A calculation of low-lying collective states in odd-even nuclei

Volume 144B, number 3,4 PHYSICS LETTERS 30 August 1984 A CALCULATION OF LOW-LYING COLLECTIVE STATES IN ODD-EVEN NUCLEI C.E. ALONSO, J.M. ARIAS Departamento de Fisica Atomica y Nuclear, Universidad de Sevilla, Spain R. BIJKER Kernfysisch Versneller Instituut, Ri]ksuniversiteit Groningen, The Netherlands and F. IACHELLO A.W. Wright Nuclear Structure Laboratory, Yale University, New Haven, CT 06520, USA Received 19 April 1984 Revised manuscript received 4 June 1984 We present results of a calculation of properties of low-lying collective quadrupole states in odd-even nuclei within the framework of the proton-neutron interacting boson-fermion model. The importance of proton-neutron degrees of freedom in low-lying collective states of nuclei has been emphasized in recent years by the interacting boson model [ 1 ]. In addition to introducing specific proton -neutron effects, such as the occurrence ofK P = 1 + bands in deformed nuclei [2], this model has the advantage that, being closely related to the microscopic shell model, it allows one to make extrapolations and predictions for properties of nuclei hitherto unknown. The next logical step in the direction of a detailed understanding of nuclear properties is that of performing similar proton-neutron calculations for odd-even nuclei. Because of the large number of low-lying states, these calculations present a major challenge. Previous calculations of odd-even nuclei within the framework of the interacting boson model have been done with the version of the model in which no distinction is made between proton and neutron degrees of freedom [3]. In this letter, we present the results of the first systematic calculations performed using the protonneutron interacting boson-fermion model. The corresponding computer program was originally written by Otsuka and Yoshida [4], and applied to the study of 79Rb and 79Kr in ref. [5]. The calculations presented in this article are based on an improved version of this program obtained by one of us (R.B.), together with B. Visscher [6]. The improvement consists of a prediagonalization and truncation of the boson spectrum before coupling the odd fermion. In the interacting boson-fermion model, spectra of odd-even nuclei are calculated by coupling the collective degrees of freedom, described by bosons, to the single-particle degrees of freedom (fermions). The hamiltonian is written as H = H (B) + H (v) + V (BE) , (1) where H (B) is the proton-neutron interacting boson hamiltonian [ 1 ], H (F) is the hamiltonian describing the single-particles degrees of freedom, and V(BF) is their interaction. The structure of the boson-fermion interaction is, in general, rather complex since the bosons are composite rather than fundamental particles and thus one needs to take into account the effects of the Pauli principle. A purely phenomenological treatment of this interaction is not possible, since it contains a large number of parameters. One must therefore rely on a microscopic derivation. Several of these have been given [7-10]. In the calculations pre141 Volume 144B, number 3,4 PHYSICS LETTERS 30 August 1984 sented here, we have used the following form v(BF) = + &,F v + Av (rid "n~), (2) for odd neutron nuclei, and V (BF) = P~r(Qv'qTr) + A,rFv, r + A,r(ndv'nTr ) , (3) for odd proton nuclei. The first terms in eqs. (2) and (3) represent a quadrupole--quadrupole interaction between bosons and fermions, Qp = {(d I" X "~ + s t X 2) (2) + x(d ~f X d)(2))o , qp={~//,QLi,(t~ul.,-t)t~,)(alt. XEj,)(2)}, p ='n, v p (4) where we have used the usual notation for boson and fermion creation and annihilation operators. The second term, representing the effects of the Pauli principle (exchange interaction), has been taken in this article to be [7] F~v =- {(sI" X d)(2)}Tr'{//.,~/, ' (10/N)1/2(2]+ 1) -1/2 × Qjr (,,iS, + 5./)Qi,,/,,/,5 + 5,,u) "t x : [(a* x x (; x 5t,)(r)] (2):/ + hermitean conjugate. (5) while Fu~ r is obtained from eq. (5) by interchanging the indices