On the quantum statistics of bound states within the Rutherford model of matter

TL;DR

This paper analyzes the quantum statistical behavior of bound states in the Rutherford model of matter, explaining their occurrence and disappearance in different temperature-density conditions using virial expansions and diagrammatic methods.

Contribution

It provides a modern quantum statistical framework for Coulomb systems, explaining bound state phenomena and transitions with new analytical results and diagrammatic calculations.

Findings

Bound states occur only in specific temperature-density regions.

Disappearance of bound states at high densities due to medium effects.

Pressure isotherms show crossover from full to partial ionization.

Abstract

The quantum statistical treatment of the Rutherford model, considering matter as a system of point charges (electrons and nuclei) is analyzed. First, in the historical context, the solutions of different fundamental problems, such as the divergence of the partition function, elaborated by Herzfeld, Planck, Brillouin and Rompe - most of the relevant papers published in the Annalen der Physik, are discussed. Beyond this, the modern state of art is presented and new results are given which explain, why bound states according to a discrete part of the spectra occur only in a valley in the temperature-density plane. Based on the actual state of the quantum statistics of Coulomb systems, virial expansions within the canonical ensemble and the grand ensemble and combinations are derived. The following transitions along isotherms are studied: (i) the formation of bound states occurring by increasing the density from low to moderate values, (ii) the disappearance of bound state effects at higher densities due to medium effects. Within the physical picture we calculate isotherms of pressure for Hydrogen in a broad density region and show that in the region between $20\, 000$ K and $100\, 000$ K and particle densities below $10^{22}$ cm$^{-3}$ the cross-over from full to partial ionization may be well described by the contributions of extended ring diagrams and ladder diagrams.