Correlativity and the origin of the T 2 difference between the Bloch–Grüneisen law and the Debye law

Abstract

To find the origin of the T 2 difference between the Bloch–Grüneisen law and the Debye law, a resistivity statistical model for ideal metals is presented. In the model, the system is regarded as a phonon system in which all phonons have the same momentum value, i.e., the mean momentum. The principal point of the simplified model is that the electrons located at the Fermi surface are scattered by these mean-momentum phonons. It is found that the electrical resistivity of an ideal metal is directly proportional to the phonon concentration and the square of the phonon mean momentum, which first related the electrical resistivity to the phonon parameters. The theoretical results from the model are consistent with experimental observations that the electrical resistivity is directly proportional to temperature T at high temperatures, and to T 5 at very low temperatures, naturally, this is consistent with the Bloch–Grüneisen law. It is found by theoretical analyses that the heat capacity of a solid at very low temperatures is only proportional to the phonon concentration. Therefore, the contribution of the square of phonon mean momentum to the electrical resistivity brings about the T 2 difference between the Bloch–Grüneisen law and the Debye law. PACS Nos.: 72.10.Di, 65.40.Ba