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原子核电荷半径的研究

曹颖逾 郭建友

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原子核电荷半径的研究

曹颖逾, 郭建友

Study of nuclear charge radius

Cao Ying-Yu, Guo Jian-You
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  • 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.
    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.
      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)
      Corresponding author: Guo Jian-You, jianyou@ahu.edu.cn
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    De Vries H, De Jager C W, De Vries C 1987 At. Data Nucl. Data Tables 36 495Google Scholar

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    Boehm E, Lee P L 1974 At. Data Nucl. Data Tables 14 605Google Scholar

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    Engfer R, Schneuwly H, Vuileumier J L, Walter H K, Zehnder A 1974 At. Data Nucl. Data Tables 14 509Google Scholar

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    Fricke G, Bernhardt C, Heilig K, Schaller L A, Schellenberg L, Shera E B, De Jager C W 1995 At. Data Nucl. Data Tables 60 177Google Scholar

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    Heilig K, Steudel A 1974 At. Data Nucl. Data Tables 14 613Google Scholar

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    Aufmuth P, Heilig K, Steudel A 1987 At. Data Nucl. Data Tables 37 455Google Scholar

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    Angeli I, Marinova K P 2013 At. Data Nucl. Data Tables 99 69Google Scholar

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    Nerlo-Pomorska B, Pomorski K 1993 Z. Phys. A 344 359Google Scholar

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    Nerlo-Pomorska B, Pomorski K 1994 Z. Phys. A 348 169Google Scholar

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    圣宗强, 樊广伟, 钱建发 2015 物理学报 64 112101Google Scholar

    Sheng Z Q, Fan G W, Qian J F 2015 Acta Phys. sin. 64 112101Google Scholar

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    曾谨言 1957 物理学报 13 357Google Scholar

    Zeng J Y 1957 Acta Phys. Sin. 13 357Google Scholar

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    曾谨言 1975 物理学报 24 151Google Scholar

    Zeng J Y 1975 Acta Phys. Sin. 24 151Google Scholar

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    张双全, 孟杰, 周善贵, 曾谨言 2002 高能物理与核物理 26 252Google Scholar

    Zhang S Q, Meng J, Zhou S G, Zeng J Y 2002 High Energy Physics and Nuclear Physics 26 252Google Scholar

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    Barzakh A E, Fedorov D V, Ivanov V S, Molkanov P L, Moroz F V, Orlov S Y, Panteleev V N, Seliverstov M D, Volkov M Y 2018 Phys. Rev. C 97 014322Google Scholar

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    Yang J J, Piekarewicz J 2018 Phys. Rev. C 97 014314Google Scholar

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    Safronova M S, Porsev S G, Kozlov M G, Thielking J, Okhapkin M V, Glowacki P, Meier D M, Peik E 2018 Phys. Rev. Lett. 121 213001Google Scholar

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    Casten R F 1985 Phys. Rev. Lett. 54 1991Google Scholar

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    Casten R F, Brenner D S, Haustein P E 1987 Phys. Rev. Lett. 58 658Google Scholar

    [19]

    Casten R F 1988 J. Phys. G: Nucl. Part. Phys. 14 S71Google Scholar

    [20]

    Casten R F, Zamfir N V 1993 Phys. Rev. Lett. 70 402Google Scholar

    [21]

    Foy B D, Casten R F, Zamfir N V, Brenner D S 1994 Phys. Rev. C 49 1224

    [22]

    Saha M, Sen S 1994 Phys. Rev. C 49 2460

    [23]

    Gangopadhyar G 2009 J. Phys. G: Nucl. Part. Phys. 36 095105Google Scholar

    [24]

    Stopkowicz S, Gauss J 2014 Phys. Rev. A 90 022507Google Scholar

    [25]

    Zhao P W, Zhang S Q, Meng J 2014 Phys. Rev. C 89 011301Google Scholar

    [26]

    Otsuka T, Suzuki T, Fujimoto R, Grawe H, Akaish Y 2005 Phys. Rev. Lett. 95 232502Google Scholar

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    Bohr A 1976 Science 48 365

    [28]

    Cheal B, Mane E, Billowes J, Bissell M L, Blaum K, Brown B A, Charlwood F C, Flanagan K T, Forest D H, Geppert C, Honma M, Jokinen A, Kowalska M, Krieger A, Kramer J, Moore I D, Neugart R, Neyens G, Nortershauser W, Schug M, Stroke H H, Vingerhoets P, Yordanov D T, Zakova M 2010 Phys. Rev. Lett. 104 202502

