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钫原子磁偶极超精细结构常数及其同位素的磁偶极矩的理论计算

娄冰琼 李芳 王沛妍 王黎明 唐永波

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钫原子磁偶极超精细结构常数及其同位素的磁偶极矩的理论计算

娄冰琼, 李芳, 王沛妍, 王黎明, 唐永波

Ab initio calculation of hyperfine-structure constant A of Fr and evaluation of magnetic dipole moments of Fr isotopes

Lou Bing-Qiong, Li Fang, Wang Pei-Yan, Wang Li-Ming, Tang Yong-Bo
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  • 应用基于B样条基组的相对论耦合簇理论方法, 计算了212Fr原子的nS (n = 7—12), nP (n = 7—12)和nD (n = 6—11)态的磁偶极超精细结构常数. 与精确实验值的比较说明这套理论方法能精确计算出磁偶极超精细结构常数, 其中7P态的磁偶极超精细常数的理论值与实验值之间的差异小于1%. 在忽略场移效应对Fr原子7P态超精细结构常数的影响下, 通过结合实验值进一步定出了207−213,220−228Fr核磁偶极矩$\mu $, 这些值与已有的测量值具有非常好的一致性. 本文报道了12S, nP (n = 9—12)和 nD (n = 10—11)态的磁偶极超精细结构常数.
    As the heaviest atom in alkali-metal elements, Fr atom has been regarded as a candidate for the search of the permanent electric dipole moment of the electron and of parity-nonconservation effects. Accurate knowledge of Fr atomic properties is of great interest. In this work, we use a relativistic coupled-cluster method to calculate the magnetic dipole hyperfine structure constants for nS (n = 7-12), nP (n = 7-12) and nD (n = 6-11) states of 212Fr. A finite B-spline basis set is used to expand the Dirac radial function, including completely the single and double excitation in correlation calculation. Our results are compared with available theoretical and experimental values. The comparison shows that our method can offer accurate calculation of magnetic dipole hyperfine structure constant. For 7P state the differences between our results and experimental values are within 1%. The magnetic dipole hyperfine structure constants for 12S, nP (n = 9-12) and nD (n = 10-11) states are reported for the first time, which are very useful as benchmarks for experimental measurements and calculations by other theoretical methods of these quantities. In the relativistic coupled-cluster theoretical framework, we study the electron correlation effect on hyperfine-structure constant A for the S, P, and D states of Fr. We observe that the electron correlation effect is very important for hyperfine-structure constant properties. The D state has a considerable correlation effect. At the same time, we also investigate contribution trends of individual electron correlation effects involving direct, core-polarization and pair-correlation ones in S, P, and D Rydberg series. It is found that the dominant contributions for the S1/2, P1/2,3/2 and nD3/2 (n = 7-11) states are to from the direct effect; however, the dominant contributions for the 6D3/2, and nD5/2 (n = 6-11) states are due to the pair-correlation and the core-polarization, respectively. For D5/2 states, there is very strong cancellation among these individual correlation effects. The knowledge of these correlation trends is useful for studying the permanent electric dipole moment and parity-nonconservation effect of Fr in future. Moreover, the magnetic dipole moment $ {\mu}$ for each of isotopes 207−213,220−228Fr is determined by combining with experimental values for magnetic dipole hyperfine structure constant of 7P state. For each of isotope 207−213Fr, our magnetic dipole moment $ {\mu}$ is perfectly consistent with the experimental value, and our uncertainties are twice smaller than those in the experiments . For each of isotope 220−228Fr, our magnetic dipole moment $ {\mu}$ has a larger uncertainty, but is still in agreement with the experimental magnetic dipole moment $ {\mu}$.
      通信作者: 唐永波, ybtang@whu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 1154094, 11774080)资助的课题.
      Corresponding author: Tang Yong-Bo, ybtang@whu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 1154094, 11774080).
    [1]

    Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules (New York: Springer) pp533−577

    [2]

    Fischer C F, Brage T, Jönsson P 1997 Computational Atomic Structure: An MCHF Approach (UK: Institute of Physics) pp1−67

    [3]

    Jönsson P, Gaigalas G, Bieroń J, Fishcher C F, Grant I 2013 Computer Physics Communications. 184 2197Google Scholar

