Nucleus density based new relationship of nuclear charge radius

Jiao Bao-Bao

doi:10.7498/aps.72.20230126

Abstract

In this paper we predict and evaluate the value of the nuclear charge radius by analyzing the relationship between nuclear mass and nuclear charge radius.We obtain 884 nuclei (Z, N ≥ 8) with known mass and known charge radii by combining AME2020 database with CR2013 database, and calculate the mass densities $ \rho_\text{m} $ of the 884 nuclei. We aim to obtain an empirical formula of one constant which is useful in describing and predicting nuclear charge radius. With the empirical formula and the AME2020 database, the root-mean-square deviation (RMSD) of the nuclear charge radius of $ \sigma = 0.093 $ fm is successfully obtained.Considering the influence of neutron numbers on $\rho_{\rm{m}}$, we use the neutron factor ${1}/{N} $ to correct the empirical formula, and the RMSD is reduced to σ = 0.047 fm (the accuracy is increased by about 50%). The second correction is shell effect of neutrons. The results show that the RMSD of nuclear charge radius is reduced to 0.034 fm based on shell effect of neutrons. We use the empirical formula with corrections to predict the nuclear charge radius (1573 nuclear charge radius with Z, N ≥ 8) which is difficult to measure experimentally. The difference between our predicted values based on AME2020 database and the experimental values measured in recent years is in the allowable range of deviation. The result shows that the new relation for nuclear charge radius is simple and reliable. In addition, the RMSD of the calculation value for 791 nuclei is reduced to σ = 0.032 fm after we have removed some nuclei with special shell effect and isotope chains. These results show that the new relation proposed in this paper can be comparable to $ A^{1/3} $ and $ Z^{1/3} $ formulas with corrections.Moreover, we study the 884 and 791 nuclear mass densities by using L-M neural network method to build description and prediction models. Comparing with CR2013, the RMSDs of nuclear charge radius are σ = 0.018 fm and σ = 0.014 fm, respectively. The RMSDs are reduced by about 50% compared with that from the empirical formula with corrections, and the predicted values are closer to the experimental values measured in recent years.

Keywords:

Authors and contacts

Jiao Bao-Bao^1,2

1.
Engineering Research Center of Nuclear Technology Application, East China University of Technology, Nanchang 330013, China
2.
School of Nuclear Science and Engineering, East China University of Technology, Nanchang 330013, China

Corresponding author: Jiao Bao-Bao, baobaojiao91@126.com

Funds: Project supported by the Open Funds of Engineering Research Center of Nuclear Technology Application, Ministry of Education, China (Grant No. HJSJYB2022-9), the Science and Technology Research Program of the Education Department of Jiangxi Province, China (Grant No. GJJ2200782), the Experimental Technology Development Program of East China University of Technology (Grant No. DHSY-202251), and the Doctoral Scientific Research Foundation of East China University of Technology (Grant No. DHBK2019151)

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References

[1]	Campbell P, Moore I D, Pearson M R 2016 Prog. Part. Nucl. Phys. 86 127 Google Scholar
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[54]	Fu G J, Bao M, He Z, Jiang H, Zhao Y M, Arima A 2012 Phys. Rev. C 86 054303

Cited By

图 1 884个原子核密度$ \rho_\text{m} $

Figure 1. The $ \rho_\text{m} $ of 884 nuclei.

DownLoad: Full-Size Img PowerPoint

图 2 884个原子核密度 (绿色折线表示的是原子核密度平均值线, 黑色光滑曲线是基于(6)式得到的拟合线, 粉色竖线分别表示中子数 N = 20, 28, 50, 82 和126的位置)

Figure 2. The green zigzag line is plotted by using the average values of $ \rho_\text{m} $ for nuclei with the same mass number N. The black curve is plotted in terms of empirical formula Eq. (6). The vertical lines are plotted at the major neutron closure N = 20, 28, 50, 82 126.

DownLoad: Full-Size Img PowerPoint

图 3 从上到下依次为884个核电荷半径的实验值与(2)式、(6)式、(7)式理论计算值的差值. 粉色竖直虚线分别表示中子数 N = 20, 28, 50, 82 和126的位置

Figure 3. The difference between the experimental value of 884 nuclear charge radii and the theoretical value calculated by Eq.(2)，Eq.(6) and Eq.(7), respectively. A dash vertical line is plotted at the major neutron closure N = 20, 28, 50, 82 and 126.

DownLoad: Full-Size Img PowerPoint

图 4 核电荷半径的计算值与CR2013数据库中实验值之间的误差. 粉色竖直虚线分别表示中子数 N = 20, 28, 50, 82 和126的位置, 黑色横虚线分别表示质子数 Z = 20, 28, 50 和 82 的位置 (单位: fm)

Figure 4. The difference between the calculated values of nuclear charge radius and experimental values in the CR2013 database. A dash vertical line is plotted at the major neutron closure N = 20, 28, 50, 82 and 126. The dash horizontal line is plotted at the major proton closure Z = 20, 28, 50 and 82 (in units of fm).

