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Bremsstrahlung, as an important radiation process in atomic physics, has significant applications in the fields of astrophysics, plasma physics, magnetic and inertial confinement fusion. In this work, the relativistic partial-wave expansion method is used to investigate the bremsstrahlung of neutral carbon atoms and different charged carbon ions scattered from intermediate- and high-energy relativistic electrons, with special attention paid to the electronic screening effect produced by the target electrons. The target wave function is obtained from the Dirac-Hartree-Fock self-consistent calculations, and the electron-atom scattering interaction potential is constructed in the central-field approximation. By solving the partial-wave Dirac equation, the continuum wave functions of the relativistic electron are obtained, from which the bremsstrahlung single and double differential cross sections can be calculated via the multipole free-free transitions between the incident and exit free electrons. The target electronic screening effects on the bremsstrahlung single and double differential cross sections are analyzed under a variety of conditions of incident electron energy and emitted photon energy. It is shown that the target electronic screening effect will significantly suppress the cross sections both at low incident energy and in the soft-photon region. Such a suppressing effect decreases with the incident electron energy and the emitted photon energy gradually increasing. Overall, the electronic screening effect has no significant influence on the shape function of bremsstrahlung.
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Keywords:
- bremsstrahlung /
- C atoms and ions /
- differential cross sections /
- shape function /
- electronic screening effect
[1] Buǐmistrov V M, Trakhtenberg L I 1975 J. Exp. Theor. Phys. 42 54
[2] Amus'ya M Y, Baltenkov A S, Paiziev A A 1976 JETP Lett. 24 332
[3] Korol A V, Obolensky O I, Solov'yov A V, Solovjev I A 2001 J. Phys. B: At. Mol. Opt. Phys. 34 1589
Google Scholar
[4] Lea S M, Silk J, Kellogg E, Murray S 1973 Astrophys. J. 184 L105
Google Scholar
[5] Cavaliere A, Fusco-Femiano R 1976 Astron. Astrophys. 49 137
[6] Hannestad S, Raffelt G 1998 Astrophys. J. 507 339
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[7] Wang W Y, Lu J G, Tong H, Ge M Y, Li Z S, Men Y P, Xu R X 2017 Astrophys. J. 837 81
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[8] Steane A M 2024 Phys. Rev. D 109 063032
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[9] Maydanyuk S P, Zhang P M, Zou L P 2016 Phys. Rev. C 93 014617
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[10] Omar A, Andreo P, Poludniowski G 2018 Radiat. Phys. Chem. 148 73
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[11] Jakubassa-Amundsen D H 2021 arXiv: 2103.06034 [physics. atom-ph]
[12] Li L, An Z, Zhu J J, Tan W J, Sun Q, Liu M T 2019 Phys. Rev. A 99 052701
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[13] Kornev A S, Zon B A, Chernov V E, Amusia M Y, Kubelík P, Ferus M 2022 Atoms 10 86
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[14] Milstein A I, Salnikov S G, Kozlov M G 2023 Nucl. Instrum. Methods Phys. Res., Sect. B 539 9
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[15] Guazzotto L, Betti R 2019 Plasma Phys. Control. Fusion 61 085028
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[16] Bhaskar B S, Koivisto H, Tarvainen O, Thuillier T, Toivanen V, Kalvas T, Izotov I, Skalyga V, Kronholm R, Marttinen M 2021 Plasma Phys. Control. Fusion 63 095010
