215 

This work is licensed under a Creative Commons Attribution 4.0 International License 
 

 IHJPAS. 37 (2) 2024 

Ibn Al-Haitham Journal for Pure and Applied Sciences Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

PISSN: 1609-4042, EISSN: 2521-3407 

   

Safa Mohsin Ibrahim1* ,  Abdallah Ben Rhaiem2  and Sameera Ahmed Ebrahiem3  

 1 Department of Physics, Faculty of Sciences of Sfax, Sfax, Tunisia. 
2 Department of Physics, Faculty of Sciences, Sfax, Tunisia. 

3Department of Physics, College of Education for Pure Sciences (Ibn Al-Haitham), Baghdad, Baghdad, Iraq 

*Corresponding Author. 

 

 

 

 

Abstract 

In the current research, the properties of even-even nuclei with a mass number (A≥100) 

for ( 𝐶𝑑48
100−124  ) isotopes have been studied. This included distortion parameters (𝛿) derived ,from an 

Intrinsic, electric, quadrupole, moments (𝑄0)and deformation, parameters (𝛽2) originate from a 

reduced, electric, transition probability 𝐵(𝐸2) ↑  based on the energy of the first excited state 

(2+).,Roots ,mean ,square radii < 𝑟2 >1/2 , major and minor of oval axes (a, b) in addition to the 

difference between them (ΔR) calculated for Cadmium isotopes. All of these parameters were 

calculated for cadmium isotopes by deforming the model of the shell equation in Matrix laboratory 

software. Major (a) and minor (b) axes were utilized to plot three-dimensional shapes (axially 

symmetric) and sketch two-dimensional shapes of the isotopes, allowing for the differentiation of 

various single-element isotopes. According to the latest research, the deformation parameters get 

smaller as the neutron count gets closer to the enchanted number of neutrons.  

Keywords: transition probability B(E2;0+→2+ )↑, electric quadrupole moment Qo, deformation 

parameters (𝛽2, δ), mean-squared charge distribution radius < 𝑟2 >1/2. 

 

1. Introduction 

    Previously, a hypothesis was put forward to account for the crustal structure observed in cores [1]. 

The closed shells evolve when big differences occur in the orbitals of the nuclei, such as at the neutron 

and proton "magic" numbers of 2, 8, 20, 28, 50, 82, and 126 [2]. The bonding energy for the final 

nucleon at shell closures is significantly higher than the value encountered in the neighboring cores 

[1,3]. Because they have a preferred axis, non-spherical nuclei can rotate. For deformed nuclei, this 

implies that they are axially symmetric, either oblate or prolate, as shown in Figure 1 below, and that 

the nuclear 2 volume is constant (incompressibility) for actual solutions [1,4]. The main reason for the 

deformation of the nuclei is the arrangement of the valence nucleons in the unfilled nucleus shell; in 

other words, the deformation only occurs when the proton (Z) and neutron (N) shells are more or less 

packed [2, 5,6]. 

Calculating of the Electric Transition Forces and Radii of Even-even- Nuclei of 

Cadmium  ( 𝑪𝒅𝟒𝟖
𝟏𝟎𝟎−𝟏𝟐𝟒  ) Cd Isotopes 

 

doi.org/10.30526/37.2.3468 

Received: 21 May 2023                     Accepted: 25 June  2023                Published: 20 April 2024 

 

https://creativecommons.org/licenses/by/4.0/
https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042
https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407
https://orcid.org/0009-0000-3729-1188
mailto:Physicist_safa@yahoo.com
https://orcid.org/0000-0002-3703-8104
mailto:abdallah.benrhaiem@fss.usf.tn
https://orcid.org/0000-0002-4797-4078
mailto:sameera.a.i@ihcoedu.uobaghdad.edu.iq


IHJPAS. 37 (2) 2024 

 

216 
 

 
Figure 1. A graphic, representation to three nuclear: (a) spherical, (b) oblate, and (c) prolate z-axis contributes to axis of 

symmetry to a flattened & elongated shape [6]. 

