Properties of superdeformed bands in A ∼ 190 region within the framework of three parameters nuclear softness model

Chinese Journal of Physics xxx (2016) 1e9

Contents lists available at ScienceDirect

Chinese Journal of Physics journal homepage: http://www.journals.elsevier.com/ chinese-journal-of-physics/

Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model A.M. Khalaf a, M.M. Sirag b, *, K.E. Abdelmageed c a

Department of Physics, Faculty of Science, Al-Azhar University, Cairo, Egypt Department of Physics, Faculty of Women, Ain Shams University, Cairo, Egypt c Department of Physics, Faculty of Science, Benha University, Benha, Egypt b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 February 2016 Accepted 15 March 2016 Available online xxx

The three parameters nuclear softness model (NS3) have been applied to eleven superdeformed (SD) bands of A z190 mass region namely: 191Hg(SD2,SD3), 192Hg(SD1), 194 Hg(SD1,SD2), 193Tl(SD1,SD2), 194Tl(SD1,SD3, SD5) and 194Pb(SD1). The level spins and the model parameters are determined by fitting the electric quadrupole transition energies Eg with the experimental ones, by using a computer simulated search program. Rotational frequencies ħu, kinematic J(1) and dynamic J(2)moments of inertia and the quadrupole deformation parameter b2 have been calculated. In all cases of our selected SD bands the results obtained are in good agreement with experimental ones. J(2) reveal a steady increase with ħu, this can be attributed to the successive alignment of pi13/2 (N ¼ 6) and yj15/ 2 (N ¼ 7) intruder orbitals in the presence of dynamical pairing correlations. The presence of DI ¼ 2 staggering in the two SD bands 1,2 of 194Hg and in the three SD bands 1,3,5 of 194 Tl have been investigated by calculating the staggering parameter which represent the fourth order derivative of the gamma-ray transition energies. Also the DI ¼ 1 staggering in SD bands of odd nuclei has been investigated in the signature partner pairs 191Hg (SD2, SD3) and 193Tl (SD1, SD2). The yrast SD bands in the isotones 194Pb and 192Hg have almost identical transition energies which implies that their J(2) are also identical. This indicates that either the two occupied 5/2þ[642] protons have a very weak influence on Eg and J(2). © 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

1. Introduction Recently the study of superdeformrd (SD) shapes in nuclei form an interesting research areas in nuclear spectroscopy both experimentally and theoretically. Such SD nuclei are associated with extremely large quadrupole deformations, typically b2 ¼ 0.6 that is located at a major to minor axis ratio equal to 2:1 in mass region A ~ 150 and b2 z 0.47 corresponding to ratio axes 1.7:1 in A ~ 190. Hence, they are expected to have a different structure to normal deformed nuclei. Over the past two decades, many SD nuclei have been observed across the nuclear periodic table in different mass regions A z 60, 80, 130, 150, 190 [1e3]. SD structures in A ~ 190 region possess many surprising features. One unexpected experimental discovery was identical bands (IB's) in SD nuclei [4e8]. The gamma transitional energies in the two yrast bands of 192Hg and 194 Pb are identical to within ±3 KeV [8]. This is a remarkable and unexpected phenomenon in nuclear structure physics. It is generally

* Corresponding author. E-mail addresses: [email protected] (M.M. Sirag), [email protected] (K.E. Abdelmageed). http://dx.doi.org/10.1016/j.cjph.2016.03.018 0577-9073/© 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

Please cite this article in press as: A.M. Khalaf et al., Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.03.018

