Questions: Radial Distribution Function and Liquid Structure

Question 1

For a simple liquid at equilibrium, g(r) = 2.8 at r ≈ 3.4 Å (the first peak) and g(r) → 1 as r → ∞. What do these two values tell you about the liquid structure?

• A. There are 2.8 particles per cubic ångstrom at 3.4 Å, and the density drops to 1 particle per cubic ångstrom at large distances
• B. The probability of finding a neighbor at 3.4 Å is 2.8 times higher than it would be in an ideal gas; at large r, correlations die out and local density matches the bulk average
• C. The coordination number of the liquid is 2.8, meaning each atom has on average 2.8 nearest neighbors
• D. The liquid has 2.8 times the density of a gas at the first coordination shell distance, and becomes a uniform gas at long range

Answer: (B)

g(r) is normalized by the ideal gas expectation. g(r) = 2.8 at the first peak means the probability of finding a neighbor at that distance is 2.8 times what you would expect in a uniform (ideal gas) distribution at the same bulk density. It does not give absolute particle count per volume — you need to multiply g(r) by the bulk density ρ and the shell volume 4πr²dr to get the actual number of neighbors. The large-r limit g(r) → 1 reflects that at large separations, correlations fade and the local density matches the bulk average — which is the definition of having no long-range order.

Question 2

You measure g(r) for two materials: Material X shows sharp, non-decaying peaks at fixed distances that persist for large r. Material Y shows a first peak at r ≈ σ, then damped oscillations that relax to g(r) = 1 by r ≈ 4σ. What do these patterns indicate?

• A. Material X is a liquid with short correlation length; Material Y is an ideal gas with correlated fluctuations
• B. Material X is a crystal with long-range periodic order; Material Y is a liquid with finite structural correlation length
• C. Both materials are liquids, but Material X is at lower temperature where order persists longer
• D. Material X is a gas at high pressure; Material Y is a supercritical fluid above the critical point

Answer: (B)

The g(r) fingerprint distinguishes phases clearly. Non-decaying peaks at lattice spacings that persist for all r is the signature of a crystal: long-range periodic order means correlations persist at any distance. Damped oscillations that relax to 1 are the signature of a liquid: short-range order exists (coordination shells at 1σ, 2σ, etc.) but correlations decay exponentially over a finite structural correlation length. An ideal gas would show g(r) = 1 everywhere (no correlations at all). This is why g(r) measured by X-ray or neutron diffraction is a primary tool for phase characterization.

Question 3

If g(r) = 1 for most values of r in a fluid, this means the fluid is at its maximum density — most shells are equally and fully occupied.

• T. True
• F. False

Answer: False

g(r) = 1 everywhere is the signature of an ideal gas — a fluid with no interparticle correlations or interactions. It does not mean maximum density; it means that the local density at any distance from a reference particle is exactly equal to the bulk average density ρ. Real fluids at any density show deviations from g(r) = 1: a hard core at small r (g = 0 where atoms cannot overlap) and peaks at the coordination shells. A high-density fluid actually shows more pronounced structure (larger peaks and deeper troughs in g(r)) than a low-density one, as packing constraints create stronger local order.

Question 4

The internal energy and pressure of a fluid can be computed directly from g(r) and the pair potential u(r), without simulating individual particle trajectories.

• T. True
• F. False

Answer: True

This is the central power of the radial distribution function. The energy equation states U = N⟨KE⟩ + (N²/2V) ∫ u(r) g(r) 4πr² dr, and the pressure equation gives P = ρkT − (ρ²/6) ∫ r(du/dr) g(r) 4πr² dr. Both integrals require only g(r) (the structural information, measurable by scattering) and u(r) (the pair potential, known from theory or fitting). This means a single scattering experiment can yield thermodynamic data for the fluid without tracking individual particles — a remarkable bridge between structure and thermodynamics.

Question 5

Explain why g(r) must equal zero for small r in any real liquid, and what physical property of matter enforces this constraint.

Model answer: g(r) = 0 at small r because real atoms have a repulsive core — a short-range repulsion (from Pauli exclusion of electron clouds) that prevents two atoms from occupying the same space. The pair potential u(r) rises steeply to very large positive values as r decreases below the atomic diameter σ. This makes it thermodynamically impossible for two atoms to overlap, so the probability of finding a neighbor at r < σ is essentially zero. In g(r) terms, the strong repulsion ensures that shells at short distances contribute zero to the running sum ρ g(r) 4πr² dr.

The hard-core exclusion at small r is the most fundamental structural feature of matter — it is why solids and liquids have finite volume and cannot be compressed indefinitely. The distance at which g(r) first rises from zero to its first peak is approximately the effective diameter of the atoms or molecules in the liquid. For liquid argon, this is about 3.4 Å (the Lennard-Jones σ parameter). For water, the first peak in the oxygen-oxygen g(r) is at about 2.8 Å, reflecting the shorter hydrogen-bonded contact distance. Reading the position of g(r)'s first rise from zero directly gives you the effective atomic size.