Microsoft Word - 10-19 Physics | 10 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 Study the Shapes of Nuclei for Heavy Elements with Mass Number Equal to (226≀A≀252) Through Determination of Deformation Parameters for two Elements (U&Cf) Haider A. Zghaier Council of Ministers / Iraqi Radioactive Sources Regulatory Authority Sameera A. Ebrahiem Hadeel Abdul_Jabbar Department of Physics, Collage of Education Ibn Al Haithem, University of Baghdad, Iraq, Baghdad samaeb85@gamil.com Article history: Received 1 October 2018, Accepted 10 October 2018, Published December 2018 Abstract The current paper focuses on the studying the forms of (even-even) nuclei for the heavy elements with mass numbers in the range from (A=226 - 252) for U and Cf isotopes. This work will consist of studying deformation parameters 𝛽 which is deduced from the "Reduced Electric Transition Probability" 𝐡 𝐸2 ↑ which is in its turn dependent on the first Excited State 2 . The "Intrinsic Electric Quadrupole Moments" (non-spherical charge distribution) 𝑄 were also calculated. In addition to that the Roots Mean Square Radii (Isotope Shift) π‘Ÿ / are accounted for in order to compare them with the theoretical results. The difference and variation in shapes of nuclei for the selected isotopes were detected using 3D-plots for them (with symmetric axes); a 2D-plot were also used for each isotope to discriminate between them by the values of semi-major is equal (a) axes and semi minor is equal (b) axes. Keyword: Nuclear Shape, Isotopes, Electric Transition, Semi minor, Deformation. 1. Introduction The state of the atomic of nucleus usually reflects the structure of the protons and neutrons shells from which it is shaped. In the case when the shells are completely filled, we attain a "magic" nucleus, which is spherical in form. Most nuclei have a tendency to be deformation on the basis that the shells are partially filled. Much of the shapes we encountered are either elongated (prolate) or flattened (oblate). These shapes can be modified between adjacent nucleus by capturing or give-away a proton or neutron. In some cases, it is sufficient re-configure the protons or neutrons within the same nucleus to change its shape. The same nucleus can therefore assume different shapes corresponding to states of different energy. 2.Theoretical 2.1. Nuclear Shape In the stable state, the natural shape of nuclei is spherical. this configuration is the optimum shape to minimize the surface energy. Nevertheless, some small deformations can be Physics | 11 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 observed, for instance, in area150 𝐴 190. The shape irregularities can be presented using the ratio [1]: 𝛿 βˆ†π‘… 𝑅 1 𝑅 Is radius average of the nu βˆ†π‘… is the differences between semi- minor and semi- major axes. βˆ†π‘… 𝑏 π‘Ž 2 But the sphere βˆ†π‘… is equal zero. 2.2. Nuclear Surface Deformations The collective motion can be clarified as vibrations and rotations of nuclear surface of the (collective model) that was initially suggested by Bohr and Mottelson [2]. The quadrupole deformation parameter 𝛽 is related to the spheroid axes [3]: 𝛽 4 3 πœ‹ 5 βˆ†π‘… 𝑅 1.06 βˆ†π‘… 𝑅 3 Where: The average radius𝑅 𝑅 𝐴 / . βˆ†R: is the difference between both of the semi- major and minor axes. As long as the value of 𝛽 is larger, the nucleus becomes more deformed. 