The paradox of the partition function

Abstract

The electronic partition function for the hydrogen atom will be analyzed, where we will discuss the old problem of the paradox in statistical mechanics: the apparent divergence of the partition function, first analyzed by Herzfeld in 1913. This problem has not been discussed in most textbooks statistical mechanics, since it has little relevance in the low temperature regime (i. e., k_BT ≪ ε_o ≃ 13,6 eV, where ε_o is the ionization energy). In the high temperature regime (of the order of 5000 K), this contribution is important in the studies of plasma physics, spectroscopy and astrophysics. We will present some methods of truncating the partition function in order to leave the finite series, analyzing the behavior of the specific heat. We observe that due to the series truncation, the system will have finite levels and will be reflected in the Schottky type behavior in the specific heat.

Keywords
partition function divergence; specific heat; hydrogen atom.