Critical Point Symmetry, X (5), in ^{154}Gd
Salah A. Eid^{1}, Sohair M. Diab^{2}
^{1}Physics Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt
^{2}Physics Department, Faculty of Education, Ain Shams University, Cairo, Egypt
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To cite this article:
Salah A. Eid, Sohair M. Diab. Critical Point Symmetry, X (5), in ^{154}Gd. American Journal of Modern Energy. Vol. 2, No. 6, 2016, pp. 43-47. doi: 10.11648/j.ajme.20160206.12
Received: September 29, 2016; Accepted: October 28, 2016; Published: November 21, 2016
Abstract: The positive-negative parity states, potential energy surfaces, V(β, γ), transition probabilities, B(E1), B(E2), staggering effect and electric monopole strength, X (E0/E2), values of ^{154}Gd have been calculated within the frame work of the interacting boson approximation model (I BA − 1). The results obtained are compared to the available experimental, theoretical data and reasonable agreement has achieved. The potential energy surfaces, levels energy and transition probability ratios show that ^{154}Gd is an X (5) candidate.
Keywords: Levels Energy, Transition Probability, B(E1), B(E2), Electric Monopole Strength, X (E0/E2)
1. Introduction
The even-even ^{154}Gd isotope has subjected to an extensive study in the past few years where the region of N= 90 shows rapid changes of nuclear deformation parameters as a function of the particle number. Many authors have studied this isotope experimentally and theoretically.
Experimentally [1-10] different techniques were applied. Silicon and germanium detector arrays, (α, 2n), (α, 4n) reactions, AFRODITE γ-ray spectrometer, combination of γ -ray scattering experiments and γ - γ coincidence following the electron capture of ^{154m}Tb were used. They detected levels energy, quadrupole moment, two neutron separation energies, B(E1), B(E2) values and magnetic moment.
Theoretically authors [11-22] have studied all the important probes for ^{154}Gd nucleus as energy spectra, moment of inertia, quadrupole moments, octupole and hexadecupole degrees of freedom, B(E1), B(E2),B(E3), magnetic moment and quadrupole moment using different theoretical approaches and models.
The aim of the present work is to use the IBA-1 model [23-26] for the following tasks:
1. Calculating the potential energy surfaces, V(β, γ);
2. Calculating levels energy, electromagnetic transition rates B(E1) and B(E2);
3. Studying the back bending;
4. Calculating staggering effect;
5. Detect any interactions between the +ve and –ve parity States and calculating the electric monopole strength X(E0/E2).
2. Levels Energy in IBA-1 Model
IBA-1 model was applied to the positive and negative parity states of ^{154}Gd isotope. The Hamiltonian employed in the present calculation is:
(1)
where
(2)
(3)
(4)
(5)
(6)
In the previous formulas, n_{d} is the number of bosons; P.P, L.L, Q.Q, T_{3}.T_{3} and T_{4}.T_{4} represent pairing, angular momentum, quadrupole, octupole and hexadecupole interactions between the bosons; EPS is the boson energy; and PAIR, ELL, QQ, OCT, HEX is the strengths of the pairing, angular momentum, quadrupole, octupole and hexadecupole interactions.
3. Transition Rates
The electric quadrupole transition operator employed in this study is given by:
(7)
where:
T^{(E2)}: absolute transition probability of the electric quadrupole (E2) transition, and E2SD and E2DD: adjustable parameters
The reduced electric quadrupole transition rates between I_{i}→I_{f} states are given by
(8)
where
I_{i}: the initial state of the electric quadrupole transition, and
I_{f}: the final state of the electric quadrupole transition.
4. Results and Discussion
4.1. The Potential Energy Surfaces
The potential energy surfaces, V(β, γ), for ^{154} Gd nucleus as a function of the deformation parameters β and γ has been calculated using [27]^{}equation:
E N_{υ} N_{¶} (β, γ) = < N_{¶} N_{υ};βγ | H_{np} | N_{¶} N_{υ;} βγ >
= ε_{d }(N_{υ}+ N_{¶})β^{2}(1+ β^{2})^{-1} +
β^{2}(1+β^{2})^{-2} {k N_{¶} N_{υ}[4-( +)βcos3 γ+
β^{2}] +N_{υ(}N_{υ}-1) (+) β^{2}} (9)
where
= (2/7)^{0.5} X_{ρ} ρ=_{¶ }or υ (10)
The calculated potential energy surfaces, V (β, γ), are Presented in Figure 1. The deviation from spherical, an harmonic vibrator U (5), to rotational characters, SU (3), at the critical symmetry point has supported quite well as well as the energy and transition probability ratios the X(5) characters to ^{154}Gd nucleus, Table. 1.
