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## Study of nuclear charge radius

Cao Ying-Yu, Guo Jian-You
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• #### 摘要

结合已有的原子核半径的实验数据, 对先前的核电荷半径公式进行验证和探讨. 比较单参数核电荷半径公式, 验证了$Z^{1/3}$律公式要优于$A^{1/3}$律公式. 对两参数公式和三参数公式进行验证, 得到两参数和三参数公式要优于单参数公式. 考虑到原子核电四极矩与形变的关系, 在原有的三参数公式中加入电四极矩因子项, 得出核电荷半径新公式. 拟合该公式发现核电荷半径理论值与实验值符合较好. 再考虑总自旋与电四极矩的关系, 求出内禀电四极矩, 代入公式中进行拟合, 均方根偏差进一步下降. 最后加入能反映奇偶摆动现象的$\delta$项, 用公式得到的均方根偏差为0.369 fm, 较好地反映出了形变与核电荷半径的关系.

#### Abstract

Based on the existing experimental data of nuclear radius, the previous formula of nuclear charge radius is verified and discussed. Comparing the formula of the single-parameter nuclear charge radius, it is proved that the formula of $Z^{1/3}$ law is better than the formula of $A^{1/3}$ law. We refitted the two-parameter formula and the three-parameter formula that have been proposed and confirmed that the two-parameter and three-parameter formula fit better than the single-parameter formula. It is shown that show that the deformation plays a key role in the nuclear charge radius. The electric quadrupole moment is an important physical quantity representing the properties of the nucleus. Its appearance indicates the deviation from spherical symmetry and also reflects the size of the nuclear deformation. The electric quadrupole moment is also one of the basic observations to understand the distribution of matter within the nucleus, to examine the nuclear model, and to observe nucleon-nuclear interactions. Taking into account the relationship between the nuclear quadrupole moment and the deformation, the electric quadrupole moment factor is added to the original three-parameter formula to obtain a new formula for the nuclear charge radius. Fitting the four-parameter formula, it is found that the theoretical value of the nuclear charge radius is in good agreement with the experimental value, the root-mean-square deviation is 0.0397 fm. Considering the relationship between the total spin and the electric quadrupole moment, the intrinsic electric quadrupole moment is obtained and brought into the formula for fitting, and the root-mean-square deviation further decreases,the root-mean-square deviation is 0.0372 fm. Finally, considering the universality of odd-even staggering, we add the $\delta$ term that can reflect the odd and even oscillation phenomenon, and the root-mean-square deviation obtained by the formula is 0.369 fm, which better reflects the relationship between the deformation and the nuclear charge radius. Compared with the formulas already proposed, the new formula can better reflect the variation trend of nuclear deformation, shell effect, odd-even staggering, etc., and the calculation accuracy is also improved, which can provide a useful reference for future experiments.

#### 作者及机构信息

###### 通信作者: 郭建友, jianyou@ahu.edu.cn
• 基金项目: 国家级-国家自然科学基金(11575002)

#### 施引文献

• 图 1  $R_{\rm c} = r_0 A^{1/3}$$R_{\rm c} = r_0 Z^{1/3}$的拟合曲线(左图是$R_{\rm c} = r_0 A^{1/3}$的拟合曲线, 右图是$R_{\rm c} = r_0 Z^{1/3}$的拟合曲线)

Fig. 1.  The fitting curve of the Eqs. (2) and (4).(The left picture is the fitting curve of the Eq. (2) and the right picture is the fitting curve of the Eq. (4))

图 2  $R_{\rm c}=r_0\left( 1-a\dfrac{N-Z}{A}+b\dfrac{1}{A}+c\dfrac{Q^*}{A} \right)A^{1/3}$的拟合曲线图

Fig. 2.  The fitting curve of the Eq. (13).

图 3  $R_{\rm c}=r_0\left( 1-a\dfrac{N-Z}{A}+b\dfrac{1}{A}+c\dfrac{Q_0^*}{A} \right)A^{1/3}$的拟合曲线图

Fig. 3.  The fitting curve of the Eq. (14).

图 4  $R_{\rm c}=r_0\left( 1-a\dfrac{N-Z}{A}+b\dfrac{1}{A}+c\dfrac{Q_0^*}{A}+d\frac{\delta}{A} \right)A^{1/3}$的拟合曲线图

Fig. 4.  The fitting curve of the Eq. (15).

图 5  Ba和Fr, Ho和Lu四个同位素链核电荷半径的实验值与(13)式—(15)式计算的核电荷半径理论值的对比

Fig. 5.  The experimental values of the nuclear charge radii of Ba and Fr, Ho and Lu isotopic chains are compared with the theoretical values calculated by Eqs. (13)–(15).

图 6  计算368个核电荷半径的实验值分别与(13)式—(15)式计算的理论值的差值. (从上到下依次为核电荷半径的实验值与(13)式的差值图, 核电荷半径的实验值与(14)式的差值图, 核电荷半径的实验值与(15)式的差值图)

Fig. 6.  The difference between the experimental value of 368 nuclear charge radii and the theoretical value calculated by Eqs. (13)–(15) , respectively.