The importance of the Debye bosons (sound waves) for the lattice dynamics of solids

Year 2020, Volume: 23 Issue: 2, 59 - 79, 28.05.2020

Ulrich Köbler

https://doi.org/10.5541/ijot.649929

Cited By: 6

Abstract

For a
number of materials with cubic lattice structure the dispersion relations of
the Debye bosons (sound waves) and of the acoustic phonons along [ζ 0 0]
direction have been analyzed quantitatively. When all phonon modes are excited,
that is, for temperatures of larger than the Debye temperature the dispersion
of the mass-less Debye bosons exhibits a pronounced non-linearity, which is
explained by interactions with the phonon background. For the exponent x in the
dispersion relation ~q^x of the Debye bosons, the rational values of
x=1/4, 1/3, 1/2, 2/3 and 3/4 could be established firmly. The discrete values
of x show that there are distinct modes of interaction with the phonons only. It
is furthermore shown that for many materials the dispersion of the acoustic
phonons along [ζ 0 0] direction follows a perfect sine function of wave vector,
which is known to be the dispersion of the linear atomic chain. This dispersion
is unlikely to be the intrinsic behavior of three-dimensional solids. It is
argued that the sine-function is induced by the Debye boson-phonon interaction.
Quantitative analyses of the temperature dependence of the heat capacity show
that the heat capacity can be described accurately over a large temperature
range by the expression c_p=c₀-B‧T^-ε. The
constants c₀ and B are material specific and define the absolute
value of the heat capacity. However, for the exponent ε the same rational value
occurs for materials with different chemical compositions and lattice structures.
The temperature dependence of the heat capacity therefore exhibits
universality. This universality must be considered as a non-intrinsic dynamic
property of the atomistic phonon system, arising from the Debye boson-phonon
interaction. The discrete modes of boson-phonon interaction are essential for
the observed universality classes of the heat capacity. Safely identified
values for ε are ε=1, 5/4 and 4/3. The fit values for c₀ are
generally larger than the theoretical Dulong-Petit value. Universal exponents
are identified also in the temperature dependence of the coefficient of the
linear thermal expansion, α(T). Since the universality in α(T) holds for the
same thermal energies (temperatures) as for the ~q^x functions in the
dispersion of the Debye bosons it can be concluded that the Debye bosons also determine
the temperature dependence of α(T). Our results show that the dynamics of the
atomic lattice is modified for all temperatures by the Debye bosons. Atomistic
models restricting on inter-atomic interactions therefore are neither
sufficient to explain the phonon dispersion relations nor the detailed
temperature dependence of the heat capacity.