4.2. Energy Spectra
The energy of the positive and negative parity states of ^{154} Gd isotope are calculated using computer code PHINT [28] with the parameters EPS = 0.424, P AIR = 0.000, ELL = 0.0084, QQ = −0.0244, OCT = 0.000, HEX = 0.000 all in MeV while, E2SD =0.1450(eb), E2DD = −0.4289(eb). A comparison between the experimental spectra [29] and our calculations, using values of the model parameters given in Table 1 for the ground state, β1, β2, γ1, γ2 and (−ve) parity bands are illustrated in Figure 2. The agreement between the calculated levels energy and their correspondence experimental values are fairly good in low-lying states and slightly higher for the higher excited states. We believe this is due to the change of the projection of the angular momentum which is due mainly to band crossing.
4.3. Transition Rates
Unfortunately there is no enough measurements of electromagnetic transition rates B(E1) or B(E2) for ^{154} Gd nucleus. The only measured value is B(E2, 0^{+}_{1} → 2^{+}_{1}) = 3.89(7) [30] which presented in Table 2. The parameters E2SD and E2DD are used in the computer code NPBEM [28] for calculating the electromagnetic transition rates and then normalized to the experimental value, B(E2, 0^{+}_{1} → 2^{+}_{1}).
No new parameters are introduced for calculating electromagnetic transition rates B(E1) and B(E2) of intra band and inter band. It is clear that the electromagnetic transition of the γ transition within any band has a large value while it is small between intra bands which is due to either a mixture of M 1 or it is forbidden.
4.4. Back Bending
The moment of inertia J and energy parameters ω are calculated using Eq. (11, 12):
2J/ħ^{2} = (4I-2) /ΔE(I→ I-2) (11)
(ħω)^{2} = (I^{2} – I + 1)[ΔE(I→ I-2)/(2I-1)] (12)
Nucleus | E4_{1}^{+}/E2_{1}^{+} | E6_{1}^{+}/E2_{1}^{+} | E8_{1}^{+}/E2_{1}^{+} | E0_{2}^{+}/E2_{1}^{+} | E6_{1}^{+}/E0_{2}^{+} | E0_{3}^{+}/E2_{1}^{+} | BE2(4_{1}^{+}-- 2_{1}^{+}) / BE2(2_{1}+-- 0_{1}+) |
^{154}Gd | 3.00 | 5.83 | 9.30 | 5.52 | 1.05 | 9.60 | 1.52 |
X(5) | 3.02 | 5.83 | 9.29 | 5.65 | 1.53 | 6.03 | 1.58 |
Figure 3. shows forward bending for^{154} Gd at I ^{+} = 6 and that bending has explained as due to partial rotational alignment of a pair of neutrons near the Fermi surface.
I_{i}^{+} | I_{f}^{+} | B(E2) _{Exp*.} | B(E2) _{IBA} | Ii^{+} | If^{+} | B(E1) _{Exp.} | B(E1) _{IBA} |
0_{1} | 2_{1} | 3.89(7) | 3.8965 | 1_{1} | 0_{1} | —— | 0.1087 |
2_{2} | 0_{1} | —— | 0.0005 | 1_{1} | 0_{2} | —— | 0.0786 |
2_{2} | 0_{2} | —— | 0.4922 | 3_{1} | 2_{1} | —— | 0.2465 |
2_{3} | 0_{1} | —— | 0.0168 | 3_{1} | 2_{2} | —— | 0.0826 |
2_{3} | 0_{2} | —— | 0.0123 | 3_{1} | 2_{3} | —— | 0.0083 |
2_{3} | 0_{3} | —— | 0.0498 | 3_{2} | 2_{1} | —— | 0.0352 |
2_{4} | 0_{3} | —— | 0.3081 | 3_{2} | 2_{2} | —— | 0.0214 |
2_{4} | 0_{4} | —— | 0.0358 | 3_{2} | 2_{3} | —— | 0.0564 |
2_{2} | 2_{1} | —— | 0.0628 | 5_{1} | 4_{1} | —— | 0.3710 |
2_{3} | 2_{1} | —— | 0.0165 | 5_{1} | 4_{2} | —— | 0.0733 |
2_{3} | 2_{2} | —— | 0.4976 | 5_{1} | 4_{3} | —— | 0.0043 |
4_{1} | 2_{1} | —— | 1.1878 | 7_{1} | 6_{1} | —— | 0.4943 |
4_{1} | 2_{2} | —— | 0.0694 | 7_{1} | 6_{2} | —— | 0.0631 |
6_{1} | 4_{1} | —— | 1.3454 | 9_{1} | 8_{1} | —— | 0.6193 |
6_{1} | 4_{2} | —— | 0.0587 | 9_{1} | 8_{2} | —— | 0.0545 |
8_{1} | 6_{1} | —— | 1.4041 | 11_{1} | 10_{1} | —— | 0.7469 |
8_{1} | 6_{2} | —— | 0.0467 | ||||
8_{1} | 6_{3} | —— | 0.0234 | ||||
10_{1} | 8_{1} | —— | 1.3954 |
Ref. [30]
4.5. The Staggering
The presence of odd-even parity states has encouraged us to study staggering eﬀect for ^{154}Gd. Staggering pattern between the energies of the ground state and the (−ve) parity octupole bands have been calculated, ∆I = 1, using staggering function Eq. (13, 14) with the help of the available experimental data [29].