2

A.M. Khalaf et al. / Chinese Journal of Physics xxx (2016) 1e9

believed that these SD bands can be described by the coupling between the nucleon single particle degrees of freedom and a highly deformed core. In mass region A ~ 190, the high N intruder orbitals such as i13/2 and j15/2 are expected to be important for describing the configurations of SD states. It has been found that the behavior of the dynamic moments of inertia for many SD near A ~ 190 are very similar to each other and show the same smooth rise with increasing the rotational frequency, because of the high e N intruder orbital configurations changes very lightly throughout the region [9]. Observation of a regular D I ¼ 2 staggering pattern of the transition energies in 194Hg (SD1, SD2) and 194Tl (SD1,SD3, SD5), where states differing by four unite of angular momentum show an energy shift relative to a smooth rotor sequence. The curve found by smoothly interpolating the band energy of the spin sequence I ¼ I0, I0 þ 4, I0 þ 8, … is somewhat displaced from the corresponding curve of the sequence I ¼ I0 þ2, I0 þ6, I0 þ10, …. The magnitude of the displacement is found to be in the range of some hundred eV to a few KeV [10e17]. Most of SD bands observed in odd e A nuclei in the mass region A ~ 190 are signature partner pairs, each pair show a large amplitude DI ¼ 1 staggering [16e22], and also the bandhead moments of inertia of each pair are almost identical. The gamma ray energies are the only spectroscopic information universally available, because the non-observation of the discrete linking transitions between the SD states and the low lying states at normal deformation. Despite the rather large amount of experimental information on SD bands and besides these more general features of SD bands near A ~ 190, there are still a number of problems which have not yet been solved. Among them are the spin, parity and absolute excitation energy relative to known normal deformed (ND) states. Many theoretical methods have been employed to assign the spin of the SD states [23e29]. The present work is an extension of a previous paper by this group [30]. In this work, we introduced the third stiffness parameter corresponding to the strength of potential energy. The paper is organized as follows: following this introduction in section 2 we develop the nuclear softness model NS2 of our previous papers [30] by introducing a third parameter, this three parameter, formula NS3 is used to predict the spins of the band heads and to examine the main properties of the SD bands. The appearance of identical bands (IB's) are examined in section 3. Section 4 is devoted to explore the DI ¼ 2 staggering. Section 5 concerns the origin of DI ¼ 1 staggering in signature partner pairs in odd-A SD bands. Quadrupole deformation parameter is examined in section 6. Numerical calculations are performed in section 7 for 11 SD bands in A ~ 190 region, discussions are also included. Finally conclusion and remarks are given in section 8. 2. Sketch of the model The energy levels of the ground state bands in deformed nuclei can be interpreted by the variable moment of inertia (VMI) model [31]. In this model the excitation energy for each state with angular momentum I contain a kinetic rotational term and a potential term

EðIÞ ¼

Z2 C IðI þ 1Þ þ ðJI  J0 Þ2 2 2JI

(1)

where: J0 is the ground state moment of inertia and C is the restoring force constant (stiffness parameter). The VMI JI as a function of I is determined from the equilibrium condition

vEðIÞ ¼0 vJI

(2)

which leads us to the expression

JI ¼

J0 1  IðIþ1Þ 2 C J3

(3)

I

An extension version of the VMI with the concept of nuclear softness is considered [32]. The nuclear softness parameter s is introduced by treating the variation of JI with I in a more generalized form by using Taylor series expansion of JI about J0.


 JI ¼ I0 1 þ s1 I þ s2 I 2 þ /

(4)

where, the softness parameter sn is given by

sn ¼

 1 vn Jn  n! JI vI n I¼0

(5)

Substituting from equation (4) into equation (1) to first order, one obtain Please cite this article in press as: A.M. Khalaf et al., Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.03.018

A.M. Khalaf et al. / Chinese Journal of Physics xxx (2016) 1e9

EðIÞ ¼

Z2 IðI þ 1Þ 1 2 22 CJ s I þ 2 0 2J0 ð1 þ sIÞ

3

(6)

Since the bandhead energy and spin are generally not known for SD bands, one may get the transition energy

