Elastic Deformation and Elastic Wave Velocity for Cubic Metal and Binary Interstitial Alloy’s Thin Films at Different Temperatures, Pressures, Interstitial Atomic Concentrations and Film Thicknesses
Main Article Content
Abstract
The theory of elastic deformation and elastic wave for cubic metal and binary interstitial alloy’s thin films is builded by a statistical moment method (SMM). The obtained results have been applied to perform numerical calculations for W, Au, WSi and AuSi materials. Some SMM calculations for W, Au are compared with experiments and other calculation methods. Other SMM calculations are new and predict experimental results.
Keywords:
Binary interstitial alloy, thin film, elastic deformation, elastic wave velocity, SMM
References
[1] N. Q. Hoc, B. D. Tinh, N. D. Hien, Elastic Moduli and Elastic Constants of Interstitial Alloy AuCuSi with FCC Structure under Pressure, High Temperature Materials and Processes, Vol. 38, 2019, pp. 264-272,
https:// doi.org/10.1515/htmp-2018-0027.
[2] N. Q. Hoc, N. T. Hoa, N. D. Hien, D. Q. Thang, Study on Nonlinear Deformation of Binary Interstitial Alloy with BCC Structure under Pressure, HNUE Journal of Science, Natural Sciences, Vol. 63, No. 6, 2018, pp. 57-65, https://doi.org/10.18173/2354-1059.2018-0029.
[3] N. Q. Hoc, N. D. Hien, D. Q. Thang, Elastic Deformation of Alloy AuSi with BCC Structure under Pressure, HNUE Journal of Science, Natural Sciences, Vol. 63, No. 6, 2018, pp. 74-83,
https://doi.org/10.18173/2354-1059.2018-0031.
[4] N. Q. Hoc, T. D. Cuong, N. D. Hien, Study on Elastic Deformation of Interstitial Alloy FeC with BCC Structure under Pressure, VNU Journal of Science: Mathematics-Physics, Vol. 35, No. 1, 2019, pp. 1-12,
https://doi.org/10.25073/2588-1124/vnumap.4293.
[5] N. Q. Hoc, N. D. Hien, T. D. Nam, V. L.Thanh, Study on Elastic Deformation of Stainess Steel under Pressure, HNUE Journal of Science, Natural Sciences, Vol. 66, No. 2, 2021, pp. 83-99, https://doi.org/10.18173/2354-1059.2021-0031.
[6] B. D. Tinh, N. Q. Hoc, D. Q. Vinh, T. D. Cuong, N. D. Hien, Thermodynamic and Elastic Properties of Interstitial Alloy FeC with BCC Structure at Zero Pressure, Advanced Materials Science and Engineering, Vol. 2018, 2018, Article No. 5251741, https://doi.org/10.1155/2018/5251741.
[7] B. Wess, V. Groger, G. Khatibi, A. Kotas, P. Zimprich, R. Stickler, B. Zagar, Sensors and Actuactors A- Physical, Vol. 99, No. 1-2, 2002, pp. 172-182, https://doi.org/10.1016/S0924-4247(01)00877-9.
[8] V. V. Hung, D. D. Phuong, N. T. Hoa, H. K. Hieu, Theoretical Investigation of the Thermodynamic Properties of Metallic Thin Films, Thin Solid Film, Vol. 583, 2015, pp.7-12, https://doi.org/10.1016/j.tsf.2015.03.040
[9] V. V. Hung, D. D. Phuong, N. T. Hoa, Investigation of Thermodynamic Properties of Metal Thin Film by Statistical Moment Method, Communications in Physics, Vol. 23, No. 4, 2013, pp. 301-311, https://doi.org/10.15625/0868-3166/23/4/3351.
[10] V. V. Hung, D. D. Phuong, N. T. Hoa, Thermodynamic Properties of Free Standing Thin Metal Films Investigated by Using Statistical Moment Method: Temperature and Pressure Dependence, Communications in Physics,
Vol. 24, No. 2, 2014, pp.177-191, https:// doi.org/10.15625/0868-3166/24/2/3731.