DownLoad: Full-Size Img PowerPoint

图 5 (a) 和 (b)分别表示 Eu, Gd, Tb, Dy 同位素链核电荷半径与核质量的实验值; (c) 和 (d)分别表示Tm, Yb, Lu, Hf同位素链核电荷半径与核质量的实验值

Figure 5. (a) and (b) represent the nuclear charge radii and nuclear mass of Eu, Gd, Tb, Dy elements, respectively; (c) and (d) represent the nuclear charge radii and nuclear mass of Tm, Yb, Lu, Hf elements, respectively.

DownLoad: Full-Size Img PowerPoint

图 6 基于神经网络模型I和II得到的核电荷半径计算值与实验值之间的差值 (单位: fm)

Figure 6. The difference between the calculated values of nuclear charge radius (obtained by neural network model I and II) and experimental values in CR2013 database (in units of fm).

DownLoad: Full-Size Img PowerPoint

图 7 利用不同模型结合CR2013数据库预得到的预言值与近几年测得的实验值$ ^{37}\text{K} $^[9], $ ^{48}\text{K} $^[9], $ ^{64}\text{Cu} $^[8,10], $ ^{65}\text{Zn} $^[11], $ ^{69}\text{Zn} $^[11]之间的差值

Figure 7. The difference between the predicted values of nuclear charge radius (obtained by the CR2013 database) and experimental values $ ^{37}\text{K} $^[9], $ ^{48}\text{K} $^[9], $ ^{64}\text{Cu} $^[8,10], $ ^{65}\text{Zn} $^[11], $ ^{69}\text{Zn} $^[11] in recent years.

DownLoad: Full-Size Img PowerPoint

表 1 不同壳层范围的拟合参数 (单位: u/fm³)

Table 1. The parameter of different shell regions (in units of u/fm³).

Region	$C_n$	$\delta1$	$a_n$	$\delta2$
8 ≤ N ≤ 20	0.276	0.010	–0.828	0.153
21 ≤ N ≤ 28	0.363	0.007	–2.86	0.171
29 ≤ N ≤ 39	0.206	0.007	1.649	0.256
40 ≤ N ≤ 50	0.383	0.005	–5.387	0.217
51 ≤ N ≤ 66	0.242	0.005	1.812	0.316
67 ≤ N ≤ 82	0.345	0.007	–4.754	0.557
83 ≤ N ≤ 104	0.227	0.006	4.774	0.573
105 ≤ N ≤ 126	0.371	0.003	–9.614	0.433
127 ≤ N	0.230	0.003	8.088	0.516

