1. Introduction
Recently, Ruthenium isotope has been a focus of the nuclear structure of many theoretical and experimental investigations. The low-lying even nuclei had been successfully explained nuclear collective characters using the interacting boson model-1 (IBM-1) [1]. In the first beginning the collective states can be described by a system of identical bosons N_{B}. These are S-boson L = 0 and d-boson L = 2. There is no discrepancy between neutron and proton in IBM-1. There are three dynamical symmetries indicated by U(5), SU(3) and O(6) analogous to spherical vibrator, deformed rotor, and γ-soft respectively. The microscopic a harmonic vibrator approach (MAVA) used in investigating the lower level collective states in Ruthenium isotopes [2].
The Ruthenium isotopes have atomic number Z = 44. It belongs near to closed shell Sn (magic number Z = 50). The external forms of even ^{104-106}Ru isotopes have
${g}_{9/2}^{-6}$ (6 proton holes) and
${g}_{9/2}^{10,12}$ (10 and 12 neutron particles) close to magic number 50. This configuration has been investigated the ground state structure from spherical to deformed symmetry. The edifice of yrast levels and electromagnetic strength of Ru isotopes studied by many scientists [3] [4] [5] [6] [7].
Recently, the properties of the yrast level were studied in Pd isotopes with even neutron N = 54 - 64 [8]. The electromagnetic reduced transition strength of Cd isotopes with N = 66 - 74 were investigated [9]. The B(E2) value of yrast band of even ^{102-112}Pd and ^{96-102}Ru isotopes [10] [11] [12] were investigated by interacting boson model (IBM-1). The low-lying level of ^{184}W and ^{184}Os nuclei were investigated [13] [14].
The present aim particularly focuses on the structure of the ground state band and the potential energy surfaces to find the dynamical symmetry of even ^{104-106}Ru isotopes by the application of IBM.
2. Method of Calculation
The Interacting Boson Model (IBM) gives occupation to truncated model space for nuclei with N number of nucleons. It provides a quantitative description of identical particles with forming pairs of angular momentum 0 and 2.
The Hamiltonian of IBM-1 [15]:
$H={\displaystyle \underset{i=1}{\overset{N}{\sum}}{\epsilon}_{i}}+{\displaystyle \underset{i\langle j}{\overset{N}{\sum}}{V}_{i}{}_{j}}$ (1)
Here
$\epsilon $ is energy of boson and V_{ij} is the potential energy of boson between i and j.
Hamiltonian is from multi-pole form [16]
$H=\epsilon {\stackrel{^}{n}}_{d}+{a}_{0}\left(\stackrel{^}{P}\cdot \stackrel{^}{P}\right)+{a}_{1}\left(\stackrel{^}{L}\cdot \stackrel{^}{L}\right)+{a}_{2}\left(\stackrel{^}{Q}\cdot \stackrel{^}{Q}\right)+{a}_{3}\left({\stackrel{^}{T}}_{3}\cdot {\stackrel{^}{T}}_{3}\right)+{a}_{4}\left({\stackrel{^}{T}}_{4}\cdot {\stackrel{^}{T}}_{4}\right)$ (2)
Here
${\stackrel{^}{n}}_{d}=\left({d}^{\u2020}\cdot \stackrel{\u02dc}{d}\right)$,
$\stackrel{^}{P}=\frac{1}{2}\left(\stackrel{\u02dc}{d}\cdot \stackrel{\u02dc}{d}\right)-\frac{1}{2}\left(\stackrel{\u02dc}{s}\cdot \stackrel{\u02dc}{s}\right)$
$\stackrel{^}{L}=\sqrt{10}{\left[{d}^{\u2020}\times \stackrel{\u02dc}{d}\right]}^{(1)}$
$\stackrel{^}{Q}={\left[{d}^{\u2020}\times \stackrel{\u02dc}{s}+{s}^{\u2020}\times \stackrel{\u02dc}{d}\right]}^{\left(2\right)}-\frac{1}{2}\sqrt{7}{\left[{d}^{\u2020}\times \stackrel{\u02dc}{d}\right]}^{(2)}$
${\stackrel{^}{T}}_{3}={\left[{d}^{\u2020}\times \stackrel{\u02dc}{d}\right]}^{\left(3\right)}$,
${\stackrel{^}{T}}_{4}={\left[{d}^{\u2020}\times \stackrel{\u02dc}{d}\right]}^{(4)}$
Here P is the pairing operator for s and d bosons, Q is quadrupole operator,
${\stackrel{^}{n}}_{d}$ is number of d boson, L is operator of angular momentum, and T_{3} octuplet operators and T_{4} is hexadecapole operators.