[11] H. Huang, F. Spaepen, Tensile Testing of Free-standing Cu, Ag and Al Thin Films and Ag/Cu Multilayers, Acta Materialia, Vol. 48, No. 12, 2000, pp.3261-3269, https://doi.org/10.1016/S1359-6454(00)00128-2.
[12] R. Knepper, S. P. Baker, Coefficients of Thermal Expansion and Biaxial Elastic Modulus of Phase Tantalum Thin Films, Applied Physics Letters, Vol. 90, No. 18, 2007, Article No. 181908,
https:// doi.org/10.1063/1.2734468.
[13] A. R. Vaz, M. C. Salvadori, M. Cattani, Journal of Metallic Nano Materials, Vol. 20-21, 2004, pp. 758-762,
https:// doi.org/10.4028/www.scientific.net/JMNM.20-21.758.
[14] B. Wess, V. Groger, G. Khatibi, A. Kotas, P. Zimprich, R. Stickler, B. Zagar, Sensors and Actuactors A- Physical, Vol. 99, No. 1-2, 2002, pp. 172-182, https://doi.org/10.1016/S0924-4247(01)00877-9.
[15] Y. Kuru, M. Wohlschlogel, U. Welzel, E. J. Mittemeijer, Coefficients of Thermal Expansion of Thin Metal Films Investigated by Non-ambient X-ray Diffraction Stress Analysis, Surface and Coating Technology, Vol. 202,
No. 11, 2008, pp.2306-2309, https://doi.org/10.1016/j.surfcoat.2007.08.002.
[16] D. Fuks, S. Dorfman, Y. F. Zhukovskii, E. A. Kotomin, A. M. Stoneham, Theory of the Growth Mode for a Thin Metallic Film on an Insulating Substrate, Surface Science, Vol. 499, No. 1, 2002, pp. 24-40, https://doi.org/10.1016/S0039-6028(01)01692-2.
[17] V. V. Hung, Statistical Moment Method in Studying Elastic and Thermodynamic Properties of Crystals, HNUE Publishing House, Hanoi, 2009 (in Vietmamese).
[18] A. J. Kalkman, G. C. Verbruggen, A. M. Janssen, Young's Modulus Measurements and Grain Boundary Sliding in Free-standing Thin Metal Films, Applied Physics Letters, Vol. 78, No. 18, 2001, pp. 2673-2675,
https:// doi.org/10.1063/1.1367896.
[19] D. Errandonea, B. Schwager, R. Ditz, C. Gessmann, R. Boehler, M. Ross, Systematics of Transition-metal Melting, Physical Review B, Vol. 63, No. 13, 2001, Article No. 132104, https://doi.org/10.1103/PhysRevB. 63. 132104.
[20] R. D. Nyilas, S. Frank, R. Spolenak, Revealing Plastic Deformation Mechanisms in Polycrystalline Thin Films with Synchrotron XRD, JOM: The Journal of the Minerals, Metals & Materials Society, Vol. 62, No. 12, 2010,
pp. 44-51, https://doi.org/10.1007/s11837-010-0179-3.
[21] N. Tang, V. V. Hung, Investigation of the Thermodynamic Properties of Anharmonic Crystals by the Momentum Method. I. General Results for Face‐Centred Cubic Crystals, Physica Status Solidi (b), Vol. 149, No. 2, 1988,
pp. 511-519, https://doi.org/10.1002/pssb.2221490212.
[22] D. R. Olsen, H. M. Berg, Journal of Applied Physics, Vol. 44, 1973, pp. 314-324, https://doi.org/10.1063/1.1661879.
[23] J. Tobon, C. P. S. Giraldo, H. Sanchez, Manufacture of Au-Si Alloys for Use in the Soldering of Gold Alloys, Welding International, Vol. 29, No. 8, 2015, pp. 594-599, https://doi.org/10.1080/09507116.2014.932986.