Stag (I) = 6∆E(I) − 4∆E(I − 1) − 4∆E(I + 1) + ∆E(I + 2) +∆E(I − 2) (13)
with
∆E(I) = E(I + 1) − E(I) (14)
The calculated staggering pattern has illustrated in Figure 4. It shows an interaction between the (−ve) and (+ve) parity states.
4.6. Electric Monopole Transitions, Xif ′ f (E0/E2)
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, Xif ′ f (E0/E2), [31] can be calculated using Eq. (15, 16) and results are presented in Table 3.
I_{i}^{+} | I_{f}^{+} | I_{'f}^{+} | Xif ′ f (E0/E2)_{Exp*} | Xif ′ f (E0/E2)_{IBA} |
0_{2} | 0_{1} | 2_{1} | ------ | 5.11 |
0_{3} | 0_{1} | 2_{2} | ------ | 0.67 |
0_{3} | 0_{2} | 2_{2} | ------ | 3.11 |
0_{4} | 0_{1} | 2_{3} | ------ | 1.13 |
0_{4} | 0_{2} | 2_{3} | ------ | 0.67 |
0_{4} | 0_{3} | 2_{3} | ------ | 4.54 |
2_{3} | 2_{2} | 2_{1} | ------ | 1.04 |
2_{4} | 2_{2} | 2_{1} | ------ | 6.28 |
2_{4} | 2_{3} | 2_{1} | ------ | 1.77 |
The small values of the Xif ′ f (E0/E2) indicate that the transition has small contribution of E0 transition, while the large valeue means that the decay of the state is mainly E0 which is by turn responsible for the shape of the nucleus.
Xif ′ f (E0/E2) = B(E0, I_{i }– I_{f}) / B(E2, I_{i} – I_{'f}) (15)
Where
I_{i }= I_{f} = 0, I_{'f} =2 or I_{i }= I_{f} ≠ 0, I_{f} = I_{'f}
Xif ′ f (E0/E2 = 2.54x10^{9} A^{3/4} [E^{5}γ MeV/ Ω_{KL}]x α (E2) x[T_{e}(E0, I_{i }– I_{f}) / T_{e}(E2, I_{i} – I_{'f})] (16)
A: mass number; depopulating it;
I_{f}: spin of the ﬁnal state of E0 transition;
I_{′ f}: spin of the ﬁnal state of E2 transition;
Eγ: gamma ray energy;
Ω_{KL}: electronic factor for K, L shells [32];
α(E2): conversion coeﬃcient of the E2 transition;
Te (E0, I_{i }− I_{f}): absolute transition probability of the E0 transition
between I_{i} and I_{f} states, and
Te (E2, I_{i} − I_{′ f}): absolute transition probability of the E2 transition between I_{i} and I_{′ f} states.
5. Conclusions
IBA-1 model has been applied successfully to ^{154}Gd isotope and:
1. The levels energy are successfully reproduced;
2. The levels energy, transition probability ratios and potential energy surfaces are calculated and show X(5) characters to ^{154}Gd;
3. Electromagnetic transition rates B(E1) and B(E2) are calculated;
4. Forward bending has observed at angular momentum I^{+} = 6;
5. Strength of the electric monopole transitions Xif ′ f (E0/E2) are calculated; and
6. Staggering eﬀect has been calculated and beat pattern has obtained which show an interaction between the (−ve) and (+ve) parity states.
References