The Hamiltonian starting with U(6) and finishing with group O(2) as given in Equation (2) is bringing to a lower state of three limits, γ-soft O(6), the vibration U(5) and the rotational SU(3) nuclei [17]. We know that in the SU(3) limits, the effective parameter is the quadrupole
${a}_{2}$, in the O(6) limit the effective parameter is the pairing
${a}_{0}$, in U(5) limits, the effective parameter is
$\epsilon $.
The Hamiltonian and eigen-values for the three limits [18]:^{ }
U(5):
$\begin{array}{l}{\stackrel{^}{H}}_{U\left(5\right)}=\epsilon {\stackrel{^}{n}}_{d}+{a}_{1}\left(\stackrel{^}{L}\cdot \stackrel{^}{L}\right)+{a}_{3}\left({\stackrel{^}{T}}_{3}\cdot {\stackrel{^}{T}}_{3}\right)+{a}_{4}\left({\stackrel{^}{T}}_{4}\cdot {\stackrel{^}{T}}_{4}\right)\\ E\left({n}_{d},\upsilon ,L\right)=\epsilon {n}_{d}+{K}_{1}{n}_{d}\left({n}_{d}+4\right)+{K}_{4}\upsilon \left(\upsilon +3\right)+{K}_{5}L\left(L+1\right)\end{array}\}$ (3)
with
${K}_{1}=1/12{a}_{1}$ _{ }
${K}_{4}=-1/10{a}_{1}+1/7{a}_{3}-3/70{a}_{4}$ _{ }
${K}_{5}=-1/14{a}_{3}+1/14{a}_{4}$ _{ }
O(6):
$\begin{array}{l}{\stackrel{^}{H}}_{0\left(6\right)}={a}_{0}\stackrel{^}{P}\cdot \stackrel{^}{P}+{a}_{1}\stackrel{^}{L}\cdot \stackrel{^}{L}+{a}_{3}{\stackrel{^}{T}}_{3}\cdot {\stackrel{^}{T}}_{3}\\ E\left(\sigma ,\tau ,L\right)={K}_{3}\left[{N}_{b}\left({N}_{b}+4\right)-\sigma \left(\sigma +4\right)\right]+{K}_{4}\tau \left(\tau +3\right)+{K}_{5}L\left(L+1\right)\end{array}\}$ (4)
with
${K}_{3}=1/4{a}_{0}$ _{ }
${K}_{4}=1/2{a}_{3}$
${K}_{5}=-1/10{a}_{3}+{a}_{1}$
SU(3):
$\begin{array}{l}{\stackrel{^}{H}}_{SU\left(3\right)}={a}_{1}\stackrel{^}{L}\cdot \stackrel{^}{L}+{a}_{2}\stackrel{^}{Q}\cdot \stackrel{^}{Q}\\ E\left(\lambda ,\mu ,L\right)={K}_{2}\left({\lambda}^{2}+{\mu}^{2}+3\left(\lambda +\mu \right)+\lambda \mu \right)+{K}_{5}L\left(L+1\right)\end{array}\}$ (5)
with
${K}_{2}=1/2{a}_{2}$ _{ }
${K}_{5}={a}_{1}-3/8{a}_{2}$ _{ }
K_{1}, K_{2}, K_{3}, K_{4}, and K_{5} are other forms of strength parameters.