 

The present work aims to determine the shape of the nucleus of even isotopes with an atomic mass 

(A) greater than or equal to 100 (A≥100) by calculating the quaternary nucleus distortion coefficient 

(β2) from the low transition potential B (E2) ↑ of 0+ → 2+ obtained from the energy of the first excited 

state of the isotopes, and comparing it to the distortion coefficient (β2) from the expected value B 

(E2). ↑ (SSANM) and calculate the quadruple nucleus deformation coefficients (δ) from the quadruple 

moment Qₒ. 

 

2.  Theoretical Part 

     For spherically symmetric charge-distribution nuclei, an intrinsic quadrupole moment (Q0) equals 

zero for negative to oblate nuclei, and positive to prolate nuclei. In terms of the inherent frame of 

reference [7,8]. The nucleus' inherent moment (Qo) is linked to a decreased transition probability. 

Ɓ(E2), which is connected to deformation nuclei or departure from spherical shape [9,10]. 

 

Where e: denotes an electric charge to proton,  

(𝐸2) ↑ is electric quadrupole transition probability in the unit of (𝑒2𝑏2) [11]. 

An electric quadrupole transition probability B(E2) of a nucleus contains information about the energy 

of low-lying levels of nuclei [12.13]. The even-even nuclide's initial excited states are 2+[12]. Thus, 

it is crucial that this state transitions to a 0+ ground state [14,15]. The fundamental experimental 

quantities are B(E2) values, which are unaffected by a nuclear model [16,17]. Pursuant to Global Best 

Fit (GLOƁAL), all that is required to estimate B(E2) is Knowledge of the energy ,E (keV) of the 2+ 

state,(e2 b2 ) [18,19]. 

 

𝐸𝛾0: is the transition energy to gamma ray in (KeV) units. 

 z: atomic numbers, 

 A: mass number of nuclueus. 

 A quadrupole moment (Q0), symbolizes a homogeneous charge distribution of the nucleus, is 

connected to the deformation parameter (β2) utilized to determine the form of an axially, 

symmetrically deformed, nucleus. [15,16,20] 

 B(E2) ↑ = 2.6 𝐸𝛾0 −1𝑍 2𝐴 −2/3                                                                                         (2)                                                                                   

𝑄𝑜 = [( 16 𝜋 /5 ) 𝐵(𝐸2)𝑒 2𝑏 2 /𝑒 2 ] ½ barn                                                  (1)  



IHJPAS. 37 (2) 2024 

 

217 
 

    𝛽2 =
Q˳√5π

3ZR2
                                                                                                                               (3)   

Where R2: Nuclear charge, radii calculated ꬵrom: [16]. 

 

With R0 = 1.2 fm. and the atomic mass A. This equation can be used to calculate the nuclear charge 

radii: 

 R2=0.0144A2/3 barn 

Deformation parameter (β2) is so related to reduce transition probability B(E2) via a formula [1].                                   

The equation below can be used to describe the intrinsic quadruple moment of a uniformly charged 

ellipsoid[18,14,21]: 

   

Q0 =
2Z

5
(b2 − a2)                                                                                                                        (6) 

Where a and b, the minor and major oval axes and used to express the quadrilateral distortion index δ 

[19,22]:  

𝛿 = 0.3(𝑏2 − 𝑎2) / 2(<  𝑟2   >)                                                                                              (7) 

Where the average of the mean-squared charge distribution radius is:    

<  𝑟2 >= 𝑏2 − 2𝑎2/5                                                                                                                 (8) 

It is possible to determine the values of the nucleus distortion parameter via:  

𝛿 = 0.75 Qo / (< 𝑟2  >  Z)                                                                                              (9) 

From (equation.5) and (equation.9) down argumentation are connected as [23,24,6]. 

