Physics 463, Winter 2008 Problem Set 3 and 4, due Friday, February 22, by 5pm. page 1 of 2
1. molecular solids The Lennard-Jones potential between pairs of atoms is
U(r) = 4σ
r12
−σ
r6
When considering the cohesive energy in a solid, we must sum up this interaction energy over all of the
atoms. For a monoatomic fcc structure in which each atom has 12 nearest neighbors, the corresponding
cohesive energy for a molecular solid with N atoms (neglecting the KE)
U=1
2N412.13188 σ
R12
−14.45392 σ
R6
where Ris the nearest neighbor distance and the 1/2 ensures that we count each atom only once.
(a) What is the origin of the attractive and repulsive terms in the LJ potential?
(b) Find the nearest neighbor separation in terms of the parameter σ(hint, this occurs when the energy
is minimized).
(c) for Ne, = 0.0031eV and σ= 2.74 ˚
A (this is obtained by measurements in the gas phase). For
soild Ne what is the theoretical cohesive energy per atom? what is the theoretical nearest neighbor
separation? The experimental values are 3.13 ˚
A and 0.02 eV respectively. The slight departure is
due to quantum mechanical zero point motion of the Ne atoms. Do you expect the agreement to be
better or worse as one moves down the periodic table (Ar, Kr, Xe)?
2. Ionic crystals The interaction in ionic crystals is a combination of long range electrostatic potentials and
a short range repulsive interaction. For two atoms iand j, the energy is
Uij =λe−rij/ρ ±q2
rij
where we use cgs units (as in Kittel), qis the charge of the ions, rij is the distance between the two ions,
and we choose (+) or (-) depending on whether we are considering like ions (repulsive) or unlike ions
(attractive) and λand ρare parameters to be extracted from the experimental data.
(a) Typically ρis on order of 10% of the nearest neighbor separation. Thus, show that the exponential
term can be neglected past nearest neighbors such that the total cohesive energy may be written for
Npairs of positive and negative ions as
U=Nzλe−R/ρ −αq2
R
where zis the number of nearest neighbors, and Ris the nearest neighbor separation negative ions,
and αis the Madelung constant:
α=X
j
(±)
pij
where rij =pijR.
(b) The Madelung constant depends only on the structure. For the sodium chloride structure it is
1.747565, for the cesium chloride structure it is 1.762675 and for the zinc-blende structure it is
1.6381. Calculate the lattice energy of KCl in the the zinc-blende, sodium chloride and zinc-blende
structure. Compare with the experimental value of the lattice energy which is 7.17 eV per molecular
unit. (See figure 3.10 in Kittel for a plot of the energy per KCl pair showing the Madelung and
