Consider a simple atomic liquid or colloidal suspension . The average density
at a distance *r* of a given particle is defined to be ,
i.e.

Here is the average density in the fluid. Notice that is a conditional density; it is the density at , given a particle is present in the origin. A qualitative picture of

It is clear that *g*(*r*) should go to 1 for large *r*. At very short *r*the radial distribution function must be zero, because two particles cannot
occupy the same space. Outside the van der Waals diameter, there is a peak
because the remaining *N*-1 particles try to diffuse into the region
occupied by the one at the origin. The surplus of particles in the first
peak causes a lack of particles a little bit further on, explaining the
minimum of *g*(*r*) around .

Radial distribution functions may be measured by means of scattering
experiments, for example by means of X-ray or neutron scattering in the case of atomic liquids, or by means of
SAXS, SANS or light scattering in the case of colloidal suspensions. Generally the
sample produces a scattering amplitude

(2.4) |

at the detector. Here is the scattering power at position , and is defined by

(2.5) |

where and are the wavevectors of the outgoing and incoming wave respectively. The measured intensity is proportional to

(2.6) |

Writing the scattering power like

(2.7) |

we find after some algebra

= | (2.8) | ||

= | (2.9) |

The important quantity for us is

= | |||

= | |||

= | (2.10) |

The -function only gives a contribution at , i.e. to the forward scattering. At all other

(2.11) |

i.e. essentially the Fourier transform of the total correlation function

Once we have determined the radial distribution function we may calculate
several quantities of interest, like

(2.12) |

(2.13) |

where is the interaction potential between two particles in the system, and

(2.14) |

For more information we refer to any book on liquid state theory, like for example J. -P. Hansen and I. R. McDonald , Theory of simple liquids.