The Fourth Virial Coefficient for Hard Spheres in Even Dimension

The fourth virial coefficient is calculated exactly for a fluid of hard spheres in even dimensions. For this purpose the complete star cluster integral is expressed as the sum of two three-folded integrals only involving spherical angular coordinates. These integrals are solved analytically for any even dimension d, and working with existing expressions for the other terms of the fourth cluster integral, we obtain an expression for the fourth virial coefficient B4(d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{4}(d)$$\end{document} for even d. It reduces to the sum of a finite number of simple terms that increases with d.