Pair Distribution Function¶
A normalized radial function comparing particle-pair occurrence at each separation with the occurrence expected from the system's mean number density.
Core Idea¶
The normalized pair distribution function \(g(r)\) describes how many other particles occur at distance \(r\) from a reference particle, compared with an uncorrelated same-density baseline. In a homogeneous, isotropic radial convention, \(g(r)=\rho(r)/\rho_0\): the separation-conditioned partner density divided by mean particle number density. Values above one indicate relative excess; values below one indicate depletion. It is not a probability density whose radial integral is one.[^ref-91ab4ea455d8]
Peaks can reveal favored pair separations. A hard-core exclusion gap or a large-distance approach to one is conditional on the particle model and decorrelation; neither is required for all materials.[ref-2e4cc35f08f9][ref-91ab4ea455d8]
Scope of Application¶
The function is used for atomic liquids and other materials, colloidal suspensions, and particle configurations with a declared spatial/radial convention. It can be computed from three-dimensional particle coordinates or recovered from appropriately corrected scattering representations. The source method must specify the particle population, average density, dimensionality, window and normalization. In particular, experimentally reduced \(G(r)\) is convertible to but not numerically identical with normalized \(g(r)\).[ref-91ab4ea455d8][ref-2ff9005759bf]
Clarity¶
Raw pair counts rise with shell volume even if relative particle structure does not change. Dividing by shell volume and mean density makes \(g(r)\) a comparison against a same-density reference rather than a graph of geometrical opportunity. A nearest-neighbor distribution, by contrast, records only the closest partner from each focal particle; \(g(r)\) retains occurrence at all separations.[^ref-91ab4ea455d8]
Manages Complexity¶
Radial averaging compresses many particle coordinates into one separation-resolved curve. For a homogeneous isotropic three-dimensional sample, \(4\pi\rho_0\int_{r_1}^{r_2}r^2g(r)\,dr\) estimates a coordination count within specified shell bounds. The compression hides direction and, unless partial functions are retained, species-specific structure. A curve does not uniquely reconstruct all underlying particle positions.[^ref-91ab4ea455d8]
Abstract Reasoning¶
Given coordinates, condition on a reference particle, count other particles in radial shells, convert counts to conditional density, and divide by the system's mean density. A peak now means that partners are overrepresented at a particular separation relative to that baseline, not that a unique force law or lattice has been proven. From scattering, obtain \(g(r)\) only with the stated \(S(Q)\) transform, density and weighting convention: the standard relation involves the deviation \(g(r)-1\), not an unqualified transform of raw \(g(r)\). Hard-core zeros, far-distance unity and thermodynamic inferences require further assumptions.[ref-91ab4ea455d8][ref-2e4cc35f08f9]
Knowledge Transfer¶
Peterson and coauthors' molecular-dynamics simulation of liquid Ar displays normalized \(g(r)\) alongside other nonidentical total-scattering PDF forms. Kodger and colleagues calculate a three-dimensional radial \(g(r)\) from confocal-imaged colloids and observe different peak structures in screened fluid and deionized crystal states. Both instantiate reference-conditioned pair density divided by mean density, despite different particle scales and modeling or measurement routes.[ref-91ab4ea455d8][ref-2ff9005759bf]
The general thought of judging pair occurrence against a baseline may extend farther, but the named \(g(r)\) here requires spatial particles, radial separation and number-density normalization. Live Correlation is a related prime-level idea of variable co-variation, not an asserted strict parent of this physical function.
[^ref-91ab4ea455d8]: Peter F. Peterson, Daniel Olds, Marshall T. McDonnell and Katharine Page, “Illustrated formalisms for total scattering data: a guide for new practitioners,” Journal of Applied Crystallography 54 (2021), 317–332, §2 simulation setup, §4, Figures 3–4, Table 4 and Appendix A equations (32)–(38). A corrigendum corrects some other plotted scales and multicomponent limits; the normalized monatomic \(g(r)\) account here does not use those affected ordinates. [^ref-2e4cc35f08f9]: Roger Pynn, “Lecture 1: Introduction & Neutron Scattering ‘Theory’,” NIST NCNR summer-school hosted teaching slides (2009), PDF pp. 15–16. [^ref-2ff9005759bf]: Thomas E. Kodger, Rodrigo E. Guerra and Joris Sprakel, “Precise colloids with tunable interactions for confocal microscopy,” Scientific Reports 5 (2015), 14635, Results and Figure 5.
Neighborhood in Abstraction Space¶
Pair Distribution Function sits in a sparse region of the domain-specific corpus (72nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Statistical Mechanics & Particle Phenomena (15 abstractions)
Nearest neighbors
- Laser Diffraction Analysis — 0.85
- Phi Coefficient — 0.83
- Quantum Concentration — 0.83
- Geometrical Frustration — 0.83
- Boltzmann–Matano Analysis — 0.83
Computed from structural-signature embeddings · 2026-10-08