X(5) symmetry to [sup.152]Sm.
Phase transition is one of the very interesting topic in nuclear structure physics. The even-even samarium series of isotopes have encouraged many authors to study that area extensively experimentally and theoretically.
Experimentally, authors studied levels energy with their half-lives, transition probabilities, decay schemes, multipole mixing ratios, internal conversion coefficients, angular correlations and nuclear orientation of [gamma]-rays [1-4].
Theoretically, different theoretical models have been applied to that chain of isotopes. One of the very interesting models is the interacting boson approximation model IBA . Iachello [11,12] has made an important contribution by introducing the new dynamical symmetries E(5) and X(5).
E(5) is the critical point symmetry of phase transition between U(5) and 0(6) while X(5) is between U(5) and S U(3) nuclei. The aim of the present work is to calculate:
1. The potential energy surfaces, V([beta],[gamma]);
2. The levels energy, electromagnetic transition rates B(E1) and B(E2);
3. The staggering effect, and
4. The electric monopole strength X(E0/E2).
2 IBA-1 model
2.1 Levels energy
The IBA-1 Hamiltonian [13-16] employed on [sup.152]Sm in the present calculation is:
H = EPS x [n.sub.d] + PAIR x (P x P)
+1/2 ELL x (L x L) + 2 QQ x (Q x Q)
+5 OCT x ([T.sub.3] * [T.sub.3]) + 5HEX x ([T.sub.4] x [T.sub.4]), (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
L x L = -10 [square root of 3][[[([d.sup.[dagger]][??]).sup.(1)]x[([d.sup.[dagger]][??]).sup.(1)]].sup.(0).sub.0] (3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[T.sub.3] x [T.sub.3] = - [square root of 7] [[[d.sup.[dagger]][??]).sup.(2)]x [d.sup.[dagger]][??]).sup.(2)]].sup.(0).sub.0], (5)
[T.sub.4] x [T.sub.4] = 3 [[[d.sup.[dagger]][??]).sup.(4)]x[d.sup.[dagger]][??]).sup.(4)]].sup.(0).sub.0]. (6)
In the previous formulas, [n.sub.d] is the number of bosons; PxP, LxL, QxQ, [T.sub.3] [T.sub.3] and [T.sub.4] [T.sub.4] represent pairing, angular momentum, quadrupole, octupole and hexadecupole interactions respectively between the bosons; EPS is the boson energy; and PAIR, ELL, QQ, OCT, HEX are the strengths of the pairing, angular momentum, quadrupole, octupole and hexadecupole interactions respectively (see Table 1).
2.2 Transition rates
The electric quadrupole transition operator employed is:
[T.sup.(E2)] = E2SD x [([s.sup.[dagger]][??] + [d.sup.[dagger]] s).sup.(2)] +
+ 1/[square root of 5] E2DD x [([d.sup.[dagger]][??]).sup.(2)]. (7)
E2SD and E2DD are adjustable parameters.
The reduced electric quadrupole transition rates between [I.sub.i] [right arrow] [I.sub.f] states are given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
3 Results and discussion
In this section we review and discuss the results.
3.1 The potential energy surfaces
The potential energy surfaces , V([beta], [gamma]), as a function of the deformation parameters [beta] and [gamma] are calculated using:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[[bar.X].sub.[rho]] = [(2/7).sup.0.5] [X.sub.[rho]][rho] = [pi] or v. (10)
The calculated potential energy surfaces, V([beta], [gamma]), are presented in Figures 1, 2, 3. [sup.152]Sm lies between [sup.150]Sm which is a vibrational like nucleus, U(5), Fig. 1, while [sup.154]Sm is a rotational like, S U(3), nucleus, Fig. 3. So, [sup.150]Sm can be an X(5) candidate where levels energy, transition probability ratios as well as the potential energy surfaces are supporting that assumption (see Table 2).
3.2 Energy spectra and electric transition rates
The energy of the positive and negative parity states of [sup.152]Sm isotope are calculated using computer code PHINT . A comparison between the experimental spectra  and our calculations, using values of the model parameters given in Table 1 for the ground state, [beta]1, [beta]2 and [gamma] bands are illustrated in Fig. 4. The agreement between the calculated levels energy and their corresponding experimental values are fair, but they are slightly higher especially for the higher excited states in [beta]1, [beta]2 and [gamma] bands. We believe this is due to the change of the projection of the angular momentum which is due mainly to band crossing. Fig. 5 shows the position of X(5) and E(5) between the other types of nuclei.
