The field theory of specific heat

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.