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This work is to investigate the photo-detachment cross-section (PCS) of anions in an expanding quantum well formed by two moving elastic walls. Through the study of the closed orbits of the detached electrons, we derive the analytical expression for the period of these closed orbits. We utilize the classical closed-orbit theory (COT) to deeply explore and derive the PCS of this system, which is a superposition of a smooth background term and an oscillatory term caused by collisions between electrons and the two elastic walls of the quantum well. The calculation results show that the oscillation amplitude of the photo-detachment cross-section is highly sensitive to the wall velocity of the extended quantum well. When the quantum well is static, the photo-detachment cross-section exhibits a regular saw-tooth structure. As the walls begin to move, this regular saw-tooth structure becomes irregular. As the wall velocity increases, the oscillation structure within the PCS becomes increasingly complex. Furthermore, the photo-detachment cross-section is closely related to the initial distance between the negative hydrogen ion and the two moving walls, known as the well width. And through calculations of two different scenarios involving extended quantum wells, we find that for an asymmetrically expanding quantum well, the effect of moving walls on anionic PCS is more significant than for a symmetrically expanding quantum well. The research findings also reveal that as the well width narrows, the localized space for electrons becomes smaller, leading to stronger quantum confinement and an increase in the oscillation amplitude across the cross-section. As the well width increases, the quantum confinement effect on the electrons weakens, resulting in a decrease in the oscillation amplitude across the cross-section. When the well width reaches a certain level, the quantum well no longer exhibits significant quantum confinement effects, and the photo-detachment cross-section tends to approach a smooth background term. Therefore, precise control of the photo-detachment cross-section of negative hydrogen ions in an expanding quantum well formed by two moving elastic walls can be achieved by adjusting the initial size of the quantum well and the expansion speed of the quantum well. The phenomena revealed in this study are quite intriguing, and the methods employed are universal, providing guidance for future studying the photo-detachment cross-sections in more complex dynamic quantum wells. The findings of this study have significant reference value in the field of surface physics, enriching our understanding of the photo-detachment dynamics of anions in moving quantum wells, and they also provide a theoretical basis and guidance for future experimental research on the photo-detachment dynamics of anions in dynamic quantum wells.
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Keywords:
- photo-detachment /
- quantum confinement /
- expanding quantum well
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Google Scholar
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图 1 扩展量子阱中氢负离子光剥离理论模型示意图
Figure 1. Schematic diagram of theoretical model of hydrogen negative ion photodetachment in the expanding quantum wells.
图 2 扩展量子阱中剥离电子的一些经典轨迹, 剥离电子能量$ E $ = 0.246 eV, 氢负离子到上阱壁的初始距离$ {Z_{10}} $ = 100 a.u., 到下阱壁初始距离$ {Z_{20}} $ = 200 a.u., 两个阱壁移动速率为$ v $ = 0.001 a.u., 两种不同的线表示从原点发射的不同方向, 电子的出射方向如图
Figure 2. Some typical classical trajectories of the detached electron in the expanding quantum wells. The electron energy $ E $ = 0.246 eV. The initial distances from the hydrogen negative ion to the upper and lower surface are $ {Z_{10}} $ = 100 a.u. and $ {Z_{20}} $ = 200 a.u. The surfaces are moving at a speed of $ v $ = 0.001 a.u. Different lines denote different electron trajectories. The initial outgoing angles of the electron trajectory are given in the plot.
图 3 扩展量子阱中剥离电子的一些典型的闭合轨道, 电子能$ E = 0.246$ eV, 负离子到上下阱壁的初始距离分别为$ {Z_{10}}= 100 $ a.u., $ {Z_{20}} = 200$ a.u., 阱壁以$ v =0.001$ a.u.的速率移动
Figure 3. Some typical closed orbits of the detached electron in the expanding quantum wells. The electron energy $ E=0.246 $ eV. The initial distances from the negative ion to the upper and lower surfaces are: $ {Z_{10}}= 100 $ a.u., $ {Z_{20}} = 200$ a.u.. The surfaces are moving at a speed of $ v = 0.001$ a.u.
图 4 离子处于不对称扩展量子阱中的剥离电子的闭合轨道
Figure 4. The closed orbits of detached electrons in asymmetrically expanding quantum wells.
图 5 离子处于对称扩展量子阱中的剥离电子的闭合轨道
Figure 5. Closed orbits of stripped electrons in a symmetric expanding quantum well.
图 6 系统PCS随量子阱弹性阱壁运动速率的变化, 当两阱壁的初始距离关于氢负离子对称时, $ {Z_{10}} = {Z_{20}} = 100 $ a.u. (a) $ v $ = 0 a.u.; (b) $ v $ = 0.005 a.u.; (c) $ v $ = 0.01 a.u.; (d) $ v $ = 0.03 a.u.
Figure 6. Variation of the PCS with the speed of the moving walls in the quantum well. The initial distances between the H– and the two walls are equal, $ {Z_{10}} = {Z_{20}} = 100 $a.u.: (a) $ v $ = 0 a.u.; (b) $ v $ = 0.005 a.u.; (c) $ v $ = 0.01 a.u.; (d) $ v $ = 0.03 a.u..