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[17] Bone T, Sedwick R 2024 Acta Astronaut. 220 356
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[21] Boozer A H 2005 Rev. Mod. Phys. 76 1071
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[22] Munirov V R, Fisch N J 2023 Phys. Rev. E 107 065205
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[24] Singh S, Armstrong C D, Kang N, et al. 2021 Plasma Phys. Control. Fusion 63 035004
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Google Scholar
[27] Roshan H R, Mahmoudian B, Gharepapagh E, Azarm A, Islamian J P 2016 Appl. Radiat. Isot. 108 124
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[28] Uribe C F, Esquinas P L, Gonzalez M, Celler A 2016 Phys. Med. 32 691
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Google Scholar
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Google Scholar
[44] Amusia M Y, Avdonina N B, Chernysheva L V, Kuchiev M Y 1985 J. Phys. B: Atom. Mol. Phys. 18 L791
Google Scholar
[45] Korol A V, Lyalin A G, Solovy'ov A V, Avdonina N B, Pratt R H 2002 J. Phys. B: At. Mol. Opt. Phys. 35 1197
Google Scholar
[46] Tseng H K, Pratt R H 1973 Phys. Rev. A 7 1502
Google Scholar
[47] Mangiarotti A, Lauth W, Jakubassa-Amundsen D H, Klag P, Malafronte A A, Martins M N, Nielsen C F, Uggerhøj U I 2021 Phys. Lett. B 815 136113
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[50] Yerokhin V A, Surzhykov A, Märtin R, Tashenov S, Weber G 2012 Phys. Rev. A 86 032708
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[51] García-Alvarez J A, Fernández-Varea J M, Vanin V R, Maidana N L 2018 J. Phys. B: At. Mol. Opt. Phys. 51 225003
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[54] Rodrigues G C, Indelicato P, Santos J P, Patté P, Parente F 2004 At. Data Nucl. Data Tables 86 117
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Google Scholar
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图 1 中性C原子及其各价态离子($ \rm C^{1+} $, $ \rm C^{2+} $, $ \rm C^{3+} $, $ \rm C^{4+} $, $ \rm C^{5+} $)的屏蔽函数
Figure 1. Screening function of C atom and different charged ions ($ \rm C^{+} $, $ \rm C^{2+} $, $ \rm C^{3+} $, $ \rm C^{4+} $, $ \rm C^{5+} $).
图 2 $ \rm C^{2+} $离子轫致辐射单重微分截面 (a)入射电子动能$ T_{1} $为1, 10, 100和1000 keV; (b)出射光子能量占比$ E_{{\mathrm{p}}} / T_{1} $为0.05—0.95
Figure 2. Bremsstrahlung single differential cross section for the $ \rm C^{2+} $ ion: (a) The kinetic energies of the incident electron $ T_{1} $ are 1, 10, 100 and 1000 keV; (b) the ratios of the emitted photon energy to the incident electron energy $ E_{{\mathrm{p}}}/T_{1} $ are in the range of 0.05–0.95.
图 3 中性C原子及$ \rm C^{6+} $, $ \rm C^{5+} $, $ \rm C^{4+} $, $ \rm C^{2+} $离子的轫致辐射单重微分截面随发射光子能量占比$ E_{{\mathrm{p}}}/T_{1} $的变化 (a) $ T_{1}=1 $ keV; (b) $ T_{1}=10 $ keV; (c) $ T_{1}=100 $ keV; (d) $ T_{1}=1000 $ keV
Figure 3. Bremsstrahlung single differential cross sections of neutral C atom and $ \rm C^{6+} $, $ \rm C^{5+} $, $ \rm C^{4+} $, $ \rm C^{2+} $ ions as a function of $ E_{{\mathrm{p}}}/T_{1} $: (a) $ T_{1}=1 $ keV; (b) $ T_{1}=10 $ keV; (c) $ T_{1}=100 $ keV; (d) $ T_{1}=1000 $ keV.
图 4 中性C原子及$ \rm C^{6+} $, $ \rm C^{5+} $, $ \rm C^{4+} $, $ \rm C^{2+} $离子的轫致辐射单重微分截面随入射电子能量$ T_{1} $的变化 (a) $ E_{{\mathrm{p}}} / T_{1}=0.05 $; (b) $ E_{{\mathrm{p}}} / T_{1}=0.5 $; (c) $ E_{{\mathrm{p}}} / T_{1}=0.95 $
Figure 4. Bremsstrahlung single differential cross sections of neutral C atom and $ \rm C^{6+} $, $ \rm C^{5+} $, $ \rm C^{4+} $, $ \rm C^{2+} $ ions as a function of $ T_1 $: (a) $ E_{{\mathrm{p}}} / T_{1}=0.05 $; (b) $ E_{{\mathrm{p}}} / T_{1}=0.5 $; (c) $ E_{{\mathrm{p}}} / T_{1}=0.95 $.