[24] T. Cagın, Thermal and Mechanical Properties of some fcc Transition Metals, Physical Review B, Vol. 59, No. 5, 1999, pp. 3468-3473, https://doi.org/ 10.1103/PhysRevB.59.3468.
[25] W. Li, H. Kou, X. Zhang, J. Ma, J. Li, P. Geng, X. Wu, L. Chen, D. Fang, Temperature-dependent Elastic Modulus Model for Metallic Bulk Materials, Mechanics of Materials, Vol. 139, 2019, Article No. 103194, https://doi.org/10.1016/j.mechmat.2019.103194.
[26] F. F. Zahroh, I. Sugihartono, E. D. Safitri, Young’s Modulus Calculation of Some Metals Using Molecular Dynamics Method Based on the Morse Potential, Computational and Experimental Research in Materials and Renewable Energy (CERiMRE), Vol. 2, No. 1, 2019, pp.19-34, https://doi.org/10.19184/cerimre.v2i1.20557.
[27] Y. A. Chang, L. Himmel, Temperature Dependence of the Elastic Constants of Cu, Ag, and Au above Room Temperature, Journal of Applied Physics, Vol. 37, No. 9, 1966, pp. 3567-3572, https:// doi.org/10.1063/1.1708903.
[28] E. Güler, M. Güler, Geometry Optimization Calculations for the Elasticity of Gold at High Pressure, Advances in Materials Science and Engineering, Vol. 2013, 2023, Article No. 525673, https://doi.org/10.1155/2013/525673.
[29] M. Matsui, High Temperature and High Pressure Equation of State of Gold, Journal of Physics, Vol. 215, No. 1, 2010, Article No. 012197, https://doi.org/10.1088/1742-6596/215/1/012197.
[30] M. Yokoo, N. Kawai, K. G. Nakamura, K. I. Kondo, Y. Tange, T. Tsuchiya, Ultrahigh-pressure Scales for Gold and Platinum ai Pressures up to 550 GPa, Physical Review B, Vol. 80, 2009, Article No. 104114, https://doi.org/10.1103/PhysRevB.80.104114.
[31] S. N. Biswas, P. Van't Klooster, N. J. Trappeniers, Effect of Pressure on the Elastic Constants of Noble Metals from -196 to +25 oC and up to 2500 bar: II. Silver and Gold, Physica B+C, Vol. 103, No. 2-3, 1981, pp. 235-246, https:// doi.org/10.1016/0378-4363(81)90127-3.
[32] T. S. Duffy, G. Shen, D. L. Heinz, J. Shu, Y. Ma, H. K. Mao, R. J. Hemley, A. K. Singh, Lattice Strains in
Gold and Rhenium under Nonhydrostatic Compression to 37 GPa, Physical Review B, Vol. 60, No. 22, 1999, pp.15063-15073, https://doi.org/10.1103/PhysRevB.60.15063.
[33] Y. Hiki, A. V. Granato, Anharmonicity in Noble Metals; Higher Order Elastic Constants, Physical Review, Vol. 144, No. 2, 1966, pp.411-419, https:// doi.org/10.1103/PhysRev.144.411.
[34] T. Tsuchiya, K. Kawamura, Ab initio study of pressure effect on elastic properties of crystalline Au, Journal of Chemical Physics, Vol. 116, No. 5, 2022, pp.2121-2124, https:// doi.org/10.1063/1.1429643.
[35] N. Tang, V. V. Hung, Thermodynamic Properties of the Crystals at Various Pressures, Physica Status Solidi (b), Vol. 162, No. 2, 1990, pp. 371-377, https://doi.org/10.1002/pssb.2221620206.
[36] N. T. Hoa, N. Q. Hoc, H. X. Dat, Study on the Thermodynamic Properties of Thin Film of FCC Interstitial Alloy AuSi at Zero Pressure Using the Statistical Moment Method, Physics, Vol. 5, 2023, pp. 59-68, https://doi.org/10.3390/physics5010005.
