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Theoretical investigation on Stark-induced transition probabilities of hydrogen-like ions

Wan Jian-Jie Zhao Xin-Ting Li Ji-Guang Dong Chen-Zhong

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Theoretical investigation on Stark-induced transition probabilities of hydrogen-like ions

Wan Jian-Jie, Zhao Xin-Ting, Li Ji-Guang, Dong Chen-Zhong
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  • Based on the nondegenerate perturbation theory, the Stark-induced transitions are studied for hydrogen-like isoelectronic sequences (Z = 1–92). The Stark-induced mixing coefficients and transition probabilities between the 2s1/2-1s1/2 levels of hydrogen-like ions are reported. The trend of Stark-induced transition probabilities varying with atomic number Z between 2s1/2-1s1/2 levels of hydrogen-like ions and the relativistic effect on the Stark-induced mixing coefficients and transition probabilities are discussed. The scaling relations of the nonrelativistic and relativistic Stark-induced transition probabilities with atomic number Z are obtained. The results show that the Stark-induced transition probabilities of hydrogen-like ions decrease monotonically along the isoelectronic sequence with the increase of atomic number Z. In addition, the relativistic effect reduces the Stark-induced transition probabilities of hydrogen-like ions, for example, by about 55% at Z = 92.
      Corresponding author: Wan Jian-Jie, wanjj@nwnu.edu.cn ; Li Ji-Guang, li_jiguang@iapcm.ac.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11874090, 12075193), the Natural Science Foundation of Gansu Province, China (Grant Nos. 20JR10RA084, 20JR5RA541, 1506RJYA131), the Foundation of Northwest Normal University, China (Grant No. NWNU-LKQN-10-7), and the Scientific Research Foundation of Physics of College of Physics and Electronic Engineering, Northwest Normal University, China
    [1]

    Bucksbaum P, Commins E D, Hunter L 1981 Phys. Rev. Lett. 46 640Google Scholar

    [2]

    Drell P S, Commins E D 1984 Phys. Rev. Lett. 53 968Google Scholar

    [3]

    Gilbert S L, Noecker M C, Watts R N, Wieman C E 1985 Phys. Rev. Lett. 55 2680Google Scholar

    [4]

    Maul M, Schӓfer A, Indelicato P 1998 J. Phys. B 31 2725Google Scholar

    [5]

    Hunter L R, Walker W A, Weiss D S 1986 Phys. Rev. Lett. 56 823Google Scholar

    [6]

    Lellouch L P, Hunter L R 1987 Phys. Rev. A 36 3490Google Scholar

    [7]

    Wielandy S, Sun T H, Hilborn R C, Hunter L R 1992 Phys. Rev. A 46 7103Google Scholar

    [8]

    Fan C Y, Garcia-Munoz M, Sellin I A 1967 Phys. Rev. 161 6Google Scholar

    [9]

    Leventhal M, Murnick D E 1970 Phys. Rev. Lett. 25 1237Google Scholar

    [10]

    Murnick D E, Leventhal M, Kugel H W 1971 Phys. Rev. Lett. 27 1625Google Scholar

    [11]

    Kugel H W, Leventhal M, Murnick D E 1972 Phys. Rev. A 6 1306Google Scholar

    [12]

    Leventhal M, Murnick D E, Kugel H W 1972 Phys. Rev. Lett. 28 1609Google Scholar

    [13]

    Lawrence G P, Fan C Y, Bashkin S 1972 Phys. Rev. Lett. 28 1612Google Scholar

    [14]

    Gould H, Marrus R 1978 Phys. Rev. Lett. 41 1457Google Scholar

    [15]

    周世勋 原著, 陈灏 修订 2009 量子力学教程 (北京: 高等教育出版社) 第121页

    Zhou S X, Chen H 2009 Quantum Mechanics (Beijing: Higher Education Press) p121 (in Chinese)

    [16]

    Cowan R D 1981 The Theory of Atomic Structure and Spectra (Berkeley: University of California) p400

    [17]

    Surzhykov A, Koval P, Fritzsche S 2005 Comput. Phys. Comm. 165 139Google Scholar

    [18]

    Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules (New York: Springer) p640

    [19]

    Johnson W R 1985 Atom. Data Nucl. Data Tables 33 405Google Scholar

    [20]

    Johnson W R 1972 Phys. Rev. Lett. 29 1123Google Scholar

  • 图 1  外电场取向示意图

    Figure 1.  Schematic diagram of an external electric field.

