The formalism of the neutron-photon interacting boson model, extended to include two-quasiparticle excitations, is used in a study of the high spin states of 154,156,158Dy isotropes. Good agreement is obtained between the calculated energies and transition rates and the experimental results

Alonso Alonso, Clara Eugenia

Arias Carrasco, José Miguel

Lozano Leyva, Manuel Luis

Physics Letters B

English

idUS. Depósito de Investigación Universidad de Sevilla

Volume 177, number 2 PHYSICS LETTERS B 11 September 1986 BACKBENDING OF Dy I S O T O P E S  D E S C R I B E D  W I T H  T H E  N E U T R O N - P R O T O N  IBA P L U S  T W O - Q U A S I P A R T I C L E  M O D E L  * C.E. ALONSO, J.M. ARIAS and M. LOZANO Departamento de Fisica A tdrnica v Nuclear Facuhad de Fisica, Universidad de Sevilla, Apdo 1065. 41080 Sevilla, Spain Received 18 April 1986; revised manuscri 3t received 30 June 1986 The formalism of the neutron-proton interacting boson model, extended to include two-quasiparticle excitations, is used in a study of the high spin states of ~,4 ~ ,  tS~Dy isotopes. Good agreement is obtained between the calculated energies and transition rates and the experimental results. The basic feature of  the structure of  the yrast line of  deformed heavy nuclei can be described as an interplay between collective rotations and single- particle excitations. For even-even nuclei the ground state band is due to the collective rotation of  the nucleus around an axis perpendicular to the symme- try axis, being the moment  o f  inertia approximately half of  that corresponding to a rigid rotor because of  the pairing correlations. When the angular velocity increases the Coriolis force tends to break pairs of  nucleons moving in high-j intruder orbits close to the Fermi surface which align their angular momenta  along the rotational axis. Each of  these pairs pro- vides a part of  the total angular momentum and the nucleus rotates at a lower rotational frequency. In this rotational alignment picture [1] the anomalies occurring in the yrast sequence of  deformed nuclei are interpreted as being due to the intersection of  the paired ground state band with a band involving one aligned quasipanicle pair (first backbending), this one with a band involving two pairs (second backbend- ing), etc. The most legitimate method to describe these phe- nomena seems to be the Har t ree-Fock-Bogol iubov theory in the cranking framework [2,3]. From a phe- nomenological point of  view, the decoupling model [4-7] has been extended in order to include two ~" Work partially supported by the Spanish Comisi6n Asesora de lnvestigaci6n Cientifica y T6cnica, contract number 2868-83. quasiparticle states in even-even [8] and o d d - o d d  [9] mass nuclei. Another phenomenological approach is the includion of  two quasipanicle excitations in the interacting boson model [10]. This was done by Gelberg and Zemel [ 11 ] in a schematic model, which, due to the "ad hoc" nature of  its hamiltonian, is not theoretically satisfactory. Using a model proposed by Yoshida et al. [12] high spin states have been described by allowing to change one o f  the bosons of  the IBA-2 model into a pair of  nucleons [13,14]. In the same approach Morrison et al. [5] have given another formulation for the inclusion of  two-quasi- panicle states in a boson basis, based on the philos- ophy of  the IBFA model [16]. In the same line one can include some recent works by Faessler et al. [ 17]. In this paper we describe the high spin states of  three Dy isotopes within the IBA plus two-quasipar- ticle model of  Yoshida et al. [12]. It is worth noting that these Dy isotopes have been studied recently within an alternative extension of  the IBA model [18]. In that model, called core-excited interacting boson mode, a nucleus is considered as a system with no constant number  of  bosons. The backbending phe- nomenon is produced by the crossing between the ground state band and another band with a larger number  of  bosons. The core-excited IBM overem- phasizes the role of  the collective degrees of  freedom, thus a calculation with a more realistic model which takes into account