    [4]

    Jönsson P, He X, Fishcher C F, Grant I 2007 Computer Physics Communications. 177 597Google Scholar

    [5]

    Dzuba V A, Flambaum V V, Kozlov M G 1996 Phys. Rev. A 54 3948Google Scholar

    [6]

    Dzuba V A, Johnson W R 1998 Phys. Rev. A 57 2459Google Scholar

    [7]

    Angstmann E J, Dzuba V A, Flambaum V V 2004 Phys. Rev. A 70 014102Google Scholar

    [8]

    Dinh T H, Dzuba V A, Flambaum V V, Ginges J S M 2008 Phys. Rev. A 78 054501Google Scholar

    [9]

    Kozlov M G, Porsev S G, Johnson W R 2001 Phys. Rev. A 64 052107Google Scholar

    [10]

    Pal R, Safronova M S, Johnson W R, Derevianko A, Porsev S G 2007 Phys. Rev. A 75 042515Google Scholar

    [11]

    Blundell S A, Johnson W R, Liu Z W, Sapirstein 1989 Phys. Rev. A 40 2233Google Scholar

    [12]

    Eliav E, Vikas M J, Ishikawa Y, Kaldor U 2005 Chem. Phys. 311 163Google Scholar

    [13]

    Mani B K, Angom D 2011 Phys. Rev. A 83 012501Google Scholar

    [14]

    Kallay M, Nataraj H S, Sahoo B K, Das B P, Visscher L 2011 Phys. Rev. A 83 030503Google Scholar

    [15]

    Nandy D K, Singh Y, Sahoo B K 2014 Phys. Rev. A 89 062509Google Scholar

    [16]

    Borschevsky A, Eliav E, Vilkas M J, Ishikawa Y, Kaldor U 2007 Phys. Rev. A 75 042514Google Scholar

    [17]

    Eliav E, Kaldor U, Ishikawa Y 1996 Phys. Rev. A 53 3050Google Scholar

    [18]

    Chaudhuri R K, Chattopadhyay S, Mahapatra U S 2013 J. Phys. Chem. A 117 12616Google Scholar

    [19]

    Tang Y B, Lou B Q, Shi T Y 2017 Phys. Rev. A 96 022513Google Scholar

    [20]

    Tang Y B, Gao N N, Lou B Q, Shi T Y 2018 Phys. Rev. A 98 062511Google Scholar

    [21]

    Byrnes T M R, Dzuba V A, Flambaum F F, Murray D W 1999 Phys. Rev. A 59 3082Google Scholar

    [22]

    Mukherjee D, Sahoo B K, Nataraj H S, Das B P 2009 J. Phys. Chem. A 113 12549Google Scholar

    [23]

    Sakemi Y, Harada K, Hayamizu T, Itoh M, Kawamura H, Liu S, Nataraj H S, Oikawa A, Saito M, Sato T 2011 J. Phys. Conf. Ser. 302 012051Google Scholar

    [24]

    Sahoo B K, Aoki T, Das B P, Sakemi Y 2016 Phys. Rev. A 93 032520Google Scholar

    [25]

    Atutov S N, Calabrese R, Corradi L, Dainelli A, Mauro C D, Khanbekyan A, Mariotti E, Minguzzi P, Moi L, Sanguinetti S, Stancari G, Tomassetti L 2008 Proc. SPIE 7027 70270C

    [26]

    Ekström C, Ingelman S, Wannberg G, Skarestad M 1978 Physica Scripta 18 51Google Scholar

    [27]

    Coc A, Thibault C, Touchard F, Duong H T, Juncar P, Liberman S, Pinard J, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1985 Phys. Lett. B 163 66Google Scholar

    [28]

    Coc A, Thibault C, Touchard F, Duong H T, Juncar P, Liberman S, Pinard J, Carre M, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1987 Nucl. Phys. A 468 1Google Scholar

    [29]

    Arnold E, Borchers W, Duong H T, Juncar P, Lermé J, Lievens P, Neu W, Neugart R, Pellerin M, Pinard J, Vialle J L, Wendt K, the ISOLDE Collaboration 1990 J. Phys. B 23 3511Google Scholar

