1. Introduction
In nuclear physics, the most important part of interest is the shape of nucleus. The ground state shape of nucleus is spherical and it may deviate from this shape, which is closely related to nuclear “magic numbers”. We will consider nuclear “magic numbers” and their evolution along the nuclear chart. In stable nuclei, large gaps exist between nuclear shells when the proton or neutron number is equal to 2, 8, 20, 28, 50, 82 and 126 [1] [2]. These gaps result in large transition energy values between the ground and first excited states, relatively low quadrupole moment and small neutron capture cross sections. The “magic numbers” and their values are not preserved; they evolve for unstable nuclei due to nuclear structure effects. Therefore, nuclear properties of the first excited 2^{+} states in even-even nuclei provide important information on the evolution of nuclear properties and shell model studies. The electric quadrupole reduced transition probabilities are important for the nuclear structural information. In recent years, the electric quadrupole reduced transition probabilities of even-even neutron rich nuclei ^{102-112}Pd [3] [4], ^{104-112}Cd [5] and ^{100-102}Ru [6] have been studied using Interacting Boson Model-1 (IBM-1). This model (IBM-1) was developed by Iachello and Arima and the associated quadrupole moment, deformation parameter was studied [3] [4]. All these nuclei studied the downward reduced transition probabilities using the IBM-1. In 1999, excitation energies, E2 transition probabilities, quadrupole excitation properties for even-even ^{104-114}Ru and ^{106-110}Pd nuclei have been studied [7]. After this study, Skyrme force SLy4 for even-even ^{94-110}Ru nuclei has been investigated using the Hartree-Fock-Bogoliubov (HFB) method [8] and the triaxial shapes properties for the even-even neutron-rich ^{106-108}Mo and ^{108-112}Ru nuclei have been also investigated [9]. Therefore, we have focused the even-even ^{104-114}Ru isotopes’ properties for the transition levels 0^{+} → 2^{+}, 2^{+} → 4^{+}, 4^{+} → 6^{+} and 6^{+} → 8^{+}. For this purpose, we have used the GBF model to investigate the basic information of the ^{104-114}Ru nuclei because this model describes three basic properties: mass and energy dependency with γ-ray transition probability; localization and emphasis for the anchor nucleus; regionalized by the magic number [10].
Using GBF method, we have calculated the upward transition probabilities, B(E2)↑ of neutron-rich even-even ^{104-114}Ru nuclei. Associated parameters like quadrupole moment (Q_{0}) and deformation parameter (
${\beta}_{2}$ ) have also been estimated. The study also reveals the effects of the estimated parameters on the nuclear structure. This also reveals how the estimated parameters affect the structure of the nucleus. This method presents the near magic number N = 50 region for the ^{104}Ru nucleus. In this paper, energy and mass dependencies have been showed with the calculated B(E2)↑ values. The relationship among B(E2)↑, Q_{0} and
${\beta}_{2}$ are also given in graphical. Finally, this paper is arranged as: GBF model has been described in 2.1 Section. B(E2)↑, Q_{0} and
${\beta}_{2}$ are discussed in Section 2.2, 2.3 and 2.4 respectively.
2. Theory
In this section we describe the procedure used to compute the electric quadrupole reduced transition probabilities and the corresponding electric quadrupole moment and deformation parameter. The procedure summary of this theory is described in the following flowchart (Figure 1):
2.1. Global Best Fit (GBF) Method
According to the Global Best Fit Method, a knowledge of the energy E (Kev) of the 2_{1}^{+} state is all that is required to make a prediction for the corresponding mean life time for the γ-ray,
${\tau}_{\gamma}$ (in ps) and hence, the B(E2)↑ (e^{2}b^{2}) value. Within the framework of the hydrodynamic model with irrotational flow, Bohr and Mottelson [11] have derived simple expressions for the
${\tau}_{\gamma}$ value is given by
${\tau}_{\gamma}\approx 0.6\times {10}^{14}{E}^{-4}{Z}^{-2}{A}^{1/3}$ (1)
Figure 1. Flowchart of the given procedure.
For small harmonic vibrations of spherical nucleus, the
${\tau}_{\gamma}$ value is,
${\tau}_{\gamma}\approx 1.4\times {10}^{14}{E}^{-4}{Z}^{-2}{A}^{1/3}$ (2)
For collective rotations of axially symmetric nuclei. The E^{−4}Z^{−2} dependence in the above expressions was adopted by Grodzins [12] in his empirical fits (for all even-even nuclei), but he replaced A^{1/3} with A. When the exponents of E and A were allowed to vary, we found earlier [10] [13] that the best global fit to the data [14] was obtained by
${\tau}_{\gamma}=1.25\times {10}^{14}{E}^{-4}{Z}^{-2}{A}^{0.69}$ (3)
Hence,
${\tau}_{\gamma}$ and B(E2)↑ is related by the equation [7],
${\tau}_{\gamma}=40.81\times {10}^{13}{E}^{-5}{\left[B\left(E2\right)\uparrow /{e}^{2}{b}^{2}\right]}^{-1}$ (4)
When converted to B(E2)↑, this expression led to
$B\left(E2\right)\uparrow \text{\hspace{0.17em}}=3.26{E}^{-1}{Z}^{2}{A}^{-0.69}$ (5)
We also showed that, the 1/E dependence is more important than the exact A dependence. If the exponent of A is fixed as −2/3 (instead of −0.69), the revised best fit to the data was found [15] to be,
$B\left(E2\right)\uparrow \text{\hspace{0.17em}}=2.6{E}^{-1}{Z}^{2}{A}^{-2/3}$ (6)
Here, B(E2)↑ is the electric quadrupole transition probability, E is the excitation energy, Z is the atomic number and A is the mass number.
