Atoms at finite temperatures

These are working notes written as a first step in understanding atomic potentials and atomic structure in finite-temperature systems. 1 Thomas-Fermi Theory As a first step, we consider the Thomas-Fermi theory at finite temperatures. This is my version of the discussion of temperature-dependent Thomas-Fermi theory given in the seminal work of Feynman, Metropolis, and Teller [1]. Later, we will make detailed comparisons with results from that paper. In the discussion to follow, we imagine that a material is divided into neutral cells, each containing a single nucleus (charge Z) and Z electrons. We isolate an individual neutral cell and treat it in a thermodynamic average sense, ignoring the interaction between neighboring cells. These cells are later considered to be spherical; the radius of each cell is taken to be the Wigner-Seitz radius determined from the material density ρm (g/cm) and the atomic weight A (g/mol) by: Ω = A Aρm R = ( 3Ω 4π )1/3 , where A = 6.022×1023 is Avagadro’s number and Ω is the atomic volume. 1.1 Elementary formulas We consider an atom with N electrons moving in a potential V (r). We suppose that locally the electrons are moving in a box of side L and depth V (r). The number of states in momentum interval dp is dN = 2 L (2π)3 dp ,