    [30]

    Arnold E, Borchers W, Carré M, Duong H T, Juncar P, Lermé J, Liberman S, Neu W, Neugart R, Otten W, Pellerin M, Pinard J, Pesnelle A, Vialle J L, Wendt K, the ISOLDE Collaboration 1989 J. Phys. B 22 L391Google Scholar

    [31]

    Bauche J, Duong H T, Juncar P, Liberman S, Pinard J, Coc A, Thibault C, Touchard F, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1986 J. Phys. B 19 L593Google Scholar

    [32]

    Grossman J S, Orozco L A, Simsarian J E, Sprouse G D, Zhao W Z 1999 Phys. Rev. Lett. 83 935Google Scholar

    [33]

    Sansonetti J E 2007 J. Phys. Chem. Ref. Data 36 497Google Scholar

    [34]

    Gomez E, Aubin S, Orozco L A, Sprouse G D, Iskrenova-Tchoukova E, Safronova M S 2008 Phys. Rev. Lett. 100 172502Google Scholar

    [35]

    Dzuba V A, Flambaum V V, Sushkov O P 1984 J. Phys. B: At. Mol. Phys. 17 1953Google Scholar

    [36]

    Owusu A, Dougherty R W, Gowri G, Das T P 1997 Phys. Rev. A 56 305Google Scholar

    [37]

    Safronova M S, Johnson W R, Derevianko A 1999 Phys. Rev. A 60 4476Google Scholar

    [38]

    Sahoo B K, Nandy D K, Das B P, Sakemi Y 2015 Phys. Rev. A 91 042507Google Scholar

    [39]

    Duong H T, Juncar P, Liberman S, Mueller A C, Neugart R, Otten E W, Peuse B, Pinard J, Stoke H H, Thibault C, Touchard F, Vialle J L, Wendt K, the ISOLDE Collaboration 1987 Europhys. Lett. 3 175Google Scholar

    [40]

    Barber Z W, Stalnaker J E, Lemke N D, Poli N, Oates C W, Fortier T M, Diddams S A, Hollberg L, Hoyt C W, Taichenachev A V, Yudin V I 2008 Phys. Rev. Lett. 100 103002Google Scholar

    [41]

    Kien F L, Balykin V I, Hakuta K 2005 J. Phys. Soc. Jpn. 74 910Google Scholar

    [42]

    Ingvar L 1978 Int. J. Quantum Chem. 12 33

    [43]

    Sinha D, Mukhopadhyay S, Mukherjee D 1986 Chem. Phys. Lett. 129 369Google Scholar

    [44]

    Blundell S A, Johnson W R, Sapiratein J 1991 Phys. Rev. A 43 3407Google Scholar

    [45]

    Porsev S G, Beloy K, Derevianko A 2010 Phys. Rev. D 82 036008Google Scholar

    [46]

    Sahoo B K, Sur C, Beier T, Das B P, Chaudhuri R K, Mukherjee D 2007 Phys. Rev. A 75 042504Google Scholar

    [47]

    Safronova M S, Safronova U I 2011 Phys. Rev. A 83 052508Google Scholar

  • 图 1  212Fr原子S1/2, P1/2, P3/2, D3/2和D5/2态磁偶极超精细结构常数中的电子关联效应

    Fig. 1.  Electron correlation effects in hyperfine-structure constant A for S1/2, P1/2, P3/2, D3/2 and D5/2 states of 212Fr.

    图 2  直接效应ADF、核极化效应ACP、对关联效应APC, 以及相对于CCSD的3种效应的总和AT = ADF + ACP + APC, 针对主量子数n的S, P和D态的结果A的比率 (a) ADF/A; (b) ACP/A; (c) APC/A; (d) AT/A

    Fig. 2.  Ratios of direct effect ADF, core polarization effect ACP, pair correlation effect APC, and the total of the three effects AT = ADF + APC + ACP to the CCSD, results A for S, P and D states against the principal quantum number n: (a) ADF/A; (b) ACP/A; (c) APC/A; (d) AT/A.

    表 1  212Fr原子S态的超精细结构常数A (单位: MHz)

    Table 1.  Hyperfine-structure constant A for the S states of 212Fr in MHz.

