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本文推导了BrF振动基态(X1Σ, v = 0)下J = 1←0的转动超精细光谱的跃迁偶极矩, 总结了跃迁选择定则为: ΔJ = ±1; ΔF1 = 0, ±1和ΔF = 0, ±1; 而且, 当ΔF1 = ΔF时谱线强度很强, 反之很弱. 当能级之间存在微扰相互作用时, 某些谱线由电偶极和核磁偶极跃迁共同产生, 然而磁偶极仅仅贡献大约十亿分之一的光谱强度. 计算所得光谱线宽和相对强度与实验结果一致. 同时, 在|JI1F1I2F
$\rangle $ 基矢下对Hamilton量矩阵对角化确定了转动超精细光谱的位置, 与实验误差小于1/50谱线宽度(<10–8). 最后模拟了微波转动超精细光谱, 所得结果有助于超精细分子光谱实验和其他相关应用研究.The transition dipole of the hyperfine-rotation spectrum of J = 1←0 within the vibronic ground (X1Σ, v = 0) state of BrF molecule is derived, and thus, the transition selection rules are summarized as follows: ΔJ = ±1; ΔF1 = 0, ±1 and ΔF = 0, ±1, and those of ΔF1 = ΔF are intense while those of ΔF1 ≠ ΔF are weak. Some spectral lines result from both the electric dipole transition and nuclear magnetic dipole transition due to perturbations, however, the magnetic dipole transition only contributes about one-billionth in the spectral intensity. The spectral linewidth is determined to be about 18 kHz by calculating the spectral transition probability. The obtained spectral linewidth and relative intensities are consistent with the experimental results. Additionally, the hyperfine-rotation spectral positions are determined by diagonalizing the Hamiltonian matrix in the basis of |JI1F1I2F$\rangle $ , which is also in good agreement with the experiments within 10–8 (one-fiftieth of the spectral line width). Hence, the microwave hyperfine-rotation spectrum is simulated. In addition, we find that the nuclear spin-spin interaction not only slightly shifts the hyperfine-rotation spectral positions but also changes the sequence of the spectra. As to those unavailable constants of molecules, the fairly precise molecular constants can be achieved by quantum chemical calculation, say, by employing MOLPRO program, and then the simulated spectra can guide the spectral assignment. Besides the guidance of spectral assignment, our results are also helpful for other relevant applications such as in absolute single quantum state preparation.-
Keywords:
- hyperfine structure /
- magnetic transition dipole /
- hyperfine-rotation spectrum /
- BrF
[1] Bernath P F 2020 ????? (Oxford: Oxford University Press)
[2] Kennedy C J, Oelker E, Robinson J M, Bothwell T, D. Kedar, Milner W R, Marti G E, Derevianko A, Ye J 2020 Phys. Rev. Lett. 125 201302
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[12] William D, Phillips 1998 Rev. Mod. Phys. 70 721
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Google Scholar
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[17] Mccarron D J, Norrgard E B, Steinecker M H, Demille D 2015 New J. Phys. 17 035014
Google Scholar
[18] Yeo M, Hummon M T, Collopy A L, Yan B, Hemmerling B, Chae E, Doyle J M, Ye J 2015 Phys. Rev. Lett. 114 223003
Google Scholar
[19] Ni K K, Ospelkaus S, Miranda M, Pe'Er A, Neyenhuis B, Zirbel J J, Kotochigova S, Julienne P S, Jin D S, Ye J 2008 Science 322 231
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[20] Peter, Molony K, Philip, Gregory D, Zhonghua, Ji, Bo, Lu, Michael, Köppinger P 2014 Phys. Rev. Lett. 113 255301
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[21] Takekoshi T, Reichsoellner L, Schindewolf A, Hutson J M, Sueur C, Dulieu O, Ferlaino F, Grimm R, Naegerl H C 2014 Phys. Rev. Lett. 113 205301
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[22] Park J W, Will S A, Zwierlein M W 2015 Phys. Rev. Lett. 114 205302
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[37] Müller H S, Gerry M C 1995 J. Chem. Phys. 103 577
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[38] Shao X, Gong T, Wu L, Yang X 2011 J. Quant. Spectrosc. Radiat. Transfer 112 1005
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[39] Ospelkaus S, Ni K K, Quéméner G, Neyenhuis B, Wang D, Miranda M D, Bohn J, Ye J, Jin D 2010 Phys. Rev. Lett. 104 030402
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图 1 79BrF(上)和81BrF(下)振动基态(X1Σ, v = 0)下的超精细能级. 图中还标明了各能级的能量值和量子数
Fig. 1. Hyperfine-rotation energy levels of 79BrF (upper) and 81BrF (lower) in the vibronic ground state (X1Σ, v = 0). The quantum numbers and the values of the levels are labeled as well.
