## 3.3 Exercise on Plotting

A simple model for the interaction potential between two atoms as a function of their distance, $$r$$, is that of Lennard-Jones15:

$U(r) = \dfrac{B}{ r^{12}} − \dfrac{A}{r^{6}}$

$$A$$ and $$B$$ are positive constants16.

1. Plot $$U(r)$$ for Argon atoms where $$A = 1.024 \times 10^{−23}$$ J nm$$^6$$ and $$B = 1.582 \times 10^{−26}$$ J nm$$^{12}$$.
Your plot should show the “interesting” part of this curve, which tends rapidly to very large values at small $$r$$.

2. What is the depth, $$\epsilon$$, and location, $$r_0$$, of the potential minimum for this system?17 Indicate these values by drawing horizontal and vertical lines at these locations18.

3. On a second y-axis19 on the same figure, plot the interatomic force given by: $F(r) = −\dfrac{dU}{dr} = \dfrac{12 B}{ r^{13}} − \dfrac{6 A}{r^{7}}$

4. For small displacements from the equilibrium interatomic separation (where $$F = 0$$), the potential may be approximated to the harmonic oscillator function, $$V(r) = \dfrac{1}{2}k(r-r_0)^2+ \epsilon$$ where

$k = \left|\dfrac{d^2U}{dr^2}\right|_{r_0}= \dfrac{156 B}{ {r_0}^{14}} − \dfrac{42 A}{{r_0}^{8}}$

1. Plot $$U(r)$$ and $$V(r)$$ on the same figure but as a separate subplot.

1. This was popular in the early days of computing because $$r^{−12}$$ is easy to compute as the square of $$r^{−6}$$.↩︎

2. Life is easier if you divide A and B by Boltzmann’s constant, $$1.381\times 10^{−23}$$ JK$$^{−1}$$ so as to measure U(r) in units of K.↩︎

3. Think like a scientist not a programmer!↩︎

4. Hint: Look up plt.hlines() and plt.vlines().↩︎

5. Hint: Look up twinx()↩︎