[37] M. N. Magomedov, Calculation of Debye Temperature and Gruneisen Parameter, Journal of Physical Chemistry, Vol. 61, No. 4, 1987, pp. 1003-1009 (in Russian).
[38] L. V. Tikhonov, G. Y. Koponenko, Mechanical Properties of Metals and Alloys, Nauka-Dumka, Kiev, 1986
(in Russian).
[39] N. K. Xuong, Colour Metal Material, Science and Technique Publishing House, Hanoi, 2003 (in Vietnamese).
[40] M. J. Mehl, Pressure Dependence of the Elastic Moduli in Aluminum-rich Al-Li Compounds, Physical Review.B, Vol. 47, No. 5, 1993, pp. 2493, https://doi.org/10.1103/PhysRevB.47.2493.
[41] R. L. David, CRC Handbook of Chemistry and Physics, 86th Ed., Taylor & Francis, Boca Raton, London-New York-Singapore, 2005.
[42] L. Burakovsky, C. W. Greeff, D. L. Preston, Analytic Model of the Shear Modulus at All Temperatures and Densities, Physical Review B, Vol. 67, No. 9, 2003, Article No. 094107,
https://doi.org/10.1103/PhysRevB.67.094107.
[43] M. J. Mehl, D. A. Papacnstantopoulos, Applications of a Tight-binding Total-energy Method for Transtion and Noble Metals: Elastic Constants, Vaccancies, and Surfaces of Monatomic Metals, Physical Review B, Vol. 54,
No. 7, 1996, pp. 4519, https://doi.org/10.1103/PhysRevB.54.4519.
[44] H. M. Ledbetter, E. R. Naimon, Relationship between Single-crystal and Polycrystal Elastic Constants, Journal of Applied Physics, Vol. 45, 1974, pp.66-69, https://doi.org/10.1063/1.1663019.
[45] S. Santra, H. Dong, T. Laurila, A. Paul, Role of Different Factors Affecting Interdiffusion in Cu(Ga) and Cu(Si) Solid Solutions, The Royal Society A, Vol. 470, 2014, Article No. 20130464,
https:// doi.org/10.1098/rspa.2013.0464.
[46] C. S. Smith, J. W. Burns, The Elastic Constants of Cu-4 Percent Si, Journal of Applied Physics, Vol. 24, No. 1, 1953, pp.15-18, https://doi.org/10.1063/1.1721124.
https:// doi.org/10.1515/htmp-2018-0027.
[2] N. Q. Hoc, N. T. Hoa, N. D. Hien, D. Q. Thang, Study on Nonlinear Deformation of Binary Interstitial Alloy with BCC Structure under Pressure, HNUE Journal of Science, Natural Sciences, Vol. 63, No. 6, 2018, pp. 57-65, https://doi.org/10.18173/2354-1059.2018-0029.
[3] N. Q. Hoc, N. D. Hien, D. Q. Thang, Elastic Deformation of Alloy AuSi with BCC Structure under Pressure, HNUE Journal of Science, Natural Sciences, Vol. 63, No. 6, 2018, pp. 74-83,
https://doi.org/10.18173/2354-1059.2018-0031.
[4] N. Q. Hoc, T. D. Cuong, N. D. Hien, Study on Elastic Deformation of Interstitial Alloy FeC with BCC Structure under Pressure, VNU Journal of Science: Mathematics-Physics, Vol. 35, No. 1, 2019, pp. 1-12,
https://doi.org/10.25073/2588-1124/vnumap.4293.
[5] N. Q. Hoc, N. D. Hien, T. D. Nam, V. L.Thanh, Study on Elastic Deformation of Stainess Steel under Pressure, HNUE Journal of Science, Natural Sciences, Vol. 66, No. 2, 2021, pp. 83-99, https://doi.org/10.18173/2354-1059.2021-0031.