Unfortunately there are no available measurements of electromagnetic transition rates B(E1) for [sup.152]Sm nucleus, Table 3, while some of B (E2) are measured. The measured B (E2,[2.sup.+.sub.1] [right arrow] [0.sup.+.sub.1]) is presented, in Table 4, for comparison with the calculated values . The parameters E2S D and E2DD displayed in Table 1 are used in the computer code NPBEM  for calculating the electromagnetic transition rates and the calculated values are normalized to B([2.sup.+.sub.1] [right arrow] [0.sup.+.sub.1]). No new parameters are introduced for calculating electromagnetic transition rates B (E1) and B (E2) of intraband and interband.
3.3 Staggering effect
The presence of (+ve) and (-ve) parity states has encouraged us to study the staggering effect [21-23] for [sup.152]Sm isotope using staggering function equations (11, 12) with the help of the available experimental data .
St (I) = 6[DELTA]E (I) - 4[DELTA]E (I - 1) - 4[DELTA]E (I + 1) + + [DELTA]E (I + 2) + [DELTA]E (I - 2), (11)
[DELTA]E (I) = E (I + 1) - E (I). (12)
The calculated staggering patterns are illustrated in Fig. 6 and show an interaction between the (+ve) and (-ve) parity states for the ground state band of [sup.152]Sm.
3.4 Electric monopole transitions
The electric monopole transitions, E0, are normally occurring between two states of the same spin and parity by transferring energy and zero unit of angular momentum. The strength of the electric monopole transition, [X.sub.if'f] (E0/E2),  can be calculated using equations (13, 14) and presented in Table 5.
[X.sub.if'f] (E0/E2) = B(E0, [I.sub.i] - [I.sub.f])/B(E2, [I.sub.i] - [I.sub.'f]), (13)
Salah A. Eid and Sohair M. Diab. X(5) Symmetry to [sup.152]Sm
where [I.sub.i] = [I.sub.f] = 0 , [I.sub.'f] = 2 and [I.sub.i] [I.sub.f] [not equal to] 0 , [I.sub.f] = [I.sub.'f] .
[X.sub.if'f] (E0/E2) = (2.54 x [10.sup.9]) [A.sup.3/4] x
x [E.sup.5.sub.[gamma]](MeV)/[[OMEGA].sub.KL] [alpha](E2) [T.sub.e](E0, [I.sub.i] - [I.sub.f])/[T.sub.e]([E.sub.2], [I.sub.i] - [I.sub.'f]). (14)
A: mass number;
[I.sub.i]: spin of the initial state where E0 and E2 transitions are depopulating it;
[I.sub.f]: spin of the final state of E0 transition;
[I.sub.'f]: spin of the final state of E2 transition;
[E.sub.[gamma]]: gamma ray energy;
[[OMEGA].sub.KL]: electronic factor for K,L shells ;
[alpha](E2): conversion coefficient of the E2 transition;
[T.sub.e](E0, [I.sub.i] - [I.sub.f]): absolute transition probability of the E0 transition between [I.sub.i] and [I.sub.f] states, and
[T.sub.e]([E.sub.2], [I.sub.i] - [I.sub.'f]): absolute transition probability of the E2 transition between [I.sub.i] and [I.sub.'f] states.
The IBA-1 model has been applied successfully to the [sup.152]Sm isotope and:
1. Levels energy are successfully reproduced;
2. Potential energy surfaces are calculated and show X(5) characters to [sup.152]Sm;
3. Electromagnetic transition rates B(E1) and B(E2) are calculated;
4. Staggering effect has been calculated and beat pattern observed which show an interaction between the (-ve) and (+ve) parity states, and
5. Strength of the electric monopole transitions [X.sub.if'f] (E0/ E2) are calculated.
Submitted on January 10, 2016 / Accepted on January 12, 2016
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Salah A. Eid (1) and Sohair M. Diab (2)
(1) Faculty of Engineering, Phys. Dept., Ain Shams University, Cairo, Egypt.