δ = 0.946𝛽2                                                                                                                                 (10) 

Value of < 𝑟2 >  was evaluated using the following expressions [25,18]: 

< 𝑟2 > = 0.63Ro2 (1+10/3( a0 / Ɍo) 2 ) / (1 + ( ao / Ro) 2 ) (A≤100)                      (11) 

  < 𝑟2 > = 0.63 (1.2 A1/3) 2 (A >100)                                                                           (12) 

which considers the impact of the surface diffusion properties on the light nuclei. The following are 

the potential form-factor parameters for radial Woods-Saxon [26,27]:  

(Ro = 1.07 A1/3 fm ) and (ao = 0.55 fm). Were derived from the fast electron scattering data. 

Equations 3, 4, and 5 gave us[28,29]: 

𝑎 = √<  𝑟2  > (1.66 −
2δ

0.9
)                                                                                                   (13) 

𝑏 = √𝟓 <  𝒓𝟐  > −2𝑎2                                                                                                           (14)     

The ellipsoid axis (a,b)' major and minor variance may be written as: [30,22]: 

 

 

 

3.  Results and Discussion  

     Several coefficients were calculated for multiples of  an ,nuclei with the numbers of mass greater 

than or equal ,to 100 (A≥100) for elemental cadmium (Cd-48) and its isotopes, Which are 

R =  * Ro or R = b -a 

 

R=R0A1/3                                                                                         (4) 

𝛽2 = 4𝜋 / 3𝑍𝑅𝑜 2 [(𝐸2) ↑ 𝑒 2 𝑏 2/ 𝑒 2 ] ½                                                               (5)  



IHJPAS. 37 (2) 2024 

 

218 
 

(100Cd,102Cd,104Cd,106Cd,108Cd,110Cd,112Cd,114Cd,116Cd,118Cd,120Cd,122Cd,124Cd ) in this research, 

these parameters are required for our study: 

 

Table 1. The number of mass for cadmium isotopes (A), a number of neutrons (N), the energy of gamma for first level 

Eᵞ, average nuclear radius (R0
2), low electrical transition potential (B (E2) ↑ in e2 b2 unit, moment Tetrapolar electrode 

(Qₒ) in barn unit, and the parameters of deformation (β2, δ) to Cd-48. 

 

Present Work Theoretical Value E_γ (KeV) N A 

𝑹𝟎
𝟐 𝑩(𝑬𝟐) ↑ 

(𝒆𝟐𝒃𝟐) 

𝑸ₒ (b) 𝜷𝟐 𝜹 β2 

 

B(E2) ↑) 

(e2 b2) 

for 

(SSANM)[14] 

31.02378 0.2768 1.6682 0.1480 0.1118 0.1115 0.157 1004.5 52 100 

31.43620 0.3534 1.8848 0.1650 0.1250 0.1407 257 0. 776.55 54 102 

31.84570 0.4117 2.0343 0.1758 0.1335 0.1658 0.366 658 56 104 

32.25217 0.4227 2.0615 0.1759 0.1339 0.1871 0.478 632.66 58 106 

32.65665 0.4173 2.0482 0.1726 0.1317 0.2030 0.577 632.97 60 108 

33.05905 0.3967 1.9970 0.1663 0.1371 0.2102 0.634 657.7622 62 110 

33.45812 0.4175 2.0487 0.1685 0.1291 0.2073 0.6320 617.516 64 112 

33.85610 0.4562 2.1416 0.1741 0.1336 0.2025 0.617 558.456 66 114 

34.25058 0.4905 2.2205 0.1784 0.1372 0.1974 0.600 513.50 68 116 

34.64264 0.5105 2.2654 0.1800 0.1387 0.9120 0.581 487.77 70 118 

35.03337 0.4867 2.2120 0.1738 0.1341 0.1733 0.484 505.9 72 120 

35.42154 0.4277 2.0735 0.1611 |0.1246 0.1476 0.359 569.45 74 122 

35.80825 0.3929 1.9873 0.1528 0.1183 0.1219 0.250 613.2 76 124 

 

The quaternary electrical moments B(E2) of some elements are shown in Table 1. We can see that 

these values vary depending on their mass number—the number of neutrons and protons—and that 

when values B(𝐸2) approach the magic number, they fall short of other elements for the same isotope. 