[6] B. D. Tinh, N. Q. Hoc, D. Q. Vinh, T. D. Cuong, N. D. Hien, Thermodynamic and Elastic Properties of Interstitial Alloy FeC with BCC Structure at Zero Pressure, Advanced Materials Science and Engineering, Vol. 2018, 2018, Article No. 5251741, https://doi.org/10.1155/2018/5251741.
[7] B. Wess, V. Groger, G. Khatibi, A. Kotas, P. Zimprich, R. Stickler, B. Zagar, Sensors and Actuactors A- Physical, Vol. 99, No. 1-2, 2002, pp. 172-182, https://doi.org/10.1016/S0924-4247(01)00877-9.
[8] V. V. Hung, D. D. Phuong, N. T. Hoa, H. K. Hieu, Theoretical Investigation of the Thermodynamic Properties of Metallic Thin Films, Thin Solid Film, Vol. 583, 2015, pp.7-12, https://doi.org/10.1016/j.tsf.2015.03.040
[9] V. V. Hung, D. D. Phuong, N. T. Hoa, Investigation of Thermodynamic Properties of Metal Thin Film by Statistical Moment Method, Communications in Physics, Vol. 23, No. 4, 2013, pp. 301-311, https://doi.org/10.15625/0868-3166/23/4/3351.
[10] V. V. Hung, D. D. Phuong, N. T. Hoa, Thermodynamic Properties of Free Standing Thin Metal Films Investigated by Using Statistical Moment Method: Temperature and Pressure Dependence, Communications in Physics,
Vol. 24, No. 2, 2014, pp.177-191, https:// doi.org/10.15625/0868-3166/24/2/3731.
[11] H. Huang, F. Spaepen, Tensile Testing of Free-standing Cu, Ag and Al Thin Films and Ag/Cu Multilayers, Acta Materialia, Vol. 48, No. 12, 2000, pp.3261-3269, https://doi.org/10.1016/S1359-6454(00)00128-2.
[12] R. Knepper, S. P. Baker, Coefficients of Thermal Expansion and Biaxial Elastic Modulus of Phase Tantalum Thin Films, Applied Physics Letters, Vol. 90, No. 18, 2007, Article No. 181908,
https:// doi.org/10.1063/1.2734468.
[13] A. R. Vaz, M. C. Salvadori, M. Cattani, Journal of Metallic Nano Materials, Vol. 20-21, 2004, pp. 758-762,
https:// doi.org/10.4028/www.scientific.net/JMNM.20-21.758.
[14] B. Wess, V. Groger, G. Khatibi, A. Kotas, P. Zimprich, R. Stickler, B. Zagar, Sensors and Actuactors A- Physical, Vol. 99, No. 1-2, 2002, pp. 172-182, https://doi.org/10.1016/S0924-4247(01)00877-9.
[15] Y. Kuru, M. Wohlschlogel, U. Welzel, E. J. Mittemeijer, Coefficients of Thermal Expansion of Thin Metal Films Investigated by Non-ambient X-ray Diffraction Stress Analysis, Surface and Coating Technology, Vol. 202,
No. 11, 2008, pp.2306-2309, https://doi.org/10.1016/j.surfcoat.2007.08.002.
[16] D. Fuks, S. Dorfman, Y. F. Zhukovskii, E. A. Kotomin, A. M. Stoneham, Theory of the Growth Mode for a Thin Metallic Film on an Insulating Substrate, Surface Science, Vol. 499, No. 1, 2002, pp. 24-40, https://doi.org/10.1016/S0039-6028(01)01692-2.
[17] V. V. Hung, Statistical Moment Method in Studying Elastic and Thermodynamic Properties of Crystals, HNUE Publishing House, Hanoi, 2009 (in Vietmamese).
[18] A. J. Kalkman, G. C. Verbruggen, A. M. Janssen, Young's Modulus Measurements and Grain Boundary Sliding in Free-standing Thin Metal Films, Applied Physics Letters, Vol. 78, No. 18, 2001, pp. 2673-2675,
https:// doi.org/10.1063/1.1367896.