the interplay between single-par- ticle and collective degrees of  freedom (as the one of  0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) Volume 177, number 2 ref. [ 12]) is desirable. Since this model has been presented and discussed in some of the above references we summarize here its main features. The neutron-proton IBA model yields a good description of the low-lying collective states in even-even nuclei with a set of parameters phenom- enologically fitted to the experimental data. In order to describe levels of  high spin and excitation energy it is assumed that one proton-boson and/or one neu- tron-boson is broken to form a quasiparticle pair, usually assigned to a unique parity intruder orbital with spin j. The total angular momentum of the pair can take on the values J =  4, 6 ..... 2j-  1. The couplings to angular momenta 0 and 2 are excluded in order to avoid double counting of states, because they are included through the s and d bosons respectively. The model space includes the IBA-2 space, i.e., states with N. proton-bosons and N, neutron-bosons, and states with ( N , , ~ , - 1 )  proton (neutron)-bosons,  N,,,~ neutron (proton)-bosons  and two protons (neu- trons). The mixing between states with different number of bosons is introduced through a boson- number-changing term in the quadrupole operator: Qp =Q~ + O ~ p ( a ~ a j )  ' 2 '  +t ip [  ( a } a ~ ) ( 4 ' f f p l ( 2 )  -fl,,[d~p(~j~,)'4'] '2 ' ,  p = . ,  z~ (1) where B ~ t Qa =( ~sp + sjT/,) ,2, + ; G ( ~ , ) , 2 ,  (2) with a~ -- a*,jm) being the nucleon creation operator (ciom)= ( - 1 )J "a, j_, , ) )  and, s* and d* the s and d boson creation operators. The model hamiltonian is H=H~+HF+ Vav, (3) where HB is the IBA-2 hamiltonian [ 19 ] and Hv the fermion hamiltonian that, restricted to a single j-shell, takes the form __ 3" 1 HF--ZEjaj,,,aJ,,,+~ Z VJ(a~a~)Z"(ctfi;) sM. (4) m J,,~l For the interaction between bosons and fermions, VnF, a quadrupole-quadrupole interaction is assumed: , B VBV=~C ( Q~Q~-Q,'Qff) , (5) where Qp is the generalized quadrupole operator given by eq. (1), and Q;a is the boson quadrupole operator PHYSICS LETTERS B 11 September 1986 ofeq. (2). The Dy isotopes we are intersted in have Z =  66 and 88 ~< N~< 92. Thus,valence protons are the 50-82 major shell and valence neutrons in the 82-126 major shell. The orbitals with larger angular momentum in these shells are l~hjj/2 and 1vi13/2 for protons and neutrons respectively. In addition to these ones the 1 vhg/2 orbital is very close to the Fermi surface in all the isotopes studied and it may be important in the analysis of  the backbending phenomena in these nuclei. Because of that, we have included in our study both orbits for neutrons: lVil3/2 and lvhg/2 and the lgh~ ~/2 for protons. First of all, a description of the even-even core- nuclei in terms of the IBA-2 model is needed. For instance, the study of the ~54Dy implies the IBA-2 description of 154Dy, ~52Dy and 152Gd, since the model space is given in this case by, I ~54Dy)® I ~52Dy+2 neutrons) @ I ~ 52Gd + 2 protons) . In table 1, the IBA-2 parameters used in this study for the description of the even-even nuclei implied are displayed. For 66Dy isotopes they have been obtained by extrapolating those given in ref. [ 19] for 60Nd, 625m and 64Gd. For isotopes they have been taken from the same reference and slightly read- justed in order to improve the description of the low- est states. For the two-body interaction between fermions (in 1 vh9/2, 1 vi~ 3/2 and 1 ~rht ~/2) a surface delta interaction (SDI) has been used with strength adjusted to lower the J =  0 state by 2 MeV. The sin- gle-particle energies obtained as a result of fitting (fig. 1) coincide approximately with the quasiparticle energies obtained from a standard BCS calculation for the orbitals of interest. The additional parame- ters, in VBF (eq. (5)), have been taken constant for the three isotopes: a~ = a~ = - 0.4 and flu =fl~ = 0.5, or varying smoothly from isotope to isotope: x ' =  0.04, 0.05, 0.06 MeV for N =  88, 90, 92 respectively. For the 66Dy isotopes there is also experimental information on B(E2)'s. The study of