    图 2  Stark诱导跃迁示意图 (a) 无电场; (b) 外加电场

    Figure 2.  Stark-induced transition diagram: (a) without electric field; (b) with electric fieled.

    图 3  类氢离子能级示意图

    Figure 3.  Schematic diagram of hydrogen-like ion levels.

    图 4  类氢离子2s1/2和2p1/2, 3/2之间的Stark混合系数模方(NR和R分别表示非相对论和相对论结果)

    Figure 4.  Module squares of Stark mixing coefficients between 2s1/2 and 2p1/2, 3/2 states of hydrogen-like ions (NR and R stand for nonrelativistic and relativistic cases, respectively).

    图 5  类氢Li2+离子和Ar17+离子2s1/2能级的Stark诱导跃迁寿命

    Figure 5.  Stark-induced lifetime of 2s1/2 levels for hydrogen-like Li2+ and Ar17+ ions.

    图 6  类氢离子2s1/2-1s1/2能级之间的Stark诱导跃迁几率

    Figure 6.  Stark-induced transition probability between 2s1/2-1s1/2 levels of hydrogen-like ions.

    表 1  类氢离子n = 2能级差及径向轨道矩阵元, 其中a[b]表示a × 10b

    Table 1.  Energy differences and radial orbital matrix elements for hydrogen-like ions, where a[b] stands for a × 10b

    ZEnergy difference/cm–1Matrix element
    ΔE1[19]ΔE2$\langle $2p1/2|r||2s1/2$\rangle $$\langle $2p3/2||r||2s1/2$\rangle $$\langle $2p||r||2s $\rangle $$\langle $1s1/2||r||2p1/2$\rangle $$\langle $1s1/2||r||2p3/2$\rangle $$\langle $1s||r||2p$\rangle $
    13.52868[-2]–3.65221[-1]–5.19604–5.19611–5.196151.290241.290241.29027
    24.68400[-1]–5.84353[0]–2.59785–2.59798–2.598080.645090.645080.64513
    32.09220[0]–2.95829[1]–1.73170–1.73191–1.732050.430020.430010.43009
    45.99720[0]–9.34965[1]–1.29858–1.29885–1.299040.322470.322470.32257
    62.60840[1]–4.73326[2]–0.86533–0.86575–0.866030.214900.214890.21504
    87.32500[1]–1.49594[3]–0.64860–0.64915–0.649520.161090.161080.16128
    101.62100[2]–3.65221[3]–0.51846–0.51915–0.519620.128790.128780.12903
    123.08800[2]–7.57322[3]–0.43163–0.43246–0.433010.107240.107220.10752
    145.30600[2]–1.40303[4]–0.36954–0.37051–0.371150.091830.091810.09216
    168.46400[2]–2.39351[4]–0.32291–0.32402–0.324760.080260.080240.08064
    181.27570[3]–3.83394[4]–0.28659–0.28784–0.288680.071250.071230.07168
    201.83800[3]–5.84353[4]–0.25749–0.25888–0.259810.064030.064010.06451
    243.45000[3]–1.21171[5]–0.21372–0.21539–0.216510.053180.053150.05376
    285.86400[3]–2.24485[5]–0.18232–0.18426–0.185580.045400.045370.04608
    329.28800[3]–3.82962[5]–0.15865–0.16087–0.162380.039540.039500.04032
    361.39300[4]–6.13430[5]–0.14013–0.14263–0.144340.034960.034920.03584
    402.00100[4]–9.34965[5]–0.12522–0.12799–0.129900.031270.031230.03226
    442.78800[4]–1.36888[6]–0.11292–0.11597–0.118090.028230.028190.02932
    483.78200[4]–1.93874[6]–0.10259–0.10592–0.108250.025680.025630.02688
    525.03100[4]–2.67035[6]–0.09376–0.09737–0.099930.023510.023450.02481
    566.56100[4]–3.59176[6]–0.08612–0.09001–0.092790.021620.021560.02304
    608.45400[4]–4.73326[6]–0.07942–0.08359–0.086600.019970.019910.02150
    641.08200[5]–6.12739[6]–0.07348–0.07793–0.081190.018500.018440.02016
    681.37000[5]–7.80892[6]–0.06817–0.07290–0.076410.017200.017130.01897
    721.73900[5]–9.81489[6]–0.06338–0.06839–0.072170.016010.015950.01792
    762.20000[5]–1.21846[7]–0.05902–0.06432–0.068370.014940.014870.01698
    802.79500[5]–1.49594[7]–0.05503–0.06060–0.064950.013950.013880.01613
    843.56700[5]–1.81833[7]–0.05135–0.05720–0.061860.013040.012970.01536
    884.62700[5]–2.19021[7]–0.04793–0.05406–0.059050.012190.012120.01466
    926.07300[5]–2.61642[7]–0.04473–0.05114–0.056480.011390.011330.01402
    DownLoad: CSV