图 5 中性C原子轫致辐射双重微分截面随光子发射角$ \theta $的变化, 其中出射电子动能$ T_{2} $为1 keV, 入射电子动能$ T_{1} $分别20, 50, 100, 500和2000 keV
Figure 5. Bremsstrahlung double differential cross sections of the neutral C atom as a function of the emission angle $ \theta $, where the kinetic energy of the emitted electron $ T_2 $ is 1 keV, and the kinetic energy of the incident electron $ T_1 $ is 20, 50, 100, 500, and 2000 keV.
图 6 中性C原子和$ \rm C^{2+} $, $ \rm C^{4+} $, $ \rm C^{5+} $, $ \rm C^{6+} $离子轫致辐射双重微分截面随光子发射角$ \theta $的变化, 其中入射电子动能$ T_1 $为10 keV, 发射光子能量占比$ E_{{\mathrm{p}}}/T_{1} $为(a) 0.05, (b) 0.2, (c) 0.6, (d) 0.95
Figure 6. Bremsstrahlung double differential cross sections of neutral C atom and $ \rm C^{2+} $, $ \rm C^{4+} $, $ \rm C^{5+} $ and $ \rm C^{6+} $ ions as a function of the photon emission angle $ \theta $. The kinetic energy of the incident electron $ T_1 $ is 10 keV, and the ratios of the emitted photon energy $ E_{{\mathrm{p}}}/T_{1} $ are (a) 0.05, (b) 0.2, (c) 0.6 and (d) 0.95.
图 7 中性C原子和$ \rm C^{2+} $, $ \rm C^{4+} $, $ \rm C^{5+} $, $ \rm C^{6+} $离子轫致辐射双重微分截面随光子发射角$ \theta $的变化, 发射光子能量占比$ E_{{\mathrm{p}}}/T_{1} $为0.6, 入射电子动能$ T_1 $分别为(a) 1 keV, (b) 10 keV, (c) 100 keV, (d) 1000 keV
Figure 7. Bremsstrahlung double differential cross sections of neutral C atom and $ \rm C^{2+} $, $ \rm C^{4+} $, $ \rm C^{5+} $ and $ \rm C^{6+} $ ions as a function of the photon emission angle $ \theta $. The ratio of the emitted photon energy $ E_{{\mathrm{p}}}/T_{1} $ is 0.6, and the kinetic energies of the incident electron $ T_1 $ are (a) 1 keV, (b) 10 keV, (c) 100 keV and (d) 1000 keV.
图 8 中性C原子和$ \rm C^{2+} $, $ \rm C^{4+} $, $ \rm C^{5+} $, $ \rm C^{6+} $离子的轫致辐射形状函数, 其中发射光子能量$ E_{\rm p}$均为5 keV, 入射电子动能$ T_1 $为(a) 20 keV, (b) 100 keV.
Figure 8. Bremsstrahlung shape function of neutral C atom and $ \rm C^{2+} $, $ \rm C^{4+} $, $ \rm C^{5+} $ and $ \rm C^{6+} $ ions. The emitted photon energy $ E_{\rm p} $ is 5 keV, and the incident electron kinetic energies $ T_1 $ are (a) 20 keV and (b) 100 keV.
图 9 中性C原子和$ \rm C^{2+} $, $ \rm C^{4+} $, $ \rm C^{5+} $, $ \rm C^{6+} $离子的轫致辐射形状函数, 其中入射电子动能$ T_1 $均为50 keV, 发射光子能量$ E_{\rm p} $为(a) 5 keV, (b) 45 keV.
Figure 9. Bremsstrahlung shape function of neutral C atom and $ \rm C^{2+} $, $ \rm C^{4+} $, $ \rm C^{5+} $ and $ \rm C^{6+} $ ions. The incident electron kinetic energy $ T_1 $ is 50 keV, and the emitted photon energies $ E_{\rm p} $ are (a) 5 keV and (b) 45 keV.