Then applying particular limit of symmetry (O(6), SU(3), U(5)) to determine the frame of a set of nuclei is more advantageous than full Hamiltonian of IBM-1. It comprise multi-free parameters those make it simple to fit the structure of a nuclei. A flaw chart of method of calculation is given in Figure 1.
3. Results and Discussion
The obtained results have discussed for yrast energy level, γ-band, β-band, effective charge used to reproduce B(E2) values, transition probabilities B(E2), maxing ratio and contour plots of the potential energy surfaces using IBM-1.
The γ-unstable limit has applied for ^{104,106}Ru nuclei using data of experimental energy ratios (E_{2}: E_{4}: E_{6}: E_{8} = 1:2.5:4.5:6.5). In the framework of IBM-1, the even
Figure 1. Flowchart of method of calculation.
^{104-106}Ru nuclei have three protons boson hole and five and six neutrons boson particle respectively. Therefore total bosons numbers of ^{104}Ru and ^{106}Ru nuclei are 8 and 9, respectively. The IBM-1 models carry out with no difference between the bosons of proton and neutron. The energy ratio
$\text{R}=\text{E}{4}_{1}^{+}/\text{E}{2}_{1}^{+}$ gives the information of the symmetry shapes of a nucleus. The symbol
$\text{E}{2}_{1}^{+}$ and
$\text{E}{4}_{1}^{+}$ is at the energy level
${2}_{1}^{+}$ and
${4}_{1}^{+}$ respectively. It is known that the
$R=\text{E}{4}_{1}^{+}/\text{E}{2}_{1}^{+}\approx 2$ is for U(5),
$R=\text{E}{4}_{1}^{+}/\text{E}{2}_{1}^{+}\approx 2.5$ is for O(6) and
$R=\text{E}{4}_{1}^{+}/\text{E}{2}_{1}^{+}\approx 3.33$ for SU(3) [19] [20]. The experimental R_{4/2} of ^{104}Ru and ^{106}Ru isotopes is 2.48 and 2.60, respectively. Figure 2 shows, R_{4/2} values of ^{104}Ru and ^{106}Ru isotopes are O(6) symmetry.
Figure 2. The arrows indicate the line of
$\text{E}\left({4}_{1}^{+}\right)/\text{E}\left({2}_{1}^{+}\right)$ values of the U(5), O(6) and SU(3) limits. The
$\text{E}\left({4}_{1}^{+}\right)/\text{E}\left({2}_{1}^{+}\right)$ values of experimental data^{22} of the ^{104,106}Ru isotopes are presented as function of neutrons.
The best fit was taken up to 12^{+} of Ru isotopes with neutron N = 60, 62. The parameters were determined the experimental eigen values (E(n_{d}, υ, L)) from the Equation (4), where n_{d}, υ and L are quantum numbers. The parameters in the present data are shown in Table 1.
The calculated energy levels as well as experimental data are presented in Table 2. According to the weight of fitting the Ru-104 and Ru-106 nuclei are good candidates of O(6) symmetry. The calculation of γ-bands and β-bands are compared with experimental data and presented to Table 3 and Table 4. From the tables, the IBM calculations and experimental results are in good agreements [21].
The reduced electric transition probabilities give the more information on the structure of nuclei. The E2 transition operator must be a Hermitian tensor of rank two; consequently, the number of bosons must conserve.
