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The transition of Ga+ ions from 4s2 1S0 to 4s4p 3P0 has advantages such as a high quality factor and a small motional frequency shift, making it suitable as a reference for precision measurement experiments like optical clocks. Calculating the dynamic polarizability of 4s2 1S0—4s4p 3P0 transition for Ga+ ion is of great significance for exploring the potential applications of the Ga+ ion in the field of quantum precision measurement and for testing atomic and molecular structure theories. In this paper, the dynamic polarizability of the Ga+ ion 4s2 1S0—4s4p 3P0 transition is theoretically calculated using the relativistic configuration interaction plus many-body perturbation (RCI+MBPT) method. The “tune-out” wavelengths for the 4s2 1S0 state and the 4s4p 3P0 state, as well as the “magic” wavelength of the 4s2 1S0—4s4p 3P0 transition, are also computed. It is observed that the resonant lines situated near a certain “turn-out” and “magic” wavelength can make dominant contributions to the polarizability, while the remaining resonant lines generally contribute the least. These “tune-out” and “magic” wavelengths provide theoretical guidance for precise measurements, which is important for studying the atomic structure of Ga+ ions. The accurate determination of the difference in static polarizability between the 4s2 1S0 and 4s4p 3P0 states is of significant importance. Additionally, based on the “polarizability scaling” method, this work also discusses how the theoretical calculation errors in static polarizability measurements vary with wavelength, which provides theoretical guidance for further determining the static polarizability of the 4s2 1S0 and 4s4p 3P0 states with high precision. This is crucial for minimizing the uncertainty of the blackbody radiation (BBR) frequency shift in Ga+ optical clock and suppressing the systematic uncertainty.
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Keywords:
- dynamic polarizability /
- Ga+ /
- atomic structure calculation /
- RCI+MBPT
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] -
State RCI RCI+MBPT NIST Diff./% Refs. 4s2 1S0 396252.81 412181.94 413285.38 –0.27 413285.41CICP [5] 4s4p 3P0 43174.90 47338.76 47367.55 –0.060 47367.57 CICP [5]; 47032 MCDHF [35]; 47368Expt [36] 4s4p 3P1 43584.84 47792.83 47814.114 –0.044 47469 MCDHF [35]; 47814 Expt [36] 4s4p 1P1 68389.17 70709.25 70701.427 0.011 70701.42 CICP [5]; 70455 MCDHF [35]; 70701 Expt [36] 4s5s 3S1 96653.91 102623.02 102944.595 –0.31 100749.90 CICP [5]; 102665 MCDHF [35]; 102945 Expt [36] 4s4d 3D1 106905.35 113471.29 113815.885 –0.30 113815.87 CICP [5]; 113305 MCDHF [35]; 113816 Expt [36] 4p2 3P1 109300.03 115272.94 115224.47 –0.042 115224.49 CICP [5]; 114590 MCDHF [35]; 115224 Expt [36] 4s5p 3P1 111880.88 118110.59 118518.461 –0.34 118236 MCDHF [35]; 