Quantum Mechanical Calculation of the Heat of Solution and Residual Resistance of Gold in Silver

An attempt is made to calculate the heat of solution of gold in silver on the basis of the quantum theory of metals. These two metals are chosen because they have the same atomic volume, and therefore are the simplest case. The steps in the argument are as follows: Suppose that a gold atom replaces a silver atom in the lattice. Then, to a certain approximation, one can represent the substitution of the silver ion by a gold ion, in its effect on the electrons, by a "potential hole" of depth Δ E and radius r 0 . This potential hole will alter the energy of the conduction electrons. To a first approximation the change in energy is just Δ E , which would give zero heat of mixing. A second-order term of order (Δ E ) 2 / E F always gives a positive heat of mixing; E F is here the Fermi energy. This term is calculated exactly by wave-mechanical methods; it gives 0.69 eV. per atom. The same calculation shows, however, that there is a concentration of charge in the gold atom in excess of that in the surrounding silver atoms; this alters the potential in which the electrons move, so that a self-consistent calculation is required to obtain the true energy. For this the labour required would be almost prohibitive; therefore we use instead the Thomas-Fermi method and obtain 0.45 eV. We thus find 0.15 eV per atom for the heat of solution, which compares well with the observed value 0.13 eV. With the help of the potential obtained with the Thomas-Fermi method the residual resistance of gold in silver is found to be 0.16 micro ohm cm. for 1% solution. The considerable discrepancy as compared with the experimental value 0.38 seems closely connected with very similar discrepancies found in other theoretical work on temperature resistance of the noble metals.