    表 2  类氢离子2s1/2-1s1/2之间的跃迁波长和1 V/m 电场中的Stark诱导跃迁几率, 其中a[b]表示a × 10b

    Table 2.  Transition wavelength and Stark-induced probability between 2s1/2-1s1/2 of hydrogen-like ions in electric field of 1 V/m, where a[b] stands for a × 10b

    Zλ/nmTransition probability/s–1
    Model IModel II
    ASIT (NR)ASIT(R)ASIT (NR)ASIT (R)
    1121.502872.7510[–1]2.7508[–1]2.8023[–1]2.8022[–1]
    230.375726.2450[–3]6.2439[–3]6.3253[–3]6.3242[–3]
    313.500327.0428[–4]7.0399[–4]7.1133[–4]7.1104[–4]
    47.593931.5238[–4]1.5227[–4]1.5364[–4]1.5353[–4]
    63.375081.8124[–5]1.8095[–5]1.8234[–5]1.8205[–5]
    81.898484.0858[–6]4.0740[–6]4.1054[–6]4.0936[–6]
    101.215031.3036[–6]1.2978[–6]1.3087[–6]1.3029[–6]
    120.843775.1727[–7]5.1392[–7]5.1899[–7]5.1564[–7]
    140.619912.3847[–7]2.3637[–7]2.3915[–7]2.3705[–7]
    160.474621.2240[–7]1.2099[–7]1.2271[–7]1.2130[–7]
    180.375016.8196[–8]6.7198[–8]6.8347[–8]6.7348[–8]
    200.303764.0558[–8]3.9825[–8]4.0638[–8]3.9904[–8]
    240.210941.6577[–8]1.6143[–8]1.6603[–8]1.6169[–8]
    280.154987.8098[–9]7.5301[–9]7.8204[–9]7.5406[–9]
    320.118664.0660[–9]3.8747[–9]4.0708[–9]3.8794[–9]
    360.093752.2878[–9]2.1507[–9]2.2901[–9]2.1530[–9]
    400.075941.3688[–9]1.2668[–9]1.3700[–9]1.2680[–9]
    440.062768.5316[–10]7.7563[–10]8.5387[–10]7.7630[–10]
    480.052745.5176[–10]4.9160[–10]5.5218[–10]4.9199[–10]
    520.044933.6594[–10]3.1869[–10]3.6620[–10]3.1894[–10]
    560.038742.4954[–10]2.1184[–10]2.4971[–10]2.1199[–10]
    600.033751.7254[–10]1.4233[–10]1.7265[–10]1.4243[–10]
    640.029661.1984[–10]9.5735[–11]1.1992[–10]9.5802[–11]
    680.026288.4389[–11]6.5035[–11]8.4441[–11]6.5081[–11]
    720.023445.8719[–11]4.3471[–11]5.8755[–11]4.3503[–11]
    760.021044.0878[–11]2.8934[–11]4.0905[–11]2.8956[–11]
    800.018982.8062[–11]1.8889[–11]2.8082[–11]1.8905[–11]
    840.017221.8996[–11]1.2085[–11]1.9011[–11]1.2096[–11]
    880.015691.2390[–11]7.3978[–12]1.2401[–11]7.4061[–12]
    920.014367.8610[–12]4.3696[–12]7.8695[–12]4.3757[–12]
    DownLoad: CSV