    能级${A_{{\rm{DF}}}}$${A_{{\rm{CCSD}}}}$$\varDelta /$%Ref.[38]实验值
    7S1/26001.769403.5630.939124(94)9064.2(2)[27]
    9064.4(1.5)[39]
    8S1/21538.032014.1017.371986(19)
    9S1/2631.98792.1913.66784(9)
    10S1/2321.24396.8611.95419(9)401(5)[29]
    11S1/2185.41225.7111.10212(9)225(3)[29]
    12S1/2116.42141.2510.80
    下载: 导出CSV

    表 2  212Fr原子P态的超精细结构常数A (单位: MHz)

    Table 2.  Hyperfine-structure constant A for the P states of 212Fr in MHz.

    能级${A_{{\rm{DF}}}}$${A_{{\rm{CCSD}}}}$$\varDelta /$%Ref.[38]实验值
    7P1/2642.481198.1041.961181(9)1189.1(4.6)[28]
    1187.1(6.8)[39]
    1192.0(2)[32]
    8P1/2228.04372.0433.66371(5)373.0(1)[39]
    9P1/2106.78167.2130.88
    10P1/258.3589.5329.46
    11P1/235.2653.3828.51
    12P1/222.8834.2427.68
    7P3/251.0597.8843.5596(3)97.2(1)[27]
    97.2(1)[39]
    8P3/218.6732.5137.8232(3)32.8(1)[39]
    9P3/28.8915.0035.83
    10P3/24.918.1534.75
    11P3/23.004.9234.01
    12P3/21.973.2033.33
    下载: 导出CSV

    表 3  212Fr原子D态的超精细结构常数A (单位: MHz)

    Table 3.  Hyperfine-structure constant A for the D states of 212Fr in MHz.

    能级${A_{{\rm{DF}}}}$${A_{{\rm{CCSD}}}}$$\varDelta / $%Ref.[38]实验值
    6D3/233.2592.9161.2779(5)
    7D3/216.8230.1739.6529(3)
    8D3/28.6513.8132.2013(1)13.0(6)[29]
    9D3/24.937.5028.757(1)7.1(7)[29]
    10D3/23.064.5226.64
    11D3/22.032.9324.97
    6D5/213.14–53.92126.38–54(5)
    7D5/26.32–13.64150.21–15(3)
    8D5/23.20–5.67161.12–6(1)–7.1(6)[29]
    9D5/21.81–2.96166.96–3.3(6)–3.6(4)[29]
    10D5/21.12–1.72170.59
    11D5/20.74–1.10173.16
    下载: 导出CSV

    表 4  Fr原子同位素的磁偶极矩$\mu$

    Table 4.  Magnetic dipole moment $\mu$ of Fr isotope.

    同位素核自旋7P1/27P3/2${\mu}$
    ${A_{{\rm{expt}}.}}$[33]${{\mu} _{1/2}}$${A_{{\rm{expt}}.}}$[33]${{\mu} _{3/2}}$${{\mu} _{{\rm{present}}}}$${{\mu} _{{\rm{expt}}{\rm{.}}}}$[33]
    207Fr9/290.7(6)3.85(3)3.85(3)3.89(9)
    208Fr7874.8(3)4.723(2)72.4(5)4.784(33)4.753(33)4.75(10)
    209Fr9/21127.9(2)3.914(1)93.3(5)3.963(21)3.939(22)3.95(8)
    210Fr6946.3(3)4.379(1)78.0(2)4.418(11)4.399(20)4.40(9)
    211Fr9/21142.1(2)3.964(1)94.9(3)4.031(13)3.998(34)4.00(8)
    212Fr51187(7)4.577(26)97.2(1)4.588(5)4.583(30)4.62(9)
    213Fr9/21150(8)3.991(28)95.3(3)4.047(13)4.019(30)4.02(8)
    220Fr1–73.2(5)–0.691(5)–0.691(5)–0.67(1)
    221Fr5/2808(12)1.558(23)65.5(6)1.545(14)1.552(25)1.58(3)
    222Fr233(1)0.623(19)0.623(19)0.63(1)
    223Fr3/283.3(9)1.179(13)1.179(13)1.17(2)
    224Fr142.1(7)0.397(7)0.397(7)0.40(1)
    225Fr3/277(3)1.090(42)1.090(42)1.07(2)
    226Fr17(1)0.066(9)0.066(9)0.071(2)
    227Fr1/2316(2)1.491(9)1.491(9)1.50(3)
    228Fr2–41(2)–0.77(4)–0.77(4)–0.76(2)
    下载: 导出CSV
  • [1]

    Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules (New York: Springer) pp533−577

    [2]

    Fischer C F, Brage T, Jönsson P 1997 Computational Atomic Structure: An MCHF Approach (UK: Institute of Physics) pp1−67

    [3]

    Jönsson P, Gaigalas G, Bieroń J, Fishcher C F, Grant I 2013 Computer Physics Communications. 184 2197Google Scholar

    [4]

    Jönsson P, He X, Fishcher C F, Grant I 2007 Computer Physics Communications. 177 597Google Scholar

    [5]

    Dzuba V A, Flambaum V V, Kozlov M G 1996 Phys. Rev. A 54 3948Google Scholar

    [6]

    Dzuba V A, Johnson W R 1998 Phys. Rev. A 57 2459Google Scholar

    [7]

    Angstmann E J, Dzuba V A, Flambaum V V 2004 Phys. Rev. A 70 014102Google Scholar

    [8]

    Dinh T H, Dzuba V A, Flambaum V V, Ginges J S M 2008 Phys. Rev. A 78 054501Google Scholar

    [9]

    Kozlov M G, Porsev S G, Johnson W R 2001 Phys. Rev. A 64 052107Google Scholar

    [10]

    Pal R, Safronova M S, Johnson W R, Derevianko A, Porsev S G 2007 Phys. Rev. A 75 042515Google Scholar

    [11]

    Blundell S A, Johnson W R, Liu Z W, Sapirstein 1989 Phys. Rev. A 40 2233Google Scholar

    [12]

    Eliav E, Vikas M J, Ishikawa Y, Kaldor U 2005 Chem. Phys. 311 163Google Scholar

    [13]

    Mani B K, Angom D 2011 Phys. Rev. A 83 012501Google Scholar

    [14]

    Kallay M, Nataraj H S, Sahoo B K, Das B P, Visscher L 2011 Phys. Rev. A 83 030503Google Scholar

    [15]

    Nandy D K, Singh Y, Sahoo B K 2014 Phys. Rev. A 89 062509Google Scholar

    [16]

    Borschevsky A, Eliav E, Vilkas M J, Ishikawa Y, Kaldor U 2007 Phys. Rev. A 75 042514Google Scholar

    [17]

    Eliav E, Kaldor U, Ishikawa Y 1996 Phys. Rev. A 53 3050Google Scholar

    [18]

    Chaudhuri R K, Chattopadhyay S, Mahapatra U S 2013 J. Phys. Chem. A 117 12616Google Scholar

    [19]

    Tang Y B, Lou B Q, Shi T Y 2017 Phys. Rev. A 96 022513Google Scholar

    [20]

    Tang Y B, Gao N N, Lou B Q, Shi T Y 2018 Phys. Rev. A 98 062511Google Scholar

    [21]

    Byrnes T M R, Dzuba V A, Flambaum F F, Murray D W 1999 Phys. Rev. A 59 3082Google Scholar

    [22]

    Mukherjee D, Sahoo B K, Nataraj H S, Das B P 2009 J. Phys. Chem. A 113 12549Google Scholar

    [23]

    Sakemi Y, Harada K, Hayamizu T, Itoh M, Kawamura H, Liu S, Nataraj H S, Oikawa A, Saito M, Sato T 2011 J. Phys. Conf. Ser. 302 012051Google Scholar

    [24]

    Sahoo B K, Aoki T, Das B P, Sakemi Y 2016 Phys. Rev. A 93 032520Google Scholar

    [25]

    Atutov S N, Calabrese R, Corradi L, Dainelli A, Mauro C D, Khanbekyan A, Mariotti E, Minguzzi P, Moi L, Sanguinetti S, Stancari G, Tomassetti L 2008 Proc. SPIE 7027 70270C

    [26]

