Applied Mechanics and Materials Vol. 456 (2014) pp 429-432Online available since 2013/Oct/31 at www.scientific.net (2014) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.456.429
Recent development of vacancy formation energy
Xiaohua YU 1,a, Zhaolin ZHAN 1*,b, Ju Rong 2,c , Yuan WANG 3,d
1Faculty of Materials Science and Engineering, Kunming University of Science and Technology,
Kunming 650093, China;
2
3Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016,china; School of Machinery & Transportation Engineering, Southwest Forestry University, Kunming
650224, China.
a [email protected], b [email protected],c [email protected], d [email protected]
Key words: Vacancy; Vacancy formation energy; Nano-scale; Research status; Physical properties; Thermodynamic properties
Abstract: Vacancies of the crystal is one of the core issues in solid physics, solid state chemistry, metallurgy and materials science. In this paper, we present the latest research on the vacancy formation energy and size dependence of vacancy formation energy. According to the vacancy formation energy of common methods and common description of the vacancy formation energy theory, the vcancy formation energy characterization, calculation and simulation methods are described; the nature and origin of the vacancy in Nano-scale is analyzed. finally, further research will focus on the properties mechanisms of the other physical properties, mechanical properties and thermodynamic in materials.
Introduction
Vacancy is the most basic and common defects in solid, The main parameters for describing the vacancy is its formation energy.Vacancy has important implications on materials physics, mechanics, thermodynamics, electrical and other properties, it is one of the most core issues in the solid state physics, solid state chemistry, metallurgy as well as materials science [1-3]. Detailed study the formation mechanism and the distribution of vacancy, is a very important significance for further grasping the macroscopic and microscopic properties. Typically, there are two methods to infer the distribution and availability of vacancy, namely the direct method and indirect method.The common indirect method is resistivity[4], surface energy, Young's modulus[5,6], melting[7], the lattice constant, density, thermal capacity and so on. The direct common method is fluorescence, electron spin resonance method[8], microscopic observation method, DFT method[9], pseudopotential method[10], on the potential method[11] and so on. However, all above these method is some discussion of single vacancy nature, practical and effective method for characterization of vacancies have not seen any definitive reports. In order to have more complete understanding of the relevant research results, this paper mainly review the characterization, calculation and simulation methods of the vacancy formation energy, the changes of vacancy formation energy in the nanoscale are analyzed and compared, the link between the vacancy and other physical properties, mechanical properties with thermodynamic is summed, the research in this field is discussed.
The common indirect method of vacancy formation energy
With resistivity measurements vacancy formation energy
Sun haoliang and coworkers[12], used magnetron sputtering to prepare thin film nano-W, and used four-probe tester to study resistivity of nano thin film. Study found that, the resistivity of W film decreases with increasing grain size. When the grain size decreases from 17.3 to 11.6, the performance of a significant size effect. The root cause of resistivity change, is the size becomes smaller, the required vacancy formation energy decreased, the grain boundary, the interface and the surface occupied by the increased rate, increases the degree of scattering of electrons cause. Zhang
cuiLing and his partners [13], take for ZrO2: 16% Y nano oxide ceramic for example, studied the diffusion activation energy with temperature variation, particle size and surface tension. The study found, the ion diffusion activation energy decreases with decreasing grain size. Wherein the particle size is small, the changes significantly, the size greater than 100nm, the changes small.
Surface energy, melting point, Young's modulus calculated vacancy formation energy
Wang Yongjiang [14], according to the surface of the thermodynamic theory, from the crystal lattice of a complete remove an atom, should produce a certain amount of surface area, a certain amount of specific surface Gibbs free energy. He used the material surface energy and melting point, to calculated most vacancy erengy of the metal elements. Found by calculation, the melting point and the vacancy formation energy, self diffusion activation energy and vacancy formation energy show proportional linear. This conclusion, is also confirmed by A.M. Brown[15]. Notably, Jiang Qing and his partners [16]found, in melting point and vacancy materials at nanometer scale, diffusion of nanoscale materials is still apply. Liu Zheng[17] thinks, increase a schottky defect formation inside the crystal, should be divided into three main parts: the elastic energy, binding energy and the electronic configuration can. Each of these three parts of energy were calculated, then the crystal vacancy formation energy. Using the model, Li, Na, K, Rb, Cr of five alkali metal was calculated, and the results is agree well with experiment.