As a result, the distortion values (𝛽2) are also low. On the contrary, we notice from Table 1, that the 

largest value of the distortion coefficient (𝛽2) to (48Cd 118) is equal to (𝛽2 = 0.1800), and the minimum 

value for the deformation coefficient for (48Cd100) is (𝛽2 = 0.1480). Residual values of 𝛽2 range 

between these two values, and thus we notice that the closer the number of Z&N to the magic numbers, 

the more stable the nuclei are. We also notice from Table 1 that the distortion parameter of the values 

(δ) is as large as possible at (48Cd 118) equal to (δ = 0.1387) and as low as possible at (48Cd100) equal 

to (δ = 0.1118) and the remaining values are between these two values. 

 

Table 2. the mass of the number (A), the number of neutrons (N), the gamma energy of the ground level, the transition 

probability (T) and the average half-life (τ (s)) of Cadmium (Cd) isotopes. 

A N Ei(kev) Eɤ(kev) t1/2(s) T(s) τ (s) 

100 52 1004.5 1004.5 49.1 70.8514 0.0141 

102 54 776.55 776.55 (5.5m)330 476.1905 0.0021 

104 56 658 658 (57.7m)3462 4.9957*103 2.0017*10-4 

106 58 632.64 632.66 (7.27 ps)7.27*10-12 1.0491*10-11 9.5323*1010 

108 60 632.986 632.97 (6.86ps)6.86*10-12 9.8990*10-12 1.0102*1011 

110 62 657.7638 657.7622 (5.39 ps)5.39*10-12 7.7778*10-12 1.2857*1011 



IHJPAS. 37 (2) 2024 

 

219 
 

112 64 617.520 617.516 (6.51ps) 6.51*10-12 9.3939*10-12 1.0645*1011 

114 66 558.456 558.456 (10.2ps) 10.2*10-12 1.4719*10-11 6.7941*1010 

116 68 513.490 513.50 (14.1ps) 14.1*10-12 2.0346*10-11 4.9149*1010 

118 70 487.77 487.77 (50.3m) 3018 4.3550*103 2.2962*10-4 

120 72 505.9 505.9 50.80 73.3045 0.0136 

122 74 569.45 569.45 5.24 7.5613 0.1323 

124 76 613.33 613.2 1.25 1.8038 0.5544 

 

In Table 2, we note that the highest value of transition potential T(s) is equal to 4.9957 * 103 at 104Cd 

where N = 56 and the lowest value of T(s) is 7.7778 * 10-12 at 110Cd and the highest value of the 

average half-life is 9.5323 * 1010 at 106Cd. The lowest value is 2.0017*10-4 at 102Cd, and the highest 

value for the gamma energy is 1004.5 at 100Cd, which is from the even-even nuclei with O+ spinning. 

Also, the gamma energy values shown in the above table were relied on for the purpose of calculating 

the electrical transmission probability B(E2), and through (B(E2)[23,24], it was calculated the 

deformation coefficients (𝛽2, δ) through which the shape of the nuclei was determined, depending on 

the deformation equation on which it was based on our current study. 

It was discovered by examining the rms values of the radius〈𝑟 2〉½ in Table 3 that these values rise 

with increasing mass number (𝐴). By comparing the current values with the experimental values, it 

was found that a calculated value for〈𝑟 2〉1/2,(Ҏ.w) agrees, with the experimental, value of〈𝑟 2〉

from references [25]. 

These findings confirm that when the number of nucleons approaches the magic number, the nuclei 

become more stable and close to spherical, and the motion of the nucleons in the sub-shell is a vibration 

about spherical. Figure 2 depicts the shapes of these nuclei. 

