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The influence of diffusion and defects of crystal surface on the crystal vibration mode are an important and basic subject in surface physics research. The frequency of lattice vibration corresponds to the energy band of the system. Since the vibrations of the atoms in the crystal lattice are not isolated from each other, and the crystal lattice is periodic, thereby forming a lattice wave in the crystal. The lattice wave represents that all the atoms in the crystal vibrate at an identical frequency, which is often called a vibration mode. The lattice chain model has been studied as the vibrating mode of phonon and the energy-band in solid state physics. The vibrating modes of the lattice chain model have been analyzed with the Newton equation and the Born-von-Karman boundary condition in the literaure. In general, it is difficult to solve this problem due to the complex nonlinear characteristic of the interactions between the matter particles and the environment. Noting the complicacy in directly diagonalizing quantum Hamiltonian operator of a long chain, we introduce the invariant eigenoperator method (IEO) for deriving the energy gap of a given crystal lattice without solving its eigenstates in the Heisenberg picture. The Heisenberg equation is as important as the Schrödinger equation. However, it has been seldom used for directly deriving the energy-gap in previous studies. Following the Heisenberg's original idea that most observable physical quantity in quantum mechanics is energy spectrum, Hong-yi Fan, one of the authors of the present paper, developed the IEO method. This method provides a natural result of combining both the Schrödinger operator and the Heisenberg equation. Using the IEO method, we study the vibration modes of crystal lattice, which are affected by absorbing an atom with mass m0, which is different from the mass of atom in the crystal. Moreover, the attractive potential constantβ0 of the lattice surface differs from the inner constantβ. With the help of invariant eigen-operator method, we deduce the vibration mode ω=√(2β(1-cosh α))/ħm, where α=ln[-(mβ0+m0(-2β+β0)+√β0√-4 mm0β+(m+m0)2β0)/2m0β]. Our numerical results show that vibration mode ω depends not only on the absorption potential and the mass of the absorbed atom, but also on the mass of the lattice atom and the inner potential. In general, by discussing the vibration modes via some numerical solutions or approximate methods, we show the relations between the system vibration modes with different parameters which describe the environment influences. These results can deepen our understanding of quantum Brownian motion and demonstrate the applicability of the IEO method.
[1] Zhu J G 2012 Solid State Physics (Beijing:Science Press) p3 (in Chinese)[朱建国 2012 固体物理学 (北京:科学出版社) 第3页]
[2] Wang Z H 2011 Material Surface Engineering (Beijing:Chemical Industry Press) p9 (in Chinese)[王兆华 2011 材料表面工程 (北京:化学工业出版社) 第9页]
[3] Xu B S, Zhu S H, Liu S C 2014 Material Surface Engineering Technical (Beijing:Chemical Industry Press) p11 (in Chinese)[徐滨士, 朱绍华, 刘世参 2014 材料表面工程技术 (北京:化学工业出版社) 第11页]
[4] Chen H J, Huang S Y, Zhang Z B, Liu Y H, Wang X K 2017 Acta Chim. Sin. 75 560 (in Chinese)[陈海军, 黄舒怡, 张志宾, 刘云海, 王祥科 2017 化学学报 75 560]
[5] Yu Z P, Wang Y, Liu X Y 2012 Sci. China:Inform. Sci. 42 1644 (in Chinese)[余志平, 王燕, 刘晓彦 2012 中国科学:信息科学 42 1644]
[6] Jiang P 1991 Physics 20 15 (in Chinese)[蒋平 1991 物理 20 15]
