Question

11. Saha equation for a generic atom. (a) Work out the analog of equation (11.41) for a multi-electron atom and its first stage of ionization: atom atome Couch your result in terms of two internal partition functions, Zatom(int) and Zion(int), and other, familiar quantities. Describe explicitly the situation that you take for the zero of energy, and describe (relative to that zero) the energies that enter the two partition functions. (b) Display the form to which your equation reduces if you approximate each partition function by its ground-state term.

Answer 1

"The reaction atom leftrightarrow atom^+ + e implies the relationship, mu_atom = mu_ion + mu_e where ""ion"" denotes singly charged ionized atom (1) Now, using, H rightharpoonoverleftharpoon H^+ + e, the law of mass action takes the for [H^+][e]/[H] = k(T) = lambda H^-3 +. Lambda^-3_e/lambda^-3_H z_H + (int.) Z_e (int.)/z_H(int.) (2) where, z_x (int.) is the internal partition function for atom/electron/ion 'x' Now, the masses of atom and ion are so nearly equal that their thermal de-broglie wavelengths also nearly equal and hence, can out in equation (2). i.e. lambda H + approximatelyequalto lambda_H. Also, the internal partition function for the electron is merely the spin degeneracy factor = 2. Because from eq. (2), we get [ion] [e]/[atom] = 1/lambda e^3 z_e (int.) times z_ion (int.)/z_atom (int.) = 2/lambda e^3 z_ion. (int.)/z_atom (int.) For the zero of energy, take the e^- to be at rest at infinity and ion to be at rest at infinity and its "