 Eg ðIÞ ¼ EðIÞ  EðI  2Þ ¼ A

 IðI þ 1Þ ðI  2ÞðI  1Þ  þ 4BðI  1Þ 1 þ sI 1 þ sðI  2Þ

(7)

with A ¼ ħ2/2J0 and B ¼ ½ Cs2J20 To evaluate the softness parameter s, consider the three transitions

e1 ¼ Eg ðI þ 2 /IÞ ¼ EðI þ 2Þ  EðIÞ

(8)

e2 ¼ Eg ðI þ 4/IÞ ¼ EðI þ 4Þ  EðIÞ

(9)

e3 ¼ Eg ðI þ 6/IÞ ¼ EðI þ 6Þ  EðIÞ

(10)

Substituting for Eg from equations (6) and (7), yield

e1 ¼ 2A

ð2I þ 3Þ þ IðI þ 2Þs þ 4BðI þ 1Þ ½1 þ sðI þ 2Þ½1 þ sI

(11)

e2 ¼ 4A

ð2I þ 5Þ þ IðI þ 4Þs þ 8BðI þ 2Þ ½1 þ sðI þ 4Þ½1 þ sI

(12)

e3 ¼ 6A

ð2I þ 7Þ þ IðI þ 6Þs þ 12BðI þ 3Þ ½1 þ sðI þ 6Þ½1 þ sI

(13)

By eliminating A and B from equations (11)e(13), one get a cubic equation in s in the form

a0 þ a1 s þ a2 s2 þ a3 s3 ¼ 0

(14)

where the coefficients a‘s are given by

a0 ¼ 3e1 þ 3e2  e3


 a1 ¼ 9 I 2 þ 8I þ 12 e1 þ 3 3I 2 þ 20I þ 24 e2  3I 2 þ 16I þ 20 e3

(15) (16)




 a2 ¼  12I 3 þ 117I 2 þ 312I þ 204 e1 þ 12I 3 þ 117I 2 þ 524I þ 240 e2  4I 3 þ 39I 2 þ 112I þ 84 e3

(17)




 a3 ¼ I 3I 3 þ 36I 2 þ 132I þ 144 e1 þ I 3I 3 þ 36I 2 þ 132I þ 144 e2  I I 3 þ 12I 2 þ 44I þ 48 e3

(18)

Solving the cubic equation (14) yield s. Substitute with known s in equations (11) and (12) one can get the values of A and B as an initial values for the fitting procedure. 3. Identical SD bands It was discovered that some SD nuclei with different mass numbers have bands with identical gamma-ray transition energies [4e8]. This implies that they have nearly identical dynamical moment of inertia and rotational frequencies. 4. DI ¼ 2 staggering in gamma-ray energies It has been found that some SD bands show an unexpected DI ¼ 2 staggering in gamma e ray energies [10e17]. SD rotational sequences with nuclear spins differing by two may split into two branches and a small energy displacement occurs between the two sets. The spin values of the two branches are I, Iþ4, Iþ8, …. and Iþ2, Iþ6, Iþ10, …. respectively. This is commonly called DI ¼ 4 bifurcation, because the bands divide into two branches with levels differing in spin by four. The presence of this two regular DI ¼ 4 families in SD band suggests an explanation based on a fourfold state symmetry. To explore the DI ¼ 2 staggering, the deviation of the gamma-ray energies from a smooth reference D4Eg(I) was determined [14] by calculating the finite difference approximation of the fourth order derivative of the gamma-ray energies Eg(I) at a given spin I by Please cite this article in press as: A.M. Khalaf et al., Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.03.018

4

A.M. Khalaf et al. / Chinese Journal of Physics xxx (2016) 1e9

D4 Egref ðIÞ ¼

 1  Eg ðI  4Þ  4Eg ðI  2Þ þ 6Eg ðIÞ  4Eg ðI þ 2Þ þ Eg ðI þ 4Þ 16

(19)

