Summary and Terms

Hooke's Law: A mathematical model that describes the behavior of a spring, specifically the restoring force the spring exerts when it is displaced a given distance. Defined as

$$F(x) = -k x$$.
Where $F$ is the restorative spring force, $k$ is the spring constant, and $x$ is the position the spring has been displaced from the equilibrium position of $x = 0$. Section 4.3.2

Potential Energy and Force Relations - Springs: In a spring system, the amount of mechanical energy stored by the spring as it is extended or compressed. Force is the negative spatial derivative of the potential energy

$$F(x) = - \frac{dU(x)}{dx}$$

This spring relation is a starting point to understand how we might model the interatomic bond. Section 4.3.6

Interatomic Potential and Force Relations: The interatomic potential is a function that models the energy in a system of two interacting atoms, dependent on separation distance. Just like the relationship between force and energy for a spring, we also define the force/energy relation here as the negative derivative of the potential

$$F(x) = - \frac{dU(x)}{dx}$$

There are many examples of interatomic potentials applicable to all sorts of fields, including beyond materials science (biology, physics, chemistry, etc.). Section 4.4.1

Lennard-Jones Potential: A simple but powerful empirical interatomic potential that specifies the potential energy between two atoms based on the distance between them. The Lennard Jones potential is given by:

$$U_{\text{LJ}}(r) = 4 \epsilon \left[\Big(\frac{\sigma}{r}\Big)^{12}-\Big(\frac{\sigma}{r}\Big)^6\right]$$

This potential is used to model many simple and advanced system, and governs the interactions of atoms in our simulations in this chapter. Section 4.6

Interatomic distance $r$ or $x$: The distance the two atoms are separated is represented either by $r$ or $x$. We typically use the Cartesian frame when introducing this concept and therefore show the $x$-axis when plotting interatomic energy curves.

$\epsilon$: In the Lennard-Jones potential, the $\epsilon$ parameter primarily impacts the depth of the energy well. The larger the value, the greater the depth of the function's potential energy well. In this model, it correlates with the equilibrium bond energy (or strength).

$\sigma$: In the Lennard-Jones potential, the $\sigma$ parameter primarily impacts the position (and width) of the interatomic energy curve. In this model it correlates to the equilibrium bond length. Section 4.5.1

Interionic Potential: An interatomic potential that specifies the potential energy specifically for ionic charges. Covered in lecture and problem sets explicitly

$$U_{\mathrm{II}(r)} = \frac{B}{r^n}+\frac{Z_1Z_2 e^2}{4 \pi \epsilon_0 r}$$

Where $r$ is the interionic distance, $B$ and $n$ are empirical values defining the strength of the repulsive interaction, $e$ is the charge on the electron, $Z_1$ and $Z_2$ are the charges on the bonded ions, and $\epsilon$ is the permittivity of free space.