these values will give us a good test of the model wave functions. The electric quadrupole operator is written as T~E2'= Z {e,~Q;B, +e,Z,°~p(°~*Jct,) '2' p ~,V +fl,,~t(a~a~)'4'~,] '2' - fl/,e/B [ d~p (a;aj)(4'] (2) } , (6) where e~ ( ~ )  is the proton-boson (neutron-boson) effective charge and v z e, (e,) is proportional to the 131 Volume 177, number 2 PHYSICS LETTERS B 11 September 1986 Table 1 Parameters ~, ~c in the IBA-2 hamiltonian (in MeV). All the other parameters in the IBA-2 hamiltonian are taken the same as in ref. [ 19] Nucleus Parameter N 86 88 90 92 ~6Dy e 0.717 0.620 0.592 ~c -0 .028 -0 .055 -0 .073 ,4Gd ~ 0.596 0.550 ~c -0.051 0.080 0.550 -0 .070  0.460 -0.081 2.0 2>1.0 v 0.0 I I I I I t 88 90 92. neutron number Fig. 1. Quasiparticle energies used in H~ for the single-panicle orbits included in this study. proton (neutron) effective charge. The boson effective charges in the 7 ~E2~ operator have been determined by adjusting the experimental value of B(E2; 2 ~ 0 { )  for each isotope. For the charges e~ and e~ in eq. (6) we have used e,Z=0.53 e b and e~ = 0.21 e b. These values have been obtained by using harmonic oscillator radial wave functions and proton (neutron) effective charge equal to 1.5 e (0.5 e). The level structure of the J54Dy isotope suggests a small prolate deformation [20] although there are predictions of a prolate-to-oblate transition above spin 40h [21,22]. In fig. 2a we present the results of the calculations in comparison with the experimen- tal levels of Pakkanen et al. [20]. The configuration responsible for the backbending in this case in our calculation is 152Gd®(7~hll/2) 2, being that of two quasineutrons considerably higher in energy. Our study yields a g-factor of the states around the back- bending higher than the collective g-factor due to the contribution of the two protons. The transition prob- abilities B(E2) calculated are presented in fig. 3a along with the experimental data [20]. The backbending in 156Dy is commonly inter- preted as the transition from the ground state rota- tional band to the aligned two-quasiparticle i~3/2- neutron band [23], Fig. 2b shows the result of the study done using IBA-2 plus two quasiparticles. The configurations including two-quasineutrons are lower in energy. It is important to notice that both config- urations ~ 54Dy@ (1 Ph9/2) 2 and 154Dy ® (1 vii 3/2)2 are competitive at the point of the first backbending, being that of two neutrons lh9/2 a bit lower in energy. Theoretical and experimental [24] B(E2) values are shown in fig. 3b. Our calculation follows nicely the sudden fall for I =  16fi and the rising for I =  18h. The interpretation of the third experimental point in fig. 3b ( I=  6fi) is difficult in the framework of the inter- play between two rotational bands, since the excita- tion energy of the first 6 + level is only 770 keV. As can be seen in fig. 3b we are not able to reproduce such a point. In our study an additional band cross- ing between the band with two quasineutrons and that of two quasiprotons is observed at spin 24h. This band crossing is not seen in the plot of E(1)-E(I-2), but it is clear in the behavior of the B(E2). The exper- imental data for B(E2) seem to indicate such a low- ering at I=24h.  In our calculation, in the crossing point, the bands have almost no mixing between them and therefore the jump is too exaggerated. A small amount of mixing would contribute to improve the agreement. This result is in agreement to one meas- urement of g-factors for this isotope [25]. In this experiment there is a clear negative contribution to the collective g-factor at I -  14,5 which is due to the alignment of two quasineutrons and a positive con- 132 Volume 177, number 2 PHYSICS LETTERS B 11 September 1986 900 > ~600 i W i ~ 3 0 0  ,154Dy' , , i //''' 1 , '~,1 , .//"I , .,.I 11156Dyl 1 ' I ' ,''---"1 ' "~'  .l"" I/./.~ ,.I _ a b I , l t [ I I I , I , I i I t I , I , I J I , 2 6 10 14 18 22 26 2 6 10 14 18 22 26 I I l I ' I ' I ' I ' I ' I I 158Dy / / s t I ~ I , I i 1 i I J I I I 2 6 10 14 18 22 26 Fig. 2. Comparison of the experimental energy differences in the yrast line (full line) with the results of  the present study (dashed line) for: (a) ~ D y ,  (b) 156Dy and (c) ~58Dy. I ' I ' I ' I ' I ' I ' I 154Dy o 1.6 0.8 0,0 , I I I , i ' ,  ~ : ' , :  ~ : : ~  -156Dy T b  1"6 i ~ 0.8 ¢'M W ~'0.0 . . . . . . . . . . . . .  