA-1 calculations 4.I. The IBA-1 Model Parameters The four parameters (e, a0, al and a2) of the Hamiltonian (1) needed to describe the 96-1~ and lo6 116Cd nuclei in the above presented formalism are given in Fig. 3. Their values have been obtained by looking for a best fit to the low-energy spectra, while varying relatively smoothly these parameters along each isotopic chain. The values we obtain (Fig. 3) are in the same range with those obtained in a previous study of Ru and Pd nuclei, although in that case a different (ECQF) procedure has been used [15]. In the present calculations we have used a constant value %e=-y~/2 for all nuclei, the effects of a free variation of ZQ! 273 (ECQF formalism) remaining as a further study problem Energy Spectra Figure 4 shows the low excitation levels known in the two isotopic chains and their description by the model approach discussed above. In the Mo chain (Fig. 4a) the lighter isotopes are almost spherical and are easily reproduced by IBA-1 calculations close to the U(5) dynamical symmetry. The heaviest studied isotopes 1~176 show rotational features, characteristic of y-soft deformed nuclei (near the 0(6) limit). The middle isotopes 9s, to0mo have more peculiar level schemes, especially concerning the position of the 0+ and 2 + levels. All these features, indicative of a shape coexistence, could be unitarily described only by the IBA-2 mixing calculations [8]. It can be seen in Fig. 4a that the description of the level positions by the N, ff IBA-1 calculations is reasonably good; it is, in fact, qualitatively similar to that of the IBA-2 mixing procdure [8]. The level positions in the Cd isotopes (Fig. 4b) show a more gradual change than those in Mo. Although for a long time the Cd isotopes were considered as good examples of quadrupole vibrators, the existence of additional 0 + and 2 + levels close to the two-phonon triplet, thus forming a quintuplet of states interconnected by strong E2 transitions, posed a real challenge to theoretical calculations [16, 17]. Both mixing IBA-2 calculations [7, 9, 10] and an explicit consideration of two particle - two hole excitations into a core coupling model [9, 10] explained these peculiarities, emphasizing the importance of the two-proton excitations across the Z = 50 gap. As seen in Fig. 4b, the present calculations are able to provide a similarly good description of the level schemes Electromagnetic Transitions As a more detailed test of the model, we discuss the B(E2) transitions in these nuclei, which are experimentally known for most levels of interest. Figures 5 and 6 show the comparison between the calculations and both the experimental data and the predictions of the IBA-2 mixing calculations. For both the Mo and Cd nuclei, the two parameters e~ and )~ of the E2 transition operator (2) were kept constant along the isotopic chain. The values used are eb=0.12eb, Z=--2.5 for Mo and e~ eb, % = -2.9 for Cd, respectively. In both cases the overall agreement with the data is reasonable, and the predictions are qualitatively similar with those of the IBA-2 mixing approach (Figs. 5, 6). The

4 274 G. Cata et al.: Effective Boson Number Calculations (HeY) 0,02 X,,, 0, [HEY) 0.01 I h I I I I I 1,. 1 I ,02 Q1 (HEY) (HEY) t, I I I 0.0 l I, I I I I 0.03 O (MeW - a 0 (MeV) -0,0~ I I I I 0,01 I I I I I I I, s (MeV) 0.6 s (HeVl 0, I 54 I I I I N 42M0 0.4 I I I I I I B 4aCd N Fig. 3. The parameters of the IBA-1 model Hamiltonian (1) used in the present calculation

5 G. Cata et al.: Effective Boson Number Calculations 275 A 3~ 9 21 [] 0] o 0~ a/3 + o o ~ ~ ~ o z o ~ ~o r z ~ 6{ O -O JO B ' N N Fig. 4. Results of the present calculations compared to ~2Mo experimental level schemes 4aCd largest discrepancy between the two theoretical calculations (a factor of 2) occurs in the Mo case (Fig. 5) for the ratio R2=B(E2; O2~21)/B(E2; 21 ~01) of 98Mo (N= 56). This isotope presents a strong mixing in the IBA-2 calculations [8], therefore this discrepancy, which comes from an underestimation of the experimental value by our calculations, may indicate the limitation of the Neff IBA-1 appraoch. A certain improvement in the description of the B(E2) data of this nucleus can be obtained by some changes in the Hamiltonian parameters given in Fig. 3. In Figs. 7 and 8 we present a detailed comparison of level and decay schemes for two nuclei, l~176 and 114Cd, which were found with a strong configuration mixing in the IBA-2 mixing approach [7, 8]. One can thus see, in greater detail, the similarity of the descriptions provided by the IBA-2 mixing model and the present approach, respectively. We mention only briefly that in the case of the Cd isotopes we have also compared calculated E0 transitions with existing experimental data (for the first three 0 + states in 112'114Cd). The reasonable agreement obtained constitute a further positive test of the model description of these states [18]. 5. Conclusions Low-lying positive parity levels in the 96 lo4mo and 1~ nuclei have been described in the frame of the IBA-1 model with an effective boson number. The effective values for the proton boson number have been obtained with an empirical procedure from existing IBA-2 mixing model results, and have been found consistent with those extracted from the N~ N,

