A neutron scattering experiment on a liquid shows a strong peak in S(k) at k₀ = 2π/(2.8 Å). What does this peak indicate about the liquid's structure?
AThere is an energy gap at that wavevector, analogous to a bandgap in a crystal
BThe liquid has a preferred interparticle spacing of approximately 2.8 Å — the dominant length scale of local atomic packing
CThe compressibility of the liquid diverges at that length scale
DThe pair distribution function g(r) is zero at r = 2.8 Å, meaning particles avoid that separation
S(k₀) > 1 means there is enhanced density-density correlation at the length scale λ = 2π/k₀ ≈ 2.8 Å. Physically, many pairs of atoms are separated by approximately that distance — it corresponds to the typical nearest-neighbor spacing in the liquid. Scattered waves from pairs at that separation interfere constructively, producing the peak. This is the opposite of option D: a peak in S(k) at k₀ corresponds to a peak in g(r) at r ≈ 2π/k₀, indicating preferred (not avoided) separations.
Question 2 Multiple Choice
Near a liquid-gas critical point, the isothermal compressibility κ_T diverges to infinity. What happens to S(k) as k → 0 at the critical point?
AS(k → 0) → 0, because the system becomes perfectly ordered and suppresses long-wavelength density fluctuations
BS(k → 0) → 1, the ideal-gas value, because critical fluctuations restore normal behavior at long wavelengths
CS(k → 0) → ∞, because S(0) = ρ k_B T κ_T and κ_T diverges at the critical point
DS(k → 0) develops a sharp Bragg peak, indicating a phase transition to crystalline order
The thermodynamic identity S(k → 0) = ρ k_B T κ_T directly links the long-wavelength structure factor to the isothermal compressibility. At the critical point, κ_T → ∞, so S(0) → ∞. This means enormous long-wavelength density fluctuations — the system is indifferent to redistributing mass at large scales. Physically, this divergence is the origin of critical opalescence: the sample scatters visible light intensely (at small k corresponding to visible wavelengths), turning milky-white, because density fluctuations become enormously enhanced on those length scales.
Question 3 True / False
The static structure factor S(k) and the pair distribution function g(r) contain the same physical information about structural correlations in a material — they are related by a Fourier transform.
TTrue
FFalse
Answer: True
S(k) = 1 + ρ ∫ [g(r) − 1] e^{ik·r} d³r is a Fourier relationship between S(k) and the total correlation function h(r) = g(r) − 1. Since Fourier transforms are invertible, S(k) and g(r) encode exactly the same structural information, expressed in reciprocal space versus real space respectively. g(r) is more intuitive for describing real-space pair separations; S(k) is the directly measurable quantity in X-ray, neutron, and light scattering experiments.
Question 4 True / False
An ideal gas and a crystal would both exhibit S(k) = 1 for most wavevectors k, since S(k) measures mainly density fluctuations, which are present in both phases.
TTrue
FFalse
Answer: False
An ideal gas has g(r) = 1 everywhere (no correlations), so h(r) = 0 and S(k) = 1 for all k > 0 — this part is correct. But a crystal has perfectly periodic density: g(r) consists of sharp peaks at lattice spacings. The Fourier transform of this periodic structure gives sharp delta-function peaks at the reciprocal lattice vectors — the Bragg peaks of X-ray crystallography. S(k) is featureless for an ideal gas but sharply peaked for a crystal. Real liquids fall between these extremes, showing broad oscillatory peaks that decay at large k.
Question 5 Short Answer
Explain physically why a peak in S(k) at wavevector k₀ corresponds to enhanced scattering at that scattering angle. What is the connection between S(k₀) > 1 and the real-space structure of the material?
Think about your answer, then reveal below.
Model answer: In a scattering experiment, each atom scatters radiation independently. The total scattered intensity at wavevector transfer k is determined by coherent interference among waves scattered by all pairs of atoms. When many atom pairs are separated by distance r ≈ 2π/k₀, waves scattered by these pairs have a path-length difference of approximately one wavelength — they interfere constructively, producing a peak in the scattered intensity. S(k₀) > 1 is the quantitative statement that the density-density correlation function is enhanced at that length scale: there is an above-average probability of finding pairs at separation 2π/k₀. The peak in S(k) is therefore a direct readout of the dominant structural length scale in real space, measurable without any direct imaging of the material.
The Fourier relationship means S(k) and g(r) are dual representations of the same structural reality. Peaks in S(k) are signatures of periodicity or preferred spacings in g(r). Scattering experiments measure S(k) directly because coherent interference is the physical mechanism of Bragg and diffuse scattering — making S(k) the natural language for describing how radiation probes material structure at different length scales.