表 1 中性C原子及其各价态离子($\rm C^{1+}$, $\rm C^{2+}$, $\rm C^{3+}$, $\rm C^{4+}$, $\rm C^{5+}$)的基态轨道能量及总能量, 并与Rodrigues等 [54]的计算结果和NIST [55]的数据进行对比, 表中Diff表示当前计算结果和NIST数据相对误差
Table 1. Orbital and total energies of the neutral C atom and different charged ions ($\rm C^{1+}$, $\rm C^{2+}$, $\rm C^{3+}$, $\rm C^{4+}$, $\rm C^{5+}$) in their ground states and the comparison with results of Rodrigues et al. [54] and the NIST data [55]. “Diff” represents the relative errors between the present calculations of total energies and the NIST data[55].
Target $ E/ {\rm eV }$ Diff/% $ 1 {\mathrm{s}}_{1/2} $ $ 2 {\mathrm{s}}_{1/2} $ $ 2 {\mathrm{p}}_{1/2} $ $ 2 {\mathrm{p}}_{3/2} $ Total Ref. [54] NIST [55] C –298.98 –16.86 –9.08 –9.07 –1025.12 –1026 –1030.11 0.48 $ \rm C^{1+} $ –314.60 –30.44 –22.65 –1014.57 –1015 –1018.85 0.42 $ \rm C^{2+} $ –336.99 –47.29 –990.72 –991 –994.47 0.38 $ \rm C^{3+} $ –362.47 –66.27 –945.03 –945 –946.58 0.16 $ \rm C^{4+} $ –392.48 –880.85 –882.08 0.14 $ \rm C^{5+} $ –490.04 –490.04 –489.99 0.01 
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表 2 中性C原子轫致辐射单重微分截面$ \sigma(k) $, 并与Pratt等[33]的结果进行对比
Table 2. Comparison of present bremsstrahlung single differential cross section $ \sigma(k) $ for the neutral C atom with the calculations of Pratt et al[33].
$ T_1/{\rm{keV}} $ $ E_{\mathrm{p}}/T_1 $ $ \sigma(k) /\text{mb}$ Diff/% Present work Pratt et al.[33] 1 0.2 4.923 5.587 –11.88 0.5 4.902 5.525 –11.27 0.8 4.747 5.272 –9.96 0.95 4.741 5.162 –8.16 10 0.2 7.486 8.308 –9.90 0.5 5.953 6.405 –7.06 0.8 4.829 5.060 –4.56 0.95 4.495 4.573 –1.71 100 0.2 7.896 8.130 –2.88 0.5 4.964 5.018 –1.07 0.8 2.939 2.970 –1.05 0.95 1.925 1.963 –1.95 200 0.2 7.614 7.586 0.37 0.5 4.402 4.377 0.58 0.8 2.352 2.354 –0.08 0.95 1.366 1.380 –1.02 1000 0.2 7.687 7.515 2.29 0.5 3.953 3.879 1.91 0.8 1.773 1.762 0.65 0.95 0.815 0.818 –0.36 2000 0.2 8.387 8.303 1.01 0.5 4.507 4.451 1.26 0.8 2.118 2.112 0.31 0.95 0.941 0.947 –0.68
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表 3 中性C原子轫致辐射形状函数, 并与Kissel等[57]的结果进行对比
Table 3. Comparison of the bremsstrahlung shape function for the neutral C atom with the results of Kissel et al[57].