图 9 不同阱壁速度时, 总PCS中的振荡因子$ M(E, v) $与扩展量子阱中弹性壁的速率的关系, 红线是对称扩展量子阱中的因子$ M(E, v) $, 离子与阱壁的初始距离 $ {Z_{10}} $ = $ {Z_{20}} $ = 100 a.u., 黑线是不对称扩展量子阱中的因子$ M(E, v) $, 离子与阱壁的初始距离分别为$ {Z_{10}} $ = 100 a.u., $ {Z_{20}} $ = 300 a.u. (a) $ v $ = 0 a.u.; (b) $ v $ = 0.005 a.u.; (c) $ v $ = 0.01 a.u.; (d) $ v $ = 0.03 a.u.
Figure 9. Dependence of the modulating factor $ M(E, v) $in the total PCS on the speed of the moving walls in the expanding quantum well, the red line is the factor $ M(E, v) $ in the symmetric expanding quantum well, $ {Z_{10}} $ = $ {Z_{20}} $ = 100 a.u., and the black line is the case in the asymmetric expanding quantum well, $ {Z_{10}} $ = 100 a.u., $ {Z_{20}} $ = 300 a.u.: (a) $ v $ = 0 a.u.; (b) $ v $ = 0.005 a.u.; (c) $ v $ = 0.01 a.u.; (d) $ v $ = 0.03 a.u..
图 7 对称扩展量子阱中阱壁运动速率不同时的PCS比较, 两阱壁的初始距离关于氢负离子对称, $ {Z_{10}} = {Z_{20}} = $$ 100 $a.u. (a) $ v $ = 0.005 a.u.; (b) $ v $ = 0.01 a.u.
Figure 7. Comparison of the PCS in the moving quantum well at different speed of the moving walls. The initial distances between the H– and the two walls are equal, $ {Z_{10}} = $$ l {Z_{20}} = 100 $a.u.: (a) $ v $ = 0.005 a.u, (b) $ v $ = 0.01 a.u..
图 8 阱壁的速率不同时, 氢负离子在不对称扩展量子阱中的PCS, 两阱壁的初始距离关于氢负离子不对称, $ {Z_{10}} $ = 100 a.u., $ {Z_{20}} $ = 300 a.u. (a) $ v $ = 0 a.u.; (b) $ v $ = 0.005 a.u.; (c) $ v $ = 0.01 a.u.; (d) $ v $ = 0.03 a.u.
Figure 8. Variation of the PCS with different speeds of the moving walls in the quantum well, the initial distances between the H– and the two walls are not equal, $ {Z_{10}} $ = 100 a.u., $ {Z_{20}} $ = 300 a.u.: (a) $ v $ = 0 a.u.; (b) $ v $ = 0.005 a.u.; (c) $ v $ = 0.01 a.u.; (d) $ v $ = 0.03 a.u..
图 10 光剥离截面对下阱壁-离子初始距离的依赖关系, 两阱壁的运动速率为v = 0.001 a.u., 固定上阱壁的初始距离为$ {Z_{10}} $ = 100 a.u., 下阱壁的初始距离 (a) $ {Z_{20}} $ = 100 a.u.; (b) $ {Z_{20}} $ = 200 a.u.; (c) $ {Z_{20}} $ = 500 a.u.; (d) $ {Z_{20}} $ = 1000 a.u.
Figure 10. Dependence of the PCS on the initial distance from the ion to the lower wall. Suppose that the two walls moves at a speed of v = 0.001 a.u., the initial distance between H– ion and the upper surface is fixed to be $ {Z_{10}} $ = 100 a.u., The initial distance from the ion to the lower wall: (a) $ {Z_{20}} $ = 100 a.u.; (b) $ {Z_{20}} $ = 200 a.u.; (c) $ {Z_{20}} $ = 500 a.u.; (d) $ {Z_{20}} $ = 1000 a.u..
图 11 振荡截面$ {\sigma _{{\text{osc}}}} $与离子到下阱壁的初始距离$ {Z_{20}} $以及扩展量子阱阱壁的移动速率的关系, 上阱壁的初始距离$ {Z_{10}} $ = 100 a.u., 两阱壁的移动速率为$ v $ = 0.005 a.u.
Figure 11. Dependence of the oscillating cross section $ {\sigma _{{\text{osc}}}} $ on the initial distance from the ion to the lower wall $ {Z_{20}} $ and the moving speed of the walls $ v $in the expanding quantum well. The initial position of the upper wall is at $ {Z_{10}} $ = 100 a.u.. Both walls move at a speed of $ v $ = 0.005 a.u..
