# Is there any difference between radial distribution function and pair correlation function?

I have understood the part that for solids pair correlation function is the measure of the probability of finding the center of another particle in the neighborhood of a given particle. On the other hand radial distribution function also determines the periodicity in a system. If we plot g(r) for pair correlation function we eventually get the same pattern for the radial distribution in solids.

I want to know if my understanding for these two are correct and if there is some fundamental difference.

The function $g(\mathbf{r})$, or more properly $h(\mathbf{r})=g(\mathbf{r})-1$ which tends to zero at large separation, is usually termed the pair correlation function. Only when the system is isotropic, such as a liquid, do we usually refer to $g(r)$ as the radial distribution function, since it depends only on the magnitude, not the direction, of the separation vector: $r=|\mathbf{r}|$.
You can find the full definitions in books such as Chaikin and Lubensky, Principles of condensed matter physics. I'll just give a short summary here. The general definition of $g$ takes into account its dependence on both of the positions under consideration. If the instantaneous number density of $N$ atoms is written as a sum of 3D Dirac delta functions of the atomic positions $$n(\mathbf{r}) = \sum_{i=1}^N \delta(\mathbf{r}-\mathbf{r}_i)$$ then the pair correlation function $g(\mathbf{r}_1,\mathbf{r}_2)$ is defined such that $$\langle n(\mathbf{r}_1) \rangle \langle n(\mathbf{r}_2) \rangle \, g(\mathbf{r}_1,\mathbf{r}_2) = \langle n(\mathbf{r}_1) n(\mathbf{r}_2) \rangle - \langle n(\mathbf{r}_1)\rangle \delta(\mathbf{r}_1-\mathbf{r}_2)$$ This has the interpretation that $g(\mathbf{r}_1,\mathbf{r}_2)$ is the probability density (per unit volume), given an atom at $\mathbf{r}_1$ of finding a different atom at $\mathbf{r}_2$, normalized by the number density in both places. Going from this to the radial distribution function involves two stages. Firstly, when the system is homogeneous and translationally invariant, one can express $g$ as a function of the difference in position vectors, $g(\mathbf{r}_1-\mathbf{r}_2)\equiv g(\mathbf{r})$. In this case, $\langle n(\mathbf{r})\rangle=\langle n\rangle$, independent of $\mathbf{r}$. We may integrate over one of the coordinates, and the result may be expressed $$\langle n \rangle \, g(\mathbf{r}) = \left\langle \sum_{i\neq 1}^N \delta(\mathbf{r}-\mathbf{r}_i+\mathbf{r}_1) \right\rangle$$ The probabilistic interpretation is even clearer here: both sides represent the density of (other) atoms per unit volume, at a position $\mathbf{r}$ relative to atom $1$, given that the latter is at $\mathbf{r}_1$.
Secondly, when the system is, additionally, isotropic, one can express $g$ as a function of the separation distance $g(r)$, where $r=|\mathbf{r}|$, and this is what we call the radial distribution function.
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