118518 Expt [36] 4s5p 1P1 114324.78 120211.72 120550.431 –0.28 120715.81 CICP [5]; 120322 MCDHF [35]; 120550 Expt [36] 4s6s 1S0 126011.01 132559.65 133010.30 –0.34 130793.68 CICP [5]; 133517 MCDHF [35]; 133741 Expt [36] 4s5d 3D1 129989.22 136706.55 137157.524 –0.33 137155.79 CICP [5]; 136759 MCDHF [35]; 137157 Expt [36] 4s6p 3P1 132039.60 138646.98 — — — 4s6p 1P1 132671.60 139209.74 — — — 4s7s 1S0 138282.16 145005.12 145494.205 –0.34 145176 MCDHF [35]; 145494 Expt [36] 4s6d 3D1 140241.35 147033.96 147520.34 –0.33 — 4s8s 1S0 144640.75 151383.73 151923.93 –0.36 — 4s7d 3D1 145733.76 152563.77 153064.92 –0.33 — 4s7p 3P1 141291.01 148033.38 — — — 4s7p 1P1 141504.08 148239.62 — — — 4s8p 3P1 146349.69 153163.32 — — — 4s8p 1P1 146429.38 153251.11 — — — 4s9s 1S0 148859.67 154959.21 — — — 4s8d 3D1 149030.92 155885.88 156386.7 –0.32 — 
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Method RCI RCI+MBPT Recommend Refs. Guage Length Velocity Length Velocity 4s2 1S0—4s4p 3P1 0.055752 0.059065 0.064832 0.072400 0.065 (17) 0.0744 [34]; 0.0895 MCDHF [35]; 0.0802 RRPA [38] 4s2 1S0—4s4p 1P1 3.0918 3.0507 2.8480 3.0361 2.84 (24) 2.69 CICP [5]; 2.87 [34]; 2.68 MCDHF [35]; 2.81 MP [37];
2.79 RRPA [38]; 2.71 MCHF [39]; 2.78 (11) Expt [40]4s2 1S0—4s5p 3P1 0.000595 0.000455 0.00594 0.000400 0.006 (6) — 4s2 1S0—4s5p 1P1 0.28458 0.27302 0.15426 0.23982 0.15 (13) 0.138 [34] 4s2 1S0—4s6p 3P1 0.00229 0.00237 0.00815 0.00272 0.0082 (59) — 4s2 1S0—4s6p 1P1 0.0741 0.0684 0.0868 0.0594 0.087 (27) — 4s2 1S0—4s7p 1P1 0.0264 0.0229 0.0143 0.0205 0.014 (12) — 4s2 1S0—4s7p 3P1 0.00232 0.00249 0.00803 0.00298 0.008 (6) — 4s2 1S0—4s8p 1P1 0.00986 0.00753 0.0183 0.00768 0.02 (1) — 4s2 1S0—4s8p 3P1 0.00202 0.00233 0.00787 0.003104 0.008 (6) — 4s4p 3P0—4s5s 3S1 0.93304 0.92359 0.92029 0.90827 0.920 (25) 0.974 CICP [5]; 1.00 MCDHF [35]; 0.982 MP [37] 4s4p 3P0—4s4d 3D1 2.1286 2.0871 2.0181 2.0670 2.02 (11) 2.00 CICP [5]; 2.08 [34]; 2.02 MCDHF [35]; 2.05 MP [37] 4s4p 3P0—4p2 3P1 1.8133 1.7818 1.6470 1.7695 1.65 (17) 1.64 CICP [5]; 1.64 MCDHF [35]; 1.72 MP [37] 4s4p 3P0—4s6s 3S1 0.26761 0.26392 0.26890 0.26353 0.269 (5) 0.214 CICP [5]; 0.205 MCDHF [35]; 0.217 MP [37] 4s4p 3P0—4s5d 3D1 0.66449 0.64536 0.62443 0.65590 0.62 (4) 0.461 CICP[5]; 0.442 MCDHF [35]; 0.479 MP [37] 4s4p 3P0—4s7s 3S1 0.14979 0.14750 0.15084 0.14798 0.151 (3) — 4s4p 3P0—4s6d 3D1 0.36574 0.35358 0.33986 0.36272 0.340 (26) — 4s4p 3P0—4s8s 3S1 0.10206 0.10042 0.10065 0.098613 0.1021 (34) — 4s4p 3P0—4s7d 3D1 0.24440 0.23567 0.22522 0.24281 0.225 (19) — 4s4p 3P0—4s9s 3S1 0.088839 0.087378 0.078599 0.078310 0.079 (11) — 4s4p 3P0—4s8d 3D1 0.18107 0.17433 0.16616 0.18060 0.181 (14) —