[7] Jiang Z M, Wu Z Q 1988 Acta Phys. Sin. 37 629 (in Chinese)[蒋最敏, 吴自勤 1988 37 629]
[8] Hong S L 1988 Acta Phys. Sin. 37 1450 (in Chinese)[洪水力 1988 37 1450]
[9] An Z, Li Z J, Liu Z L, Yao K L 1994 Acta Phys. Sin. 43 516 (in Chinese)[安忠, 李占杰, 刘祖黎, 姚凯伦 1994 43 516]
[10] Li H N, Xu Y B, Li H Y, He P M, Bao S N 1999 Acta Phys. Sin. 48 273 (in Chinese)[李宏年, 徐亚伯, 李海洋, 何丕模, 鲍世宁 1999 48 273]
[11] Xie Z, An Z, Li Y C 2005 Acta Phys. Sin. 54 3922 (in Chinese)[谢尊, 安忠, 李有成 2005 54 3922]
[12] Jumeau R, Bourson P, Ferriol M, Lahure F, Ponçot M, Dahoun A 2013 Int. J. Spectrosc. 72 598
[13] Michelot F 1992 Phys. Rev. A 45 4271
[14] Fan H Y 1993 Phys. Rev. A 47 2379
[15] Elliott R J, Gibson A F 1975 Phys. Today 28 58
[16] Fan H Y, Li C 2004 Phys. Lett. A 321 75
[17] Ren Y C, Fan H Y 2013 Acta Phys. Sin. 62 156301 (in Chinese)[任益充, 范洪义 2013 62 156301]
[18] Fan H Y, Wu H 2005 Mod. Phys. Lett. B 19 1361
[19] Fan H Y, Wu H, Yuan H C 2011 Invariant Eigen-Operator Method in Quantum Mechanics (Shanghai:Shanghai Jiao Tong University Press) p175 (in Chinese)[范洪义, 吴昊, 袁洪春 2011 量子力学的不变本征算符方法(上海:上海交通大学出版社) p175]
[20] Fan H Y, Wu H, Xu X F 2005 Int. J. Mod. Phys. B 19 4073
[21] Chen J H, Fan H Y 2010 Mod. Phys. B 24 2387
[22] Rubin R J 1963 Phys. Rev. 131 964
[23] Fan H Y, Wu H 2008 Commun. Theor. Phys. 49 50
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[1] Zhu J G 2012 Solid State Physics (Beijing:Science Press) p3 (in Chinese)[朱建国 2012 固体物理学 (北京:科学出版社) 第3页]
[2] Wang Z H 2011 Material Surface Engineering (Beijing:Chemical Industry Press) p9 (in Chinese)[王兆华 2011 材料表面工程 (北京:化学工业出版社) 第9页]
[3] Xu B S, Zhu S H, Liu S C 2014 Material Surface Engineering Technical (Beijing:Chemical Industry Press) p11 (in Chinese)[徐滨士, 朱绍华, 刘世参 2014 材料表面工程技术 (北京:化学工业出版社) 第11页]
[4] Chen H J, Huang S Y, Zhang Z B, Liu Y H, Wang X K 2017 Acta Chim. Sin. 75 560 (in Chinese)[陈海军, 黄舒怡, 张志宾, 刘云海, 王祥科 2017 化学学报 75 560]
[5] Yu Z P, Wang Y, Liu X Y 2012 Sci. China:Inform. Sci. 42 1644 (in Chinese)[余志平, 王燕, 刘晓彦 2012 中国科学:信息科学 42 1644]
[6] Jiang P 1991 Physics 20 15 (in Chinese)[蒋平 1991 物理 20 15]
[7] Jiang Z M, Wu Z Q 1988 Acta Phys. Sin. 37 629 (in Chinese)[蒋最敏, 吴自勤 1988 37 629]
[8] Hong S L 1988 Acta Phys. Sin. 37 1450 (in Chinese)[洪水力 1988 37 1450]
[9] An Z, Li Z J, Liu Z L, Yao K L 1994 Acta Phys. Sin. 43 516 (in Chinese)[安忠, 李占杰, 刘祖黎, 姚凯伦 1994 43 516]
[10] Li H N, Xu Y B, Li H Y, He P M, Bao S N 1999 Acta Phys. Sin. 48 273 (in Chinese)[李宏年, 徐亚伯, 李海洋, 何丕模, 鲍世宁 1999 48 273]
[11] Xie Z, An Z, Li Y C 2005 Acta Phys. Sin. 54 3922 (in Chinese)[谢尊, 安忠, 李有成 2005 54 3922]
[12] Jumeau R, Bourson P, Ferriol M, Lahure F, Ponçot M, Dahoun A 2013 Int. J. Spectrosc. 72 598
[13] Michelot F 1992 Phys. Rev. A 45 4271
[14] Fan H Y 1993 Phys. Rev. A 47 2379
[15] Elliott R J, Gibson A F 1975 Phys. Today 28 58
[16] Fan H Y, Li C 2004 Phys. Lett. A 321 75
[17] Ren Y C, Fan H Y 2013 Acta Phys. Sin. 62 156301 (in Chinese)[任益充, 范洪义 2013 62 156301]
[18] Fan H Y, Wu H 2005 Mod. Phys. Lett. B 19 1361
[19] Fan H Y, Wu H, Yuan H C 2011 Invariant Eigen-Operator Method in Quantum Mechanics (Shanghai:Shanghai Jiao Tong University Press) p175 (in Chinese)[范洪义, 吴昊, 袁洪春 2011 量子力学的不变本征算符方法(上海:上海交通大学出版社) p175]
[20] Fan H Y, Wu H, Xu X F 2005 Int. J. Mod. Phys. B 19 4073
[21] Chen J H, Fan H Y 2010 Mod. Phys. B 24 2387
[22] Rubin R J 1963 Phys. Rev. 131 964
[23] Fan H Y, Wu H 2008 Commun. Theor. Phys. 49 50
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