This formula contains five consecutive transition energies and is denoted as five-point formula and was previously used by Cedrwall [12]. We define the staggering parameter S(4)(I) as the difference between the experimental transition energies and the auxiliary reference

Sð4Þ ðIÞ ¼ D4 Eg ðIÞ  D4 Eg ðIÞ exp

ref

(20)

5. DI ¼ 1 staggering in signature partner pairs in odd SD bands It was noticed that most of SD bands observed in odd-A nuclei are signature partner SD bands. The basic signature function for DI ¼ 1 staggering in these signature partner pairs is the differences between the average transitions Iþ2 / I and I / I-2 energies in one band and the transition Iþ1 / I-1 energies in the signature partner

Sð2Þ ðIÞ ¼

 1 Eg ðI þ 2/IÞ þ Eg ðI/I  2Þ  Eg ðI þ 1/I  1Þ 2

(21)

where Eg(I) ¼ E(I) e E(I-2). 6. Quadrpole deformation parameter The bandhead moment of inertia J0 can be related to the quadrpole deformation b2 by the Grodzin formula [33] 2

J02 ¼ a A5=3 b0 where A is the mass number and a is a constant describe the calibration of this relationship between J0 and b2. The quadrpole moment may be expressed in terms of b2 as

Q0 ¼ 1:09 ZA b2 2 3

2 1þ 7

rffiffiffiffiffi 5

p

!

b2 fm2

7. Results and discussions Eleven SD bands in Hg-Tl-Pb nuclei are considered. Equation (7) is used to fit the observed gamma ray transition energies of our selected SD bands with J, s, C and the bandhead spin value I0 as adjusted parameters. A computer simulated search program is used to obtain a minimum c2 deviation.

c2 ¼

2 1 X  Eg ðcal; Ii Þ  Eg ðexp; Ii Þ N i

Table 1 The Calculated best fitted model parameters J,s,c and suggested bandhead spin Io in NS3 model for our selected SD bands in A~190 mass region. For each band, transition from Ioþ2 to Io is given in first column. The last column gives the relative root mean square deviation c. Band

Eg(Ioþ2/Io) (KeV)

J (ħ2MeV1)

sx103

cx103 (MeV3)

Io (ħ)

c

191

252.4 272.0 214.4 211.7 200.79 206.6 227.3 268.0 240.5 187.9 124.9

91.3306 91.067 82.5182 80.2799 89.6542 92.4018 93.8949 96.9764 90.7109 100.439 85.9969

2.7116 3.1642 4.8938 5.7160 3.2897 2.6622 2.0981 2.0796 3.2065 1.5095 3.6323

2.826 2.922 2.0629 1.3337 1.4255 1.4057 2.6808 3.3212 2.4533 5.3311 2.381

10.5 11.5 8 8 8 8.5 9.5 12 10 8 4

2.847 2.7804 1.1422 2.0276 0.5408 2.2564 0.3400 0.1335 0.9253 0.0936 0.2078

Hg(SD2) 191 Hg(SD3) 192 Hg(SD1) 194 Hg(SD1) 194 Hg(SD2) 193 Tl(SD1) 193 Tl(SD2) 194 Tl(SD1) 194 Tl(SD3) 194 Tl(SD5) 194 Pb(SD1)

Please cite this article in press as: A.M. Khalaf et al., Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.03.018

