Size Effect in Low Temperature Heat Capacities

It has been shown by Bolt and Maa that the distribution of normal modes of vibration of an elastic continuum is given by Aν2+B(S/V)ν+0(1) where A and B are numerical constants and S/V is the ratio of the surface area to the volume of the continuum. When this result is applied to the theory of the vibrational heat capacity of a solid at low temperatures, the Aν2-term yields the well-known T3 law and the B(S/V)ν-term leads to a contribution proportional to T2. About 0.5 to 10 percent of the heat capacity of non-metallic powders and materials with a domain structure can be attributed to the T2 term at 1°K. The importance of the surface term increases with decreasing temperature. It is well known that at low temperatures the electronic contribution to the heat capacity of a metal is proportional to the temperature. In a system of very small metallic particles, the proportionality constant is increased by a quantity proportional to (S/V⅔)n−⅓ where n is the number of conduction electrons in the metal.