[19] D. Errandonea, B. Schwager, R. Ditz, C. Gessmann, R. Boehler, M. Ross, Systematics of Transition-metal Melting, Physical Review B, Vol. 63, No. 13, 2001, Article No. 132104, https://doi.org/10.1103/PhysRevB. 63. 132104.
[20] R. D. Nyilas, S. Frank, R. Spolenak, Revealing Plastic Deformation Mechanisms in Polycrystalline Thin Films with Synchrotron XRD, JOM: The Journal of the Minerals, Metals & Materials Society, Vol. 62, No. 12, 2010,
pp. 44-51, https://doi.org/10.1007/s11837-010-0179-3.
[21] N. Tang, V. V. Hung, Investigation of the Thermodynamic Properties of Anharmonic Crystals by the Momentum Method. I. General Results for Face‐Centred Cubic Crystals, Physica Status Solidi (b), Vol. 149, No. 2, 1988,
pp. 511-519, https://doi.org/10.1002/pssb.2221490212.
[22] D. R. Olsen, H. M. Berg, Journal of Applied Physics, Vol. 44, 1973, pp. 314-324, https://doi.org/10.1063/1.1661879.
[23] J. Tobon, C. P. S. Giraldo, H. Sanchez, Manufacture of Au-Si Alloys for Use in the Soldering of Gold Alloys, Welding International, Vol. 29, No. 8, 2015, pp. 594-599, https://doi.org/10.1080/09507116.2014.932986.
[24] T. Cagın, Thermal and Mechanical Properties of some fcc Transition Metals, Physical Review B, Vol. 59, No. 5, 1999, pp. 3468-3473, https://doi.org/ 10.1103/PhysRevB.59.3468.
[25] W. Li, H. Kou, X. Zhang, J. Ma, J. Li, P. Geng, X. Wu, L. Chen, D. Fang, Temperature-dependent Elastic Modulus Model for Metallic Bulk Materials, Mechanics of Materials, Vol. 139, 2019, Article No. 103194, https://doi.org/10.1016/j.mechmat.2019.103194.
[26] F. F. Zahroh, I. Sugihartono, E. D. Safitri, Young’s Modulus Calculation of Some Metals Using Molecular Dynamics Method Based on the Morse Potential, Computational and Experimental Research in Materials and Renewable Energy (CERiMRE), Vol. 2, No. 1, 2019, pp.19-34, https://doi.org/10.19184/cerimre.v2i1.20557.
[27] Y. A. Chang, L. Himmel, Temperature Dependence of the Elastic Constants of Cu, Ag, and Au above Room Temperature, Journal of Applied Physics, Vol. 37, No. 9, 1966, pp. 3567-3572, https:// doi.org/10.1063/1.1708903.
[28] E. Güler, M. Güler, Geometry Optimization Calculations for the Elasticity of Gold at High Pressure, Advances in Materials Science and Engineering, Vol. 2013, 2023, Article No. 525673, https://doi.org/10.1155/2013/525673.
[29] M. Matsui, High Temperature and High Pressure Equation of State of Gold, Journal of Physics, Vol. 215, No. 1, 2010, Article No. 012197, https://doi.org/10.1088/1742-6596/215/1/012197.
[30] M. Yokoo, N. Kawai, K. G. Nakamura, K. I. Kondo, Y. Tange, T. Tsuchiya, Ultrahigh-pressure Scales for Gold and Platinum ai Pressures up to 550 GPa, Physical Review B, Vol. 80, 2009, Article No. 104114, https://doi.org/10.1103/PhysRevB.80.104114.
[31] S. N. Biswas, P. Van't Klooster, N. J. Trappeniers, Effect of Pressure on the Elastic Constants of Noble Metals from -196 to +25 oC and up to 2500 bar: II. Silver and Gold, Physica B+C, Vol. 103, No. 2-3, 1981, pp. 235-246, https:// doi.org/10.1016/0378-4363(81)90127-3.