图 2 BrF振动基态(X1Σ, v = 0)下J = 1←0转动超精细跃迁光谱模拟(下图), 红线代表79BrF, 黑线代表81BrF. 两同位素丰度相差很小, 使得它们的光谱强度几乎相等. 超高分辨的光谱模拟见上图(79BrF)和中图(81BrF), 其中, 谱线1.5, 1–1.5, 2, 1.5, 2–1.5, 1和2.5, 2–1.5, 2的相对强度极小, 导致它们无法观测到(蓝圈部分)
Fig. 2. Simulated hyperfine-rotation spectra (lower) of the J = 1←0 transition within the vibronic ground state (X1Σ, v = 0) of BrF of its two isotopes, 79BrF in Red and 81BrF in black. Their spectral intensities are almost the same accordingly due to their nearly equal natural abundance of the two isotopes. Details of the spectra of 79BrF (upper) and 81BrF (medium) of the unresolved spectra (lower) are plotted as well. Intensities of the spectra F1, F = 1.5, 1–1.5, 2, 1.5, 2–1.5, 1 and 2.5, 2–1.5, 2 are too small to observe, as shown in the blue circles.
表 1 BrF(X1Σ, v = 0)分子常数
Table 1. Molecular parameters of BrF(X1Σ, v = 0)
79BrF 81BrF B/MHz 10628.46302 10577.63957 D/kHz 12.028 11.956 eqQ/MHz 1086.89197 907.97681 C1/kHz 89.051 95.818 C2/kHz –24.17 –24.54 C3/kHz –7.15 –7.71 C4/kHz 4.86 5.24 
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表 2 BrF分子振动基态(X1Σ, v = 0)中J = 1←0跃迁的转动超精细光谱计算值(单位: MHz), 同时列出了其与实验值的偏差和归一化光谱强度
Table 2. Calculated hyperfine-rotation spectra (in MHz) of the J = 1←0 transition in the vibronic ground state (X1Σ, v = 0) of BrF molecule. Deviations (in MHz) from the experimental spectra and the normalized intensity are listed as well.
$F_1',F'\text{-}F''_1,F'' $ 79BrF 81BrF 强度
(归一化)计算值 误差a 计算值 误差a 0.5, 0–1.5, 1 20986.0744 0.0003 20928.7933 0 0.1428 0.5, 1–1.5, 1 20986.1035 –0.0011 20928.8236 0.0002 0.0917 1.5, 1–1.5, 1 21475.0918 0 21337.3854 0.0001 0.3571 1.5, 2–1.5, 1 21475.0730 21337.3660 0.0713 2.5, 2–1.5, 1 21203.1734 0.0001 21110.3358 0 0.6427 0.5, 1–1.5, 2 20986.0937 0 20928.8131 –0.0002 0.3570 1.5, 1–1.5, 2 21475.0820 21337.3749 0.0714 1.5, 2–1.5, 2 21475.0632 0 21337.3556 –0.0001 0.6427 2.5, 2–1.5, 2 21203.1636 21110.3253 0.0713 2.5, 3–1.5, 2 21203.1484 0 21110.3109 0 1 σb 0.0004 0.0001 a 计算值减去参考文献中的实验值[37], 误差缺失表示谱线强度太弱而实验无法观测到. b σ为计算总体方差.
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表 3 BrF振动基态下的转动超精细跃迁偶极矩
Table 3. Hyperfine-rotation transition dipoles of BrF within its vibronic ground state.