$ T_1/ {\rm keV} $ $ E_{\mathrm{p}} / T_1 $ $ \theta/ (°)$ Shape function/sr–1 Diff/% Present work Kissel et al.[57] 10 0.6 0 0.0487 0.0498 –2.21 30 0.0783 0.0796 –1.68 90 0.0968 0.0962 0.60 120 0.0611 0.0602 1.44 180 0.0232 0.0232 –0.01 50 0.6 0 0.0877 0.0843 3.98 30 0.1386 0.1389 –0.23 90 0.0747 0.0746 0.19 120 0.0363 0.0360 0.83 180 0.0150 0.0149 0.61 100 0.6 0 0.1392 0.1380 0.87 30 0.2005 0.2019 –0.68 90 0.0558 0.0556 0.42 120 0.0244 0.0243 0.36 180 0.0111 0.0110 0.65
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[1] Buǐmistrov V M, Trakhtenberg L I 1975 J. Exp. Theor. Phys. 42 54
[2] Amus'ya M Y, Baltenkov A S, Paiziev A A 1976 JETP Lett. 24 332
[3] Korol A V, Obolensky O I, Solov'yov A V, Solovjev I A 2001 J. Phys. B: At. Mol. Opt. Phys. 34 1589
Google Scholar
[4] Lea S M, Silk J, Kellogg E, Murray S 1973 Astrophys. J. 184 L105
Google Scholar
[5] Cavaliere A, Fusco-Femiano R 1976 Astron. Astrophys. 49 137
[6] Hannestad S, Raffelt G 1998 Astrophys. J. 507 339
Google Scholar
[7] Wang W Y, Lu J G, Tong H, Ge M Y, Li Z S, Men Y P, Xu R X 2017 Astrophys. J. 837 81
Google Scholar
[8] Steane A M 2024 Phys. Rev. D 109 063032
Google Scholar
[9] Maydanyuk S P, Zhang P M, Zou L P 2016 Phys. Rev. C 93 014617
Google Scholar
[10] Omar A, Andreo P, Poludniowski G 2018 Radiat. Phys. Chem. 148 73
Google Scholar
[11] Jakubassa-Amundsen D H 2021 arXiv: 2103.06034 [physics. atom-ph]
[12] Li L, An Z, Zhu J J, Tan W J, Sun Q, Liu M T 2019 Phys. Rev. A 99 052701
Google Scholar
[13] Kornev A S, Zon B A, Chernov V E, Amusia M Y, Kubelík P, Ferus M 2022 Atoms 10 86
Google Scholar
[14] Milstein A I, Salnikov S G, Kozlov M G 2023 Nucl. Instrum. Methods Phys. Res., Sect. B 539 9
Google Scholar
[15] Guazzotto L, Betti R 2019 Plasma Phys. Control. Fusion 61 085028
Google Scholar
[16] Bhaskar B S, Koivisto H, Tarvainen O, Thuillier T, Toivanen V, Kalvas T, Izotov I, Skalyga V, Kronholm R, Marttinen M 2021 Plasma Phys. Control. Fusion 63 095010
Google Scholar
[17] Bone T, Sedwick R 2024 Acta Astronaut. 220 356
Google Scholar
[18] Batani D, Antonelli L, Barbato F, et al. 2018 Nucl. Fusion 59 032012
Google Scholar
[19] Ren K, Wu J F, Dong J J, Li Y R, Huang T X, Zhao H, Liu Y Y, Cao Z R, Zhang J Y, Mu B Z, Yan J, Jiang W, Pu Y D, Li Y L, Peng X S, Xu T, Yang J M, Lan K, Ding Y K, Jiang S E, Wang F 2021 Sci. Rep. 11 14492
Google Scholar
[20] Lindl J D, Amendt P, Berger R L, Glendinning S G, Glenzer S H, Haan S W, Kauffman R L, Landen O L, Suter L J 2004 Phys. Plasmas 11 339
Google Scholar
[21] Boozer A H 2005 Rev. Mod. Phys. 76 1071
Google Scholar
[22] Munirov V R, Fisch N J 2023 Phys. Rev. E 107 065205
Google Scholar
[23] Underwood C I D, Baird C D, Murphy C D, et al. 2020 Plasma Phys. Control. Fusion 62 124002
Google Scholar
[24] Singh S, Armstrong C D, Kang N, et al. 2021 Plasma Phys. Control. Fusion 63 035004
Google Scholar
[25] Benitez J, Todd D, Xie D 2022 J. Phys.: Conf. Ser. 2244 012083
Google Scholar
[26] Rong X, Du Y, Ljungberg M, Rault E, Vandenberghe S, Frey E C 2012 Med. Phys. 39 2346
Google Scholar