 

Table  3. The mass of the number (A), the number of neutrons (N), the mean square root of the radius < 𝐫𝟐 >𝟏/𝟐, the 

minor and major axes (b,a) and the difference between them (∆R) in two ways for Cadmium (Cd-48) isotopes. 

A N Theoretical 

Value 

Present Work 

〈𝒓𝟐〉𝟏/𝟐 𝒇𝒎 

[25] 

〈𝐫𝟐〉  fm 〈𝒓𝟐〉𝟏/ 𝟐 𝒇𝒎 𝐚 (𝐟𝐦) 𝐛 (𝐟𝐦) ∆ 𝐑𝟏 ∆ 𝐑𝟐 ∆ 𝐑𝟑 

100 52 --- 23.3038 4.8274 2.6165 3.2319 0.5555 0.6155 0.7799 

102 54 4.4810 23.5578 4.8536 2.5964 3.2842 0.6250 0.6879 0.8754 

104 56 4.5122 23.8101 4.8795 2.5855 3.3208 0.6718 0.7353 0.9388 

106 58 4.5383 24.0605 4.9051 2.5915 3.3308 0.6780 0.7392 0.9453 

108 60 4.5577 24.3093 4.9304 2.6029 3.3320 0.6708 0.7291 0.9334 

110 62 4.5765 24.5563 4.9554 2.6191 3.3253 0.6515 0.7062 0.9045 

112 64 4.5944 24.8018 4.9801 2.6214 3.3402 0.6657 0.7188 0.9223 

114 66 4.6087 25.0457 5.0045 2.6182 3.3634 0.6832 0.7452 0.9585 

116 68 4.6203 25.2880 5.0287 2.6169 3.3834 0.7160 0.7666 0.9881 

118 70 4.6246 25.5289 5.0526 2.6200 3.3963 0.7277 0.7763 1.0023 

120 72 4.6300 25.7683 5.0762 2.6358 3.3892 0.7079 0.7534 0.9732 

122 74 --- 26.0062 5.0996 2.6623 3.3650 0.6611 0.7027 0.9073 

124 76 --- 26.2428 5.1227 2.6816 3.3514 0.6313 0.6698 0.8648 

 



IHJPAS. 37 (2) 2024 

 

220 
 

 
Figure 2. The 3-D Shapes for the deformation of quadrupole for 48Cd isotope from a (major) and b (minor) axes 

 

5. Conclusion 

    We conclude from the results and numbers mentioned above that the first excited state energy (2+) 

of these nuclei starts to change progressively with the emergence of the mass numbers 𝐴, and the 

nucleons are outside the core. The core is in one direction, and the cores can be of stable and non-

spherical shape with permanent deformation. 

This means that the movement of the nuclei will be outside the closed envelope when the nuclei 

(protons and neutrons) are far from the magic numbers. Thus, the rotational motion and the nucleus 

are more deformed. 

 

Acknowledgment 

    Firstly, I would like to extend my sincere thanks to the participants in this research: (Sameera 

Ahmed Ebrahiem) for being the basis and first supporter in this work, as I relied in my research on 

most of her scientific works of scientific research, dissertations and dissertations, in addition to 

((bdallah Ben Rhaiem )  for being an essential part In helping me and directing me to the appropriate 

publishing houses. In addition to relying on his scientific information in physics, I also do not forget 

the credit of the rest of the researchers and authors whose scientific publications I benefited from in 

my research (Raman S, Nestor CW, and Tikkanen GR), as well as other things mentioned in the 



IHJPAS. 37 (2) 2024 

 

221 
 

sources below. I also do not forget the greatest gratitude. To my first foundation in my studies, Ibn Al-

Haytham College of Education for Pure Sciences, for giving me the opportunity to publish my research 

in a smooth manner without complications. 

 

Conflict of Interest 

The authors declare that they have no conflicts of interest. 

 

Funding: None. 

 

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