2.2. Electric Quadrupole Reduced Transition Probability
The upward transition probabilities B(E2)↑ is the transition of a particle from lower energy state to higher energy state [10]. It can be calculated by using the following equation obtained from GBF method,
$B\left(E2;{l}_{i}\to {l}_{f}\right)\uparrow \text{\hspace{0.17em}}=2.6{E}^{-1}{Z}^{2}{A}^{-2/3}$ (7)
Here,
${l}_{i}$ is the lower energy state and
${l}_{f}$ is the higher energy state and the subscript i and f indicate the initial and final respectively.
2.3. Electric Quadrupole Moment
The nuclear electricintrinsic quadrupole moment is a parameter which describes the nuclear charge distribution. A non-zero intrinsic quadrupole moment Q_{0} indicates that the charge distribution is not spherically symmetric. By convention the value of Q_{0} is taken to be positive (Q_{0} > 0) if the ellipsoid is prolate and negative (Q_{0} < 0) if it is oblate [16]. Intrinsic quadrupole moment, Q_{0} is related to the electric quadrupole transition probabilities B(E2)↑, calculated by the following equation [17]
${Q}_{0}={\left[\frac{16\pi B\left(E2\right)\uparrow}{5{e}^{2}}\right]}^{1/2}$ (8)
Here, Q_{0} measured in barn (b) unit.
2.4. Deformation Parameter
Deformation Parameter is the parameter which measures the elongation of the axially symmetric shape of a deformed nucleus from its spherical shape. Deformation parameter is denoted by
${\beta}_{2}$ which is, related to B(E2)↑, calculated by the equation [17]
${\beta}_{2}=\left(4\pi /3Z{R}_{0}^{2}\right){\left[B\left(E2\right)\uparrow /{e}^{2}\right]}^{1/2}$ (9)
Here,
${R}_{0}$ is the average radius nuclear which can be obtained from the following equation,
${R}_{0}^{2}=0.0144{A}^{2/3}$ barn (b). (10)
3. Results and Discussion
The values of the E, R_{0}, B(E2)↑, β_{2} and Q_{0} for the even-even ^{104-114}Ru nuclei are given in Table 1. For the even-even ^{104-114}Ru nuclei, E has been obtained from the references [18] - [23] and other values B(E2)↑, Q_{0}, β_{2} and R_{0}, have been obtained using the Equations (7)-(10) respectively. Using these values, Figures 2-6 have been drawn in below where their relations and behaviors have been discussed.
Electric quadrupole transition probabilities B(E2)↑ is drawn as a function of transition levels for even-even ^{104-114}Ru nuclei in Figure 2. This shows the decrease of B(E2)↑ with increasing the transition levels for each nucleus. For the 0^{+} - 2^{+} transition level, transition probability is higher than that of the other transition levels for each nucleus. In this level, transition probability for the ^{104}Ru is the lowest than the other nuclei.
Deformation parameter is drawn as a function of B(E2)↑ in Figure 3. This figure shows the transition probability increasing with the increase of deformation parameter. For ^{104-114}Ru nuclei, deformation parameter change follows the almost linear relationship with respect to the transition levels.
Quadrupole moment variations with the change of B(E2)↑ are shown in Figure 4. Figure 4 shows that quadrupole moments increase linearly with the
Table 1. The electric quadrupole transition probabilities, deformation parameters and quadrupole moments of ^{104-114}Ru isotope.
Figure 2. The change of reduced transition probabilities with transition levels.
Figure 3. Deformation parameter change with respect to the reduced transition probabilities.
Figure 4. Quadrupole moment variations with the change of reduced transition probabilities.
Figure 5. Quadrupole moment change with respect to the transition levels.
Figure 6. The change of deformation parameter against the transition levels.
transition probabilities increasing. In the figure, we see that quadrupole moment is lower for the isotope ^{104}Ru.
Quadrupole moments are represented with the variation of transition levels for even-even ^{104-114}Ru nuclei in Figure 5. In this figure, we see the quadrupole moments decrease with increasing the transition levels for each nucleus and it reaches in the range 1.8 - 3.0 barn (b).
Figure 6 shows the deformation parameters variation with respect to the transition levels for these even-even nuclei. From this figure, we can conclude that at first transition level 0^{+} - 2^{+} the deformation of the nucleus shape will be maximum for each nucleus, gradually the deformations decrease for upper transition levels.
4. Conclusion
It is seen from the data and corresponding graphs, when the transition levels of any nuclei increased the electric quadrupole moment and reduced transition probabilities B(E2)↑ of the given nuclei are decreased. The deformation parameters also decrease with increasing transition levels. It concludes from the data and corresponding graph, the transition probabilities, quadrupole moment, and deformation parameters have comparatively lower values for the isotopes which have neutron number close to magic number 50.
Acronyms
Acronyms’ Name Acronyms’ Symbol
Upward transition probabilities B(E2)↑
Global Best Fit GBF
Quadrupole moment Q_{0}
Deformation parameter β_{2}
Interacting Boson Model-1 IBM-1
Mean life time for the γ-ray
${\tau}_{\gamma}$
Energy E
Proton number Z
Nuclear mass number A
Initial energy level l_{i}
Final energy level l_{f}
Electric charge e
Nuclear average radius R_{0}