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

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

doi:10.7498/aps.70.20210181

Abstract

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 2s_1/2-1s_1/2 levels of hydrogen-like ions are reported. The trend of Stark-induced transition probabilities varying with atomic number Z between 2s_1/2-1s_1/2 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.

Keywords:

Authors and contacts

1.
College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
2.
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

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

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References

[1]	Bucksbaum P, Commins E D, Hunter L 1981 Phys. Rev. Lett. 46 640 Google Scholar
[2]	Drell P S, Commins E D 1984 Phys. Rev. Lett. 53 968 Google Scholar
[3]	Gilbert S L, Noecker M C, Watts R N, Wieman C E 1985 Phys. Rev. Lett. 55 2680 Google Scholar
[4]	Maul M, Schӓfer A, Indelicato P 1998 J. Phys. B 31 2725 Google Scholar
[5]	Hunter L R, Walker W A, Weiss D S 1986 Phys. Rev. Lett. 56 823 Google Scholar
[6]	Lellouch L P, Hunter L R 1987 Phys. Rev. A 36 3490 Google Scholar
[7]	Wielandy S, Sun T H, Hilborn R C, Hunter L R 1992 Phys. Rev. A 46 7103 Google Scholar
[8]	Fan C Y, Garcia-Munoz M, Sellin I A 1967 Phys. Rev. 161 6 Google Scholar
[9]	Leventhal M, Murnick D E 1970 Phys. Rev. Lett. 25 1237 Google Scholar
[10]	Murnick D E, Leventhal M, Kugel H W 1971 Phys. Rev. Lett. 27 1625 Google Scholar
[11]	Kugel H W, Leventhal M, Murnick D E 1972 Phys. Rev. A 6 1306 Google Scholar
[12]	Leventhal M, Murnick D E, Kugel H W 1972 Phys. Rev. Lett. 28 1609 Google Scholar
[13]	Lawrence G P, Fan C Y, Bashkin S 1972 Phys. Rev. Lett. 28 1612 Google Scholar
[14]	Gould H, Marrus R 1978 Phys. Rev. Lett. 41 1457 Google 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. 165 139 Google 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 33 405 Google Scholar
[20]	Johnson W R 1972 Phys. Rev. Lett. 29 1123 Google Scholar

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图 1 外电场取向示意图

Figure 1. Schematic diagram of an external electric field.

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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.

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图 3 类氢离子能级示意图

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

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图 4 类氢离子2s_1/2和2p_{1/2, 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 2s_1/2 and 2p_{1/2, 3/2} states of hydrogen-like ions (NR and R stand for nonrelativistic and relativistic cases, respectively).

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图 5 类氢Li²⁺离子和Ar¹⁷⁺离子2s_1/2能级的Stark诱导跃迁寿命

Figure 5. Stark-induced lifetime of 2s_1/2 levels for hydrogen-like Li²⁺ and Ar¹⁷⁺ ions.

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图 6 类氢离子2s_1/2-1s_1/2能级之间的Stark诱导跃迁几率

Figure 6. Stark-induced transition probability between 2s_1/2-1s_1/2 levels of hydrogen-like ions.

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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 × 10^b

