Investigation of Even-Even Ru Isotopes in Interacting Boson Model-2

Transcription

1 Commun. Theor. Phys. (Beijing, China) 46 (2006) pp c International Academic Publishers Vol. 46, No. 4, October 15, 2006 Investigation of Even-Even Ru Isotopes in Interacting Boson Model-2 A.H. Yilmaz and M. Kuruoglu Department of Physics, Karadeniz Technical University, Trabzon 61080, Turkey (Received November 14, 2005; Revised February 5, 2006) Abstract The interacting boson model of Arima, Iachello, and co-workers is applied to the even ruthenium isotopes, 96 Ru 116 Ru. Excitation energies, electromagnetic transition strengths, quadrupole and magnetic dipole moments, and (E2/M1) mixing ratios have been described systematically. Mixed symmetry states are investigated. It is seen that the properties of low-lying levels in these isotopes, for which the comparison between experiment and theory is possible, can be satisfactorily characterized by the Interacting Boson Model-2. PACS numbers: Fw, Re, j Key words: nuclear structure, 96 Ru 116 Ru, energy levels, interacting boson model-2 1 Introduction The interacting boson model is a very effective phenomenological model for describing low-lying collective properties of nuclei across an entire major shell. The building blocks are the s- and d-boson, which are interpreted as the approximation to the correlated fermion pairs with L π = 0 + and 2 +, respectively. [1] In the original model (IBM-1), only one kind of s and d bosons were considered. [2] Then the distinction between protons and neutrons was made in the proton-neutron interacting boson model (IBM-2). [3] The building blocks of IBM-2 are the proton-proton and neutron-neutron fermion pairs with L π = 0 +, 2 +, approximated by the corresponding bosons in the model. [4] Ruthenium isotopes have been the subject of studies in nuclear-structure physics. [5 12] The Ru-nuclei are six proton holes away from the Z = 50 shell closure. In this work IBM-2 calculations of 96 Ru 116 Ru are carried out, the attention being paid to excitation energies, electromagnetic transition rates, quadrupole moments, magnetic dipole moments, and mixing ratios. Our main goal of this investigation is to extend the IBM-2 calculation for even-even Ru isotopes from A = 96 to 116. This broader systematic enables us to investigate the electromagnetic properties and shape change. We also pay special attention to excited 0 + 2, 2+ 2, 2+ 3, and 3+ 1 states in addition to yrast states. The general interacting boson model Hamiltonian contains several terms. The Hamiltonian generally used in phenomenological calculations can be written as H = ε π n dπ + ε ν n dν + κ Q π Q + ν M πν + ω πν L π L ν, (1) where a dipole-dipole interaction has been explicitly introduced. The Hamiltonian consists of the d-boson energy, quadrupole, dipole interaction, and Majorana terms. The parameter ω πν is the strength of the proton-neutron dipole interaction, which we consider explicitly in this calculation. The operators of Eq. (1) have the following form: Q ρ = [d ρs ρ + s d ρ ρ ] (2) + χρ[d d ρ ρ ] (2), (2) M πν = 1 2 ξ ( 2 [s ν d π d νs π] (2) [s ν dπ d ν s π ] (2)) ξ k [d νd π] (k) [ d ν dπ ] (k), (3) k=1,3 L ρ = 10 [ d ρ d ρ ] (1), (4) in terms of usual s and d boson creation and annihilation operators for neutrons (ν) and protons (π), respectively. Q ρ is the quadrupole operator. The χ π and χ ν parameters are related to the strength of the quadrupole protonneutron interaction. The Majorana term M νπ contains three parameters ξ 1, ξ 2, and ξ 3. The Majorana parameters play an important role in fixing the location of states with mixed proton-neutron symmetry relative to the totally symmetric states. L is the angular momentum operator of either kind of bosons. The magnitude and sign of the multipole mixing ratios are found to depend sensitively on ξ 2. In the interacting boson model-2, the E2 transition operator is given by T (E2) = e π Q π + e ν Q ν, (5) where e π (e ν ) is the effective charge of proton (neutron) bosons in units of eb. They may be obtained from the B(E2) values of transitions. [13] The quadrupole operator, Q ρ, has the same definition as in the Hamiltonian equation (1), and for consistency we choose the same value for χ ρ as in the Hamiltonian. The M1 transition operator can be written as 3 T (M1) = 4π (g πl π + g ν L ν ), (6) where g ρ is the proton (neutron) g-factor in units of µ N. 2 Choice of Parameters A modified version of the NPBOS code [14] has been used to diagonalize the Hamiltonian (1). The electromagnetic matrix elements between eigenstates were calculated using the programme NPBTRN. The isotopes corresponding author, hakany@ktu.edu.tr

