The Radial Distribution Function (RDF), often denoted as \( g(r) \), is a key concept in physics, chemistry, and materials science, especially for studying the structure of liquids, colloids, glasses, and other disordered systems. It describes how particle density varies as a function of distance from a reference particle, providing insight into the local arrangement and correlations between particles in a system.
Definition:
The RDF, $g(r)$ , is defined as the ratio of the local particle density at a distance $r$ from a reference particle to the average particle density of the entire system:
$$
g(r) = \frac{\text{Number of particles at distance } r \text{ per unit volume}}{\text{Average number of particles per unit volume}}.
$$
Mathematical Expression:
For a system with an average number density $\rho$ , the RDF at a distance $r$ is given by:
$$
g(r) = \frac{1}{\rho} \left\langle \frac{dN(r)}{dV(r)} \right\rangle,
$$
where:
- $\rho$ is the number density of particles,
- $dN(r)$ is the number of particles in a spherical shell of thickness $dr$ at distance $r$ ,
- $dV(r)$ is the volume of the shell, typically $4\pi r^2 dr$ ,
- $\langle \cdot \rangle$ denotes averaging over all particles.
Interpretation of RDF:
- $g(r) = 1$ : Particles are distributed randomly at distance $r$ , indicating no correlation beyond this distance.
- $g(r) > 1$ : Higher probability of finding particles at distance $r$ , suggesting structural organization or clustering at this distance.
- $g(r) < 1$ : Lower probability, indicating that the distance $r$ corresponds to a depletion zone or a region where particles are less likely to be found.
Key Features in RDF Plots:
- First Peak: Represents the most probable distance between neighboring particles, providing information about the nearest-neighbor structure (e.g., average bond length in liquids or colloids).
- Subsequent Peaks: Indicate second-nearest neighbors, third-nearest neighbors, etc., and help identify medium-range order in the system.
- Asymptotic Behavior: At larger distances, $g(r)$ approaches 1, indicating that the correlation between particles vanishes and the distribution becomes homogeneous.