Department of Materials Science and Metallurgy
University of Cambridge
M Phil
Modelling of Materials
MP2 Question Sheet
1.
Describe the pair potential approximation for interatomic interactions in a material and
discuss its range of validity. Give some examples of pair potentials employed in
atomistic simulations of materials and explain the computational advantages of using
them. Why would it be incorrect to use a pair potential to describe titanium or silicon?
According to the Lennard-Jones pair potential model, the cohesive energy per atom of a
ccp (or fcc) crystal can be expressed as:
U = 2  12.13  12 - 14.45  6
r
r
where r is the nearest neighbour separation and  and  are adjustable parameters.
(a) determine the equilibrium nearest neighbour separation r0 in terms of 
(b) hence determine the equilibrium cohesive energy U0 in terms of 
(c) if the experimental values of r0 and U0 for solid argon are 0.375nm and -0.08 eV
respectively, determine appropriate values for the parameters  and 
(d) if the bulk modulus of a ccp crystal can be expressed as:
2
 U
2
B =
9 r 0 r2 r=r
0
determine the bulk modulus of solid argon using the Lennard-Jones model and comment
on the result.
2. The following empirical pair potential has been used to perform atomistic simulations of
aluminium (ccp, a0 = 4.0270 Å):
V(r) = D { exp[-2(r – r0)] - 2exp[-(r – r0)] }
where D,  and r0 are constants given by 0.1311 eV, 2.2661 Å-1 and 2.8673 Å
respectively.
(a) Calculate the total energy necessary to break the first and second nearest neighbour
bonds in aluminium.
(b) Hence estimate the vacancy formation energy in aluminium assuming that when an
atom is removed from the crystal it goes to the surface where it reforms half of its broken
bonds.
(c) The experimental vacancy formation energy for aluminium is 0.75 eV. Suggest
possible reasons for the discrepancy between this value and the one calculated above.
3.
Many atomistic simulations of metals have used density dependent interatomic
interactions to properly model the binding energy of the crystal. In one such model the
binding energy of a ccp transition metal is given by
E =  • • V(r ij ) - g • (ni)1/2
i ji
i
where  is a parameter with dimensions of energy, g is a dimensionless parameter, V(r)
is a pair potential, rij is the distance between sites i and j, and ni is the sum of pair
potentials, (r ), from site i
ni = •  (r ij )
j
Give a physical interpretation for each of the terms in these two equations.
Let the pair potentials V(r) and (r ) be the exponential exp(-2kr) where k is a constant.
If the interactions beyond first neighbours are ignored show that the condition for the
perfect ccp crystal to be in equilibrium, with the first neighbour bond length equal to R,
requires that g = 2 12e-kR and that the cohesive energy per atom of the perfect crystal
is then equal to 12e-2kR. Calculate the bulk modulus, B = v(d2E/dv2), at equilibrium
where v is the volume per atom.
4. Describe how pair potentials used for modelling ionic materials can be refined to include
the effects of charge overlap, dipole interactions and atomic polarisation?
The bulk modulus of a binary ionic compound can be expressed as:
B
  u 
 
  
where Ω and u are the volume and energy per ion pair respectively.
Show that the bulk modulus for an ionic material with the NaCl structure is given by:
1  2u 

B
 

18r0 r2 r r0
where r0 is the equilibrium nearest neighbour separation.
If the energy per ion pair is given by:
u(r )  
q 2
r

C
rm
where q is the ionic charge and , C and m are constants, show that for the NaCl structure:
m  1 q2
B  

 18  r04
5.
In the free electron model, the density of states in a three-dimensional crystal varies as
√E where E represents the allowed electron energy spectrum. In one-dimension, this
quantity varies as 1/√E. Show that for a two-dimensional crystal, the density of states is
independent of E.
6.
Define what is meant by the Fermi energy EF of a solid at finite temperature. Why must
the definition involve Fermi-Dirac statistics? Indicate the position of EF on a schematic
band structure diagram for a simple metal and an intrinsic semiconductor.
Derive the following expression for the Fermi energy of a free electron metal where n is
the number of electrons per unit volume:
EF 
2
3 n
2m
2
2
3
Estimate the Fermi energy of silver (fcc) assuming a lattice parameter of 4.05Å. At what
temperature is there a 10% probability that electrons in silver have an energy which is
1% above the Fermi energy?
7.
Discuss the physical significance of the Brillouin zone in the electron theory of solids.
Explain why the gradient of the electron energy spectrum is zero at the zone boundary.
Consider the first Brillouin zone of a fcc lattice which is a truncated octahedron. The
furthest point from the origin () is a point W which lies along [210]. Sketch the first
Brillouin zone and label the points  and W. Show that for a trivalent metal such as
aluminium, the free electron Fermi sphere completely fills the first Brillouin zone.