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 )

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" B 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) r / 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.


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 observed, for instance, in area150  190.The shape irregularities can be presented using the ratio [1]: Is radius average of the nu ∆ is the differences between semi-minor and semi-major axes.
∆   2 But the sphere ∆ is equal zero.

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  is related to the spheroid axes [3]: Where: The average radius   / .∆R: is the difference between both of the semimajor and minor axes.As long as the value of  is larger, the nucleus becomes more deformed.

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  〈 〉 / with one of nuclear ground-state characteristics [4].The root mean square (rms) of nuclear charge radius  〈 〉 / , 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 〈 〉 / can be directly driven from scattered electrons distribution; for a uniformly charged spherical shape, the radius of square charge distribution is 〈 〉 [5]: Where:  is mass number,  is the radius of the sphere, and    / .

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]: Where:   is proton's density of radial charge and  radius of charge,  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].
If the deviation from sphericity is not very large, the average radius:  1/2   and    from Equation 2 can be presented and with  /, from Equation 1 , the quadrupole moment is [8]: The nucleus quadrupole distortion parameter values δ calculated from the Equation (8) [9]:  0.75  / 〈 〉 8 The semi-axes  and  are gained from the two following Equations (9 & 10) [10].

Quadrupole Deformations
As a rule of thumb, nuclei with charge value  or  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 () as shown in Figure 1.[11].
The Shape of nuclei deformation can be symmetrical, this can be explained by deformation factor  which in turn associated to quadrupole moment  , this particular case which indicates a homogeneous distribution of charge [11,12]: Where:  is the atomic number,  1.2  / fm.And  is the deformation parameter and  1 .

The Reduced Electric quadrupole Transition probability 𝑬𝟐 ↑
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  2 act as a decisive part in determining; the average lifetime of a nuclear state and nuclear deformation  .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]: 𝑄 12 Where:  2 ↑ is the reduced electric quadrupole transition probability in the unit of   and  is the intrinsic quadrupole moment in unit of barn (b).
The values of  2 ↑ 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  .If charge distribution is thought to be uniform reaching up to the distance  ,  and charge value at zero is below ( ) can be associated with  2 ↑ through the Equation ( 13) [17]: 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  2 ↑ (  value [15]:  2 ↑ 2.6    15

3.Result 3.1. Deformity Parameters 𝜷 𝟐
The deformations factor for the Uranium and Californium isotopes  can be derived from  2 ↑ (Miniature Electric Transition Probability) of even-even nucleus by the application of Equation (13).
-Miniature Electric Transition Probability  2 ↑: 0 → 2 from the ground 0 to the first excited of 2 states calculated by using Equation (15).The energy  KeV of the first excited states ( 2 was obtained from the references (18).-Average Nuclear Radius  calculated using Equation 14. -

Deformity Parameters: 𝜹
Another methodology to calculate deformation parameters (  ) is available by utilizing the actual quadrupole moments  in Equation (8).To assess this method, the following variables need to available: -〈 〉 which is calculated from Equation (4) for  100 .
- 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 ∆ was calculated using Equations (1 -3), respectively.The results are shown in Tables 3. and 4.

Discussion
The present study focuses on nuclei characterized by even-even form for the heavy elements with mass numbers equal to 226  252 , which were included in deformation parameters study  &  derived from the  2 ↑ and  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  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 evident〈 〉 / , it was noticed that the increase in 〈 〉 / values is comparable with (A) increase.For evaluation reasons, it was noticed that the the calculated values of present work 〈 〉 / nicely correlated with experimental values of 〈 〉 / in Ref [18].Also, the values of (∆R) were calculated using three different methods, and it was found that these results were fairly close.

Uranium isotopes 𝑼 𝟗𝟐 𝟐𝟐𝟔 𝟐𝟒𝟎
In Table 1 [19], stated that the minimum value of  0.2313 , corresponding to the energy of the first excited state  80.5 MeV for (U-226), and the highest is  0.3039 corresponding to the of the first excited state energy [20]  ), corresponds small values of the first excited state energies (the gaps between shells are low spaces), also the number of nucleons in the subshell outside closed shell are filled with many nucleons and the collective motion of these nucleons will be rotational motion.On the other hand, and  are significant.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.

4.2.Californium isotopes 𝑪𝒇 𝟗𝟖 𝟐𝟒𝟒 𝟐𝟓𝟐
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.

1 .
43.498 KeV for (U-234).Other values are ranging between these values as shown in the Figure It is also noted that these values of  are considered large values, which means large deformed shapes as shown in Figures (3.& 5.

Figure 3 .
Figure 3. Shapes of axially symmetric quadrupole deformation for  isotope from major (a) and minor (b) axes.

Figure 4 .
Figure 4. Shapes of axially symmetric quadrupole deformation for  isotope from major (a) and minor (b) axes.

Figure 5 .
Figure 5. Axially-symmetric quadrupole deformations, the prolate shape of Nucleus with   ) (where z is the minor axis (b) (symmetric axis) and (x, y) are major axes (a) for the U isotopes.

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 Qₒ in unit of barn, and Deformation Parameters β , δ for 92U Isotopes.

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 Qₒ in unit of barn, and Deformation Parameters β , δ for98Cf Isotopes.

Table 3 .
Mass number A , Neutron Number N , Root Mean Square Radii r

Table 2 .
shows that there are four isotopes chosen form this element, the highest value of Figure2.shows this behavior.Also we note that the number of nucleons  98, 244  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 (∆ R) by three methods for Cf Isotopes.