[32] T. S. Duffy, G. Shen, D. L. Heinz, J. Shu, Y. Ma, H. K. Mao, R. J. Hemley, A. K. Singh, Lattice Strains in
Gold and Rhenium under Nonhydrostatic Compression to 37 GPa, Physical Review B, Vol. 60, No. 22, 1999, pp.15063-15073, https://doi.org/10.1103/PhysRevB.60.15063.
[33] Y. Hiki, A. V. Granato, Anharmonicity in Noble Metals; Higher Order Elastic Constants, Physical Review, Vol. 144, No. 2, 1966, pp.411-419, https:// doi.org/10.1103/PhysRev.144.411.
[34] T. Tsuchiya, K. Kawamura, Ab initio study of pressure effect on elastic properties of crystalline Au, Journal of Chemical Physics, Vol. 116, No. 5, 2022, pp.2121-2124, https:// doi.org/10.1063/1.1429643.
[35] N. Tang, V. V. Hung, Thermodynamic Properties of the Crystals at Various Pressures, Physica Status Solidi (b), Vol. 162, No. 2, 1990, pp. 371-377, https://doi.org/10.1002/pssb.2221620206.
[36] N. T. Hoa, N. Q. Hoc, H. X. Dat, Study on the Thermodynamic Properties of Thin Film of FCC Interstitial Alloy AuSi at Zero Pressure Using the Statistical Moment Method, Physics, Vol. 5, 2023, pp. 59-68, https://doi.org/10.3390/physics5010005.
[37] M. N. Magomedov, Calculation of Debye Temperature and Gruneisen Parameter, Journal of Physical Chemistry, Vol. 61, No. 4, 1987, pp. 1003-1009 (in Russian).
[38] L. V. Tikhonov, G. Y. Koponenko, Mechanical Properties of Metals and Alloys, Nauka-Dumka, Kiev, 1986
(in Russian).
[39] N. K. Xuong, Colour Metal Material, Science and Technique Publishing House, Hanoi, 2003 (in Vietnamese).
[40] M. J. Mehl, Pressure Dependence of the Elastic Moduli in Aluminum-rich Al-Li Compounds, Physical Review.B, Vol. 47, No. 5, 1993, pp. 2493, https://doi.org/10.1103/PhysRevB.47.2493.
[41] R. L. David, CRC Handbook of Chemistry and Physics, 86th Ed., Taylor & Francis, Boca Raton, London-New York-Singapore, 2005.
[42] L. Burakovsky, C. W. Greeff, D. L. Preston, Analytic Model of the Shear Modulus at All Temperatures and Densities, Physical Review B, Vol. 67, No. 9, 2003, Article No. 094107,
https://doi.org/10.1103/PhysRevB.67.094107.
[43] M. J. Mehl, D. A. Papacnstantopoulos, Applications of a Tight-binding Total-energy Method for Transtion and Noble Metals: Elastic Constants, Vaccancies, and Surfaces of Monatomic Metals, Physical Review B, Vol. 54,
No. 7, 1996, pp. 4519, https://doi.org/10.1103/PhysRevB.54.4519.
[44] H. M. Ledbetter, E. R. Naimon, Relationship between Single-crystal and Polycrystal Elastic Constants, Journal of Applied Physics, Vol. 45, 1974, pp.66-69, https://doi.org/10.1063/1.1663019.
[45] S. Santra, H. Dong, T. Laurila, A. Paul, Role of Different Factors Affecting Interdiffusion in Cu(Ga) and Cu(Si) Solid Solutions, The Royal Society A, Vol. 470, 2014, Article No. 20130464,
https:// doi.org/10.1098/rspa.2013.0464.
[46] C. S. Smith, J. W. Burns, The Elastic Constants of Cu-4 Percent Si, Journal of Applied Physics, Vol. 24, No. 1, 1953, pp.15-18, https://doi.org/10.1063/1.1721124.