Hu Wangyu and his partners[18], based on brooks[6]and Lgarashis’[19,20]model, proposed a method to calculate the surface energy. They believe that, have a vacancy in the crystal, will cause the energy change of two aspects of the material. first, new surface due to the increased surface area can, the second elastic strain due to vacancy contraction, and the minimum of the two energy value is the vacancy formation energy. Accordingly, they introduced the modifying factor, to calculated the FCC, BCC, HCP metal crystal vacancy formation energy. Computing results, the theoretical calculation is consistent with the actual measurement value. Zhang Xiyan[21], based on Tiwari and Patils’[3] model, improved the calculation method of metal surface energy, calculated the formation energy of FCC, BCC, HCP structure of metal vacancy energy. The results also show that, the surface energy of the experimental value has a linear relation with the vacancy energy.
Our group[22,23], according to the thermodynamic and bond breaking theory, think the essence of vacancy formation is fracture atomic bond energy, and should cause the specific surface area of material changes. thus proposed a vacancy formation energy mode, which can be calculated by bond energy. Then, we choose 21 kinds of different crystal structure (FCC, BCC, HCP) of the metal, to validate the model, as shown in Figure 1, 2. Can be seen from the graph, the formation energy of 21 metal ,is proportional with the vacancy and bond energy.
E f (e v ) E f (e v )
Figure 1 The relationship with the vacancy formation energy and FCC structural metal bond
energy
Figure2 The relationship with the vacancy formation energy and BCC structural metal bond Em (kJ/mol) Em (kJ/mol) Experimental direct determination of the vacancy formation energy
Qian Kailu,et,al[24], extruded the purified cadmium into filaments sample( the diameter of the sample is in millimeter scale), according to the concentration of vacancies after quenching will decrease, the decrease in vacancy concentration will lead to change in this sample volume. He used micro elongation apparatus of the reflected light with a resolution of 0.05µm, tested the change of the length of the sample after quenching. The experimentally derived the vacancy formation energy of
cadmium agree well with calculated values 0.29ev. Zhao Xinchun et,al [8] base on the diffractive dynamics, used the scattering matrix method to study the diffraction intensity of the vacancy. S Dannefaer and coworks [25], according to the electrons and positrons annihilate contact with each other,thereby obtaining the principle of electron density inside the material, by positron annihilation lifetime measurements inside the material, can analyze the existence and distribution of vacancies. 300K to 1523K experiments within the range, and gives the vacancy formation energy of 3.6 ± 0.2ev, experimental and calculated values of the same.
Computer simulation for vacancies
The crux of the vacancy formation energy and migration energy simulation by using the computer is the interaction energy between the selected atoms. In the past years, molecular dynamics simulations on the vacancy use two and three-body form of Stillinger-Weber potential model [26]; With the continuous development of the computer, there has been much closer to the embedded atom method (EAM) of the actual potential,that is many-body potential model [27], as well as first-principles pseudopotential model [28]. Tomonori Kitashima with his partners [29], use two and three-body form of Stillinger-Weber pair potential model to analyze vacancy diffusion in gallium arsenide.the study show that even at the melting point, there are still weak solid diffusion, and in the self-interstitial diffusion, the diffusion coefficient of arsenic atoms is larger than that of gallium atoms. Baskes[30], using a more accessible and fix potential embedded atom method (MWAM) which including the angle of force, study on the simulated Mo / Si system and predict the lattice constant, elastic constants, vacancy formation energy, the phase interface stability. MJMehl with his companders [31], using first-principles, according to the all-electronic linearized augmented plane wave (LAPW) method, resecach on the vacancy formation energy of aluminum, and calculate the vacancy formation energy of 4,8,16 and 27 lattice supercell in relaxation and non- relaxation state. Size Effect of vacancy formation energy
Zhang xiyan and his partners[32] consider that, in the nanometer size range, materials vacancy formation energy is proportional to its melting point, this relationship still applies. According to the Size dependent melting temperature of metallic, They calculate the corresponding nanoscale vacancy formation energy. On the basis of the model, they introduced the shape factor, and calculated nano metal vacancy formation energy in fcc, bcc, hcp structure. The results show that, the vacancy formation energy decreases as the size is reduced, when the grain size is 20nm or less, the reduction is evident. NH March and coworks [33]also obtained the similar conclusion.Qi Weihong and his partners [34,35] , roposed a size dependence of bind energy model, and on this basis, they are calculated Au nano metals vacancy formation energy of the size effect, the calculation results agree well with the actual.