2.3. The Root Mean Square Charge Radius (Isotopes Shift) The stated efficacy of nuclear structure, (e.g. shell closures and initiation of deformity), can be referred to by one key nuclear information which can be represented by the root mean square (rms) of nuclear charge radius 𝑅 βŒ©π‘Ÿ βŒͺ / with one of nuclear ground-state characteristics [4]. The root mean square (rms) of nuclear charge radius 𝑅 βŒ©π‘Ÿ βŒͺ / , with one of nuclear ground-state properties, is considered the key nuclear materials information which refers to stated nuclear structure effectiveness, for instance: shell closures and a deformation starting. [4].This (rms) radius βŒ©π‘Ÿ βŒͺ / can be directly driven from scattered electrons distribution; for a uniformly charged spherical shape, the radius of square charge distribution is βŒ©π‘Ÿ βŒͺ [5]: βŒ©π‘Ÿ βŒͺ 3 5 𝑅 3 5 𝑅 𝐴 / 100 4 Where: 𝐴 is mass number, 𝑅 is the radius of the sphere, and 𝑅 𝑅 𝐴 / . 2.4. Electric Quadrupole Moment The electric multi-pole moments can fairly represent the spatial distribution and charge allocation in a nucleus in a straight forward application of classical electrostatic principles [6]. Constant quadruple moments can be measured experimentally for a number of nuclei. A symmetrical axis oval shape can be anticipated for these nuclei. This classic configuration has guided the way for the definition of intrinsic quadruple moment as in the Equation (5) [7]: Physics | 12 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 𝑄 𝑑 𝜌 π‘Ÿ 3𝑧 π‘Ÿ 5 Where: 𝜌 π‘Ÿ is proton's density of radial charge and π‘Ÿ radius of charge, 𝑄 can be described as per Equation (6), given that it is calculated for a homogeneously charged ellipsoid, with a charge Ze and semi-axis (a) and (b) with later pointing along the (z) axis [8]. 𝑄 2 5 𝑍 π‘Ž 𝑏 6 If the deviation from sphericity is not very large, the average radius: 𝑅 1/2 π‘Ž 𝑏 and π›₯𝑅 𝑏 π‘Ž from Equation 2 can be presented and with 𝛿 π›₯𝑅/𝑅, from Equation 1 , the quadrupole moment is [8]: 𝑄 4 5 𝑍𝑅 𝛿 7 The nucleus quadrupole distortion parameter values Ξ΄ calculated from the Equation (8) [9]: 𝛿 0.75 𝑄 / π‘βŒ©π‘Ÿ βŒͺ 8 The semi-axes π‘Ž and 𝑏 are gained from the two following Equations (9 & 10) [10]. π‘Ž βŒ©π‘Ÿ βŒͺ 1.66 2𝛿 0.9 9 𝑏 5βŒ©π‘Ÿ βŒͺ 2π‘Ž 10 2.5. Quadrupole Deformations As a rule of thumb, nuclei with charge value 𝑍 or 𝑁 deviating from the magic number are usually deformed. The most abundant type of deformation is in quadrupole. Accordingly, the nuclei shape may either be prolate or oblate where the quadrupole deformation has one symmetry axis (𝑧) as shown in Figure 1. [11]. The Shape of nuclei deformation can be symmetrical, this can be explained by deformation factor 𝛽 which in turn associated to quadrupole moment 𝑄 , this particular case which indicates a homogeneous distribution of charge [11,12]: 𝛽 √5πœ‹ 3 𝑄 𝑍𝑅 11 Where: 𝑍 is the atomic number, 𝑅 1.2 𝐴 / fm. And 𝛽 is the deformation parameter and 𝛽 1 . 2.6. The Reduced Electric quadrupole Transition probability π‘¬πŸ ↑ In isotopes, the conversion between different nuclear states through radioactive electromagnetic transformation is the ideal way attains stable nuclear structure, and an opportunity to study different structure models [14]. The transmission of 𝐡 𝐸2 act as a decisive part in determining; the average lifetime of a nuclear state and nuclear deformation 𝛽 . It is also responsible for the volume of essential electric quadrupole moment and the energy levels of low-lying nuclei. Highest moments and transmission forces of quadrupole indicate the participation of many nucleons in the combined effects [15]. In this case, the probability of reduced electric quadrupole transition B (E2) ↑ from the spin 0 ground state to the first excited spin 2 state is specified by [16]: Physics | 13 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 