Abstract
Finite temperature quantum field theory in the heat kernel method is used to study the heat capacity of condensed matter. The lattice heat is treated à la P. Debye as energy of the elastic (sound) waves. The dimensionless functional of free energy is re-derived with a cut-off parameter and used to obtain the specific heat of crystal lattices. The new dimensionless thermodynamical variable is formed as Planck’s inverse temperature divided by the lattice constant. The dimensionless constant, universal for the class of crystal lattices, which determines the low temperature region of molar specific heat, is introduced and tested with the data for diamond lattice crystals. The low temperature asymptotics of specific heat is found to be the fourth power in temperature instead of the cubic power law of the Debye theory. Experimental data for the carbon group elements (silicon, germanium) and other materials decisively confirm the quartic law. The true low temperature regime of specific heat is defined by the surface heat, therefore, it depends on the geometrical characteristics of the body, while the absolute zero temperature limit is geometrically forbidden. The limit on the growth of specific heat at temperatures close to critical points, known as the Dulong–Petit law, appears from the lattice constant cut-off. Its value depends on the lattice type and it is the same for materials with the same crystal lattice. The Dulong–Petit values of compounds are equal to those of elements with the same crystal lattice type, if one mole of solid state matter were taken as the Avogadro number of the composing atoms. Thus, the Neumann–Kopp law is valid only in some special cases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Kittel, Introduction to Solid State Physics, 7th ed. (New York: John Wiley & Sons, Inc., 1996).
L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics. Part 1 (Oxford: Pergamon, 1980).
A. Einstein, “Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme” [Planck’s Theory of Radiation and the Theory of Specific Heat], Ann. Phys. 22, 180–190 (1907).
P. Drude, “Zur Elektronentheorie der Metalle” [On the Electron Theory of Metals], Ann. Phys. 306 (3), 566–613 (1900).
L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. 7: The Theory of Elasticity (Pergamon: Oxford, UK, 1986).
D. Royer and E. Dieulesaint, Elastic Waves in Solids, Vol. 1: Free and Guided Propagation (Springer: Berlin, 2000).
A.E. Love, A Treatise on the Mathematical Theory of Elasticity (Dover: Mineola, NY, 1944). Russian transl. of the 4th ed. (ONTI: Moscow, 1935).
L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Pergamon Oxford, 1987).
P. Debye, “Zur Theorie der spezifischen Wärmen” [On the Theory of Specific Heats], Ann. Phys. 39, 789–839 (1912).
J.C. Maxwell, “On the Dynamical Theory of Gases,” Philos. Trans. R. Soc. Lond. Ser. A 157, 49–88 (1867).
B.S. DeWitt, Global Approach to Quantum Field Theory, Vol. 1 and 2 (Oxford University Press: Oxford, 2003).
G.A. Vilkovisky, “Expectation Values and Vacuum Currents of Quantum Fields,” Lecture Notes in Phys. 737, 729–784 (2008).
Yu.V. Gusev, “Finite Temperature Quantum Field Theory in the Heat Kernel Method,” Russ. J. Math. Phys. 22 (1), 9–19 (2015).
A.O. Barvinsky and G.A. Vilkovisky, “The Generalized Schwinger-Dewitt Technique in Gauge Theories and Quantum Gravity,” Phys. Rep. 119, 1–74 (1985).
H.J. McSkimin and P. Andreatch Jr., “Elastic Moduli of Silicon vs Hydrostatic Pressure at 25.0 C and -195.8 C,” J. Appl. Phys. 35 (7), 2161–2165 (1964).
H.J. McSkimin and P. Andreatch,, “Elastic Moduli of Diamond As a Function of Pressure and Temperature,” J. Appl. Phys. 43 (8), 2944–2948 (1972).
H.J. McSkimin and P. Andreatch., “Elastic Moduli of Germanium vs Hydrostatic Pressure at 25.0 C and -195.8 C,” J. Appl. Phys. 34 (3), 651–655 (1963).
H.J. McSkimin and P. Andreatch, Jr., “Third-Order Elastic Moduli of Gallium Arsenide,” J. Appl. Phys. 38, 2610–2611 (1967).
V.S. Vladimirov, Equations of Mathematical Physics (New York, NY: Marcel Dekker Inc., 1971) Russian 4th ed. (Nauka: Moscow, 1981).