Summary and Outlook
The being of vacancy not only have a direct influence on the bond energy of the material, the melting point, resistance, flexibility and other physical properties,it can effect on the binding energy, activation energy, Gibbs free energy, interfacial energy, interfacial stability, thermodynamic characteristics, but also become the important parameters of the computer simulation,so that it will become more and more attention by scholars. From size effect of vacancy formation energy, We can find that the vacany not only be able to respond various macroscopic properties of materials ,but also play an important role in the micro studies. However, the current research on the nature of vacancy is not enough, so in future studies on vacancy will remain its main mechanism ,but simulate observations is supplement.
Acknowledgements
This work was supported by the Chinese National Science Foundation (grant 51165016). References
[1] L.A. Girifalco: Scripta Metallurgica, Vol. 1 (1967) No.1, p. 5.
[2] Francesco Delogu: Materials Chemistry and Physics, Vol. 115(2009) p. 361.
[3] P.A. Korzhavyi, I.A.Abrikosov and B.Johansson: Physical Review B, Vol.59(1999) No.18, p. 11693.
[4] W changjun, Zhang zhenqi and Wang yanqin: Physics Examination and Testing, Vol. 2 (1986) p.
45.(In Chinese).
[5] HU Wangyu, Q I Wei-hong and ZHANG Bangwei: Journal of Hunan University Vol. 5 (1999) No. 26, p. 10.(In Chinese).
[6] H Brooks: Impurities and imperfections (Cleveland: American society for metals.1955).
[7] G.P Tiwari, R.V Patil: scripta metal lugriea. Vol. 19 (1975) p. 833.
[8] Zhao xinchun: The Characterization and calculation of vocancy(MS, guangxi University, china 2008).
[9] M W Finnis: J Phys: Condens Matter, Vol. 2 (1990) p. 331.
[10] B.Chakraborty, R.SIEGEL: Physical Review B, Vol. 27 (1983) p. 4535.
[11] R A Johnson: Phys Rev,A, Vol. 134 (1964) p. 1329.
[12] Sun haoliang, Xu kewei: Scientific Journal of Materials Science, Vol. 1 (2006) p. 335. (In Chinese).
[13] ZHANG Cuiling, ZHENG Ruilun: Journal of Southwest University, Vol. 31 (2009) No.3,p.
38.(in chinese)
[14] Wang yongjiang: Physics, Vol. 15 (1959) No. 9, p. 469. (in chinese)
[15] A.M. Brown, M.F. Ashby : Acta Metallurgica, Vol. 28 (1980) No. 8, p. 1085.
[16] Zhang sihua. Size-dependent diffusion activation energy(Ph.D, JILIN University, china 2004).
[17] Liu zheng,Lv zhenjia: Chinese Science Bulletin, Vol. 13 (1992) p. 1239.(in chinese).
[18] Shu Xiaolin, Chen Ziyu, and Hu Wangyu: BeiJing University of Aeronautics & Astronautics Vol. 32 (2006) No. 10, p. 1259.(in chinese)
[19] M Lgarashi, M Khantha and V Vitek: Philo Mag, Vol. B63 (1991) No. 8, p. 603.
[20] R Pasianot, E J Savino: Phy Rev, Vol. B45 (1992) p. 12704.