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    Ohtsubo T, Stone N J, Stone J R, Towner I S, Bingham C R, Gaulard C, Koster U, Muto S, Nikolov J, Nishimura K, Simpson G S, Soti G, Veskovic M, Walters W B, Wauters F 2012 Phys. Rev. Lett. 109 032504Google Scholar

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    Snyder J B, Reviol W, Sarantites D G, Afanasjev A V, Janssens R V F, Abusara H, Carpenter M P, Chen X, Chiara C J, Greene J P, Lauritsen T, McCutchan E A, Seweryniak D, Zhu S 2013 Phys. Rev. B 723 61

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    Heyde K, Wood J L 2011 Rev. Mod. Phys. 83 1467Google Scholar

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    Yordanov D T, Balabanski D L, Bieron J, Bissell M L, Blaum K, Budincevic I, Frizsche S, Frommgen N, Georgiev G 2013 Phys. Rev. Lett. 110 192501Google Scholar

    [33]

    陈春杏 2015 硕士学位论文 (广西: 广西师范大学)

    Chen C X 2015 M. S. Thesis (Guangxi: Guangxi Normal University) (in Chinese)

    [34]

    Stone N J 2016 At. Data Nucl. Data Tables 111-112 1Google Scholar

  • 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.

    表 1  各种核电荷半径公式

    Table 1.  The mentioned equations for nuclear charge radius $R_{\rm c}$.

    公式参数σ/fm
    ${R_{\rm c}} = {r_0}A^{1/3}$r0 = 1.2269 fmσ = 0.1224
    ${R_{\rm c}} = {r_0}Z^{1/3}$r0 = 1.6394 fmσ = 0.0939
    ${R_{\rm c}} = {r_0}\left( {1 - a\dfrac{ {N - Z} }{A} } \right){A^{1/3} }$r0 = 1.2827 fm, a = 0.2700σ = 0.0855
    ${R_{\rm c}} = {r_0}\left( {1 - a\dfrac{ {N - Z} }{A} + b\dfrac{1}{A} } \right){A^{1/3} }$r0 = 1.2331 fm, a = 0.1461, b = 2.3301σ = 0.0510
    ${R_{\rm c}} = {r_0}\left( {1 + a\dfrac{ {N - {N^*} } }{Z} } \right){Z^{1/3} }$r0 = 1.6312 fm, a = 0.0627σ = 0.0618
    ${R_{\rm c}} = {r_0}\left( {1 - a\dfrac{ {N - Z} }{A} + b\dfrac{1}{A} + c\dfrac{ { {Q^*} } }{A} } \right){A^{1/3} }$r0 = 1.2221 fm, a = 0.1350, b = 2.4698, c = 0.8976σ = 0.0397
    ${R_{\rm c}} = {r_0}\left( {1 - a\dfrac{ {N - Z} }{A} + b\dfrac{1}{A} + c\dfrac{ {Q_0^*} }{A} } \right){A^{1/3} }$r0 = 1.2220 fm, a = 0.1410, b = 2.4200, c = 0.3643σ = 0.0372
    ${R_{\rm c}} = {r_0}\left( {1 - a\dfrac{ {N - Z} }{A} + b\dfrac{1}{A} + c\dfrac{ {Q_0^*} }{A} + d\dfrac{\delta }{A} } \right){A^{1/3} }$r0 = 1.2223 fm, a = 0.1421, b = 2.4577,
    c = 0.3660 d = 0.1705
    σ = 0.0369
    下载: 导出CSV
  • [1]

    De Vries H, De Jager C W, De Vries C 1987 At. Data Nucl. Data Tables 36 495Google Scholar

    [2]

    Boehm E, Lee P L 1974 At. Data Nucl. Data Tables 14 605Google Scholar

    [3]

    Engfer R, Schneuwly H, Vuileumier J L, Walter H K, Zehnder A 1974 At. Data Nucl. Data Tables 14 509Google Scholar

    [4]

    Fricke G, Bernhardt C, Heilig K, Schaller L A, Schellenberg L, Shera E B, De Jager C W 1995 At. Data Nucl. Data Tables 60 177Google Scholar

    [5]

    Heilig K, Steudel A 1974 At. Data Nucl. Data Tables 14 613Google Scholar

    [6]

    Aufmuth P, Heilig K, Steudel A 1987 At. Data Nucl. Data Tables 37 455Google Scholar

    [7]

    Angeli I, Marinova K P 2013 At. Data Nucl. Data Tables 99 69Google Scholar

    [8]

    Nerlo-Pomorska B, Pomorski K 1993 Z. Phys. A 344 359Google Scholar

    [9]

    Nerlo-Pomorska B, Pomorski K 1994 Z. Phys. A 348 169Google Scholar

    [10]