H.-Y. Hao and H. J. Maris, “Dispersion of the Long-Wavelength Phonons in Ge, Si, GaAs, Quartz, and Sapphire,” Phys. Rev. B 63, 224301 (2001).
J. de Launay, “The Theory of Specific Heats and Lattice Vibrations, in F. Seitz and D. Turnbull (eds.),” Solid State Physics. Advances in Research and Applications, Vol. 2 (Academic Press: New York, 1956), pp. 219–303.
N.G. Szwacki and T. Szwacka, Basic Elements of Crystallography (Pan Stanford Publishing Pte. Ltd.: Singapore, 2010).
A.T. Petit and P.L. Dulong, “Recherches sur quelques points importants de la Théorie de la Chaleur,” Annales de Chimie et de Physique 10, 395–413 (1819).
M. Laing and M. Laing, “Dulong and Petit Law: We Should Not Ignore Its Importance,” J. Chem. Educ. 83, 1499–1504 (2006).
R.P. Stoffel, C. Wessel, M.-W. Lumey, and R. Dronskowski, “Ab Initio Thermochemistry of Solid-State Materials,” Angewandte Chemie Int. Ed. 49 (31), 242–5266 (2010).
P.D. Desai, “Thermodynamic Properties of Iron and Silicon,” J. Phys. Chem. Ref. Data 15 (3), 967–983 (1986).
V.P. Maslov, “The Mathematical Theory of Classical Thermodynamics,” Math. Notes 93 (1), 102–136 (2013).
Yu.V. Gusev, “On the Integral Law of Thermal Radiation,” Russ. J. Math. Phys. 21 (4), 460–471 (2014).
J.E. Desnoyers and J.A. Morrison, “The Heat Capacity of Diamond between 12.8 and 278 K,” Philosophical Magazine 3, 42–48 (1958).
P. Flubacher, A.J. Leadbetter, and J.A. Morrison, “The Heat Capacity of Pure Silicon and Germanium and Properties of Their Vibrational Frequency Spectra,” Phil. Magazine 4 (39), 273–294 (1959).
W.T. Berg and J.A. Morrison, “The Thermal Properties of Alkali Halide Crystals. I. The Heat Capacity of Potassium Chloride, Potassium Bromide, Potassium Iodide, and Sodium Iodide between 2.8 and 270 Degrees K,” Proc. R. Soc. Lond. A 242 (1231), 467–477 (1957).
M.E. Schlesinger, “Thermodynamic Properties of Solid Binary Antimonides,” Chem. Rev. 113, 8066–8092 (2013).
N.N. Greenwood and A. Earnshaw, Chemistry of the Elements, Second ed. (Oxford, U.K.: Butterworth- Heinemann, 1998).
D.L. Price, J.M. Rowe, and R.M. Nicklow, “Lattice Dynamics of Grey Tin and Indium Antimonide,” Phys. Rev. B 3, 1268–1279 (1971).
H.O. Pierson, Handbook of Carbon, Graphite, Diamond and Fullerenes. Properties, Processing and Applications (Noyes Publications: Park Ridge, NJ, 1993).
S. Adachi, Properties of Semiconductors: Group-IV, III-V and II-VI Semiconductors (John Wiley & Sons: Chichester, UK, 2005).
V.M. Glazov and A.S. Pashinkin, “Thermal Expansion and Heat Capacity of GaAs and InAs,” Inorganic Materials 36 (3), 225–231 (2000).
Ioffe Physico-Technical Institute, New Semiconductor Materials. Characteristics and Properties, Section “Physical Properties of Semiconductors”.
A.C. Victor, “Heat Capacity of Diamond at High Temperature,” J. Chem. Phys. 36, 1903–1911 (1962).
I. Barin, Thermochemical Data of Pure Substances, Third ed. (VCH: Weinheim, FRG, 1995).
V.M. Glazov, A.S. Pashinkin, “The Thermophysical Properties” (Heat Capacity and Thermal Expansion of Single-Crystal Silicon,) High Temperature 39 (3), 413–419 (2001).
L.V. Gurvich, I.V. Vetyts, and C.B. Alcock (eds.), Thermodynamic Properties of Inorganic Substances, Vol. 2 (Hemisphere: New York, 1990).