𝐡 𝐸2 ∢ 0 β†’ 2 5 16πœ‹ 𝑒 𝑄 12 Where: 𝐡 𝐸2 ↑ is the reduced electric quadrupole transition probability in the unit of 𝑒 𝑏 and 𝑄 is the intrinsic quadrupole moment in unit of barn (b). The values of 𝐡 𝐸2 ↑ are required as an experimental measures independent on nuclear model used. Nevertheless, if the model in hand believed to be dependent on the measured quantity is very useful and it represents the deformation parameter 𝛽 . If charge distribution is thought to be uniform reaching up to the distance 𝑅 πœƒ, πœ‘ and charge value at zero is below (𝛽 ) can be associated with 𝐡 𝐸2 ↑ through the Equation (13) [17]: 𝛽 4πœ‹ / 3𝑍𝑅 𝐡 𝐸2 ↑/𝑒 13 𝑅 1.2 𝐴 fm 0.0144𝐴 / 𝑏 14 In accordance with the global system, the energy acknowledgement E (KeV) of the 2 state is whole that is required of creating a prediction for the corresponding 𝐡 𝐸2 ↑ (𝑒 𝑏 value [15]: 𝐡 𝐸2 ↑ 2.6 𝐸 𝑍 𝐴 15 3.Result 3.1. Deformity Parameters 𝜷𝟐 The deformations factor for the Uranium and Californium isotopes 𝛽 can be derived from 𝐡 𝐸2 ↑ (Miniature Electric Transition Probability) of even-even nucleus by the application of Equation (13). - Miniature Electric Transition Probability 𝐡 𝐸2 ↑: 0 β†’ 2 from the ground 0 to the first excited of 2 states calculated by using Equation (15). The energy 𝐸 KeV of the first excited states ( 2 was obtained from the references (18). - Average Nuclear Radius 𝑅 calculated using Equation 14 . - 3.2. Deformity Parameters: 𝜹 Another methodology to calculate deformation parameters ( 𝛿 ) is available by utilizing the actual quadrupole moments 𝑄 in Equation (8). To assess this method, the following variables need to available: - βŒ©π‘Ÿ βŒͺ which is calculated from Equation (4) for 𝐴 100 . - 𝑄 of nuclei were calculated from the Equation (11). All these values were classify in Tables 1 and 2. The major and minor axes (a and b) were calculated using Equations (9 and 10), respectively, The difference between them βˆ†π‘… was calculated using Equations (1 - 3), respectively. The results are shown in Tables 3. and 4. Physics | 14 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 Table 𝟏. Mass Number of Isotopes (A) , Neutron Number (N), Gamma Energy of the First Excited State 2 E , Nuclear Average Radius (R ), Reduced Electric Transition Probability B(E2)↑ in unit of e b , Intrinsic Quadrupole Moment Qβ‚’ in unit of barn, and Deformation Parameters Ξ² , Ξ΄ for 92U Isotopes. 𝐙 𝐀 𝐍 𝐄𝛄 πŠπžπ• [19] Theoretical Value Present Work B(E2)↑ e2b2 For SSANM [32] π›ƒπŸ π‘πŸŽ 𝟐 B(E2)↑ e2b2 𝐐ₒ (b) π›ƒπŸ 𝛅 92 226 134 80.5 10 7.161 0.2280 0.5343 7.3680 8.6064 0.2313 0.2084 228 136 59 10 7.966 0.2391 0.5374 9.9940 10.0235 0.2678 0.2413 230 138 51.72 4 8.793 0.2498 0.5406 11.3346 10.6746 0.2836 0.2555 232 140 47.522 7 9.580 0.2592 0.5437 12.2520 11.0982 0.2931 0.2641 234 142 43.498 1 10.310 0.2674 0.5468 13.3230 11.5731 0.3039 0.2739 236 144 45.242 3 10.880 0.2731 0.5499 12.7370 11.3157 0.2955 0.2663 238 146 44.91 3 11.210 0.2756 0.5530 12.7591 11.3256 0.2941 0.2650 240 148 45 1 11.490 0.2775 0.5561 12.6628 11.2827 0.2913 0.2625 Table 𝟐. Mass Number of Isotopes (A) , Neutron Number (N), Gamma Energy of the First Excited State 2 E , Nuclear Average Radius (R ), Reduced Electric Transition Probability B(E2)↑ in unit of e b , Intrinsic Quadrupole Moment Qβ‚’ in unit of barn, and Deformation Parameters Ξ² , Ξ΄ for98Cf Isotopes. 