Analysis of the anharmonic thermal expansion coefficient of crystalline silver



  • Tong Sy Tien (Corresponding Author) Faculty of Fundamental Sciences, University of Fire Prevention & Fighting
  • Nguyen Thi Minh Thuy Faculty of Fundamental Sciences, University of Fire Prevention & Fighting
  • Nguyen Thi Viet Chinh Institute of Science and Technology, TNU-University of Sciences
  • Nguyen Cong Toan Department of Physics, VNU University of Science
  • Nguyen Bao Trung Department of Physics, VNU University of Science
  • Nguyen Van Nghia Faculty of Energy, Thuyloi University



Anharmonic thermal expansion coefficient; Crystalline silver; Anharmonic correlated Debye model.


The anharmonic thermal expansion (TE) coefficient of crystalline silver (Ag) has been calculated and analyzed in the temperature-dependent. Based on the anharmonic effective potential, the calculation model is developed using the correlated Debye model and the many-body perturbation approach. Thermodynamic parameters of the crystal lattice are derived from the influence of thermal vibrations of all atoms. The anharmonicity results from phonon-phonon interactions, with each thermal vibration can be quantized and treated as a phonon. The obtained expression of the anharmonic TE coefficient of Ag can satisfy all their temperature-dependent fundamental properties. The numerical results of Ag agree well with those obtained from the other theoretical models and experimental data at various temperatures in the range from 0 K to 1000 K. The obtained results indicate the effectiveness of the present model in investigating the TE coefficient of Ag.


[1]. H. Liu, W. Sun, Z. Zhang, L. Lovings, and C. Lind, “Thermal Expansion Behavior in the A2M3O12 Family of Materials,” Solids, vol. 2, no. 1, pp. 87-107, (2021). DOI:

[2]. J. W. Hwang, “Thermal expansion of nickel and iron, and the influence of nitrogen on the lattice parameter of iron at the Curie temperature,” Masters Theses, Missouri: University of Missouri-Rolla, (1972).

[3]. V. A. Drebushchak, “Thermal Expansion of Solids: Review on Theories,” Journal of Thermal Analysis and Calorimetry, vol. 142, pp. 1097-1113, (2020). DOI:

[4]. G. Laplanche, P. Gadaud, O. Horst, F. Otto, G. Eggeler, and E. P. George, “Temperature Dependencies of the Elastic Moduli and Thermal Expansion Coefficient of an Equiatomic, Single-Phase CoCrFeMnNi High-Entropy Alloy,” Journal of Alloys and Compounds, vol. 623, pp. 348-353, (2015). DOI:

[5]. P. Eisenberger and G. S. Brown, “The study of disordered systems by EXAFS: Limitations,” Solid State Communications, vol. 29, no. 6, pp. 481-484, (1979). DOI:

[6]. Z. K. Liu, Y. Wang, and S. Shang, “Thermal Expansion Anomaly Regulated by Entropy,” Scientific Reports, vol. 4, p. 7043, (2014). DOI:

[7]. J. Emsley, “Nature’s Building Blocks: An A-Z Guide to the Elements,” 2nd edition, New York: Oxford University Press, (2011).

[8]. C. R. Hammond, “The Elements, in Handbook of Chemistry and Physics,” 81st edition, Boca Raton: CRC Press, (2004).

[9]. J. Y. Maillard and P. Hartemann, “Silver as an antimicrobial: Facts and gaps in knowledge,” Critical Reviews in Microbiology, vol. 39, no. 4, pp. 373-83, (2012). DOI:

[10]. N. V. Hung, C. S. Thang, N. B. Duc, D. Q. Vuong, and T. S. Tien, “Temperature Dependence of Theoretical and Experimental Debye-Waller Factors, Thermal Expansion and XAFS of Metallic Zinc,” Physica B: Condensed Matter, vol. 521, pp. 198-203, (2017). DOI:

[11]. N. V. Hung, C. S. Thang, N. B. Duc, D. Q. Vuong, and T. S. Tien, “Advances in theoretical and experimental XAFS studies of thermodynamic properties, anharmonic effects and structural determination of fcc crystals,” European Physical Journal B, vol. 90, p. 256, (2017). DOI:

[12]. T. S. Tien, “Advances in Studies of the Temperature Dependence of the EXAFS Amplitude and Phase of FCC Crystals,” Journal of Physics D: Applied Physics, vol. 53, no. 11, p. 315303, (2020). DOI:

[13]. T. S. Tien, “Investigation of the anharmonic EXAFS oscillation of distorted HCP crystals based on extending quantum anharmonic correlated Einstein model,” Japanese Journal of Applied Physics, vol. 60, no. 11, p. 112001, (2021). DOI:

[14]. N. V. Hung, T. S. Tien, N. B. Duc, and D. Q. Vuong, “High-order expanded XAFS Debye-Waller factors of HCP crystals based on classical anharmonic correlated Einstein model,” Modern Physics Letters B, vol. 28. no. 21, p. 1450174, (2014). DOI:

[15]. T. S. Tien, “Analysis of EXAFS oscillation of FCC crystals using classical anharmonic correlated Einstein model,” Radiation Physics and Chemistry, vol. 186, p. 109504, (2021). DOI:

[16]. T. S. Tien, N. T. M. Thuy, V. T. K. Lien, N. T. N. Anh, D. N. Bich, and L. Q. Thanh, “Calculation of Temperature-Dependent Thermal Expansion Coefficient of Metal Crystals Based on Using Anharmonic Correlated Debye Model,” Advances in Technology Innovation, vol. 8, no. 1, pp. 73-80, (2023). DOI:

[17]. T. S. Tien, “Analysis of EXAFS oscillation of monocrystalline diamond-semiconductors using anharmonic correlated Debye model,” European Physical Journal Plus, vol. 136, p. 539, (2021). DOI:

[18]. Y. S. Touloukian, R. K. Kirby, R. E. Taylor, and P. D. Desai, “Thermophysical Properties of Matter,” New York: Plenum, vol. 12, p. 298, (1975).

[19]. N. V. Hung and J. J. Rehr, “Anharmonic correlated Einstein-model Debye-Waller factors,” Physical Review B, vol. 56, no. 1, pp. 43-46, (1997). DOI:

[20]. A. I. Frenkel and J. J. Rehr, “Thermal expansion and x-ray-absorption fine-structure cumulants,” Physical Review B, vol. 48, no. 1, pp. 585-588, (1993). DOI:

[21]. S. H. Simon, “The Oxford Solid State Basics,” 1st edition, Oxford: Oxford University Press, (2013).

[22]. L. A. Girifalco and V. G. Weizer, “Application of the Morse Potential Function to Cubic Metals,” Physical Review, vol. 114, no. 3, pp. 687-690, (1959). DOI:

[23]. N. V. Hung, N. B. Trung, and B. Kirchner, “Anharmonic correlated Debye model Debye-Waller factors,” Physica B: Condensed Matter, vol. 405, pp. 2519-2525, (2010). DOI:

[24]. G. Beni and P. M. Platzman, “Temperature and polarization dependence of extended x-ray absorption fine-structure spectra,” Physical Review B, vol. 14, no. 4. pp. 1514-1518, (1976). DOI:

[25]. G. D. Mahan, “Many-Particle Physics,” 2nd edition, New York: Plenum, (1990). DOI:

[26]. C. Y. Ho, R. E. Taylor, “Thermal Expansion of Solids,” Materials Park: ASM International, (1998).

[27]. L. Tröger, T. Yokoyama, D. Arvanitis, T. Lederer, M. Tischer, and K. Baberschke, “Determination of bond lengths, atomic mean-square relative displacements, and local thermal expansion by means of soft-x-ray photoabsorption,” Physical Review B, vo. 49, no. 2, pp. 888-903, (1994). DOI:

[28]. N. W. Ashcroft and N. D. Mermin, “Solid State Physics,” 1st edition, New York: Holt-Rinehart & Winston, (1976).

[29]. I. V. Pirog, T. I. Nedoseikina, I. A. Zarubin, A. T. Shuvaev, “Anharmonic pair potential study in face-centered-cubic structure metals,” Journal of Physics: Condensed Matter, vol. 14, pp. 1825-1832, (2002). DOI:

[30]. J. Haug, A. Chassé, R. Schneider, H. Kruth, and M. Dubiel, “Thermal expansion and interatomic potentials of silver revealed by extended x-ray absorption fine structure spectroscopy using high-order perturbation theory,” Physical Review B, vol. 77, p. 184115, (2008). DOI:




How to Cite

Tong Sy, T., T. Nguyen Thi Minh, C. Nguyen Thi Viet, T. Nguyen Cong, T. Nguyen Bao, and N. . Nguyen Van. “Analysis of the Anharmonic Thermal Expansion Coefficient of Crystalline Silver”. Journal of Military Science and Technology, vol. 89, no. 89, Aug. 2023, pp. 103-9, doi:10.54939/1859-1043.j.mst.89.2023.103-109.



Research Articles