DownLoad: CSV

[1]	Campbell P, Moore I D, Pearson M R 2016 Prog. Part. Nucl. Phys. 86 127 Google Scholar
[2]	Cheal B, Flanagan K T 2010 J. Phys. G: Nucl. Par. 37 113101 Google Scholar
[3]	Blaum K, Dilling J, Nötershäser M 2013 Phys. Scr. T152 014017 Google Scholar
[4]	Angeli I, Marinova K P 2013 At. Data Nucl. Data Tables 99 69 Google Scholar
[5]	Marinova K P, Angeli I https://www-nds.iaea.org/radii/.
[6]	Angeli I 2004 At. Data Nucl. Data Tables 87 185 Google Scholar
[7]	Angeli I 1999 International Nuclear Data Commitee online: http://iaeand.iaea.or.at/indcsel.html.
[8]	De Groote R P, Billowes J, Binnersley C L, et al. 2020 Nat. Phys. 16 620 Google Scholar
[9]	Koszorús Á, Yang X F, Jiang W G, et al. 2021 Nat. Phys. 17 439 Google Scholar
[10]	Bissell M L, Carette T, Flanagan K T, et al. 2016 Phys. Rev. C 93 064318 Google Scholar
[11]	Xie L, Yang X F, Wraith C, et al. 2019 Phys. Lett. B 797 134805 Google Scholar
[12]	Nerlo-Pomorska B, Pomorski K 1993 Z. Phys. A 344 359 Google Scholar
[13]	Nerlo-Pomorska B, Pomorski K 1994 Z. Phys. A 348 169 Google Scholar
[14]	圣宗强, 樊广伟, 钱建发 2015 物理学报 64 112101 Google Scholar Sheng Z Q, Fan G W, Qian J F 2015 Acta Phys. Sin. 64 112101 Google Scholar
[15]	曹颖逾, 郭建友 2020 物理学报 69 162101 Google Scholar Cao Y Y, Guo J Y 2020 Acta Phys. Sin. 69 162101 Google Scholar
[16]	Goriely S, Chamel N, Pearson J M 2016 Phys. Rev. C 93 034337 Google Scholar
[17]	Zhao P W, Li Z P, Yao J M, Meng J 2010 Phys. Rev. C 82 054319 Google Scholar
[18]	Xia X W, Lim Y, Zhao P W, Liang H Z, Qu X Y, Chen Y, Liu H, Zhang L F, Zhang S Q, Kim Y, Meng J 2018 At. Data Nucl. Data Tables 121-122 1 Google Scholar
[19]	Iimura H, Buchinger F 2008 Phys. Rev. C 78 067301 Google Scholar
[20]	Buchinger F, Pearson J M 2005 Phys. Rev. C 72 057305 Google Scholar
[21]	Dieperink A E L, Van Isacker P 2009 Eur. Phys. J. A 42 269 Google Scholar
[22]	Wang N, Li T 2013 Phys. Rev. C 88 011301R Google Scholar
[23]	Stoitsov M V, Dobaczewski J, Nazarewicz W, et al. 2003 Phys. Rev. C 68 054312 Google Scholar
[24]	Goriely S, Chamel N, Pearson J M 2010 Phys. Rev. C 82 035804 Google Scholar
[25]	Bao M, Zong Y Y, Zhao Y M, Arima A 2020 Phys. Rev. C 102 014306 Google Scholar
[26]	Garvey G T, Gerace W J, Jaffe R L, Talmi I, Kelson I 1969 Rev. Mod. Phys. 41 S1 Google Scholar
[27]	Sun B H, Lu Y, Peng J P, Liu C Y, Zhao Y M 2014 Phys. Rev. C 90 054318 Google Scholar
[28]	Bao M, Lu Y, Zhao Y M, Arima A 2016 Phys. Rev. C 94 064315 Google Scholar
[29]	焦宝宝 2022 物理学报 71 152101 Google Scholar Jiao B B 2022 Acta Phys. Sin. 71 152101 Google Scholar
[30]	Ma C, Zong Y Y, Zhao Y M, Arima A 2021 Phys. Rev. C 104 014303 Google Scholar
[31]	Von Weizsäcker C F 1935 Z. Phys 96 431 Google Scholar
[32]	Möller P, Myers W D, Sagawa H, Yoshida S 2012 Phys. Rev. Lett. 108 052501 Google Scholar
[33]	Geng L, Toki H, Meng J 2005 Prog. Theor. Phys. 113 785 Google Scholar
[34]	Goriely S, Chamel N, Pearson J M 2009 Phys. Rev. Lett. 102 152503 Google Scholar
[35]	Duflo J, Zuker A P 1995 Phys. Rev. C 52 R23 Google Scholar
[36]	Qi C 2015 J. Phys. G: Nucl. Par. 42 045104 Google Scholar
[37]	Wang N, Liang Z, Liu M, et al. 2010 Phys. Rev. C 82 044304 Google Scholar
[38]	Garvey G T, Kelson I 1966 Phys. Rev. Lett. 16 197 Google Scholar
[39]	Fu G J, Lei Y, Jiang H, Zhao Y M, et al. 2011 Phys. Rev. C 84 034311 Google Scholar
[40]	Jiang H, Fu G J, Zhao Y M, et al. 2010 Phys. Rev. C 82 054317 Google Scholar
[41]	Jiao B B 2018 Mod. Phys. Lett. A 33 1850156 Google Scholar
[42]	Jiao B B 2020 Int. J. Mod. Phys. E 29 2050024 Google Scholar
[43]	Yang G L, Qi B, Wang X D, Qi C 2022 Phys. Rev. C 106 024325 Google Scholar
[44]	Utama R, Piekarewicz J 2017 Phys. Rev. C 96 044308 Google Scholar
[45]	Utama R, Piekarewicz J 2018 Phys. Rev. C 97 014306 Google Scholar
[46]	Zhang H F, Wang L H, Yin J P, Chen P H, Zhang H F 2017 J. Phys. G: Nucl. Par. 44 045110 Google Scholar
[47]	Niu Z M, Liang H Z 2022 Phys. Rev. C 106 L021303 Google Scholar
[48]	Shang T S, Li J, Niu Z M 2022 Nucl. Sci. Tech. 33 153 Google Scholar
[49]	Z M Niu and H Z Liang 2018 Phys. Lett. B 778 48 Google Scholar
[50]	Ma Y F, Su C, Liu J, Ren Z Z, Xu C, Gao Y H 2020 Phys. Rev. C 101 014304 Google Scholar
[51]	Audi G, Kondev F G, Wang M, Huang W J, Naimi S 2017 Chin. Phys. C 41 030001 Google Scholar
[52]	Wang M, Huang W J, Kondev F G, Audi G, Naimi S 2021 Chin. Phys. C 45 030003 Google Scholar
[53]	卢希庭, 江栋兴, 叶沿林 2000 原子核物理 (北京: 原子能出版社) 第7—9 页 Lu X T, Jiang D X, Ye Y L 2000 Nuclear Physics (Beijing: Atomic Energy Press) pp7–9 (in Chinese)
[54]	Fu G J, Bao M, He Z, Jiang H, Zhao Y M, Arima A 2012 Phys. Rev. C 86 054303

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