﻿Theory of the Metallic State. 133 



ductivity of crystals at low temperatures *, namely, that the 

 reciprocal of the heat conductivity, the thermic resistance, 

 is approximately proportional to the temperature. Accord- 

 ing to Debye the heat is transported in the form of elastic 

 waves. These are scattered by inhomogeneity in the 

 elastic constants of the space-lattice, caused by variations 

 in density due to heat-motion. If a metal is composed of 

 two interleaved space-lattices, as assumed in this paper, its 

 measured heat conductivity will be the sum of the con- 

 ductivity of the atomic space-lattice and that of the electron 

 space-lattice. At ordinary temperatures the conductivity of 

 the atomic space-lattice may be neglected, as it will be 

 of the same order as that of a crystal. The conductivity of 

 the electron space-lattice will be comparatively very great, 

 for it corresponds to a crystal at a very low temperature. 

 Now the formula for the conduction of heat developed by 

 Debye is only valid for temperatures of the order T>/3v m , 

 v m being the limiting frequency. The electron space-lattice 

 will have a very high limiting frequency according to 



/ 9 \V S N 1/3 

 Debye's formula v m — \ -. — ) ,,» ,.., , on account of its small 

 \dt7r/ p*-i*Kri* 



mass and comparatively small compressibility. For N one 



can put — , p being the proportion of atoms dissociated ; 



p, the density, is Nm= - — , in being the mass of an electron, 



whilst the compressibility k depends upon N and the dielectric 

 constant. Now there is no reason why, if one atom expels 

 an electron, all the others should not do the same ; therefore 

 p is probably one, perhaps two or more. In this case k is 

 of the same order as it is for the solid f, though its exact 

 value depends upon the distance at which the attraction of 

 the ions becomes noticeable and upon the dielectric constant 



D of the material. Therefore v m is of the order a / — v' m , 



M being the mass of the atom and v' m the limiting frequency 



/M 



of the atomic space-lattice ±. As \ / — is between 100 and 



V in 



600, the electron space-lattice at 300° corresponds to the 



* Ann. d. Phys. (4) xxxiv. p. 185 (1911). 



t According to Haber's empirical formula, it should be exactly the 

 same (vide Verli. d. d. Phys. Ges. xiii. 24. p. 1 1 17 (1911)). 



X For the sake of simplicity only the compressibility has been taken 

 into account. In other words, the velocity of a transversal wave is 

 assumed proportional to the velocity of a longitudinal wave. 