𝐙 𝐀 𝐍 𝐄𝛄 πŠπžπ• [19] Theoretical Value Present Work B(E2)↑ e2b2 For SSANM [32] π›ƒπŸ π‘πŸŽ 𝟐 B(E2)↑ e2b2 𝐐ₒ (b) π›ƒπŸ 𝛅 98 244 146 40 2 14.970 0.2941 0.5623 15.9872 12.6776 0.3039 0.2739 248 150 41.53 6 15.510 0.2961 0.5684 15.2322 12.3746 0.2935 0.2645 250 152 42.722 15.680 0.2962 0.5715 14.7281 12.1681 0.2870 0.2587 252 154 45.72 5 15.840 0.2961 0.5745 13.6894 11.7312 0.2753 0.2481 Table 3. Mass number A , Neutron Number N , Root Mean Square Radii r / , Major and minor axes (a, b) and the difference between them (βˆ† R) by three methods for U Isotopes 𝒁 𝑨 𝑡 Theoretical Value Present Work βŒ©π’“πŸβŒͺ𝟏 πŸβ„ π’‡π’Ž [18] βŒ©π’“πŸβŒͺ𝟏/𝟐fm 𝒂 π’‡π’Ž 𝒃 π’‡π’Ž βˆ† π‘ΉπŸ (fm) βˆ† π‘ΉπŸ (fm) βˆ† π‘ΉπŸ‘ (fm) 92 226 134 ------- 5.8017 2.6424 3.8787 1.3585 1.2363 1.5951 228 136 ------- 5.8188 2.5646 3.9924 1.5776 1.4278 1.8523 230 138 ------- 5.8357 2.5323 4.0440 1.6752 1.5117 1.9669 232 140 ------- 5.8526 2.5138 4.0774 1.7366 1.5636 2.0390 234 142 5.8291 5.8694 2.4920 4.1142 1.8058 1.6222 2.1202 236 144 5.8431 5.8860 2.5154 4.0958 1.7606 1.5804 2.0672 238 146 5.8571 5.9026 2.5222 4.0975 1.7572 1.5753 2.0632 240 148 ------- 5.9191 2.5322 4.0953 1.7457 1.5631 2.0496 Physics | 15 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 4. Discussion The present study focuses on nuclei characterized by even-even form for the heavy elements with mass numbers equal to 226 𝐴 252 , which were included in deformation parameters study 𝛽 & 𝛿 derived from the 𝐡 𝐸2 ↑ and 𝑄 values. It is found that the first excited state energy levels 2 of these nuclei (which show a collective behavior), begin to change smoothly when the mass number 𝐴 increases, and the nucleons outside the core polarize either the whole or partial vibrations of the core to one direction and permanent deformation of the nucleus can be acquired. Also from evidentβŒ©π‘Ÿ βŒͺ / , it was noticed that the increase in βŒ©π‘Ÿ βŒͺ / values is comparable with (A) increase. For evaluation reasons, it was noticed that the the calculated values of present work βŒ©π‘Ÿ βŒͺ / nicely correlated with experimental values of βŒ©π‘Ÿ βŒͺ / in Ref [18]. Also, the values of (βˆ†R) were calculated using three different methods, and it was found that these results were fairly close. 4.1. Uranium isotopes π‘ΌπŸ—πŸ πŸπŸπŸ” πŸπŸ’πŸŽ In Table 1 [19], stated that the minimum value of 𝛽 0.2313 , corresponding to the energy of the first excited state 𝐸 80.5 MeV for (U-226), and the highest is 𝛽 0.3039 corresponding to the of the first excited state energy [20] 𝐸 43.498 KeV for (U-234). Other values are ranging between these values as shown in the Figure 1. It is also noted that these values of 𝛽 are considered large values, which means large deformed shapes as shown in Figures (3. & 5.), corresponds small values of the first excited state energies (the gaps between shells are low spaces), also the number of nucleons in the sub- shell outside closed shell are filled with many nucleons and the collective motion of these nucleons will be rotational motion. On the other hand, and 𝑄 are significant. 4.2.Californium isotopes π‘ͺπ’‡πŸ—πŸ– πŸπŸ’πŸ’ πŸπŸ“πŸ Table 2. shows that there are four isotopes chosen form this element, the highest value of the deformation parameter belongs to 𝐢𝑓 and equals to 𝛽 0.3039 , corresponding to the first excited state energy 𝐸 40 KeV .And the lowest value of the deformation parameter belongs to 𝐢𝑓 equals to 𝛽 0.2753 corresponds to the energy of the first excited state 𝐸 45.72 KeV . Figure 2. shows this behavior. Also we note that the number of nucleons 𝑍 98, 244 𝑁 252 that fill the shells outside closed shell are large, the energy values of the first excited state of the selected Californium isotopes are considered very small if compared with the energy of the closed shell (the distances between the ground Table 4. A Mass number, N Neutron Number, r / Root Mean Square Radii, (a, b) Major and minor axes and the difference between them (βˆ† R) by three methods for Cf Isotopes. 