[21] ZHANG Xiyan, ZHAO Xinchun and JIA Chong: Journal of Chong qing University, Vol. 31 (2008) No. 12, p. 1342.(in chiese)
[22] Yu XiaoHua, Rong Ju and Zhan zhaolin: Journal of Northeastern University, Vol. 33 (2012) No. n2, p. 35.
[23] Zhan zhaolin. Accelerating Formation of Nanocrystallite Al-coatings by Ball Peening Process(Ph.D, Beijing University of Science and Technology, china 2005).
[24] Qian kailv, Wang xinsheng and Wang yongjiang: Physics, Vol. 21 (1965) No. 12, p. 2033. (in chinese)
[25] S Dannefaer, P Mascher and D KERR: Rev.Let, Vol. 56 (1986) p. 2195.
[26] Inder P Batra, Farid F Abraham: Phys. Rep. B, Vol. 35 (1986) p. 9552.
[27] S Murray. Daw , M.I. Baskes: Phy.Rev. Lett, Vol. 50 (1983) p. 1285.
[28] R Car, M Parrinello: Phys.Rev. Lett, Vol. 60 (1988) p. 204.
[29] T Kitashima, K Kakimoto and H Ozoe: J. Elec. Soci, Vol. 150 (2003) No. 3, p. 6198.
[30] I. Baskes: Mater. Sci.Engin.A, Vol. 261 (1999) p. 165.
[31] M J Mchl, B M Klein : Physica B, Vol. 172 (1991) p. 211.
[32] Zhao XinChun, Jia Chong and Zhang XiYan: JOURNAL OF NANJING UNIVERSITY, Vol. 45 (2009) No. 2, p. 310.(in chinese)
[33] N.H.March: Solid State Communication, Vol. 63 (1987) No. 11, p. 1075.
[34] W.H.QI, M.P.WANG: Journal of Materials Science, Vol. 39 (2004) p. 2529.
[35] W.H.Qi, M.P.Wang: Physica B, Vol. 334 (2003) p. 432.
Research in Mechanical Engineering and Material Science 10.4028/www.scientific.net/AMM.456
Recent Development of Vacancy Formation Energy 10.4028/www.scientific.net/AMM.456.429
Applied Mechanics and Materials Vol. 456 (2014) pp 429-432Online available since 2013/Oct/31 at www.scientific.net (2014) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.456.429
Recent development of vacancy formation energy
Xiaohua YU 1,a, Zhaolin ZHAN 1*,b, Ju Rong 2,c , Yuan WANG 3,d
1Faculty of Materials Science and Engineering, Kunming University of Science and Technology,
Kunming 650093, China;
2
3Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016,china; School of Machinery & Transportation Engineering, Southwest Forestry University, Kunming
650224, China.
a [email protected], b [email protected],c [email protected], d [email protected]
Key words: Vacancy; Vacancy formation energy; Nano-scale; Research status; Physical properties; Thermodynamic properties
Abstract: Vacancies of the crystal is one of the core issues in solid physics, solid state chemistry, metallurgy and materials science. In this paper, we present the latest research on the vacancy formation energy and size dependence of vacancy formation energy. According to the vacancy formation energy of common methods and common description of the vacancy formation energy theory, the vcancy formation energy characterization, calculation and simulation methods are described; the nature and origin of the vacancy in Nano-scale is analyzed. finally, further research will focus on the properties mechanisms of the other physical properties, mechanical properties and thermodynamic in materials.
Introduction
Vacancy is the most basic and common defects in solid, The main parameters for describing the vacancy is its formation energy.Vacancy has important implications on materials physics, mechanics, thermodynamics, electrical and other properties, it is one of the most core issues in the solid state physics, solid state chemistry, metallurgy as well as materials science [1-3]. Detailed study the formation mechanism and the distribution of vacancy, is a very important significance for further grasping the macroscopic and microscopic properties. Typically, there are two methods to infer the distribution and availability of vacancy, namely the direct method and indirect method.The common indirect method is resistivity[4], surface energy, Young's modulus[5,6], melting[7], the lattice constant, density, thermal capacity and so on. The direct common method is fluorescence, electron spin resonance method[8], microscopic observation method, DFT method[9], pseudopotential method[10], on the potential method[11] and so on. However, all above these method is some discussion of single vacancy nature, practical and effective method for characterization of vacancies have not seen any definitive reports. In order to have more complete understanding of the relevant research results, this paper mainly review the characterization, calculation and simulation methods of the vacancy formation energy, the changes of vacancy formation energy in the nanoscale are analyzed and compared, the link between the vacancy and other physical properties, mechanical properties with thermodynamic is summed, the research in this field is discussed.