lr and v. Other, more complex forms, could be used, if needed. The third term represents a monopole-monopole interaction ndp t m Jo ' n o (2] + 1) "]m"]m ' p = rr, v. (6) P The microscopic structure of the bosons enters in eqs. (4)-(5) through the coefficients [11 ] (2i/' = (l { ill g(2)l[l' t "' I ), (7) where the o~ are the structure coefficients of the Spairs and N is the number of pairs. Finally, one needs the fermion hamiltonian, H (F). When only one odd particle is present, it is sufficient to consider the single-particle part of H (F), n (F) = ~E.a ~f a (8) jm I ]m jm ' where the E/are the single particle energies in the presence of N pairs. In the calculations performed so far, the coefficients uj, o/and the energies E] have been obtained using the BCS approximation E/= [(6/. - X) 2 + A 21 1/2, v2 = { [1 - (e/- ~)/E/] , (9) rather than the full generalized seniority scheme. In conclusion, according to the model described here, spectra of odd-even nuclei can be calculated in terms of three parameters Pv, Av, Av (or FTr, An, A~). In order to study specifically proton-neutron effects, we have applied this model to odd-neutron Xe nuclei and odd-proton Cs nuclei in the 50-82 major shell. Both isotopic chains have the same even-even core. Thus, differences in the spectra can arise only from the boson-fermion interaction V (BF) and the single-particle hamiltonian, H (F). The boson hamiltonian, H (B) was taken from a previous calculation [12]. The unperturbed single-particle energies, ej, were taken as in table 1. The values shown in this table were extracted from ref. [13] for proton levels and from the experimental spectrum of 131~ for neutron levels. In the 50bn81 major whell 50-82 there are 5 single-particle levels, 4 with positive parity (1 g7/2, 2 d5/2, 3 Sl/2 and 2 d3/2) and 1 with negative parity (1 hll/2 ). Since they do not mix, calculations can be done independently for each parity. In this letter, we report only results for negative parity states. Those for positive parity states will Table 1 Unperturbed single particle energies (MeV) used in the present calculation. Level lg7/2 2ds/2 lh11/2 3Sl/2 2d3/2 proton levels 0 0.60 1.50 3.35 3.00 neutron levels 0 0.80 2.00 2.10 2.50 142 Volume 144B, number 3,4 PHYSICS LETTERS 30 August 1984 Table 2 Boson-fermion interaction parameters for negative parity states (MeV). F A A protons 0.6 1.3 -0.3 neutrons 0.6 1.3 0 be presented in a longer publication [14]. The values of the parameters used in the calculation are given in table 2. Fig. 1 shows the results for the Xe isotopes, while fig. 2 shows the results for the Cs isotopes. The proton-neutron effects are clearly displayed in these figures. In cesium, the ~rhll/2 level is almost complete2 ly empty (Vll/2 ~ 0). As a result, the exchange interE I I 23/'2I 54Xe i i i 54 xe I I (MeV) 54Xe 5 F23/227/2- / T21/~\ 25/225/2- , , L ~. ~'~/~ ~ ~ ~"~ 2'/~ I --~I9/2 1,9/=- ~ /,9/21 \ ,~,~v /,7,2- ,5/~- I 1:5/2- - 11/2- /,5/2- ~ 21 ---...~. /,3/2 v /o7~- / ~ / 9/2-~ {~AA 9/27'2ooo ,50 610 710 8&150 ---6J0 710 810 50 610 10 80 Neutron Number Fig. 1. Calculated negative parity spectra in the odd-A Xe isotopes. The state JP = 11/2is taken as zero of the energy. The experimental values (circles, squares, ...) are from ref. [ 15 ]. E i -h i i i i (MeV) 55Cs 55Cs 3 L _ 27/223/225z2: /25/2 _ 27/2, / / 21/2- ~ _ i25/2- \ ~ / ~ 21/2 2! 2~9 2~ 7/29/ /~ 17/ I ~ [] r~G4~"~ [3 15/~ i 1512- ~ 9/2-- ._ ~ /~ ~ _ 15/2~ AA3F~A z~ . 9/2 "~2::::2~::"~"~^ 3/2o!,,,2_ o~~4,,,2 l ' I I i/ _ I I i 55 cS J d i ,,% 7/2~ ~/2i I 50 60 70 80 50 60 70 80 50 60 70 8~0 Neutron Number Fig. 2. Calculated negative parity spectra in the odd-A Cs isotopes. The state JP = 11/2is taken as zero of the energy. The experimental values (circles, squares, ...) are from ref. [16]. 