    圣宗强, 樊广伟, 钱建发 2015 物理学报 64 112101Google Scholar

    Sheng Z Q, Fan G W, Qian J F 2015 Acta Phys. sin. 64 112101Google Scholar

    [11]

    曾谨言 1957 物理学报 13 357Google Scholar

    Zeng J Y 1957 Acta Phys. Sin. 13 357Google Scholar

    [12]

    曾谨言 1975 物理学报 24 151Google Scholar

    Zeng J Y 1975 Acta Phys. Sin. 24 151Google Scholar

    [13]

    张双全, 孟杰, 周善贵, 曾谨言 2002 高能物理与核物理 26 252Google Scholar

    Zhang S Q, Meng J, Zhou S G, Zeng J Y 2002 High Energy Physics and Nuclear Physics 26 252Google Scholar

    [14]

    Barzakh A E, Fedorov D V, Ivanov V S, Molkanov P L, Moroz F V, Orlov S Y, Panteleev V N, Seliverstov M D, Volkov M Y 2018 Phys. Rev. C 97 014322Google Scholar

    [15]

    Yang J J, Piekarewicz J 2018 Phys. Rev. C 97 014314Google Scholar

    [16]

    Safronova M S, Porsev S G, Kozlov M G, Thielking J, Okhapkin M V, Glowacki P, Meier D M, Peik E 2018 Phys. Rev. Lett. 121 213001Google Scholar

    [17]

    Casten R F 1985 Phys. Rev. Lett. 54 1991Google Scholar

    [18]

    Casten R F, Brenner D S, Haustein P E 1987 Phys. Rev. Lett. 58 658Google Scholar

    [19]

    Casten R F 1988 J. Phys. G: Nucl. Part. Phys. 14 S71Google Scholar

    [20]

    Casten R F, Zamfir N V 1993 Phys. Rev. Lett. 70 402Google Scholar

    [21]

    Foy B D, Casten R F, Zamfir N V, Brenner D S 1994 Phys. Rev. C 49 1224

    [22]

    Saha M, Sen S 1994 Phys. Rev. C 49 2460

    [23]

    Gangopadhyar G 2009 J. Phys. G: Nucl. Part. Phys. 36 095105Google Scholar

    [24]

    Stopkowicz S, Gauss J 2014 Phys. Rev. A 90 022507Google Scholar

    [25]

    Zhao P W, Zhang S Q, Meng J 2014 Phys. Rev. C 89 011301Google Scholar

    [26]

    Otsuka T, Suzuki T, Fujimoto R, Grawe H, Akaish Y 2005 Phys. Rev. Lett. 95 232502Google Scholar

    [27]

    Bohr A 1976 Science 48 365

    [28]

    Cheal B, Mane E, Billowes J, Bissell M L, Blaum K, Brown B A, Charlwood F C, Flanagan K T, Forest D H, Geppert C, Honma M, Jokinen A, Kowalska M, Krieger A, Kramer J, Moore I D, Neugart R, Neyens G, Nortershauser W, Schug M, Stroke H H, Vingerhoets P, Yordanov D T, Zakova M 2010 Phys. Rev. Lett. 104 202502

    [29]

    Ohtsubo T, Stone N J, Stone J R, Towner I S, Bingham C R, Gaulard C, Koster U, Muto S, Nikolov J, Nishimura K, Simpson G S, Soti G, Veskovic M, Walters W B, Wauters F 2012 Phys. Rev. Lett. 109 032504Google Scholar

    [30]

    Snyder J B, Reviol W, Sarantites D G, Afanasjev A V, Janssens R V F, Abusara H, Carpenter M P, Chen X, Chiara C J, Greene J P, Lauritsen T, McCutchan E A, Seweryniak D, Zhu S 2013 Phys. Rev. B 723 61

    [31]

    Heyde K, Wood J L 2011 Rev. Mod. Phys. 83 1467Google Scholar

    [32]

    Yordanov D T, Balabanski D L, Bieron J, Bissell M L, Blaum K, Budincevic I, Frizsche S, Frommgen N, Georgiev G 2013 Phys. Rev. Lett. 110 192501Google Scholar

    [33]

    陈春杏 2015 硕士学位论文 (广西: 广西师范大学)

    Chen C X 2015 M. S. Thesis (Guangxi: Guangxi Normal University) (in Chinese)

    [34]

    Stone N J 2016 At. Data Nucl. Data Tables 111-112 1Google Scholar

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出版历程
  • 收稿日期:  2019-10-08
  • 修回日期:  2020-05-08
  • 上网日期:  2020-05-20
  • 刊出日期:  2020-08-20

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