A.M. Khalaf et al. / Chinese Journal of Physics xxx (2016) 1e9

5

where N is the number of data points considered. The procedure of fitting is repeated for several trail values of I0. The results of model parameters and the bandhead spin determinations for each of the studied SD bands deduced by the present work are listed in Table 1. Also included in Table 1 are the root mean square deviation c and the lowest transition energy Eg (I0þ2/I0). Using the assigned spin values, the transition energies Eg(I), rotational frequencies ħu, kinematic J(1) and dynamic J(2) moments of inertia are calculated and systematically examined. Fig. 1 compare the measured and calculated gamma e ray transition energies Eg as given by our proposed model using the parameters listed in Table 1. We display four SD bands in even-even nuclei 192Hg(SD1),194Hg(SD1, SD2), 194Pb(SD1)in (a), three SD bands in odd-odd nuclear 194Tl(SD1, SD2, SD5) in (b) and four SD bands in odd-A nuclei 191Hg(SD2, SD3), 193Tl(SD1, SD2) in (c). The solid curves represent the model predications while the closed circles are labeled to experimental ones (all experimental data are taken from Refs. [1e3]). The agreement between theory and experiment is excellent. For 194Tl (SD5) a rigid rotor reference transition energy of 3.8995 (4I-2) has been subtracted to increase clarity of the plot, this is illustrated in Fig. 2. Fig. 3 presents the evolution of kinematic and dynamic moments of inertia J (1) and J(2) with rotational frequency ħu for our selected SD bands. The experimental data are marked by closed circles for J (2), while the corresponding theoretical points are joined by solid curves. In all cases J (2) is seen to increase with ħu. The theory-experimental agreements are very well. Furthermore, the average values of J (2) for bands 2 and 3 in 191Hg are 110ħ2MeV1 and 113ħ2MeV1 respectively and are close to the corresponding value for the yrast SD band in 192Hg (113ħ2MeV1) since all these bands have the same content in intruder orbitals (i13/2)4 protons and (j15/2)4 neutrons. Also a

Fig. 1. Comparison between the experimental (solid circles) and calculated (soid curves) gamma-ray transition energies as a function of nuclear spin I. Experimental values comes from Ref. [3]. (a) For four SD bands in even-even nuclei 192Hg, 194Hg and 194Pb. (b) For three SD bands in odd-odd nucleus 194Tl. (c) For four SD bands in odd-A nuclei 191Hg and 193Tl.

Please cite this article in press as: A.M. Khalaf et al., Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.03.018

6

A.M. Khalaf et al. / Chinese Journal of Physics xxx (2016) 1e9

Fig. 2. Comparison between the experimental (closed circles) and calculated (soid curve) gamma-ray transition energies relative to a rigid rotor with a moment of inertia J ¼ 128.221 ħ2MeV1 as a function of nuclear spin I for 194Tl(SD5).

Fig. 3. Comparison between experimental (closed circles) and theoretical (solid curves) kinematic J(1) and dynamic J(2) moments of inertia as a function of rotational frequency ħu. (a) For the yrast SD bands in 192Hg and 194Hg nuclei. (b) For yrast SD band in zero quasiparticle core 194Pb and neighboring single blocked band 193Tl(SD1) (unpaired proton) and doubly blocked band 194Tl(SD1) (unpaired proton and neutron). (c) For the excited SD bands in 191Hg (SD2, SD3), 194 Hg(SD2), 193Tl(SD2) and 194Tl(SD3,SD5).

saturation of J (2) moments of inertia at rotational frequency ħu > 0.320 MeV is observed for the two signature partners SD2 and SD3 in 193Tl. This feature reflects the combined effects of the proton pairing blocking and the complete j15/2 neutron alignment. Please cite this article in press as: A.M. Khalaf et al., Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.03.018

A.M. Khalaf et al. / Chinese Journal of Physics xxx (2016) 1e9

Fig. 4. The differences in the gamma-ray transition energies DEY between the yrast SD bands in the two pairs

7

194

Pb-192Hg and

194

Hg-192Hg.