6 ~,2Mo 0.2~ B(E2; 2~~0 T ) /,, R~= B(E2; 4~ 24) B(E2; 2.1 ~ 0'~ ) B(E2;02 ~ 21) R2- BIE2;2~ 01) 0.2l, 0.2C 0.1~ 0 12 I A z \ / \ i I I I I 1 1 I I I l I 1 t 0,06 B(E2 2+--*OT) R3-- B(E2; 2.~ +~ 0 + i ) 3 B(E2; 2~ ~2T) R&= B(E2; 2+ 0~) 0.~ /2"",, I / \ o -0,2 D -- % -0.4 I I I, I I o 6~ C ~2 ;~ Fig. 5. Comparison of B(E2) values (in e2b 2) and ratios in Mo with model predictions, Continuous lines: the present calculations; dashed lines: IBA-2 mixing calculations of [8] 4oCd (.;/+ i s 9 RI= B(E2; 2T~ 0~} R2= B(E2; 02~24) B(E2; 21 ~ 0,~) 0.09 O.OE I i } ] I t 1 ] I 1 I I I I I I I I R 3 = BIE2; 2 +~0~) R&= B(E2; 2~-~ 0~-) (~ 2 + feb) 0.1; 0.; , + * , / /" 1 I I I )4 # I I I I ) /, 66 6s Fig. 6. Same as Fig. 5, but for Cd isotopes (with IBA-2 mixing results from [7]) I I I I I J & 66 68

7 G. Cata et al.: Effective Boson Number Calculations ~2Mu EX PERIMENT T H E ORY THEORY IBA-2 (MIXINGI IBA-1 {Neff) ~ % 18,~6 ~ ' J m -",~ ' a 9 9 4, 4 i' I q2~.=-o z,1 i! e2t =-o.~2 % I lq 1 I I 0.8 1~1~5 0i < z"~' 10- ":~ , ',..., ; Fig. 7. A detailed comparison between experimental data, IBA-2 mixing model [8] and the present N.ff IBA-1 model for l~176 11~, (~rl EXPERIHENT THEORY THEORY 1.5 IBA-2 (HIXINGI IBA-1 (Neff) 1,0 0.5 O. 2~ :2 : ~ ~ ~ o~ ' '~'~ l t'lll I ~ "- 13_.:_ <3.10 -L" , ) I 11.s ', t I 9 / / I~ ~ ~ 0'.3 71o -2 I Q2~---o 3~ I I ''+ '~ Q2~o~---- I Q2ff_ W... I j Fig. 8 Same as Fig. 7 but for 114Cd scheme. The description on both yrast and non-yrast low energy levels and of their B(E2) transition probabilities is reasonable, and, generally, of the same quality with that provided by the IBA-2 mixing calculations. The present results give us a hint that numerical IBA calculations based on the N~ N. scheme prescriptions can be used to describe the structure of nuclei near the closed shells, which exhibit abnormal lowenergy level schemes, due to the presence of 2p-2h core excited states. Such calculations, having the ad- vantage of a small number of free parameters, could be used to parametrize even-even nuclei, entering as cores into IBFM and IBFFM descriptions of odd-a and odd-odd neighbouring nuclei, respectively. At the phenomenological level, our calculations emphasize the importance of an adequate consideration of the details of the neutron-proton interaction. A deeper justification of the effects of the n- p interaction as reflected in the effective boson number formalism requires a microscopic (shell model) analysis such as that performed by Scholten E19].

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