143 Volume 144B, number 3,4 PHYSICS LETTERS 30 August 1984 action vanishes, and the spectra are dominated by the quadrupole interaction. On the contrary, in Xenon, the vh 11/2 level is empty for smaller neutron numbers (~ 54)but becomes increasingly populated foriarger neutron numbers (~ 76). A consequence of this is that the state with jt" = 9/2becomes lower and lower in excitation energy and it crosses the state with JP = 11/2at neutron numbers -~ 66. The results shown in figs. 1 and 2 indicate that it is possible to provide a unified description of long odd-A isotopic chains in terms of few parameters, although a closer inspection reveals that discrepancies between calculations and experiment still remain. Particularly notable is that concerning the location of the lowest JP = 7/2state in Xe. This discrepancy was also present in the calculations performed within the framework of the interacting boson-fermion model without distinction between proton and neutron degrees of freedom [ 17], and is due to the restriction of the single-particle negative parity space to lh/11/2Introduction of the 2f7/2 and lh9/2 levels eliminates the difference between calculated and experimental states [17]. The same conclusions can be drawn by analyzing the results of the calculations of the positive parity states and of electromagnetic transition rates. It appears thus that we have now at our disposal a tool capable of analyzing properties of odd-even nuclei that include isospin effects. The scope and limitations of this tool remain to be seen and will be determined by a comparison between calculations and experiment in other isotopic chains. We thank A.E.L. Dieperink for discussions. One of us (F.I.) thanks R. Leonardi for his hospitality at the University of Trento where this article was completed. This work was in part supported by the Stichting FOM which receives financial support from the "Stichting Voor Zuiver-Wetenschappelijk Onderzoek" and in part by the Department of Energy Contract No. DE-AC0276ER 03074. References [2] F. lachello, Nucl. Phys. A358 (1981) 89; A.E.L. Dieperink, Prog. Part. Nucl. Phys. 9 (1983) 121 ; A. Richter, Proc. Intern. Conf. on Nuclear physics (Florence, Italy, 1983). [3] F. lachello and O. Scholten, Phys. Rev. Lett. 43 (1979) 679. [4] T. Otsuka and N. Yoshida, Computer Program TWC, University of Tokyo, Japan; Yoshida, A. Arima and T. Otsuka, Phys. Lett. 114B (1982) 86. [5] J. Panqueva, H.P. Hellmeister, L. Ltihmann, F.J, Bergmeister and K.P. Lieb, Nucl. Phys. A389 (1982) 424. [6] B. Visscher and R. Bijker, Computer Program BOSON and ODD PAR, University of Groningen, The Netherlands. [7] I. Talmi, in: Interacting Bose-Fermi systems in nuclei, ed. F. lachello (Plenum, New York, 1981) p. 329. [8] O. Scholten and A.E.L. Dieperink, in: Interacting Bose -Fermi systems in nuclei, ed. F. Iachello (Plenum, New York, 1981) p. 343. [9] A. Gelberg, Z. Phys. A310 (1983) 117. [10] U. Kaup, Prog. Part. Nuclear Phys. 9 (1983) 561. [ 11] T. Otsuka and A. Arima, Phys. Lett. 77B (1978) 1. [ 12] G. Puddu, O. Scholten and T. Otsuka, Nucl. Phys. A348 (1980) 109. [13] P. Mukherjee, R. Bhattacharya and I. Mukherjee, Phys. Rev. C24 (1981) 1810. [14] C.E. Alonso, J.M. Arias and R. Bijker, to be published. [15] A. Kerek, A. Lunkko, M. Grecescu and J. Sztarkier, Nucl. Phys. A172 (1971) 603; I. Rezanka, A. Kerek, A. Luukko and C.J. Herrlander, Nucl. Phys. A141 (1970) 130; H. Helppi, J. Hattula and A. Luukko, Nucl. Phys. A332 (1979) 183; V. Bacri, A. Gizon, J. Crawford, J. Genevy, J. Gizon and A. Plochocki, Proc. Intern. Conf. on Nuclei far from stability (Helsing!br, Denmark, 1981) pp. 479,480; P. Chowdhury, U. Garg, T.P. Sjoreen and D.B. Fossan, Phys. Rev. C23 (1981) 733; A. Luukko, J. Hattula, H. Helppi, O. Knuuttila and F. D6nau, Nucl. Phys. A357 (1981) 319; H. Helppi, J. Hattula, A. Luukko, M. Jaaskelainen and F. D6nau, Nucl. Phys. A357 (1981) 333. [16] U. Garg, T.P. Sjoreen and D.B. Fossan, Phys. Rev. C19 (1979) 207,217; Ch. Droste, D. Chlebowska, J. Dobaczewski, F. D/Snau, A. Kerek, G. Leander, J. Srebrny and W. Walus, Nucl. Phys. A341 (1980) 98. [17] M.A. Cunningham, Nucl. Phys. A385 (1982) 204,221. [1] A. Arima, T. Otsuka, F. IacheUo and I. Talmi, Phys. Lett. 66B (1977) 205; T. Otsuka, A. Arima, F. lachello and I. Talmi, Phys. Lett. 76B (1978) 139. 144