A proposed generalization of Emden's differential equation to embrace the ground-state electronic structure of extremely heavy positive atomic ions with and without applied magnetic fields

The differential equation for the dimensionless screening function φ(x) of a point nucleus in an extremely heavy positive atomic ion is shown to have the form φ'' = {φ(1 + γ φ/x)}n/2 / xm/2. This reduces to Emden's equation for γ = 0 and m = n - 2. The above differential equation embraces four physical regimes: non-relativistic atomic ions with (i) magnetic field strength B = 0, (ii) B very large, cases (iii) and (iv) being the relativistic analogues of (i) and (ii). The pairs of n and m are restricted in regimes (i) - (iv) delineated above to [3, 1] and [1, -1]. In the proposed differential equation above, γ is a dimensionless quantity. In zero magnetic field corresponding to [n, m] = [3, 1], γ measures the ratio of a characteristic Coulomb energy Ze2/b, with Z the atomic number of the ion and with b a scaling length ∝ Z-1/3, to the electron rest mass energy. γ involves also the magnetic field strength B in the intense magnetic field limit corresponding to [n, m] = [1, -1]. (author)