A particularly unexpected feature of SD nuclei is the observation of two bands with nearly almost identical transition energies in nuclei differing by one or two mass units. To investigate the problem of identical bands (IB‘s) the differences in gamma-ray transition energies DEg between transition in the yrast SD band in 194Pb and the corresponding transition in 192Hg versus the transition energy Eg are plotted in Fig. 4. Up to ħu ~ 0.250 MeV, the DEg values are small and constant (±2 KeV). Therefore, there two bands have been considered as IB‘s. This means that in spite of the assigned bandhead spin estimated by fitting procedure of the yrast SD of 194Pb (I0 ¼ 4) is lower than that of 192Hg (I0 ¼ 8), the two additional protonms in Pb compared with Hg does not effect the transition energies. This behavior can also be seen in dynamical moment of inertia plot, up to ħu ~ 0.250, the J (2) values are identical. It is interesting to note that DEg between the yrast SD bands in 194Hg and 192Hg (also plotted in Fig. 4)are too large to consider these two bands as IB‘s. A typical example of the DI ¼ 2 staggering in our selected SD bands is presented in Fig. 5 for 194Hg (SD1, SD2) and 194Tl (SD1, SD3, SD5), where the staggering parameter S (4) (I) defined by equation (20) are plotted as a function of spin I. A significant staggering has been observed. Now, we like to focus another kind of staggering, the DI ¼ 1 staggering in our signature partner pairs in odd SD nuclei 191Hg (SD2, SD3) and 193Tl (SD1, SD2). The S(4) (I) values equation (21) have been calculated and are shown in Fig. 6. A large amplitude staggering were shown. To get some understanding of the dependence on rotation, the deformation parameter b2 can be calculated from the bandhead moment of inertia. Table 2 gives the calculated bandhead moment of inertia J0 and the corresponding deformation

Fig. 5. The resulting values of the DI ¼ 2 staggering parameter S4(I) versus spin (I) for the five SD bands in isobars

194

Hg and

194

Tl nuclei.

Please cite this article in press as: A.M. Khalaf et al., Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.03.018

8

A.M. Khalaf et al. / Chinese Journal of Physics xxx (2016) 1e9

Fig. 6. The calculated DI ¼ 1 staggering parameter D2EY(I) extracted from the difference between the average transition Iþ2/I/I-2 energies in one band and the transition Iþ1/I-1 energies in its signature partner plotted as a function of spin I.

Table 2 Calculated bandhead energies E(IO), quadrupole deformation parameter b2 and electric quadrupole moment Qo for our selected SD states in Hg-Tl - Pb nuclei. Band

E(Io) KeV

Jo (ħ2MeV1)

b2

Qo (eb)

191

633.6821 754.386 369.037 398.237 379.955 417.576 516.592 774.08 568.3073 354.459 112.8655

94.1473 94.0653 97.8950 88.9060 93.3762 95.2953 95.5589 99.6327 95.7180 101.4980 88.8964

0.5480 0.5475 0.5094 0.5108 0.5365 0.5095 0.5103 0.5298 0.5068 0.5397 0.5600

18.9749 18.9721 17.4945 17.6717 18.7059 17.7786 17.8109 18.6653 17.7309 19.0710 20.1553

Hg(SD2) 191 Hg(SD3) 192 Hg(SD1) 194 Hg(SD1) 194 Hg(SD2) 193 Tl(SD1) 193 Tl(SD2) 194 Tl(SD1) 194 Tl(SD3) 194 Tl(SD5) 194 Pb(SD1)

parameter b2 and electric quadrupole moment Q0 values for our selected SD bands in Hg-Tl-Pb nuclei. The calibration constant a for Hg, Tl and Pb nuclei are 4.6591, 5.4386 and 3.8760 respectively. 8. Conclusion In conclusion, we have analyzed the properties of elven SD bands in mass region A ~ 190 by making use of a three parameters nuclear softness model (NS3). For each SD band, the level spins are suggested and the model parameters are determined by using a computer simulated search program in order to obtain a minimum root mean square (rms) deviation between the calculated and the experimental transition energies Eg. The excellent agreement between the calculated Eg and the observed ones, give good support to our model. For each SD band, we calculated also the rotational frequency, the kinematic J(1) and dynamic J(2) moments of inertia, the deformation parameter and the quadrupole moment. At the same time the identical bands in the isotones 194Pb and 192Hg and the presence of DI ¼ 2 staggering in SD bands of 194Hg and 194Tl and the DI ¼ 1 staggering in SD bands of odd-mass nuclei have been investigated. References [1] B. Singh, R. Zymwina, R.B. Firestone, Table of Superdeformed Nuclear Bands and Fission Isomers, in: Nuclear Dates Sheets, third ed., 97, 2002, 41e295. [2] B. Singh, Nucl. Data Sheets 107 (2006) 1.