    表 3  类氢Li2+离子和Ar17+离子2s能级的相对论Stark诱导跃迁寿命, 其中a(b)[c]表示a(b) × 10c, b是实验测量不确定度

    Table 3.  Relativistic Stark-induced transition lifetime for 2s level of hydrogen-like Li2+ and Ar17+ ions, where a(b)[c] stands for a(b) × 10c and b is the experimental uncertainty.

    Zε/(V·m-1)τSIT/ns
    ExpModel IModel II
    37.425(2)[5]2.629(21)[8]2.582.55
    9.173(2)[5]1.764(35) [8]1.691.67
    185.93[7]3.86(3) [14]4.234.22
    7.14[7]2.80(6) [14]2.922.91
    8.06[7]2.28(2) [14]2.292.29
    8.60[7]2.00(2) [14]2.012.01
    DownLoad: CSV
  • [1]

    Bucksbaum P, Commins E D, Hunter L 1981 Phys. Rev. Lett. 46 640Google Scholar

    [2]

    Drell P S, Commins E D 1984 Phys. Rev. Lett. 53 968Google Scholar

    [3]

    Gilbert S L, Noecker M C, Watts R N, Wieman C E 1985 Phys. Rev. Lett. 55 2680Google Scholar

    [4]

    Maul M, Schӓfer A, Indelicato P 1998 J. Phys. B 31 2725Google Scholar

    [5]

    Hunter L R, Walker W A, Weiss D S 1986 Phys. Rev. Lett. 56 823Google Scholar

    [6]

    Lellouch L P, Hunter L R 1987 Phys. Rev. A 36 3490Google Scholar

    [7]

    Wielandy S, Sun T H, Hilborn R C, Hunter L R 1992 Phys. Rev. A 46 7103Google Scholar

    [8]

    Fan C Y, Garcia-Munoz M, Sellin I A 1967 Phys. Rev. 161 6Google Scholar

    [9]

    Leventhal M, Murnick D E 1970 Phys. Rev. Lett. 25 1237Google Scholar

    [10]

    Murnick D E, Leventhal M, Kugel H W 1971 Phys. Rev. Lett. 27 1625Google Scholar

    [11]

    Kugel H W, Leventhal M, Murnick D E 1972 Phys. Rev. A 6 1306Google Scholar

    [12]

    Leventhal M, Murnick D E, Kugel H W 1972 Phys. Rev. Lett. 28 1609Google Scholar

    [13]

    Lawrence G P, Fan C Y, Bashkin S 1972 Phys. Rev. Lett. 28 1612Google Scholar

    [14]

    Gould H, Marrus R 1978 Phys. Rev. Lett. 41 1457Google Scholar

    [15]

    周世勋 原著, 陈灏 修订 2009 量子力学教程 (北京: 高等教育出版社) 第121页

    Zhou S X, Chen H 2009 Quantum Mechanics (Beijing: Higher Education Press) p121 (in Chinese)

    [16]

    Cowan R D 1981 The Theory of Atomic Structure and Spectra (Berkeley: University of California) p400

    [17]

    Surzhykov A, Koval P, Fritzsche S 2005 Comput. Phys. Comm. 165 139Google Scholar

    [18]

    Grant I P 2007 Relativistic Quantum Theory of Atoms and Molecules (New York: Springer) p640

    [19]

    Johnson W R 1985 Atom. Data Nucl. Data Tables 33 405Google Scholar

    [20]

    Johnson W R 1972 Phys. Rev. Lett. 29 1123Google Scholar

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Publishing process
  • Received Date:  25 January 2021
  • Accepted Date:  25 April 2021
  • Available Online:  07 June 2021
  • Published Online:  05 September 2021

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