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The Boltzmann Equation Of Phonon Thermal Transport Solved In the Relaxation Time Approximation – I – Theory Journal Article Mechanics, Materials Science & Engineering, 3 (1), pp. 35-46, 2016, ISSN: 2412-5954. |

**Author: Amelia Carolina Sparavigna**

**ABSTRACT. **The thermal transport of a dielectric solid can be determined by means of the Boltzmann equation regarding its distribution of phonons subjected to a thermal gradient. After solving the equation, the thermal conductivity is obtained. A largely used approach for the solution is that of considering a relaxation time approximation, where the collisions of phonons are represented by relaxation times. This approximation can be questionable, but its use is able of providing reliable information on thermal conductivity and on the role of impurities and lattice defects in the thermal transport. Here we start a discussion on the thermal conductivity in dielectric solids. The discussion is divided in two parts. In the first part, which is proposed in this paper, we analyse the Boltzmann equation and its solution in the relaxation time approximation. In a second part, which will be the subject of a next paper, we will show comparison of calculated and measured thermal conductivities.

**Keywords: **thermal conductivity, phonons, Boltzmann equation

DOI 10.13140/RG.2.1.1001.1923

**References**

[1] Ziman, J.M. (1962). Electrons and phonons: The theory of transport phenomena in solids. Clarendon Press. Reprint Edition in 2001. ISBN-10: 0198507798, ISBN-13: 978-0198507796

[2] Gurevich, V.L. (1986). Transport in phonon systems. North-Holland. ISBN-10: 044487013X, ISBN-13: 978-0444870131

[3] Srivastava, G.P. (1990). The physics of phonons. Taylor and Francis. ISBN ISBN-10: 0852741537, ISBN-13: 978-0852741535

[4] Omini, M., & Sparavigna, A. (1995). An iterative approach to the phonon Boltzmann equation in the theory of thermal conductivity. Physica B, 212(2), 101-112. DOI: 10.1016/0921-4526(95)00016-3

[5] Omini, M., & Sparavigna, A. (1997). Effect of phonon scattering by isotope impurities on the thermal conductivity of dielectric solids. Physica B, 233(2-3), 230-240. DOI: 10.1016/s0921-4526(97)00296-2

[6] Omini, M., & Sparavigna, A. (1996). Beyond the isotropic-model approximation in the theory of thermal conductivity. Physical Review B, 53(14), 9064-9073. DOI: 10.1103/physrevb.53.9064

[7] Omini, M., & Sparavigna, A. (1997). Heat transport in dielectric solids with diamond structure. Nuovo Cimento D, 19(10), 1537-1564.

[8] Sparavigna, A. (1997). Thermal conductivity of solid neon: an iterative analysis. Physical Review B, 56(13), 7775-7778. DOI: 10.1103/physrevb.56.7775

[9] Sparavigna, A.C. (2015). Thermal conductivity of zincblende crystals. Mechanics, Materials Science & Engineering Journal, October 2015, 1-8. DOI: 10.13140/RG.2.1.1758.0565

[10] Broido, D.A., Ward, A., & Mingo, N. (2005). Lattice thermal conductivity of silicon from empirical interatomic potentials. Physical Review B, 72(1), 014308. DOI: 10.1103/physrevb.72.014308

[11] Kaburaki, H., Li, J., Yip, S., & Kimizuka, H. (2007). Dynamical thermal conductivity of argon crystal. Journal of Applied Physics, 102(4), 043514. DOI: 10.1063/1.2772547

[12] Turney, J.E., Landry, E.S., McGaughey, A.J.H., & Amon, C.H. (2009). Predicting phonon properties and thermal conductivity from anharmonic lattice dynamics calculations and molecular dynamics simulations. Physical Review B, 79(6), 064301. DOI: 10.1103/physrevb.79.064301

[13] McGaughey, A.J., & Kaviany, M. (2004). Quantitative validation of the Boltzmann transport equation phonon thermal conductivity model under the single-mode relaxation time approximation. Physical Review B, 69(9), 094303. DOI: 10.1103/physrevb.69.094303

[14] Li, W., Carrete, J., Katcho, N.A., & Mingo, N. (2014). ShengBTE: A solver of the Boltzmann transport equation for phonons. Computer Physics Communications, 185(6), 1747-1758. DOI: 10.1016/j.cpc.2014.02.015

[15] Ashcroft, N.W., & Mermin, N.D. (1976). Solid state physics. Saunders College Publishing. ISBN-10: 0030839939, ISBN-13: 978-0030839931

[16] Callaway, J. (1961). Low-temperature lattice thermal conductivity. Physical Review, 122(3), 787-790. DOI: 10.1103/physrev.122.787

[17] Callaway, J., & von Baeyer, H.C. (1960). Effect of point imperfections on lattice thermal conductivity. Physical Review, 120(4), 1149-1154. DOI: 10.1103/physrev.120.1149

[18] Callaway, J. (1959). Model for lattice thermal conductivity at low temperatures. Physical Review, 113(4), 1046-1051. DOI: 10.1103/physrev.113.1046

[19] Geballe, T.H., & Hull, G.W. (1958). Isotopic and other types of thermal resistance in Germanium. Physical Review, 110, 773-775. DOI: 10.1103/physrev.110.773

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