${T}^{E2}={\alpha}_{\text{2}}{\left[{d}^{\u2020}s+{s}^{\u2020}d\right]}^{\left(2\right)}+{\beta}_{\text{2}}{\left[{d}^{\u2020}d\right]}^{\left(2\right)}$ (6)
Here T^{E}^{2} is the operator of reduced matrix elements of the E2. (s^{†}, d^{†}) are creation and (s, d) are annihilation operators for s and d bosons. α_{2} indicated the effective quadrupole charge and β_{2} is dimensionless coefficient, β_{2} = χα_{2}
$B\left(E2,{J}_{i}\to {J}_{f}\right)=\frac{1}{2{J}_{i}+1}{\left|\langle {J}_{f}|{T}^{E2}|{J}_{i}\rangle \right|}^{2}$ (7)
The parameters, α_{2} and β_{2} of Equation (6), were adjusted to reproduce the experimental
$B\left(E2,{2}_{1}^{+}\to {0}_{1}^{+}\right)$. The effective charge (e_{B}) in present calculation is shown in Table 5. The values of e_{B} were estimated to reproduce experimentally
$B\left(E2,{2}_{1}^{+}\to {0}_{1}^{+}\right)$. The values β_{2} = 0 for ^{104,106}Ru isotopes because these nuclei have the O(6) property. The calculated values of B(E2) transitions with experimental data are presented in Table 6 for Ru isotopes with neutron N = 60, 62 in this study [21]. The calculated data of IBM-1 is good agreements with the available experimental results.
Table 1. Adopted values for the parameters used for IBM-1 calculations. All parameters are given in MeV, excepted N and CHQ.
Table 2. g-band (in MeV) for even ^{104-106}Ru nuclei.
Table 3. γ-band (in MeV) for even ^{104-106}Ru nuclei.
Table 4. β-band (in MeV) for even ^{104-106}Ru nuclei.
Table 5. Effective charge used to reproduce B(E2) values for even ^{104-106}Ru nuclei.
Table 6. Experimental and the IBM-1 values of B(E2) for even ^{104-106}Ru nuclei (in e^{2}b^{2}).
The application of potential energy surface (PES) gives the information to find microscopic and geometric shapes such as spherical, prolate, oblate and γ independent (γ soft). It gives us about symmetry, the shape of nuclei, the minimum deepness and the change of the shape. The PES of the IBM Hamiltonian was drawn by the Skyrme mean with
$|N,\beta ,\gamma \rangle =\frac{1}{\sqrt{N!}}{\left({b}_{c}^{+}\right)}^{N}|0\rangle ,$ (8)
${b}_{c}^{+}={\left(1+{\beta}^{2}\right)}^{-1/2}\left\{{S}^{+}+\beta \left[\mathrm{cos}\gamma \left({d}_{0}^{+}\right)+\sqrt{1/2}\mathrm{sin}\gamma \left({d}_{2}^{+}+{d}_{-2}^{+}\right)\right]\right\},$ (9)
The energy surface E(N, β, γ) for O(6) limits as a function of β and γ, has been calculated [1] [22]. Here, β were indicated the total deformation of a nucleus. Figure 3 shows the contour plots in the γ-β plane resulting from E(N, β, γ) for ^{104}Ru and ^{106}Ru isotopes. The potential surfaces are approximately independent of gamma only. In this figure, the color lines show the values of the potential energy surface in MeV. The mapped IBM energy surfaces of ^{104}Ru and ^{106}Ru are O(6) characters.
Figure 3. Contour map of potential energy surfaces for Ru isotopes with N = 60, 62.
4. Conclusion
The yrast band, gamma band and beta band, electromagnetic transition and potential energy surface of ^{104}Ru and ^{106}Ru isotopes calculated in terms of O(6) limit of interacting boson model-1. The energy levels up to 12^{+} of ^{104,106}Ru nucleus found by the best fitted of the parameters in the Hamiltonian of the IBM-1. The analyses of the IBM-1 results for the ground state band suggest a satisfactory agreement with the experimental data. The nobility and contribution of this work included that the framework of interacting boson approximations shows the Ru with neutron numbers 60 and 62 considered gamma soft O(6) symmetry.
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant No. G-223-662-37. The authors, therefore, acknowledge with thanks DSR for technical and financial support. The authors are very grateful to the University of Mosul/College of Education for Pure Science for their provided facilities, which helped to improve the quality of this work.