2 698 A.H. Yilmaz and M. Kuruoglu Vol Ru 116 Ru have N π = 3 and N ν varies from 1 to 8, while the parameters κ, χ ρ, ω ρ, and ε d, as well as the Majorana parameters ξ k, with k = 1, 2, 3, were treated as free parameters and their values were estimated by fitting to the measured level energies. We applied this procedure by choosing the traditional values of the parameters used and then allowing one parameter to change while keeping the others constant until a best fit was gained. It was carried out iteratively up to an overall fit was achieved. Figure 1 gives these two of the examples. We investigate even-even Ru isotopes in the IBM-2 framework assuming that the single-particle energies of the neutron and proton boson are the same, ε = ε π = ε ν. We also assume to use hole-neutron bosons after 110 Ru. The best fit values for the Hamiltonian parameters are given in Table 1 and the calculated energy levels are compared with the experimental data and are shown in Figs. 2 and 3 for 96 Ru 116 Ru isotopes. We see that the agreement is good for members of the ground state, γ and β bands, as well as for the higher bands. Fig. 1 The excitation energy of the lowest 2 + level in 98 Ru as a function of ε d and χ ν. Table 1 Adopted values of the parameters used for Ru isotopes in IBM-2 calculations. All parameters are given in MeV except χ ν (dimensionless). The values χ π = 0.6 and ξ 1 = 1.00 MeV were chosen for the two parameters, not varied along the isotopic chain. A ε κ χ ν ξ 2 ξ 3 ω πν Fig. 2 The energies of and levels are given for Ru isotopes. Symbols and lines are given for experimental and calculated values, respectively. Fig. 3 Comparison between experimental (symbols) and calculated (lines) energies for even-even Ru nuclei in gammaband.

3 No. 4 Investigation of Even-Even Ru Isotopes in Interacting Boson Model Mixed-Symmetry States Mixed-symmetry states have been observed in even-even nuclei in a mass range A = This event is coming from the out-of-phase collective motion of protons and neutrons and they have been observed in vibrational, [15] rotational, [16] and γ-unstable [17] collective nuclei. One of the important achievement of the IBM-2 is the prediction of the mixed-symmetry states. [18] Nonsymmetric or mixed-symmetry states are those with less than maximum neutronproton symmetry. In IBM-2, nuclear states are characterized by the F spin, which is the isospin for proton and neutron bosons. By coupling the collective proton and neutron degrees of freedom in a nucleus with N π proton pairs and N ν neutron pairs, the F spin assumes values from F max = (N π + N ν )/2 to F min = (N π N ν )/2. The fully symmetric states have maximum F spin, i.e. F = F max, while mixed-symmetry states are those with F < F max. The IBM-2 predicts enhanced M1 transitions between the mixed-symmetry states with F = F max 1 and symmetric states with F = F max and the same number of phonons, with matrix elements of the order of 1 µ N. The IBM-2 also predicts weakly collective E2 transitions between the symmetric and mixed-symmetric states following the phonon selection rule N ph = 1. The existence of mixed-symmetry states was confirmed by the discovery of the scissors mode with I π = 1 + in electron scattering experiments. [16,19] If the IBM-2 Hamiltonian were an F spin scalar, there would be no M 1 transitions between the lowest collective states, which have maximum F spin. Actually the Hamiltonian is not an F spin scalar, and its eigenstates have components with F < F max. Since the Hamiltonian, we used, has one and two-body components with F = F max, F = F max 1 and F = F max 2 in the first order. 4 Mixing Ratios The reduced E2 and M 1 matrix elements have been evaluated for a selection of transitions in ruthenium isotopes. The ratio (E2/M1) is defined as the ratio of the reduced E2 matrix element to the reduced M1 matrix element. This quantity is related to the usual δ-mixing ratio by ( E2 ) ( E2 ) δ = 0.835E γ. (7) M1 M1 5 Results and Discussion The comparisons between calculated and experimental energy levels for 96 Ru 116 Ru are shown in Figs. 2 and 3. The energies of 2 + 1, 4+ 1, 6+ 1, 8+ 1, and levels are given for Ru isotopes in Fig. 2. Experimental data are taken from Refs. [20] [33]. In general, the agreement is quite good, especially for the low-lying band levels with J π The calculated yrast 8 + and 10 + excitation energies are slightly higher than the experimental ones, which is a general feature of this type of model. 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. The results of the calculation for the J π = 2 + 2, 3+ 1, 4+ 2, 5+ 1, 6+ 2, 7+ 1, and 8+ 2 states are compared with the experimental data for Ru nuclei in Fig. 3. Most of the calculated energy levels also have a good agreement with experiment. The experimental data for energy level almost exactly match the calculated result. Spectrum of the quasi-gamma band shows staggering effect, that means the energy levels in that gamma-band are not evenly spaced, the and 4+ 2 are close together, the and 6+ 2 are close together, and a large space exists between them.[34] This event occurs in the Ru isotopes given in Fig. 3. The calculated energy levels of the and 8+ 2 are lower than the experimental data. The predicted energy of the state has found slightly higher than experimental one. This is a result of the presence of a Majorana term M πν in the Hamiltonian. Majorana interaction pushes up states of less proton-neutron symmetry relative to the states of maximal symmetry. We compared the J π = 0 + 2, 2+ 3, 0+ 3, 2+ 4, and 4+ 3 states between calculations and experimental data in Fig. 4. The energy spectra given in Fig. 4 can show a first standard for classifying the intruder states. In most deformed nuclei the excitation energy of the β band, 0 + 2, is higher than the γ band. The occurrence of the low-lying 0+ 2 states in the spectrum points to a complex dynamical structure, which is generally associated with the shape coexistence phenomenon. The spherical and deformed states should be mixed in order to account for the lowering of the 0+ 1 state within the ground state band. However, no conclusion can be drawn from the energies alone, since it is very likely that both intruder and collective states will occur in the same energy region. When comparing Fig. 3 and Fig. 4 for Ru 54 58, the lies below 2+ 2 suggesting shape coexistence, while in Ru the lies above 2+ 2 state. It has to be emphasized that in a number of cases, more firm conclusions can be determined from electromagnetic properties. Since the energy levels are not as sensitive as the B(E2) values and B(E2) ratios to the shape parameters, a comparison of the experimental and calculated B(E2) s and B(E2) ratios will give us better insight into the shape and nature of the shape transition for the ruthenium isotopes. We calculated the electromagnetic properties of the Ru isotopes using Eq. (6). Experimental data are taken from Refs. [22] and [33]. To calculate the E2 transition probabilities for intraband and interband we did not introduce any new parameters except for proton- and neutron-boson effective charges. The values which we used for neutron-boson effective charges and proton-boson effective charges are fixed