[27] Roshan H R, Mahmoudian B, Gharepapagh E, Azarm A, Islamian J P 2016 Appl. Radiat. Isot. 108 124
Google Scholar
[28] Uribe C F, Esquinas P L, Gonzalez M, Celler A 2016 Phys. Med. 32 691
Google Scholar
[29] Porter C A, Bradley K M, Hippeläinen E T, Walker M D, McGowan D R 2018 EJNMMI Res. 8 1
Google Scholar
[30] Tijani S A, Al-Hadeethi Y, Sambo I, Balogun F A 2018 J. Radiol. Prot. 38 N44
Google Scholar
[31] Bethe H, Heitler W 1934 Proc. R. Soc. London, Ser. A 146 83
Google Scholar
[32] Tseng H K, Pratt R H 1971 Phys. Rev. A 3 100
Google Scholar
[33] Pratt R H, Tseng H K, Lee C M, Kissel L 1977 At. Data Nucl. Data Tables 20 175
Google Scholar
[34] Poškus A 2018 Comput. Phys. Commun. 232 237
Google Scholar
[35] Poškus A 2022 Comput. Phys. Commun. 278 108414
Google Scholar
[36] Poškus A 2019 At. Data Nucl. Data Tables 129-130 101277
Google Scholar
[37] Poškus A 2021 Nucl. Instrum. Methods Phys. Res., Sect. B 508 49
Google Scholar
[38] Wu J Y, Wu Y, Qi Y Y, Wang J G, Janev R K, Zhang S B 2019 Phys. Rev. A 99 012705
Google Scholar
[39] Wu J Y, Wu Y, Qi Y Y, Wang J G, Janev R K, Zhang S B 2019 Mon. Not. R. Astron. Soc. 486 141
Google Scholar
[40] Wu J Y, Qi Y Y, Cheng Y J, Wu Y, Wang J G, Zhang S B 2020 Phys. Plasmas 27 043301
Google Scholar
[41] Wu J Y, Cheng Y J, Poškus A, Wu Y, Wang J G, Zhang S B 2021 Phys. Rev. A 103 062802
Google Scholar
[42] Avdonina N B, Pratt R H 1999 J. Phys. B: At. Mol. Opt. Phys. 32 4261
Google Scholar
[43] Korol A V 1992 J. Phys. B: At. Mol. Opt. Phys. 25 L341
Google Scholar
[44] Amusia M Y, Avdonina N B, Chernysheva L V, Kuchiev M Y 1985 J. Phys. B: Atom. Mol. Phys. 18 L791
Google Scholar
[45] Korol A V, Lyalin A G, Solovy'ov A V, Avdonina N B, Pratt R H 2002 J. Phys. B: At. Mol. Opt. Phys. 35 1197
Google Scholar
[46] Tseng H K, Pratt R H 1973 Phys. Rev. A 7 1502
Google Scholar
[47] Mangiarotti A, Lauth W, Jakubassa-Amundsen D H, Klag P, Malafronte A A, Martins M N, Nielsen C F, Uggerhøj U I 2021 Phys. Lett. B 815 136113
Google Scholar
[48] Groshev M E, Zaytsev V A, Yerokhin V A, Hillenbrand P M, Litvinov Y A, Shabaev V M 2022 Phys. Rev. A 105 052803
Google Scholar
[49] Yerokhin V A, Surzhykov A 2010 Phys. Rev. A 82 062702
Google Scholar
[50] Yerokhin V A, Surzhykov A, Märtin R, Tashenov S, Weber G 2012 Phys. Rev. A 86 032708
Google Scholar
[51] García-Alvarez J A, Fernández-Varea J M, Vanin V R, Maidana N L 2018 J. Phys. B: At. Mol. Opt. Phys. 51 225003
Google Scholar
[52] Li L, An Z, Zhu J J, Lin W P, Williams S 2021 Nucl. Instrum. Methods Phys. Res., Sect. B 506 15
Google Scholar
[53] Gu M F 2008 Can. J. Phys. 86 675
Google Scholar
[54] Rodrigues G C, Indelicato P, Santos J P, Patté P, Parente F 2004 At. Data Nucl. Data Tables 86 117
Google Scholar
[55] Kramida A, Yu Ralchenko, Reader J, and NIST ASD Team 2024 NIST Atomic Spectra Database (Ver. 5.12) https://physics.nist.gov/asd [2025-1-6]
[56] Lee C M, Kissel L, Pratt R H, Tseng H K 1976 Phys. Rev. A 13 1714
Google Scholar
[57] Kissel L, Quarles C A, Pratt R H 1983 At. Data Nucl. Data Tables 28 381
Google Scholar
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