The common indirect method of vacancy formation energy
With resistivity measurements vacancy formation energy
Sun haoliang and coworkers[12], used magnetron sputtering to prepare thin film nano-W, and used four-probe tester to study resistivity of nano thin film. Study found that, the resistivity of W film decreases with increasing grain size. When the grain size decreases from 17.3 to 11.6, the performance of a significant size effect. The root cause of resistivity change, is the size becomes smaller, the required vacancy formation energy decreased, the grain boundary, the interface and the surface occupied by the increased rate, increases the degree of scattering of electrons cause. Zhang
cuiLing and his partners [13], take for ZrO2: 16% Y nano oxide ceramic for example, studied the diffusion activation energy with temperature variation, particle size and surface tension. The study found, the ion diffusion activation energy decreases with decreasing grain size. Wherein the particle size is small, the changes significantly, the size greater than 100nm, the changes small.
Surface energy, melting point, Young's modulus calculated vacancy formation energy
Wang Yongjiang [14], according to the surface of the thermodynamic theory, from the crystal lattice of a complete remove an atom, should produce a certain amount of surface area, a certain amount of specific surface Gibbs free energy. He used the material surface energy and melting point, to calculated most vacancy erengy of the metal elements. Found by calculation, the melting point and the vacancy formation energy, self diffusion activation energy and vacancy formation energy show proportional linear. This conclusion, is also confirmed by A.M. Brown[15]. Notably, Jiang Qing and his partners [16]found, in melting point and vacancy materials at nanometer scale, diffusion of nanoscale materials is still apply. Liu Zheng[17] thinks, increase a schottky defect formation inside the crystal, should be divided into three main parts: the elastic energy, binding energy and the electronic configuration can. Each of these three parts of energy were calculated, then the crystal vacancy formation energy. Using the model, Li, Na, K, Rb, Cr of five alkali metal was calculated, and the results is agree well with experiment.
Hu Wangyu and his partners[18], based on brooks[6]and Lgarashis’[19,20]model, proposed a method to calculate the surface energy. They believe that, have a vacancy in the crystal, will cause the energy change of two aspects of the material. first, new surface due to the increased surface area can, the second elastic strain due to vacancy contraction, and the minimum of the two energy value is the vacancy formation energy. Accordingly, they introduced the modifying factor, to calculated the FCC, BCC, HCP metal crystal vacancy formation energy. Computing results, the theoretical calculation is consistent with the actual measurement value. Zhang Xiyan[21], based on Tiwari and Patils’[3] model, improved the calculation method of metal surface energy, calculated the formation energy of FCC, BCC, HCP structure of metal vacancy energy. The results also show that, the surface energy of the experimental value has a linear relation with the vacancy energy.
Our group[22,23], according to the thermodynamic and bond breaking theory, think the essence of vacancy formation is fracture atomic bond energy, and should cause the specific surface area of material changes. thus proposed a vacancy formation energy mode, which can be calculated by bond energy. Then, we choose 21 kinds of different crystal structure (FCC, BCC, HCP) of the metal, to validate the model, as shown in Figure 1, 2. Can be seen from the graph, the formation energy of 21 metal ,is proportional with the vacancy and bond energy.
