CALC The one-dimensional calculation of Example 42.4

Chapter 42, Problem 42.54

(choose chapter or problem)

CALC The one-dimensional calculation of Example 42.4 (Section 42.3) can be extended to three dimensions. For the threedimensional fcc NaCl lattice, the result for the potential energy of a pair of and ions due to the electrostatic interaction with all of the ions in the crystal is where is the Madelung constant. Another contribution to the potential energy is a repulsive interaction at small ionic separation r due to overlap of the electron clouds. This contribution can be represented by where A is a positive constant, so the expression for the total potential energy is (a) Let be the value of the ionic separation r for which is a minimum. Use this definition to find an equation that relates and A, and use this to write in terms of For NaCl, Obtain a numerical value (in electron volts) of for NaCl. (b) The quantity is the energy required to remove a ion and a ion from the crystal. Forming a pair of neutral atoms from this pair of ions involves the release of 5.14 eV (the ionization energy of Na) and the expenditure of 3.61 eV (the electron affinity of Cl). Use the result of part (a) to calculate the energy required to remove a pair of neutral Na and Cl atoms from the crystal. The experimental value for this quantity is 6.39 eV; how well does your calculation agree?