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Transition Contributions Refs. $ \alpha \left(0\right)( $4s2 1S0$ ) $ — — 4s2 1S0—4s4p 3P1 0.013 (7) — 4s2 1S0—4s4p 1P1 16.69 (2.82) 16.601[5] 4s2 1S0—4s5p 3P1 4.4 (4.4)×10–5 — 4s2 1S0—4s5p 1P1 0.027 (27) 0.016[5] 4s2 1S0—4s6p 3P1 7.1 (7.1)×10–5 — 4s2 1S0-—4s6p 1P1 0.008 (5) — 4s2 1S0—4snp 3P1, n = 7—8 0.00012 (12) — 4s2 1S0—4snp 1P1, n = 7—9 0.0006 (4) — Core 1.24 (1)[5] 1.24 (1)[5] Total 17.98 (2.82) 17.95 (34)[5] $ \alpha \left(0\right)( $4s4p 3P0$ ) $ — — 4s4p 3P0—4s5s 3S1 2.23 (12) 2.257[5] 4s4p 3P0—4s4d 3D1 8.98 (98) 8.668 [5] 4s4p 3P0—4p2 3P1 5.87 (1.21) 5.945[5] 4s4p 3P0—4s6s 3S1 0.124 (5) — 4s4p 3P0—4s5d 3D1 0.62 (8) — 4s4p 3P0—4sns 3S1, n = 7—9 0.057 (13) — 4s4p 3P0—4snd 3D1, n = 6—8 0.283 (29) — Core 1.24 (1) [5] 1.24 (1)[5] Total 19.41 (1.56) 19.58 (38)[5] $ {{\Delta }}\alpha \left(0\right) $ 1.43 (3.2) 1.63 (72) [5]
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Transition ‘Tune-out’ wavelengths ‘Magic’ wavelengths 209.101 176.42 148.61 117.197 113.09 209.286 168.1 148.27 116.38 106.7 4s2 1S0—4s4p 3P1 0.025 0.085 <0.1 <0.001 <0.001 <0.1 4s2 1S0—4s4p 1P1 0.0076 0.049 2.7 0.049 0.16 2.5 4s2 1S0—4s5p 1P1 <0.001 <0.01 <0.1 <0.001 <0.01 0.11 4s2 1S0—4s6p 1P1 <0.001 <0.01 <0.1 <0.001 <0.01 <0.1 4s4p 3P0—4s5s 3S1 0.17 <0.001 0.0017 0.0073 <0.01 0.27 <0.001 <0.01 <0.1 4s4p 3P0—4s4d 3D1 0.19 0.077 0.034 0.0095 0.026 1.6 0.047 0.036 0.68 4s4p 3P0—4p2 3P1 0.22 0.15 0.048 0.13 0.030 1.7 0.16 0.047 0.96 4s4p 3P0—4s6s 3S1 <0.01 <0.001 <0.001 0.0065 <0.01 <0.1 <0.001 <0.01 <0.1 4s4p 3P0—4s5d 3D1 <0.01 <0.001 <0.001 <0.001 <0.01 <0.1 <0.001 <0.01 <0.1 Others <0.001 <0.01 <0.01 <0.001 <0.01 <0.01 <0.1 <0.01 <0.01 <0.1 Total 0.026 0.34 0.16 0.059 0.16 0.11 3.6 0.17 0.17 2.8
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Transition ‘Tune-out’ wavelengths ‘Magic’ wavelengths 209.101 176.42 148.61 117.197 113.09 209.286 168.1 148.27 116.38 106.7 4s2 1S0—4s4p 3P1 –32.06 –0.03 –0.01 –0.01 –0.01 9.51 –0.02 –0.01 –0.01 0.00 4s2 1S0—4s4p 1P1 30.77 46.73 177.29 –36.57 –29.58 30.72 57.11 184.64 –34.99 –22.00 4s2 1S0—4s5p 1P1 0.03 0.03 0.04 0.05 0.06 0.03 0.04 0.04 0.06 0.06 4s2 1S0—4s6p 1P1 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 Others 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 Core 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 Total 0.00 47.99 178.58 –35.27 –28.27 41.52 58.39 185.93 –33.68 –20.68 4s4p 3P0—4s5s 3S1 8.59 –55.40 –4.78 –1.64 –1.46 8.54 –15.34 –4.71 –1.60 –1.21 4s4p 3P0—4s4d 3D1 18.64 33.00 –352.26 –13.85 –11.66 18.61 45.19 –297.43 –13.37 –9.07 4s4p 3P0—4p2 3P1 11.66 19.42 353.37 –10.10 –8.41 11.64 25.33 484.39 –9.73 –6.46 4s4p 3P0—4s6s 3S1 0.18 0.22 0.32 16.76 –1.87 0.18 0.24 0.33 –18.78 –0.62 4s4p 3P0—4s5d 3D1 0.87 1.0 1.43 6.46 20.77 0.87 1.12 1.44 7.43 –6.93 Others 0.44 0.52 0.68 1.13 1.39 0.44 0.61 0.68 1.13 2.37 Core 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 Total 41.62 0.00 0.00 0.00 0.00 41.52 58.39 185.93 –33.68 –20.68 Diff. 0.00 0.00 0.00 0.0 0.0
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