In: Physics
A common approximation to the interatomic forces in a material
is the Lennard-Jones potential between neighboring atoms in a
solid:
U(r) = A/r12 - B/r6
This can be used to find various information about a solid
material.
Assume that for the lattice of a particular alloy of copper, the
Lennard-Jones constants are:
A = 9.08 ⨯ 10-133 J·m12
B = 3.96 ⨯ 10-76 J·m6
Find the following:
(a) The equilibrium distance between neighboring atoms in the
copper lattice (this is called the lattice constant
a).
(b) Model this as an approximate potential around the equilibrium
position
Ueff(r) = (1/2)k(r-a)2 + C
(i) What is the effective spring stiffness k?
(ii) What is the energy minimum C?
(c) Find the resonance frequency of a copper atom in a linear chain
of atoms? [Assume only nearest neighbor interactions]
(d) Find the vibration amplitude at the temperature T = 5⁰C. Assume
that the oscillator has one degree of freedom.
[ Use kB = 1.38⨯10-23 J/K/atom as Boltzmann's
constant. ]