1.6 _ 1 5 ~  0.8 C - 0 . 0  1 , I , r , , I ~ , i 2 6 10 14 18 22 26 I Fig. 3. Comparison of the experimental B(E2) values for transi- tions in the yrasi line (points) with the results of the present study (full line) for: (a) 154Dy, (b) I~"Dy and (c) 15~Dy. tribution to it at I=  24h due to the alignment of a couple of quasiprotons. It is important to note that this experiment is not conclusive about the back- bending nature since the (HI, xn) reaction employed populates states on and above the yrast line. Because of that, a new measurement of g-factors is desirable in order to clarify if there is an additional band cross- ing at around I=24h, as the experiment of ref. [25] and our calculation indicate. The calculated and experimental [26,27] energy differences and B(E2) values for ~58Dy are presented in figs. 2c and 3c. In the case of this isotope the con- figuration responsible for the backbending is known, since a measurement of the average g-factors in this region exists [26] and demonstrates that the align- ment of an i~3/2-neutron pair is the dominant effect in the intrinsic structure of the yrast band around the backbending region. In our calculation the configu- rations with two quasineutrons are considerably lower than that with two quasiprotons, being the responsible for the first backbending, in agreement with the results of ref. [27]. In this case the configu- ration with two neutrons li~3/2 is the most important in the backbending region, due basically to the behavior of the quasiparticle energies (see fig. 1). Because of that, this configuration is expected to be the basic one for the study of heavier isotopes. In fig. 3c the B(E2) values for this isotope are displayed along with the experimental data [26,27]. The basic 133 Volume 177, number 2 PHYSICS LETTERS B 11 September 1986 b e h a v i o r  of E(1)-E(l-2) and  B(E2)  versus  I is well  r ep roduced .  S o m e  data  on side bands  are also known  for these  isotopes.  T h e i r  b e h a v i o r  is reasonably  well  repro-  duced  in our  study. T h e  analysis  o f  these  bands  together  wi th  a c o m p a r i s o n  be tween  our  resul ts  and  those  o f  ref. [ 18] will  be  p resen ted  in deta i l  in a for th-  c o m i n g  paper .  In conclus ion ,  we show that  the IBA-2 plus two  quas ipar t ic le  m o d e l  is succesful  in r ep roduc ing  the  ma in  features  o f  the energy spectra  and  electr ic  q u a d r u p o l e  t rans i t ion  rates in the region o f  the  first backbend ing  p h e n o m e n o n  in the t rans i t iona l  iso- topes  ~ 54Dy, 1S6Dy and  ~ 5SDy. It is also a resul t  o f  the present  analysis  tha t  the respons ib le  for  the  back-  bend ing  in these nucle i  is a two-p ro ton  quas ipar t i c le  conf igura t ion  for 154Dy, and  the  a l i gnmen t  o f  two quas ineu t rons  for  ~56"~ 58Dy" In  ~56Dy an add i t i ona l  band crossing with  a two-pro ton  conf igura t ion  at spin 24h is d e d u c e d  f rom the p resen t  analysis.  Exper i -  men ta l  m e a s u r e m e n t s  o f  g-factors  for  ~54,~56Dy are highly des i rable  in o rde r  to k n o w  the  na tu re  o f  the  yrast  l ine at h igh spin in these nuclei .  The  au thors  acknowledge  helpful  d i scuss ions  wi th  F. Iachel lo.  References [l] R.M. Lieder and H. Ryde, Advances in nuclear physics, Vol.10, M. Baranger and E. Vogt (Plenum, New York, 1978) p.1. [2] H.J. Mang, Phys. Rep. 18 (1975) 325. [3] M. Diebel, A.N. Mantri and U. Mosel, Nucl. Phys. A 345 (1980) 72. [4] F.S. Stephens, R.M. Diamond, J.R. Leigh, T. Kammuri and K. Nakai, Phys. Rev. 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Backbending of Dy isotopes described with the neutron-photon IBA plus two-quasiparticle model

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Backbending of Dy isotopes described with the neutron-photon IBA plus two-quasiparticle model

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