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[1] Milner V, Hanssen J L, Campbell W C, Raizen M G 2001 Phys. Rev. Lett. 86 1514
Google Scholar
[2] Kaplan A, Friedman N, Andersen M, Davidson N 2001 Phys. Rev. Lett. 87 274101
Google Scholar
[3] Andersen M F, Kaplan A, Friedman N, Davidson N 2002 J. Phys. B: At. Mol. Opt. Phys. 35 2183
Google Scholar
[4] Friedman N, Kaplan A, Davidson N 2002 Adv. At. Mol. Opt. Phys. 48 99
[5] Stone A D 2010 Nature 465 10
Google Scholar
[6] Chuu D S, Hsiao C M, Mei W N 1992 Phys. Rev. B 46 3898
Google Scholar
[7] Du M L Delos J B 1988 Phys. Rev. A 38 1896
Google Scholar
[8] Du M L 1989 Phys. Rev. A 40 4983
Google Scholar
[9] Du M L, Delos J B 1987 Phys. Rev. Lett. 58 1731
Google Scholar
[10] Du M L, Delos J B 1988 Phys. Rev. A 38 1913
Google Scholar
[11] Du M L, Delos J B 1989 Phys. Rev. A 134 476
[12] Yang G C, Rui K K, Zheng Y Z 2009 Physica B: Condens. Matter. 404 1576
Google Scholar
[13] Zhao H J, Ma Z J, Du M L 2015 Physica B: Condens. Matter. 466 54
Google Scholar
[14] Du M L 2006 Eur. Phys. J. D 38 533
Google Scholar
[15] Zhao H J, Du M L 2009 Phys. Rev. A 79 023408
Google Scholar
[16] Wang D H, Li S S, Wang Y H, Mu H F 2012 J. Phys. Soc. Jpn. 81 114301
Google Scholar
[17] Novick J, Delos J B 2012 Phys. Rev. E 85 016206
Google Scholar
[18] 唐田田, 王德华, 黄凯云, 王姗姗 2012 物理学报 61 063202
Google Scholar
Tang T T, Wang D H, Huang K Y, Wang S S 2012 Acta Phys. Sin. 61 063202
Google Scholar
[19] 唐田田, 王德华, 黄凯云 2011 物理学报 60 053203
Google Scholar
Tang T T, Wang D H, Huang K Y 2011 Acta Phys. Sin. 60 053203
Google Scholar
[20] 唐田田, 张朝民, 张敏 2013 物理学报 62 123201
Google Scholar
Tang T T, Zhang C M, Zhang M 2013 Acta Phys. Sin. 62 123201
Google Scholar
[21] Wang D H 2014 Chin. J. Phys. 52 138
Google Scholar
[22] Tang T T, Zhu Z L, Yao J G, Wang D H 2017 Can. J. Phys. 95 38
Google Scholar
[23] 唐田田, 朱子亮, 姚建刚 2016 光子学报 45 1202002
Google Scholar
Tang T T, Zhu Z L, Yao J G 2016 Acta Photonica Sin. 45 1202002
Google Scholar
[24] Afaq A, Azmat I, Amin U R, Naveed K, Ansari M M 2016 Braz. J. Phys. 46 489
Google Scholar
[25] Zhao H J, Du M L 2018 Physica B: Condens. Matter. 530 121
[26] Wang D H, Pang Z H, Zhuang K Z, Li Y F, Xie L 2017 Prama. J. Phys. 89 71
Google Scholar
[27] Azmat I, Kiran H, Sana M, Saba J, Afaq A 2019 Chin. Phys. B 28 023201
Google Scholar
[28] 李洋阳, 孙世艳, 赵海军 2019 原子与分子物理学报 36 799
Google Scholar
Li Y Y, Sun S Y, Zhao H J 2019 J. Atom. Mol. Phys. 36 799
Google Scholar
[29] Tong S, Wang D H, Sun X Y 2021 Indian J. Phys. 95 551
Google Scholar
[30] Wang D H 2021 Z. Naturforsch. A. 76 407
[31] Feng W, Deng D 2021 Proceeding of the 2021 International Conference on Management of Data New York, USA, June 20–25, 2021 p541
[32] 唐田田, 祝庆利 2021 原子与分子物理学报 38 053001
Google Scholar
Tang T T, Zhu Q L 2021 J. Atom. Mol. Phys. 38 053001
Google Scholar
[33] Welander J, Navarro Navarrete J E, Rohlén J, Leopold T, Thomas R D, Pegg D J, Hanstorp D 2022 Rev. Sci. Instrum. 93 065004
Google Scholar
[34] Zhang L, Li C, Wang X, Feng W, Yu Z, Chen Q, Leng J, Guo M, Yang P 2023 IEEE International Parallel and Distributed Processing Symposium Milan, Italy, May 29–June 2, 2023 p864
[35] Fermi E 1949 Phys. Rev. 75 1169
Google Scholar
[36] Ulam S M, 1961 Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability University of California Berkeley, June 20–30, 1961 p315
[37] Dosecher S W, Rice M H 1969 Am. J. Phys. 37 1246
Google Scholar
[38] da Luz M G E, Cheng B K 1992 J. Phys. A: Math. Gen. 25 L1043
Google Scholar
[39] Martino S D, Anza F, Facchi P, Kossakowski A, Marmo G, Messina A, Militello B, Pascazio S 2013 J. Phys. A: Math. Theor. 46 365301
Google Scholar
[40] Wang D H 2018 Phys. Rev. A 98 053419
Google Scholar
[41] Yang B C, Delos J B, Du M L 2014 Phys. Rev. A 89 013417
Google Scholar
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