A researcher measures g(r) for a fluid and finds g(r) = 1 for all values of r. What does this tell them about the fluid?
AThe fluid is perfectly ordered, like a crystal
BThe particles are spatially uncorrelated — the fluid behaves like an ideal gas
CThe fluid has strong repulsive interactions at all distances
DThe measurement failed; g(r) = 1 is physically impossible in any real fluid
g(r) is defined relative to a random (uncorrelated) distribution: g(r) = ρ_local(r) / ρ_bulk. When g(r) = 1 everywhere, the local density at every distance exactly equals the bulk average, meaning no spatial correlations exist — particles are placed as if randomly. This is the ideal gas limit. A crystal would show sharp peaks at lattice spacings persisting to long range; a liquid shows oscillating peaks that decay over a few particle diameters.
Question 2 Multiple Choice
For a hard-sphere fluid with particle diameter σ, a student claims that g(r) must be small but positive for r < σ due to thermal fluctuations occasionally driving particles to overlap. Is this correct, and what does g(r) actually equal for r < σ?
ACorrect — thermal energy allows occasional overlaps, so g(r) is small but positive for r < σ
BIncorrect — g(r) = 0 exactly for r < σ because hard spheres cannot overlap under any circumstances
DIncorrect — g(r) = 1 for r < σ, then drops to zero at the particle surface
Hard spheres are defined by an infinite repulsive potential for r < σ — two hard spheres simply cannot overlap regardless of thermal energy. Therefore g(r) = 0 exactly for r < σ. The sharp rise from 0 at r = σ corresponds to the excluded-volume boundary, followed by the first peak at r ≈ σ representing the densely packed nearest-neighbor shell. This is not a thermal effect that could be overcome; it is a hard geometric constraint.
Question 3 True / False
A liquid shows oscillating peaks in g(r) that decay over a few particle diameters, while an ideal gas has g(r) = 1 everywhere. This means the pair distribution function can distinguish between liquid and gas phases.
TTrue
FFalse
Answer: True
True. The oscillating decay in g(r) for a liquid reflects the short-range order of neighbor shells — particles pack around each other in shells at ~σ, ~2σ, ~3σ, etc., but this order fades over a few particle diameters (unlike the infinite-range order in a crystal). The ideal gas shows no such structure. g(r) is precisely the tool that quantifies this difference in spatial correlation, making it a diagnostic of the structural character of a phase.
Question 4 True / False
The static structure factor S(k) and the pair distribution function g(r) are independent quantities — S(k) measures momentum-space structure while g(r) measures real-space structure — and neither can be derived from the other.
TTrue
FFalse
Answer: False
False. S(k) and g(r) are Fourier transform pairs: S(k) = 1 + ρ∫[g(r) − 1]e^{ik·r}d³r. They encode exactly the same structural information in reciprocal and real space respectively. This is why X-ray and neutron scattering experiments — which directly measure S(k) as a diffraction pattern — give you g(r): you simply Fourier-transform the scattering data. They are not independent; knowing one fully determines the other.
Question 5 Short Answer
Why is g(r) = 1 for a completely uncorrelated system, and what does a peak with g(r) > 1 at some distance r* physically signify?
Think about your answer, then reveal below.
Model answer: g(r) = 1 when the local particle density at distance r equals the bulk average density — the probability of finding a neighbor is exactly what you would expect from a random (Poisson) distribution. A peak with g(r) > 1 at r* means the local density there is enhanced above the bulk average, indicating that particles preferentially sit at that separation. For a liquid, this corresponds to a nearest-neighbor shell: interactions (repulsion at close range, attraction or packing geometry at r*) cause particles to cluster preferentially at that distance.
The ratio g(r) = ρ(r)/ρ_bulk normalizes out the trivial effect of bulk density, so deviations from 1 purely reflect correlations. A peak above 1 means particles are more likely to be found at that separation than chance alone predicts — a signature of structure imposed by interactions. The height and position of the first peak directly quantifies nearest-neighbor packing, while subsequent peaks reveal further shells. This normalization is what makes g(r) comparable across different densities and systems.