S. Adachi, Physical Properties of III-V Semiconductor Compounds: InP, InAs, GaAs, GaP, InGaAs, and InGaAsP (John Wiley & Sons: New York, 1992).
T.C. Cetas, C.R. Tilford, and C.A. Swenson, “Specific Heats of Cu, GaAs, GaSb, InAs, and InSb from 1 to 30K,” Phys. Rev. 174 (3), 835–844 (1968).
E.S.R. Gopal, Specific Heats at Low Temperatures (Plenum Press: New York, 1966).
P.B. Gilkey, Asymptotic Formulae in Spectral Geometry (Boca Raton: Chapman & Hall/CRC, 2004).
H. Weyl, “Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Körpers” [The Asymptotic Distribution Law of Eigen-Oscillations of an Arbitrarily Shaped Elastic Body], Rend. Circ. Mat. Palermo 39, 1–49 (1915)
V.I. Arnold (ed.), Hermann Weyl. Selected Works. Mathematics and Theoretical Physics (Nauka: Moscow, 1984), pp. 9–57.
Bureau International des Poids et Mesures, The International System of Units, 8th ed. (BIPM: Sèvres, France, 2006).
T. Atake, S. Takai, A. Honda, Y. Saito, and K. Saito, “Low Temperature Heat Capacity and Lattice Dynamics Studies of Synthetic Diamond and Cubic Boron Nitride,” Rep. Res. Laboratory of Engineering Materials, Tokyo Institute of Technology 16, 15–25 (1991).
W.T. Berg, “Heat Capacity of Aluminum between 2.7 and 20 K,” Phys. Rev. 167 (3), 583–586 (1968).
W.T. Berg, “Low-Temperature Heat Capacities of Silver Chloride and Lithium Iodide,” Phys. Rev. B 13, 2641–2645 (1976).
A. Einstein and L. Infeld, The Evolution of Physics. The Growth of Ideas from Early Concepts to Relativity and Quanta (Cambridge University Press: London, 1967), Russian transl. 3rd ed. (Nauka: Moscow, 1965).
M. Planck and M. Masius, The Theory of Heat Radiation (Philadelphia: Blakinston’s, 1914).
M. Born and Th. von Kármán, “Zur Theorie der speziwischen Wärmer” [On the Theory of Specific Heats], Phys. Z. 14, 15–19 (1913).
M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press: Oxford, 1962).
C.V. Raman, “The Specific Heats of Crystals. Part 1. General Theory,” Proc. Indian Acad. Sci. A 44 (4), 153–159 (1956).
C.V. Raman, “The Specific Heats of Crystals. Part 2. The Case of Diamond,” Proc. Indian Acad. Sci. A 44 (4), 160–164 (1956).
Y. Matsuda, H. Kawaji, T. Atake, Y. Yamamura, S. Yasuzuka, K. Saito, and S. Kojima, “Non-Debye Excess Heat Capacity and Boson Peak of Binary Lithium Borate Glasses,” J. Non-Crystalline Solids 357, 534–537 (2011).
A.D. Styrkas, “Preparation of Shaped Gray Tin Crystals,” Inorganic Materials 41 (6), 580–584 (2005).
L. Finegold and N.E. Phillips, “Low-Temperature Heat Capacities of Solid Argon and Krypton,” Phys. Rev. 177 (3), 1383–1391 (1969).
P. Flubacher, A.J. Leadbetter, and J.A. Morrison, “A Low Temperature Adiabatic Calorimeter for Condensed Substances. Thermodynamic Properties of Argon,” Proc. Phys. Soc. 78, 1449–1461 (1961).
R. Pässler, “Exponential Series Representation for Heat Capacities of Semiconductors and Wide- Bandgap Materials,” Phys. Stat. Sol. (b) 245 (6), 1133–1146 (2008).
R. Pässler, “Dispersion-Related Theory for Heat Capacities of Semiconductors,” Phys. Stat. Sol. (b) 244 (12), 4605–4623 (2007).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gusev, Y.V. The field theory of specific heat. Russ. J. Math. Phys. 23, 56–76 (2016). https://doi.org/10.1134/S1061920816010040
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920816010040