$ (J' = 1)F_1', F' $ $(J'' = 0) F_1'', F''$ 0.5, 0 0.5, 1 1.5, 1 1.5, 2 2.5, 2 2.5, 3 1.5, 1 0.2247 0.1124 0.5616 0.1123 1.0110 0 1.5, 2 0 0.5617 0.1123 1.0110 0.1122 1.5728
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[1] Bernath P F 2020 ????? (Oxford: Oxford University Press)
[2] Kennedy C J, Oelker E, Robinson J M, Bothwell T, D. Kedar, Milner W R, Marti G E, Derevianko A, Ye J 2020 Phys. Rev. Lett. 125 201302
Google Scholar
[3] Changala P B, Weichman M L, Lee K F, Fermann M E, Ye J 2019 Science 363 49
Google Scholar
[4] Denis M, Pi A, Timmermans R, Eliav E, Borschevsky A 2019 Phys. Rev. A 99 042512
Google Scholar
[5] Hudson J J, Sauer B E, Tarbutt M R, Hinds E A 2002 Phys. Rev. Lett. 89 023003
Google Scholar
[6] Cairncross W B, Gresh D N, Grau M, Cossel K C, Roussy T S, Ni Y, Zhou Y, Ye J, Cornell E A 2017 Phys. Rev. Lett. 119 153001
Google Scholar
[7] Bouchendira R, Cladé P, Biraben F 2011 Phys. Rev. Lett. 106 080801
Google Scholar
[8] Webb J K, Flambaum V V, Churchill C W, Drinkwater M J, Barrow J D 1999 Phys. Rev. Lett. 82 884
Google Scholar
[9] Liang Q, Chan Y C, Changala P B, Nesbitt D J, Ye J, Toscano J 2021 Proc. Natl. Acad. Sci. 118 e2105063118
Google Scholar
[10] Kolkowitz S P I, Langellier N, Lukin M D, Walsworth R L and Ye J 2016 Phys. Rev. D 94 124043
Google Scholar
[11] Valtolina G, Matsuda K, Tobias W G, Li J R, Marco L D, Ye J 2020 Nature 588 239
Google Scholar
[12] William D, Phillips 1998 Rev. Mod. Phys. 70 721
Google Scholar
[13] Bethlem H L, Berden G, Meijer G 1999 Phys. Rev. Lett. 83 1558
Google Scholar
[14] Barry F J, McCarron J D, Norrgard B E, Steinecker H M, DeMille D 2014 Nature 512 286
Google Scholar
[15] Marco L D, Valtolina G, Matsuda K, Tobias W G, Covey J P, Ye J 2019 Science 363 853
Google Scholar
[16] Hummon M T, Yeo M, Stuhl B K, Collopy A L, Xia Y, Ye J 2013 Phys. Rev. Lett. 110 143001
Google Scholar
[17] Mccarron D J, Norrgard E B, Steinecker M H, Demille D 2015 New J. Phys. 17 035014
Google Scholar
[18] Yeo M, Hummon M T, Collopy A L, Yan B, Hemmerling B, Chae E, Doyle J M, Ye J 2015 Phys. Rev. Lett. 114 223003
Google Scholar
[19] Ni K K, Ospelkaus S, Miranda M, Pe'Er A, Neyenhuis B, Zirbel J J, Kotochigova S, Julienne P S, Jin D S, Ye J 2008 Science 322 231
Google Scholar
[20] Peter, Molony K, Philip, Gregory D, Zhonghua, Ji, Bo, Lu, Michael, Köppinger P 2014 Phys. Rev. Lett. 113 255301
Google Scholar
[21] Takekoshi T, Reichsoellner L, Schindewolf A, Hutson J M, Sueur C, Dulieu O, Ferlaino F, Grimm R, Naegerl H C 2014 Phys. Rev. Lett. 113 205301
Google Scholar
[22] Park J W, Will S A, Zwierlein M W 2015 Phys. Rev. Lett. 114 205302
Google Scholar
[23] Wang F, He X, Li X, Zhu B, Chen J, Wang D 2015 New J. Phys. 17 035003
Google Scholar
[24] Matsuda K, Marco L D, Li J R, Tobias W G, Ye J 2020 Science 370 1324
Google Scholar
[25] Gu Y, Chen K, Huang Y, Yang X 2019 Chin. Phys. B 28 43702
Google Scholar
[26] Huang Y, Shao X, Yang X 2016 J. Phys. B 49 135101
Google Scholar
[27] Chen K, Huang Y, Yang X 2017 Chin. J Chem. Phys. 30 418
Google Scholar
[28] Smith D F, Tidwell M, Williams D V P 1950 Phys. Rev. 77 420
Google Scholar
[29] Calder V, Hansen D, Hoffman D, Ruedenberg K 1972 J Chem. Phys. 49 5399
[30] Nair K, Hoeft J, Tiemann E 1979 J Mol. Spectrosc. 78 506
Google Scholar
[31] Clyne M A A, Curran A H, Coxon J A 1976 J Mol. Spectrosc. 63 43
Google Scholar
[32] Aldegunde J, Hutson J M 2008 Phys. Rev. A 78 033434
Google Scholar
[33] Wang D, Shao X, Huang Y, Li C, Yang X 2021 Chin. Phys. B 30 113301
Google Scholar
[34] Yang Q S, Li S C, Yu Y, Gao T 2018 J Phys. Chem. A 122 3021
Google Scholar
[35] Brown J M, Carrington A 2003 Cambridge University Press
[36] Arima A, Horie H, Sano M 1957 Prog. Theor. Exp. Phys. 17 567
Google Scholar
[37] Müller H S, Gerry M C 1995 J. Chem. Phys. 103 577
Google Scholar
[38] Shao X, Gong T, Wu L, Yang X 2011 J. Quant. Spectrosc. Radiat. Transfer 112 1005
Google Scholar
[39] Ospelkaus S, Ni K K, Quéméner G, Neyenhuis B, Wang D, Miranda M D, Bohn J, Ye J, Jin D 2010 Phys. Rev. Lett. 104 030402
Google Scholar
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