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

Z	Energy difference/cm^–1		Matrix element
Z	ΔE₁^[19]	ΔE₂	$\langle $2p_1/2\|r\|\|2s_1/2$\rangle $	$\langle $2p_3/2\|\|r\|\|2s_1/2$\rangle $	$\langle $2p\|\|r\|\|2s $\rangle $	$\langle $1s_1/2\|\|r\|\|2p_1/2$\rangle $	$\langle $1s_1/2\|\|r\|\|2p_3/2$\rangle $	$\langle $1s\|\|r\|\|2p$\rangle $
1	3.52868[-2]	–3.65221[-1]	–5.19604	–5.19611	–5.19615	1.29024	1.29024	1.29027
2	4.68400[-1]	–5.84353[0]	–2.59785	–2.59798	–2.59808	0.64509	0.64508	0.64513
3	2.09220[0]	–2.95829[1]	–1.73170	–1.73191	–1.73205	0.43002	0.43001	0.43009
4	5.99720[0]	–9.34965[1]	–1.29858	–1.29885	–1.29904	0.32247	0.32247	0.32257
6	2.60840[1]	–4.73326[2]	–0.86533	–0.86575	–0.86603	0.21490	0.21489	0.21504
8	7.32500[1]	–1.49594[3]	–0.64860	–0.64915	–0.64952	0.16109	0.16108	0.16128
10	1.62100[2]	–3.65221[3]	–0.51846	–0.51915	–0.51962	0.12879	0.12878	0.12903
12	3.08800[2]	–7.57322[3]	–0.43163	–0.43246	–0.43301	0.10724	0.10722	0.10752
14	5.30600[2]	–1.40303[4]	–0.36954	–0.37051	–0.37115	0.09183	0.09181	0.09216
16	8.46400[2]	–2.39351[4]	–0.32291	–0.32402	–0.32476	0.08026	0.08024	0.08064
18	1.27570[3]	–3.83394[4]	–0.28659	–0.28784	–0.28868	0.07125	0.07123	0.07168
20	1.83800[3]	–5.84353[4]	–0.25749	–0.25888	–0.25981	0.06403	0.06401	0.06451
24	3.45000[3]	–1.21171[5]	–0.21372	–0.21539	–0.21651	0.05318	0.05315	0.05376
28	5.86400[3]	–2.24485[5]	–0.18232	–0.18426	–0.18558	0.04540	0.04537	0.04608
32	9.28800[3]	–3.82962[5]	–0.15865	–0.16087	–0.16238	0.03954	0.03950	0.04032
36	1.39300[4]	–6.13430[5]	–0.14013	–0.14263	–0.14434	0.03496	0.03492	0.03584
40	2.00100[4]	–9.34965[5]	–0.12522	–0.12799	–0.12990	0.03127	0.03123	0.03226
44	2.78800[4]	–1.36888[6]	–0.11292	–0.11597	–0.11809	0.02823	0.02819	0.02932
48	3.78200[4]	–1.93874[6]	–0.10259	–0.10592	–0.10825	0.02568	0.02563	0.02688
52	5.03100[4]	–2.67035[6]	–0.09376	–0.09737	–0.09993	0.02351	0.02345	0.02481
56	6.56100[4]	–3.59176[6]	–0.08612	–0.09001	–0.09279	0.02162	0.02156	0.02304
60	8.45400[4]	–4.73326[6]	–0.07942	–0.08359	–0.08660	0.01997	0.01991	0.02150
64	1.08200[5]	–6.12739[6]	–0.07348	–0.07793	–0.08119	0.01850	0.01844	0.02016
68	1.37000[5]	–7.80892[6]	–0.06817	–0.07290	–0.07641	0.01720	0.01713	0.01897
72	1.73900[5]	–9.81489[6]	–0.06338	–0.06839	–0.07217	0.01601	0.01595	0.01792
76	2.20000[5]	–1.21846[7]	–0.05902	–0.06432	–0.06837	0.01494	0.01487	0.01698
80	2.79500[5]	–1.49594[7]	–0.05503	–0.06060	–0.06495	0.01395	0.01388	0.01613
84	3.56700[5]	–1.81833[7]	–0.05135	–0.05720	–0.06186	0.01304	0.01297	0.01536
88	4.62700[5]	–2.19021[7]	–0.04793	–0.05406	–0.05905	0.01219	0.01212	0.01466
92	6.07300[5]	–2.61642[7]	–0.04473	–0.05114	–0.05648	0.01139	0.01133	0.01402

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表 2 类氢离子2s_1/2-1s_1/2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 × 10^b

Table 2. Transition wavelength and Stark-induced probability between 2s_1/2-1s_1/2 of hydrogen-like ions in electric field of 1 V/m, where a[b] stands for a × 10^b

