Crystal structures IISpacefilling and crystal stability

1. Space filling of crystal structures (Analytical task)By assuming hard spheres touching each other, calculate the space filling of the

• simple cubic (sc) structure,

• face centered cubic (fcc) structure,

• hexagonal closed packed (hcp) structure with an ideal ratio ca

=√

83.

2. Stability of a crystal lattice (Tool task)In the following task, we determine the lowest energy configuration of a crystal. The potentialenergy is here modelled by a modified Lennard-Jones potential

V = 4ε(σ

r)12 − (

σ

r)6 + A

( rσ

)2+B

( rσ

)+ C, (1)

with ε = 0.0104 eV and σ = 3.4 A. The polynomial A(rσ

)2+B r

σ+C is used to bring this potential

smoothly to zero for a given potential cutoff radius rcut. The potential is intended to model theinteraction between Argon atoms and was implemented in the code MiniMol, a minimal MolecularStatics / Molecular Dynamics tool. A detailed description of how to handle MiniMol can be foundon our webpage in the tool description“HandsOn MiniMol”.

Prepare rectangular simulation cells for Argon in the

• face centered cubic (fcc) structure,

• body centered cubic (bcc) structure,

• hexagonal closed packed (hcp) structure with an ideal ca

=√

83

ratio.

Vary the lattice constant a from of 3.0 A to 6.0 A and plot the energy per atom given by MiniMolwith respect to a. Which is, according to this model, the most stable crystal structure of Argon?Determine the lattice constant and the cohesive energy of this model.

Please, turn page −→

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Hint 1: The rectangular cell for hcp with the lattice parameters a and c is given by

~a1 = (a, 0, 0) ~a2 = (0,√

3a, 0) ~a3 = (0, 0, c) (2)

with four atoms in the cell located at the fractional coordinates ~b1 = (0, 0, 0), ~b2 = (12, 12, 0),

~b3 = (12, 16, 12), and ~b4 = (0, 2

3, 12).

Hint 2: The cohesive energy is defined as

Ecoh = Eatom, isolated − Eatom, crystal (3)

The potential used in MiniMol sets Eatom, isolated to zero.

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