Please cite this article in press as: A.M. Khalaf et al., Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.03.018

A.M. Khalaf et al. / Chinese Journal of Physics xxx (2016) 1e9 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

National Nuclear Data Center nndc, Brookhaven National Laboratory (cited on July 2012). T. Byrski, et al., Phys. Rev. Lett. 64 (1990) 1650. C. Bzacktosh, B. Haas, N. Nazarewicz, Rev. Nucl. Part. Sci. 45 (1995) 485. A.M. Khalf, M.M. Taha, M. Kotb, Prog. Phys. 4 (2012) 39. A.M. Khalf, M. Sirag, M. Taha, Turk. J. Phys. 37 (2013) 49. A.M. Khalf, M.O. Okasha, Prog. Phys. 10 (2014) 246. M.A. Riley, et al., Nucl. Phys. A512 (1990) 178. S. Flibotte, et al., Phys. Rev. Lett. 7 (1993) 4299. A.O. Macchiavelli, et al., Phys. Rev. C512 (1995) R1. B. Cederwall, Phys. Scr. 72 (1994) 3150. I. Hamamoto, B. Mottelson, Phys. Lett. B333 (1994) 2947. A.M. Khalaf, M. Sirag, Egypt. J. Phys. 35 (2004) 359. Madiha O. Okasha, Prog. Phys. 10 (2014) 41. A.M. Khalaf, K.E. Abdelmageed, Eman Saber, Int. J. Theor. Appl. Sci. 6 (2) (2014) 56. A.M. Khalaf, et al., Prog. Phys. 3 (2013) 39. M.P. Carpenter, et al., Phys. Rev. C56 (1995) 2400. I.M. Hilbent, et al., Phys. Rev. C54 (1996) 2253. Go Hackman, et al., Phys. Rev. C55 (1997) 148. Xo T. He, et al., Nucl. Phys. A760 (2005) 263. A.M. Khalaf, A. Allam, Eman Sabar, Egypt. J. Phys. 39 (2008) 41. J.E. Draper, et al., Phys. Rev. C42 (1990) R1791. J.A. Becker, et al., Nucl. Phys. A520 (1990) 187. A.M. Hegazi, M.H. Ghoniem, A.M. Khalaf, Egypt. J. Phys. 30 (3) (1999) 293. A.M. Khalaf, et al., Egypt. J. Phys. 33 (2002) 585, 34(2003) 159, 34 (2003) 195,35(2004)79. A.M. Khalaf, M.A. Allam, M.M. Sirag, Egypt. J. Phys. 441 (2) (2010) 151. A.M. Khalaf, M. Kotb, K.E. Abdelmageed, J. Adv. Phys. 6 (3) (2014) 1251. A.M. Khalaf, F.A. Altalhi, J. Adv. Phys. 7 (2) (2015) 1414. A. Khalaf, K. Abdelmageed, M.M. Sirag, Turk. J. Phys. 39 (2015) 178. M.A.J. Mariscotti, G. Scharf-Goldhaber, B. Buck, Phys. Rev. 178 (1969) 1864. J.S. Batra, R.K. Gupta, Phys. Rev. C43 (1991) 1725. L. Grodzins, Phys. Lett. 2 (1962) 88e91.

Please cite this article in press as: A.M. Khalaf et al., Properties of superdeformed bands in A ~ 190 region within the framework of three parameters nuclear softness model, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.03.018

9