Question

4. Atomic interactions can be modeled using a variety of potential energy approximations. One very common potential form is the Lennard-Jones 6:12 potential: U(r)=4ε[(σ/r)12 − (σ/r)6]. Where ε and σ are constants specific to a given material (note: these terms are NOT equivalent to stress and strain, but this is the standard notation for the L-J parameters). Here, r is the interatomic spacing given in units of Angstroms, and U(r) is given in units of eV/atom. A molecular dynamics simulation was performed by Zhang and coworkers [1] to study the properties of Al thin films in which the authors proposed a Lennard-Jones potential of the form above to model Al-Al interactions. The values used for the material parameters were: ε=0.368 and σ =2.548.

Ref [1] H. Zhang and Z. N. Xia, Nuclear Instruments & Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 160 (2000) 372-376.

1. (a) Using the given material parameters and the form of the interatomic potential energy curve, plot U(r) for aluminum from r =2 to 4 Angstroms in increments of 0.25 Angstroms.

2. (b) Determine the equation for and graph the interatomic forces F as a function of interatomic separation r for Al over the same range of r used in part (a), indicating units of F(r). Also, analytically and graphically determine the equilibrium interatomic spacing, ro. Mark this point on the graph produced in (a) as well.

Answer 1

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above fig is plot of U vs r

fig(2) plot of F vs r

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