E f (e v ) E f (e v )
Figure 1 The relationship with the vacancy formation energy and FCC structural metal bond
energy
Figure2 The relationship with the vacancy formation energy and BCC structural metal bond Em (kJ/mol) Em (kJ/mol) Experimental direct determination of the vacancy formation energy
Qian Kailu,et,al[24], extruded the purified cadmium into filaments sample( the diameter of the sample is in millimeter scale), according to the concentration of vacancies after quenching will decrease, the decrease in vacancy concentration will lead to change in this sample volume. He used micro elongation apparatus of the reflected light with a resolution of 0.05µm, tested the change of the length of the sample after quenching. The experimentally derived the vacancy formation energy of
cadmium agree well with calculated values 0.29ev. Zhao Xinchun et,al [8] base on the diffractive dynamics, used the scattering matrix method to study the diffraction intensity of the vacancy. S Dannefaer and coworks [25], according to the electrons and positrons annihilate contact with each other,thereby obtaining the principle of electron density inside the material, by positron annihilation lifetime measurements inside the material, can analyze the existence and distribution of vacancies. 300K to 1523K experiments within the range, and gives the vacancy formation energy of 3.6 ± 0.2ev, experimental and calculated values of the same.
Computer simulation for vacancies
The crux of the vacancy formation energy and migration energy simulation by using the computer is the interaction energy between the selected atoms. In the past years, molecular dynamics simulations on the vacancy use two and three-body form of Stillinger-Weber potential model [26]; With the continuous development of the computer, there has been much closer to the embedded atom method (EAM) of the actual potential,that is many-body potential model [27], as well as first-principles pseudopotential model [28]. Tomonori Kitashima with his partners [29], use two and three-body form of Stillinger-Weber pair potential model to analyze vacancy diffusion in gallium arsenide.the study show that even at the melting point, there are still weak solid diffusion, and in the self-interstitial diffusion, the diffusion coefficient of arsenic atoms is larger than that of gallium atoms. Baskes[30], using a more accessible and fix potential embedded atom method (MWAM) which including the angle of force, study on the simulated Mo / Si system and predict the lattice constant, elastic constants, vacancy formation energy, the phase interface stability. MJMehl with his companders [31], using first-principles, according to the all-electronic linearized augmented plane wave (LAPW) method, resecach on the vacancy formation energy of aluminum, and calculate the vacancy formation energy of 4,8,16 and 27 lattice supercell in relaxation and non- relaxation state. Size Effect of vacancy formation energy
Zhang xiyan and his partners[32] consider that, in the nanometer size range, materials vacancy formation energy is proportional to its melting point, this relationship still applies. According to the Size dependent melting temperature of metallic, They calculate the corresponding nanoscale vacancy formation energy. On the basis of the model, they introduced the shape factor, and calculated nano metal vacancy formation energy in fcc, bcc, hcp structure. The results show that, the vacancy formation energy decreases as the size is reduced, when the grain size is 20nm or less, the reduction is evident. NH March and coworks [33]also obtained the similar conclusion.Qi Weihong and his partners [34,35] , roposed a size dependence of bind energy model, and on this basis, they are calculated Au nano metals vacancy formation energy of the size effect, the calculation results agree well with the actual.
Summary and Outlook
The being of vacancy not only have a direct influence on the bond energy of the material, the melting point, resistance, flexibility and other physical properties,it can effect on the binding energy, activation energy, Gibbs free energy, interfacial energy, interfacial stability, thermodynamic characteristics, but also become the important parameters of the computer simulation,so that it will become more and more attention by scholars. From size effect of vacancy formation energy, We can find that the vacany not only be able to respond various macroscopic properties of materials ,but also play an important role in the micro studies. However, the current research on the nature of vacancy is not enough, so in future studies on vacancy will remain its main mechanism ,but simulate observations is supplement.
Acknowledgements
This work was supported by the Chinese National Science Foundation (grant 51165016). References
[1] L.A. Girifalco: Scripta Metallurgica, Vol. 1 (1967) No.1, p. 5.
[2] Francesco Delogu: Materials Chemistry and Physics, Vol. 115(2009) p. 361.
[3] P.A. Korzhavyi, I.A.Abrikosov and B.Johansson: Physical Review B, Vol.59(1999) No.18, p. 11693.
[4] W changjun, Zhang zhenqi and Wang yanqin: Physics Examination and Testing, Vol. 2 (1986) p.
45.(In Chinese).
[5] HU Wangyu, Q I Wei-hong and ZHANG Bangwei: Journal of Hunan University Vol. 5 (1999) No. 26, p. 10.(In Chinese).
