The driving mechanism behind the metallic bond formation is the liberation of valence electrons. The screening effect of the Coulomb potential of the positively charged atomic nuclei originated by the liberated, free moving valence electrons helps reducing the repulsive interaction between them. Required:
a. Calculate the kinetic energy of a bound valence electron assuming its wavelength is 10 times the Bohr radius.
b. Now consider the specific case of gold with an fcc crystal structure and a lattice parameter of 4.07 A, a free carrier concentration of 5.9x10^22 cm^-3, and Thomas Fermi screening length rTF of 0.58 Ả. Plot the Coulomb potential Pc(r) and the screened Coulomb potential ps(r) for a range of the separation distances r-1.0 to 4.0 Å using an increment of 0.05 Å. Note that Au ions in the metal are carrying one elemental charge.
c. Calculate the difference of the Coulomb repulsion energy for the unscreened and screened case for the nearest neighbor distance and the second nearest neighbor distance in Au. Compare the energy difference to the energy gained by liberating the valence electrons. Make an argument how critical the screening effect is to stabilize the crystal.