Keywords

Lattice dynamics, boson fields, universality

Supporting Institution

none

Project Number

none

References

• [1] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford, 1956.
• [2] P. Debye, Zur Theorie der spezifischen Wärme, Ann. Physik, 39, 789-839, 1912.
• [3] K.G. Wilson, J. Kogut, The renormalization group and the ε expansion, Physics Reports, 12C, 75-200, 1974.
• [4] G.A. Alers, Use of Sound Velocity Measurements in Determining the Debye Temperature of Solids, in: Physical Acoustics, ed. by W.P. Mason, vol III B, Academic Press, New-York,1-42, 1965.
• [5] W.S. Corak, M.P. Garfunkel, C.B. Satterthwaite, A. Wexler, Atomic Heats of Copper, Silver and Gold from 1 0K to 5 0K, Phys. Rev. 98, 1699-1708, 1955.
• [6] U. Köbler, A. Hoser, Experimental Studies of Boson Fields in Solids, World Scientific, Singapore, 2018.
• [7] U. Köbler, On the Thermal Conductivity of Metals and of Insulators, JIoT, 20, 210-218, 2017.
• [8] Y.S. Touloukian, E.H. Buyco, Thermophysical Properties of Matter, vol 5: Specific Heat of Nonmetallic Solids, IFI/Plenum, New-York, 1970.
• [9] T.H.K. Barron, W.T. Berg, J.A. Morrison, On the heat capacity of crystalline magnesium oxide, Proc. Roy. Soc. (London) A250, 70-83,1959.
• [10] J.C. Ho, D.P. Dandekar, Low-temperature heat capacities of RbCl, RbBr and CsCl, Phys. Rev. B 30, 2117-2119, 1984.
• [11] W.T. Berg, J.A. Morrison, The heat capacity of potassium chloride, potassium bromide potassium iodide and sodium iodide, Proc. Roy. Soc. (London) A242, 467-477, 1957.
• [12] R.F.S. Hearmon, The elastic constants of crystals and other anisotropic materials, in: Landolt-Börnstein, ed. by K.-H. Hellwege, vol. III/11, 1-286, Springer, Berlin 1979.
• [13] U. Köbler, On the Distinction between Debye Bosons and Acoustic Phonons, IJoT, 18, 277-284, 2015.
• [14] Y.S. Touloukian, E.H. Buyko, Thermophysical Properties of Matter, vol. 4: Specific Heat of Metallic Elements and Alloys, IFI/Plenum, New-York, 1970.
• [15] I. Barin, Thermochemical Data of Pure Substances, 3th edition, VCH, Weinheim, 1995.
• [16] U. Köbler, V.Yu. Bodryakov, On the Melting Process of Solids, IJoT, 18, 200-204, 2015.
• [17] M.E. Fisher, A.N. Berker, Scaling for first-order phase transitions in thermodynamic and finite systems, Phys. Rev. B 26, 2507-2513, 1982.
• [18] P. Heller, Experimenal investigations of critical phenomena, Rep. Prog. Physics 30, 731-826, 1967.
• [19] R.K. Pathria, Statistical Mechanics, 2th edition, Butterworth-Heinemann, Oxford 1996.
• [20] Y. Fujii, N.A. Lurie, R. Pynn, G. Shirane, Ineaastic neutron scattering from solid 36Ar, Phys. Rev. B 10, 3647-3659, 1974.
• [21] Y. Endoh, G. Shirane, J. Skalyo, Jr., Lattice dynamics of solid neon at 6.5 and 23.7 K, Phys. Rev. B 11 (1975) 1681-1688, 1975.
• [22] J. Skalyo, Jr., Y. Endoh, G. Shirane, Inelastic neutron scattering from solid krypton at 10 oK, Phys. Rev. B 9 1797-1803, 1974.
• [23] N.A. Lurie, G. Shirane, J. Skalyo, Jr., Phonon dispersion relation in xenon at 10 K, Phys. Rev. B 9, 5300-5306, 1974.
• [24] C. Stassis, G. Kline, W. Kamitakahara, S.K. Sinha, Lattice dynamics of fcc 4He, Phys. Rev. B 17 1130-1135, 1978.
• [25] H. Bilz, W. Kress: Phonon Dispersion Relations in Insulators, Springer, Berlin, 1979.
• [26] R.W.G. Wyckoff, Crystal Structures, vol.1, R.E. Krieger Publ. Comp. Malabar, Florida 1982.
• [27] A. Hoser, U. Köbler, Functional crossover in the dispersion relations of magnons and phonons, J. Phys: Conf. Series 746, 012062-8, 2016.
• [28] A. Hoser, U. Köbler, Boson Fields in Ordered Magnets, Acta Phys. Pol. A 127, 350-352, 2015.
• [29] E.C. Svensson, B.N. Brockhouse, J.M. Rowe, Crystal Dynamics of Copper, Phys. Rev. 155, 619-632, 1967.
• [30] G. Dolling, H.G. Smith, R.M. Nicklow, P.R. Vijayaraghavan, M.K. Wilkinson, Lattice Dynamics of Lithium Fluoride, Phys. Rev. 168, 970-979, 1968.
• [31] L. Pintschovius, W. Reichardt, B. Scheerer, Lattice dynamics of TiC, J. Phys. C: Solid State Phys. 11, 1557-1562, 1978.
• [32] A.A.Z. Ahmad, H.G. Smith, N. Wakabayashi, M.K. Wilkinson, Lattice Dynamics of Cesium Chloride, Phys. Rev. B 6, 3956-3961, 1972.
• [33] M. Steiner, B. Dorner, J. Villain, Inelastic neutron investigation of the anisotropy of the spin wave linewidth in the one-dimensional easy-plane ferromagnet CsNiF3, J. Phys. C: Solid State Phys. 8, 165-175, 1975.
• [34] A. Larose, B.N. Brockhouse, Lattice vibrations in tungsten at 22 oC studied by neutron scattering, Can. J. Phys. 54 1819-1823, 1976.
• [35] N. Vagelatos, D. Wehe, J.S. King, Phonon dispersion and phonon density of states for ZnS an ZnTe, J. Chem. Phys. 60, 3613-3618, 1974.
• [36] M.M. Elcombe, The lattice dynamics of strontium fluoride, J. Phys. C: Solid State Phys. 5, 2702-2710, 1972.
• [37] Y.L. Yarnell, J.L. Warren, S.H. Koenig, Experimental Dispersion Curves for Phonons in Aluminum, in: Lattice Dynamics, Proc. Int. Conf. Copenhagen, ed. by R.F. Wallis, 57-61, Pergamon Press, 1963.
• [38] G. Raunio, S. Rolandson, Lattice Dynamics of NaCl, KCl, RbCl and RbF, Phys. Rev. B 2, 2098-2103, 1970.
• [39] Y.S. Touloukian, R.K. Kirby, R.E. Taylor, P.D. Desai, Thermophysical Properties of Matter, vol.12, Thermal Expansion of Metallic Elements and Alloys, IFI/Plenum, New York-Washington, 1975.
• [40] Y.S. Touloukian, R.K. Kirby, R.E. Taylor, T.Y.R. Lee, Thermophysical Properties of Matter, vol.13, Thermal Expansion of Nonmetallic Solids, IFI/Plenum, New York-Washington, 1977.