(2) Faculty of Education, Phys. Dept., Ain Shams University, Cairo, Egypt. E-mail: firstname.lastname@example.org
Table 1: Parameters used in I BA-1 Hamiltonian (all in MeV). nucleus EPS PAIR ELL QQ [sup.152]Sm 0.3840 0.000 0.0084 -0.0244 nucleus OCT HEX E2SD(eb) E2DD(eb) [sup.152]Sm 0.0000 0.0000 0.1450 -0.4289 Table 2: Energy and transition probability ratios. nucleus [MATHEMATICAL [MATHEMATICAL [MATHEMATICAL EXPRESSION NOT EXPRESSION NOT EXPRESSION NOT REPRODUCIBLE REPRODUCIBLE REPRODUCIBLE IN ASCII] IN ASCII] IN ASCII] [sup.152]Sm 3.02 5.83 9.29 X(5) 3.02 5.83 9.29 nucleus [MATHEMATICAL [MATHEMATICAL EXPRESSION NOT EXPRESSION NOT REPRODUCIBLE REPRODUCIBLE IN ASCII] IN ASCII] [sup.152]Sm 5.66 1.03 X(5) 5.65 1.53 nucleus [MATHEMATICAL BE2([4.sup.+.sub.1] - EXPRESSION NOT [2.sup.+.sub.1])/BE2 REPRODUCIBLE ([2.sup.+.sub.1] - IN ASCII] [0.sup.+.sub.1]) [sup.152]Sm 8.92 1.53 X(5) 6.03 1.58 Table 3: Calculated B(E1) in 152Sm. [I.sup.-.sub.i] B(E1)Exp. B(E1)IBA-1 [I.sup.+.sub.f] [1.sub.1][0.sub.1] -- - 0.0979 [1.sub.1][0.sub.2] -- - 0.0814 [3.sub.1][2.sub.1] -- - 0.2338 [3.sub.1][2.sub.2] -- - 0.0766 [3.sub.1][2.sub.3] -- - 0.0106 [3.sub.2][2.sub.1] -- - 0.0269 [3.sub.2][2.sub.2] -- - 0.0291 [3.sub.2][2.sub.3] -- - 0.0434 [5.sub.1][4.sub.1] -- - 0.3579 [5.sub.1][4.sub.2] -- - 0.0672 [5.sub.1][4.sub.3] -- - 0.0050 [7.sub.1][6.sub.1] -- - 0.4815 [7.sub.1][6.sub.2] -- - 0.0574 [9.sub.1][8.sub.1] -- - 0.6075 [9.sub.1][8.sub.2] -- - 0.0490 [11.sub.1][10.sub.1] -- - 0.7367 [11.sub.1][10.sub.2] -- - 0.0413 Table 4: Calculated B(E2) in [sup.152]Sm (x from Ref.) [I.sup.+.sub.i] B(E2)Exp *. B(E2)IBA-1 [I.sup.+.sub.f] [2.sub.1] [0.sub.1] 0.670(15) 0.6529 [3.sub.1] [2.sub.1] -- - 0.0168 [4.sub.1] [2.sub.1] 0.1.017(4) 1.0014 [6.sub.1] [4.sub.1] 1.179(33) 1.1304 [0.sub.2] [2.sub.1] 0.176(1) 0.3363 [2.sub.2] [2.sub.1] 0.0258(26) 0.0610 [2.sub.2] [4.sub.1] 0.091(11) 0.1057 [4.sub.2] [2.sub.1] 0.0035(35) 0.0003 [4.sub.2] [4.sub.1] 0.037(23) 0.0458 [2.sub.3] [0.sub.1] 0.0163(11) 0.0141 [2.sub.3] [2.sub.1] 0.0417(42) 0.0125 [2.sub.3] [4.sub.1] 0.0416(32) 0.0296 [4.sub.3] [2.sub.1] 0.0035(13) 0.0038 [4.sub.3] [4.sub.1] 0.037(13) 0.0084 [4.sub.3] [4.sub.2] -- - 0.1235 [4.sub.3] [2.sub.2] -- - 0.0070 [4.sub.3] [2.sub.3] -- - 0.3110 [4.sub.2] [2.sub.2] -- - 0.6418 [8.sub.1] [6.sub.1] -- - 1.1681 [8.sub.1] [6.sub.2] -- - 0.0376 [10.sub.1] [8.sub.1] -- - 1.1421 Table 5: [X.sub.if'f] (E0/E2) ratios in [sup.152]Sm (* from Ref ). [I.sup.+.sub.i] X(E0/E2)Exp *. X(E0/E2)IBA-1 [I.sup.+.sub.f] [0.sub.2] [0.sub.1] 0.7(0.1) 0.85 [0.sub.3] [0.sub.2] -- - 3.68 [0.sub.3] [0.sub.1] -- - 0.72 [0.sub.4] [0.sub.3] -- - 4.39 [0.sub.4] [0.sub.2] -- - 0.64 [0.sub.4] [0.sub.1] -- - 1.27 [2.sub.2] [2.sub.1] 4.5(0.5) 3.52 [2.sub.3] [2.sub.1] -- - 12.23 [2.sub.3] [2.sub.2] -- - 11.19 [4.sub.3] [4.sub.1] -- - 1.76 [4.sub.3] [4.sub.2] -- - 1.40 [4.sub.4] [4.sub.1] -- - 0.44 [4.sub.4] [4.sub.2] -- - 3.15 [4.sub.2] [4.sub.1] 6.6(2.10) 2.02 [6.sub.2] [6.sub.1] -- - 1.46 [8.sub.2] [8.sub.1] -- - 1.20 [10.sub.2] [10.sub.1] -- - 1.07
Please note: Some tables or figures were omitted from this article.
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|Author:||Eid, Salah A.; Diab, Sohair M.|
|Publication:||Progress in Physics|
|Date:||Apr 1, 2016|
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