𝐙 𝐀 𝐍 Theoretical Value Present Work 〈𝐫𝟐βŒͺ𝟏 πŸβ„ 𝐟𝐦 [18] 〈𝐫𝟐βŒͺ𝟏/𝟐 fm 𝐚 𝐟𝐦 𝐛 𝐟𝐦 βˆ† π‘πŸ (fm) βˆ† π‘πŸ (fm) βˆ† π‘πŸ‘ (fm) 98 244 146 ---- 5.9518 2.5094 4.1430 1.8313 1.6336 2.1501 248 150 ---- 5.9841 2.5410 4.1240 1.7779 1.5830 2.0874 250 152 ---- 6.0002 2.5596 4.1107 1.7435 1.5512 2.0471 252 154 ---- 6.0161 2.5905 4.0816 1.6765 1.4911 1.9683 Physics | 16 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 state and the first excited state are small so that the transition of the nucleons between these two states is very easy). Therefore, the values of the reduced electrical transition will be large. In addition to that, the values of intrinsic electric Quadrupole moments are large too. All these factors work to make the deformation parameters large, which in turn make the nuclei of these isotopes permanently deformed, and elliptical shapes as shown in Figure 4. In addition, the collective motion of nucleons in the external orbits is rotational motion. Figure 1. Deformation Parameter (Ξ²2) value of the (92U) Isotopes as function of neutron Number. Figure 2. Deformation Parameter (Ξ²2) value of the (98Cf) Isotopes as function of neutron Number. 144 146 148 150 152 154 156 0.27 0.275 0.28 0.285 0.29 0.295 0.3 0.305 0.31 0.315 0.32 Neutron Number (N) D e fo rm a tio n P a ra m e te r (  2 )  2 for P.W. Physics | 17 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 Figure 3. Shapes of axially symmetric quadrupole deformation for π‘ˆisotope from major (a) and minor (b) axes. Figure 4. Shapes of axially symmetric quadrupole deformation for 𝐢𝑓 isotope from major (a) and minor (b) axes. Figure 5. Axially-symmetric quadrupole deformations, the prolate shape of Nucleus with π‘₯ 𝑦 𝑧) (where z is the minor axis (b) (symmetric axis) and (x, y) are major axes (a) for the U isotopes. -4 -2 0 2 4 -4 -3 -2 -1 0 1 2 3 4 Major axis (a) M in o r a xi s (b ) 92 U isotopes 226 92 U 228 92 U 230 92 U 232 92 U 234 92 U 236 92 U 238 92 U 240 92 U -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Major axis (a) M in o r a x is ( b ) 92 U isotopes 226 92 U 228 92 U 230 92 U 232 92 U 234 92 U 236 92 U 238 92 U 240 92 U -3 -2.5 -2 -1.5 -1 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Major axis (a) M in o r a x is ( b ) 98 Cf Isotopes 244 98 Cf 246 98 Cf 248 98 Cf 250 98 Cf -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 Major axis (a) M in o r a xi s ( b ) 98 Cf Isotopes 244 98 Cf 246 98 Cf 248 98 Cf 250 98 Cf -2 0 2 -2 0 2 -3 -2 -1 0 1 2 3 x 234 92 U Isotope y z -2 0 2 -2 0 2 -3 -2 -1 0 1 2 3 x 226 92 U Isotope y z Physics | 18 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 Figure 6. Axially-symmetric quadrupole deformations, the prolate shape of Nucleus with π‘₯ 𝑦 𝑧) (where z is the minor axis (b) (symmetric axis) and (x, y) are major axes (a) for Cf isotopes. 5. Conclusions From the results and the Figures of 2D and 3D shapes of the nuclei we conclude the following: - It is found that the energy of the first excited state 2 of these nuclei begin to change smoothly when the mass number 𝐴 increases, and the nucleons outside the core polarize either the whole or partial vibrations of the core to one direction and permanent deformation of the nucleus can be acquired and the nuclei are stable and non-spherical shape. - Far from magic numbers, the motion of the nucleons out of closed shell will be rotational motion and the nucleus will be more distorted especially in the region (242β©½Aβ©½252). - The prolate shape is the dominant form of deformed nuclei. References 1. Wong, S. S. M. Introductory Nuclear Physics; Second Ed.; WILEY-VCH Verlag GmbII Co. KGaA, Weinheim 2004. 