[6] H Brooks: Impurities and imperfections (Cleveland: American society for metals.1955).
[7] G.P Tiwari, R.V Patil: scripta metal lugriea. Vol. 19 (1975) p. 833.
[8] Zhao xinchun: The Characterization and calculation of vocancy(MS, guangxi University, china 2008).
[9] M W Finnis: J Phys: Condens Matter, Vol. 2 (1990) p. 331.
[10] B.Chakraborty, R.SIEGEL: Physical Review B, Vol. 27 (1983) p. 4535.
[11] R A Johnson: Phys Rev,A, Vol. 134 (1964) p. 1329.
[12] Sun haoliang, Xu kewei: Scientific Journal of Materials Science, Vol. 1 (2006) p. 335. (In Chinese).
[13] ZHANG Cuiling, ZHENG Ruilun: Journal of Southwest University, Vol. 31 (2009) No.3,p.
38.(in chinese)
[14] Wang yongjiang: Physics, Vol. 15 (1959) No. 9, p. 469. (in chinese)
[15] A.M. Brown, M.F. Ashby : Acta Metallurgica, Vol. 28 (1980) No. 8, p. 1085.
[16] Zhang sihua. Size-dependent diffusion activation energy(Ph.D, JILIN University, china 2004).
[17] Liu zheng,Lv zhenjia: Chinese Science Bulletin, Vol. 13 (1992) p. 1239.(in chinese).
[18] Shu Xiaolin, Chen Ziyu, and Hu Wangyu: BeiJing University of Aeronautics & Astronautics Vol. 32 (2006) No. 10, p. 1259.(in chinese)
[19] M Lgarashi, M Khantha and V Vitek: Philo Mag, Vol. B63 (1991) No. 8, p. 603.
[20] R Pasianot, E J Savino: Phy Rev, Vol. B45 (1992) p. 12704.
[21] ZHANG Xiyan, ZHAO Xinchun and JIA Chong: Journal of Chong qing University, Vol. 31 (2008) No. 12, p. 1342.(in chiese)
[22] Yu XiaoHua, Rong Ju and Zhan zhaolin: Journal of Northeastern University, Vol. 33 (2012) No. n2, p. 35.
[23] Zhan zhaolin. Accelerating Formation of Nanocrystallite Al-coatings by Ball Peening Process(Ph.D, Beijing University of Science and Technology, china 2005).
[24] Qian kailv, Wang xinsheng and Wang yongjiang: Physics, Vol. 21 (1965) No. 12, p. 2033. (in chinese)
[25] S Dannefaer, P Mascher and D KERR: Rev.Let, Vol. 56 (1986) p. 2195.
[26] Inder P Batra, Farid F Abraham: Phys. Rep. B, Vol. 35 (1986) p. 9552.
[27] S Murray. Daw , M.I. Baskes: Phy.Rev. Lett, Vol. 50 (1983) p. 1285.
[28] R Car, M Parrinello: Phys.Rev. Lett, Vol. 60 (1988) p. 204.
[29] T Kitashima, K Kakimoto and H Ozoe: J. Elec. Soci, Vol. 150 (2003) No. 3, p. 6198.
[30] I. Baskes: Mater. Sci.Engin.A, Vol. 261 (1999) p. 165.
[31] M J Mchl, B M Klein : Physica B, Vol. 172 (1991) p. 211.
[32] Zhao XinChun, Jia Chong and Zhang XiYan: JOURNAL OF NANJING UNIVERSITY, Vol. 45 (2009) No. 2, p. 310.(in chinese)
[33] N.H.March: Solid State Communication, Vol. 63 (1987) No. 11, p. 1075.
[34] W.H.QI, M.P.WANG: Journal of Materials Science, Vol. 39 (2004) p. 2529.
[35] W.H.Qi, M.P.Wang: Physica B, Vol. 334 (2003) p. 432.
Research in Mechanical Engineering and Material Science 10.4028/www.scientific.net/AMM.456
Recent Development of Vacancy Formation Energy 10.4028/www.scientific.net/AMM.456.429