    Ekström C, Ingelman S, Wannberg G, Skarestad M 1978 Physica Scripta 18 51Google Scholar

    [27]

    Coc A, Thibault C, Touchard F, Duong H T, Juncar P, Liberman S, Pinard J, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1985 Phys. Lett. B 163 66Google Scholar

    [28]

    Coc A, Thibault C, Touchard F, Duong H T, Juncar P, Liberman S, Pinard J, Carre M, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1987 Nucl. Phys. A 468 1Google Scholar

    [29]

    Arnold E, Borchers W, Duong H T, Juncar P, Lermé J, Lievens P, Neu W, Neugart R, Pellerin M, Pinard J, Vialle J L, Wendt K, the ISOLDE Collaboration 1990 J. Phys. B 23 3511Google Scholar

    [30]

    Arnold E, Borchers W, Carré M, Duong H T, Juncar P, Lermé J, Liberman S, Neu W, Neugart R, Otten W, Pellerin M, Pinard J, Pesnelle A, Vialle J L, Wendt K, the ISOLDE Collaboration 1989 J. Phys. B 22 L391Google Scholar

    [31]

    Bauche J, Duong H T, Juncar P, Liberman S, Pinard J, Coc A, Thibault C, Touchard F, Lermé J, Vialle J L, Büttgenbach S, Mueller A C, Pesnelle A, the ISOLDE Collaboration 1986 J. Phys. B 19 L593Google Scholar

    [32]

    Grossman J S, Orozco L A, Simsarian J E, Sprouse G D, Zhao W Z 1999 Phys. Rev. Lett. 83 935Google Scholar

    [33]

    Sansonetti J E 2007 J. Phys. Chem. Ref. Data 36 497Google Scholar

    [34]

    Gomez E, Aubin S, Orozco L A, Sprouse G D, Iskrenova-Tchoukova E, Safronova M S 2008 Phys. Rev. Lett. 100 172502Google Scholar

    [35]

    Dzuba V A, Flambaum V V, Sushkov O P 1984 J. Phys. B: At. Mol. Phys. 17 1953Google Scholar

    [36]

    Owusu A, Dougherty R W, Gowri G, Das T P 1997 Phys. Rev. A 56 305Google Scholar

    [37]

    Safronova M S, Johnson W R, Derevianko A 1999 Phys. Rev. A 60 4476Google Scholar

    [38]

    Sahoo B K, Nandy D K, Das B P, Sakemi Y 2015 Phys. Rev. A 91 042507Google Scholar

    [39]

    Duong H T, Juncar P, Liberman S, Mueller A C, Neugart R, Otten E W, Peuse B, Pinard J, Stoke H H, Thibault C, Touchard F, Vialle J L, Wendt K, the ISOLDE Collaboration 1987 Europhys. Lett. 3 175Google Scholar

    [40]

    Barber Z W, Stalnaker J E, Lemke N D, Poli N, Oates C W, Fortier T M, Diddams S A, Hollberg L, Hoyt C W, Taichenachev A V, Yudin V I 2008 Phys. Rev. Lett. 100 103002Google Scholar

    [41]

    Kien F L, Balykin V I, Hakuta K 2005 J. Phys. Soc. Jpn. 74 910Google Scholar

    [42]

    Ingvar L 1978 Int. J. Quantum Chem. 12 33

    [43]

    Sinha D, Mukhopadhyay S, Mukherjee D 1986 Chem. Phys. Lett. 129 369Google Scholar

    [44]

    Blundell S A, Johnson W R, Sapiratein J 1991 Phys. Rev. A 43 3407Google Scholar

    [45]

    Porsev S G, Beloy K, Derevianko A 2010 Phys. Rev. D 82 036008Google Scholar

    [46]

    Sahoo B K, Sur C, Beier T, Das B P, Chaudhuri R K, Mukherjee D 2007 Phys. Rev. A 75 042504Google Scholar

    [47]

    Safronova M S, Safronova U I 2011 Phys. Rev. A 83 052508Google Scholar

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出版历程
  • 收稿日期:  2019-01-21
  • 修回日期:  2019-03-09
  • 上网日期:  2019-05-01
  • 刊出日期:  2019-05-05

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