Z	λ/nm	Transition probability/s^–1
		Model I		Model II
		A^SIT (NR)	A^SIT(R)	A^SIT (NR)	A^SIT (R)
1	121.50287	2.7510[–1]	2.7508[–1]	2.8023[–1]	2.8022[–1]
2	30.37572	6.2450[–3]	6.2439[–3]	6.3253[–3]	6.3242[–3]
3	13.50032	7.0428[–4]	7.0399[–4]	7.1133[–4]	7.1104[–4]
4	7.59393	1.5238[–4]	1.5227[–4]	1.5364[–4]	1.5353[–4]
6	3.37508	1.8124[–5]	1.8095[–5]	1.8234[–5]	1.8205[–5]
8	1.89848	4.0858[–6]	4.0740[–6]	4.1054[–6]	4.0936[–6]
10	1.21503	1.3036[–6]	1.2978[–6]	1.3087[–6]	1.3029[–6]
12	0.84377	5.1727[–7]	5.1392[–7]	5.1899[–7]	5.1564[–7]
14	0.61991	2.3847[–7]	2.3637[–7]	2.3915[–7]	2.3705[–7]
16	0.47462	1.2240[–7]	1.2099[–7]	1.2271[–7]	1.2130[–7]
18	0.37501	6.8196[–8]	6.7198[–8]	6.8347[–8]	6.7348[–8]
20	0.30376	4.0558[–8]	3.9825[–8]	4.0638[–8]	3.9904[–8]
24	0.21094	1.6577[–8]	1.6143[–8]	1.6603[–8]	1.6169[–8]
28	0.15498	7.8098[–9]	7.5301[–9]	7.8204[–9]	7.5406[–9]
32	0.11866	4.0660[–9]	3.8747[–9]	4.0708[–9]	3.8794[–9]
36	0.09375	2.2878[–9]	2.1507[–9]	2.2901[–9]	2.1530[–9]
40	0.07594	1.3688[–9]	1.2668[–9]	1.3700[–9]	1.2680[–9]
44	0.06276	8.5316[–10]	7.7563[–10]	8.5387[–10]	7.7630[–10]
48	0.05274	5.5176[–10]	4.9160[–10]	5.5218[–10]	4.9199[–10]
52	0.04493	3.6594[–10]	3.1869[–10]	3.6620[–10]	3.1894[–10]
56	0.03874	2.4954[–10]	2.1184[–10]	2.4971[–10]	2.1199[–10]
60	0.03375	1.7254[–10]	1.4233[–10]	1.7265[–10]	1.4243[–10]
64	0.02966	1.1984[–10]	9.5735[–11]	1.1992[–10]	9.5802[–11]
68	0.02628	8.4389[–11]	6.5035[–11]	8.4441[–11]	6.5081[–11]
72	0.02344	5.8719[–11]	4.3471[–11]	5.8755[–11]	4.3503[–11]
76	0.02104	4.0878[–11]	2.8934[–11]	4.0905[–11]	2.8956[–11]
80	0.01898	2.8062[–11]	1.8889[–11]	2.8082[–11]	1.8905[–11]
84	0.01722	1.8996[–11]	1.2085[–11]	1.9011[–11]	1.2096[–11]
88	0.01569	1.2390[–11]	7.3978[–12]	1.2401[–11]	7.4061[–12]
92	0.01436	7.8610[–12]	4.3696[–12]	7.8695[–12]	4.3757[–12]

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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) × 10^c, 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) × 10^c and b is the experimental uncertainty.

Z	ε/(V·m^-1)	τ^SIT/ns
Z	ε/(V·m^-1)	Exp	Model I	Model II
3	7.425(2)[5]	2.629(21)^[8]	2.58	2.55
	9.173(2)[5]	1.764(35) ^[8]	1.69	1.67
18	5.93[7]	3.86(3)^[14]	4.23	4.22
	7.14[7]	2.80(6) ^[14]	2.92	2.91
	8.06[7]	2.28(2) ^[14]	2.29	2.29
	8.60[7]	2.00(2) ^[14]	2.01	2.01

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[1]	Bucksbaum P, Commins E D, Hunter L 1981 Phys. Rev. Lett. 46 640 Google Scholar
[2]	Drell P S, Commins E D 1984 Phys. Rev. Lett. 53 968 Google Scholar
[3]	Gilbert S L, Noecker M C, Watts R N, Wieman C E 1985 Phys. Rev. Lett. 55 2680 Google Scholar
[4]	Maul M, Schӓfer A, Indelicato P 1998 J. Phys. B 31 2725 Google Scholar
[5]	Hunter L R, Walker W A, Weiss D S 1986 Phys. Rev. Lett. 56 823 Google Scholar
[6]	Lellouch L P, Hunter L R 1987 Phys. Rev. A 36 3490 Google Scholar
[7]	Wielandy S, Sun T H, Hilborn R C, Hunter L R 1992 Phys. Rev. A 46 7103 Google Scholar
[8]	Fan C Y, Garcia-Munoz M, Sellin I A 1967 Phys. Rev. 161 6 Google Scholar
[9]	Leventhal M, Murnick D E 1970 Phys. Rev. Lett. 25 1237 Google Scholar
[10]	Murnick D E, Leventhal M, Kugel H W 1971 Phys. Rev. Lett. 27 1625 Google Scholar
[11]	Kugel H W, Leventhal M, Murnick D E 1972 Phys. Rev. A 6 1306 Google Scholar
[12]	Leventhal M, Murnick D E, Kugel H W 1972 Phys. Rev. Lett. 28 1609 Google Scholar
[13]	Lawrence G P, Fan C Y, Bashkin S 1972 Phys. Rev. Lett. 28 1612 Google Scholar
[14]	Gould H, Marrus R 1978 Phys. Rev. Lett. 41 1457 Google 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. 165 139 Google 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 33 405 Google Scholar
[20]	Johnson W R 1972 Phys. Rev. Lett. 29 1123 Google Scholar

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