2. Bohr, A.; Mottelson, B. R.; Nuclear structure,.II Nuclear Deformations ; World Scientific Publishing Co. Pte. Ltd 1998. 3. Roy, R. R.; Nigam, B.P.; Nuclear Physics Theory and Experiment; John Wiley Sons: INC. 1967. 4. Greiner, W. and Maruhn, J. A.; Nuclear Models.Springer- Verlag Berlin Heidelberg. 1996. 5. Boboshin. I.; Boboshin. B.; Orlin, S. V.; Peskov. N.; Varlamov. V.; Investigation of Quadrupole Deformation of Nucleus and its Surface Dynamic Vibrations International Conference on Nuclear Data for Science and Technology. 2007 doi: 10.1051/ndata:07103.65-68. 6. Neyens. G. Nuclear magnetic and quadrupole moments for nuclear structure research on exotic nuclei; IOP Publishing Ltd. Printed in the UK Rep. Prog. Phys. 2003, 66, 633–689. -2 0 2 -2 0 2 -3 -2 -1 0 1 2 3 x 244 98 Cf Isotope y z -2 0 2 -2 0 2 -3 -2 -1 0 1 2 3 x 252 98 Cf Isotope y z Physics | 19 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/ 31.3.2022 Vol. 31 (3) 2018 7. Mohammadi. S. Quadrupole Moment Calculation of Deformed Nuclei; Journal of Physics: Conference Series. 2012. 381, 012129. doi:10.1088/1742- 6596/381/1/012129. 8. Henley, E. M.; Garcia, A. Subatomic Physics; 3rd ed.; World Scientific Publishing Co. Pte. Ltd. 2007. 9. Boboshin, I.; Ishkhanov, B.; Komarov,S. Investigation of quadrupole deformation of nucleus and its surface dynamic vibrations ; International Conference on Nuclear Data for Science and Technology : CEA, published by EDP Sciences 2008. 10. Abdul wahab. R. A. Deformation parameters and nuclear radius of Zirconium (Zr) isotopes using the Deformed Shell Model. Wasit Journal for Science & Medicine. 2009, 2, 1, 115 - 125 11. Basdevant, J. L; Rich, J.; Spiro, M. Fundamentals in Nuclear Physics; from Nuclear Structure to Cosmology; Springer Science+Business Media, Inc: 2005. 12. Ertugrala. F.; Guliyev. E.; Kuliev. A.A. Quadrupole Moments and Deformation Parameters of the 166-180Hf, 180-186W and 152-168Sm Isotopes. 2015, doi: 10.12693/A Phys. Pola. 128.B-254, Acta Physica Polonica A. 13. Margraf. J; Heil, R.D; Kneissl, U. and Maier, U. Deformation dependence of low lying M1 strengths in even isotopes, physical review. 1993 ,47, 4. 14. Bohr, A.; Mottelson, B. R. Nuclear Structure, Volume II: Nuclear Deformations. World Scientific, Singapore 1998. 15. Raman. S; Nestor. C. W; Tikkanen. P. At. Data Nucl. Tables. 2001, 78, 1. 16. Haberichter, M; Lau, P. H. C.; Manton, N. S. Electromagnetic Transition Strengths for Light Nuclei in the Skyrme model. Kent Academic Repository. 2015. 17. Raman, S. A. Tale of Two Compilations: Quadrupole Deformations and Internal Conversion Coefficients Journal of Nuclear Science and Technology, 2002. 450-454, doi: 10.1080/00223131.10875137. 18. Angeli. a. I; Marinova. K.P. Table of experimental nuclear ground state charge radii; An update. Atomic Data and nuclear Data Table 99, Elsevier Inc: 2013, 69-95. 19. Fireston. R.B. an Shirly. V.S.; "Table of Isotopes eighth edition", Newyork. 1999, 99. 20. Mjebal. H. J.; Jarallah. N. T.; Ebrahiem S. A.; Rabee. R. F. Theoretical Calculation of The Binding And Excitation Energies For 30 Using Shell Mode And Perturbation Theory. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 2013, 26, 3. 21. Ebrahiem. S. A.; Zghaier. H. A. Estimation of geometrical shapes of mass-formed nuclei (A=102-178) from the calculation of deformation parameters for two elements (Sn & Yb); 2018 IOP Publishing, 1003, 1, 012095.