4 700 A.H. Yilmaz and M. Kuruoglu Vol eb and 0.14 eb, respectively. The value of e π was determined from the experimental B(E2; ) in Ru nuclei. The calculated and experimental B(E2) transition strengths are shown in Fig. 5. In this figure B(E2; ), B(E2; ), B(E2; ) and B(E2; ) values are also given, which are of the same order magnitude and display a typical increase towards the middle of the shell. It is found that there is no experimental data for Ru nuclei, so it is not possible to say that there could be a decreasing trend of those values. In the even even Ru isotopes, there is a good agreement between experimental and calculated B(E2) values. Fig. 4 Comparison between experimental (symbols) and calculated (lines) energies for even-even Ru nuclei in the side bands. Fig. 5 Relation between several B(E2) values and neutron numbers. Symbols denote experiments and lines calculations. Fig. 6 Quadrupole moments. Data of a, b, c, and d are taken from Refs. [35] [37] and [11], respectively. Fig. 7 The magnetic dipole moment of the first excited state for the ruthenium isotopes. Experimental data are taken from Ref. [37]. In order to conclude on the E2 properties, the results for the quadrupole moments Q 2 + of the first excited 2 + state 1 in Ru isotopes are given as a function of neutron number in Fig. 6. The calculated Q 2 + s are found consistent with 1 the experimental ones both in sign and in magnitude. A very peculiar feature of the IBM-2 is the sudden change in the quadrupole moment around neutron numbers 62 and 66, and the occurrence of higher quadrupole moments in the upper half of the shell. However, it is clear that there is no experimental data to confirm this case. In order to examine B(M1) ratios and the magnetic dipole moment of state we employed Eq. (6). For the parameters in the M1 operator the values of g ν = 0.15µ N and g π = 0.7µ N are used. We showed the B(M1) results in Table 2. It is seen that there is a good agreement between experimental and calculated ones in IBM-2. The magnetic dipole moment of the first excited state for the ruthenium isotopes are given in Fig. 7. It is seen that a very good agreement among the values is obtained.

