Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

The potential energy of interaction between two non-bonding atoms is sometimes modeled by the Lenard-Jones (also known as the "6-12") potential:

1. Sketch a qualitative graph of the L-J potential and explain what features of the curve each term is responsible for. What do these terms mean physically?
2. The equation for V is sometimes written V(r) = (A/r¹²) - (B/r⁶). Discuss why one form or the other might be more advantageous.
3. Find the location of the minimum of the potential, r₀. Before doing the calculation, discuss whether you expect this distance to be greater than, equal to, or smaller than the distance σ. (Hint: Refer to your graph in part (a).)
4. Find the location of the maximum of the force between the atoms, r_F0. Before doing the calculation, discuss whether you expect this distance to be greater than, equal to, or smaller than the distance you found in the previous part. (Hint: Refer to your graph in part (a).)
5. This function is messy to deal with. For small displacements around the minimum it's often useful to approximate the L-J potential by a shifted harmonic oscillator potential
   

   Discuss, using a graph of the original and approximate functions, what we hope to accomplish by choosing to do this. To get the best approximation in the neighborhood of the minimum of the potential, what should we choose for V₀, k and r₀?

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   This page prepared by
   Edward F. Redish
   Department of Physics
   University of Maryland
   College Park, MD 20742
   Phone: (301) 405-6120
   Email: redish@umd.edu

   Last revision 6. September, 2004.