5 No. 4 Investigation of Even-Even Ru Isotopes in Interacting Boson Model Table 2 B(M1) values for some ruthenium isotopes. Experimental data a, b, and c are taken from Refs. [31], [32], and [6], respectively Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal. Exp. Cal. 96 Ru a Ru a Ru b < Ru c Although the lowest levels of even even nuclei are well known signatures of structure, there has been little study of their energy relationships over the entire nuclear chart. Of course, it is well known that E(2 + 1 ) and E(4+ 1 )decrease through the course of a vibrator rotor shape transition. Also, the ratio R 4/2 = E(4 + 1 )/E(2+ 1 ) increases from near doubly magic nuclei to 2.0 in vibrational nuclei, in transitional species, and 3.33 in welldeformed symmetric rotor nuclei. In order to classify even-even Ru-nuclei, a behavior of E(4 + 1 )/E(2+ 1 ) to neutron number is plotted. In Fig. 8, one sees that the energy ratios of E(4 + 1 )/E(2+ 1 ) indicate transitional species for these isotopes. The agreement between the experimental values and the calculated ones is very good. The variation of both the experimental and the calculated ratios is rather small. We showed the squared F -spin amplitudes of 2 + 1, 2+ 2, 2+ 3, 3+ 1, and 4+ 1 states of even 96 Ru 116 Ru nuclei in Figs One can see that while 2 + 1, 2+ 2, and 4+ 1 eigenstates are strongly dominated by the F = F max component, the strongest contribution to the and 3+ 1 states are the one with F = F max 1. Since the probability for the F = F max component is at least 0.6, one can conclude the and 3+ 1 states as having mixed symmetry in 96, Ru and 96,100, Ru, respectively. The 2 + 1, 2+ 2, and 4+ 1 eigenstates show a rather pure fully symmetric structure all along the isotopic chain, since the amount of the squared amplitudes of its F = F max component is Fig. 8 Relation between E(4 + 1 )/E(2+ 1 ) and neutron number. Fig. 9 F -spin amplitudes of 2+ 1 levels. Fig. 10 F -spin amplitudes of levels. Fig. 11 F -spin amplitudes of 2+ 3 levels. The multipole mixing ratio, (E2/M 1), of ruthenium isotopes was calculated. Comparison between experimental and calculated values for this quantity is given in Table 3. The results are very agreed with the experimental data.

6 702 A.H. Yilmaz and M. Kuruoglu Vol. 46 Table 3 Multipole mixing ratio for the ruthenium nuclei. See Ref. [33] exp. cal. exp. cal. exp. cal. exp. cal. 96 Ru Ru < Ru Ru Ru Ru Fig. 12 F -spin amplitudes of levels. Fig. 13 F -spin amplitudes of 4+ 1 levels. 6 Conclusions In this work the results of a study for even-even Ru isotopes are given in the framework of the interacting boson model-2. The results of this work showed that the IBM-2 provides a good description for even-even Ru isotopes. The parameters used in the computation are chosen in the right way to get perfect fit. The calculated excitation energies and the experimental ones are in good agreement. The calculated yrast and excitation energies are slightly higher than the experimental ones, which is a general feature of this type of model. Because of a Majorana term in the Hamiltonian, energy state is obtained slightly higher than experimental value. It is clear that in the middle of the shell all the energy levels drops. The B(E2) and B(M 1) transition probabilities are well reproduced by calculation. Q 2 + is not a monotonic function of the mass number. Q depends sensitively on κ and χ ν as well as on 1 boson number in ruthenium isotopes. Investigation of E(4 + 1 )/E(2+ 1 ) to neutron number state that a localization of the Ru nuclei is in the transitional limit. Magnetic dipole moments µ of the ruthenium nuclei are also found in the framework of the IBM-2. We see and 3+ 1 states as having mixed symmetry. We have also calculated (E2/M1) multipole mixing ratio for the ruthenium isotopes where the data are available. The results are agreement with the experimental values. References [1] F. Iachello and A. Arima, The Interacting Boson Model, Cambridge University Press, Cambridge (1987). [2] R.F. Casten and D.D. Warner, Rev. Mod. Phys. 60 (1988) 389. [3] A. Arima and F. Iachello, The Interacting Boson Model, in Advanced in Nuclear Physics 13 (1984) 139. [4] A. Arima, T. Otsuka, F. Iachello, and I. Talmi, Phys. Lett. B 66 (1977) 205. [5] P. van Isacker and G. Puddu, Nucl. Phys. A 348 (1980) 125. [6] A. Giannatiempo, A. Nannini, P. Sona, and D. Cutoiu, Phys. Rev. C 52 (1995) [7] A. Giannatiempo, A. Nannini, and P. Sona, Phys. Rev. C 58 (1998) [8] K. Zajac, L. Prochiniak, K. Pomorski, S.G. Rohozinski, and J. Srebrny, Acta Phys. Pol. B 30 (1999) 765. [9] K. Zajaç, L. Prochiniak, K. Pomorski, S.G. Rohozinski, and J. Srebrny, Nucl. Phys. A 653 (1999) 71. [10] J.L.M. Duarte, Borello-Lewin, G. Maino, and L. Zuffi, Phys. Rev. C 57 (1998) [11] A.J. Singh and P.K. Raina, Phys. Rev. C 53 (1996) [12] J. Kotila and J. Suhonen, Phys. Rev. C 68 (2003)

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