Pair distribution functions#

The pair distribution function describes the spatial correlation between particles.

Two-dimensional (planar) pair distribution function#

Here, we present the two-dimensional pair distribution function \(g_{\text{2d}}(r)\), which restricts the distribution to particles which lie on the same surface \(S_\xi\).

Let \(g_1\) be the group of particles which are centered, and \(g_2\) be the group of particles whose density around a \(g_1\) particle is calculated. Furthermore, we define a parametric surface \(S_\xi\) as a function of \(\xi\),

\[S_\xi = \{ \mathbf{r}_{\xi} (u, v) | u_{\text{min}} < u < u_{\text{max}}, v_{\text{min}} < v < v_{\text{max}} \}\]

which consists of all points \(\mathbf{r}_\xi\). By varying \(u, v\) we can reach all points on one surface \(\xi\). Let us additionally consider a circle on that plane \(S_{i, r}\) with radius \(r\) around atom \(i\) given by

\[S_{i, r} = \{ \mathbf{r}_{i, r} | \; || ( \mathbf{r}_{i, r} - \mathbf{x_i} || = r ) \land ( \mathbf{r}_{i, r} \in S_{\xi, i} ) \}\]

where \(S_{\xi, i}\) is the plane in which atom \(i\) lies.

Then the two-dimensional pair distribution function is

\[g_{\text{2d}}(r) = \left \langle \sum_{i}^{N_{g_1}} \frac{1}{L(r, \xi_i)} \frac{\sum_{j}^{N_{g_2}} \delta(r - r_{ij}) \delta(\xi_{ij})} {\vert \vert \frac{\partial \mathbf{f}_i}{\partial r} \times \frac{\partial \mathbf{f}_i}{\partial \xi} \vert \vert _{\phi = \phi_j}} \right \rangle\]

where \(L(r, \xi_i)\) is the contour length of the circle \(S_{i, r}\). \(\mathbf{f}_i(r, \gamma, \phi)\) is a parametrization of the circle \(S_{i, r}\).

Discretized for computational purposes we consider a volume \(\Delta V_{\xi_i}(r)\), which is bounded by the surfaces \(S_{\xi_i - \Delta \xi}\), \(S_{\xi_i + \Delta \xi}\) and \(S_{r - \frac{\Delta r}{2}}, S_{r + \frac{\Delta r}{2}}\). Then our two-dimensional pair distribution function is

\[g_{\text{2d}}(r) = \left \langle \frac{1}{N_{g_1}} \sum_i^{N_{g_1}} \frac{\text{count} \; ({g_2}) \; \text{in} \;\Delta V_{\xi_i}(r)} {\Delta V_{\xi_i}(r)} \right \rangle\]

Derivation#

Let us introduce cylindrical coordinates \(r, z, \phi\) with the origin at the position of atom \(i\).

\[\begin{split}\begin{aligned} x &= r \cdot \cos \phi \\ y &= r \cdot \sin \phi \\ z &= z \\ \end{aligned}\end{split}\]

Then the two-dimensional pair distribution is given by

\[g_{\text{2d}}(r, z=0) = \left \langle \sum_{i}^{N_{g_1}} \frac{1}{2 \pi r} \sum_{j}^{N_{g2}} \delta(r - r_{ij}) \delta(z_{ij}) \right \rangle\]

where we have followed the general derivations given above.

For discretized calculation we count the number of atoms per ring as illustrated below

Sketch of the discretizationSketch of the discretization

The sketch shows an atom \(i\) from group \(g_1\) at the origin in blue. Around the atom a ring volume with average distance \(r\) from atom i is shaded in light red. Atoms \(j\) from group \(g_2\) are counted in this volume.

One-dimensional (cylindrical) pair distribution functions#

Here, we present the one-dimensional pair distribution functions \(g_{\text{1d}}(\phi)\) and \(g_{\text{1d}}(z)\), which restricts the distribution to particles which lie on the same cylinder along the angular and axial directions respectively.

Let \(g2\) be the group of particles whose density around a \(g1\) particle is to be calculated and let \(g1, g2\) lie in a cylinderical coordinate system \((R, z, \phi)\).

Then the angular pair distribution function is

\[g_{\text{1d}}(\phi) = \left \langle \sum_{i}^{N_{g_1}} \sum_{j}^{N_{g2}} \delta(\phi - \phi_{ij}) \delta(R_{ij}) \delta(z_{ij}) \right \rangle\]

And the axial pair distribution function is

\[g_{\text{1d}}(z) = \left \langle \sum_{i}^{N_{g_1}} \sum_{j}^{N_{g2}} \delta(z - z_{ij}) \delta(R_{ij}) \delta(\phi_{ij}) \right \rangle\]

Discretized for computational purposes we consider a volume \(\Delta V_{z_i,R_i}(\phi)\), which is bounded by the surfaces \(S_{z_i - \Delta z}\), \(S_{z_i + \Delta z}\), \(S_{R_i - \Delta R}\), \(S_{R_i + \Delta R}\) and \(S_{\phi - \frac{\Delta \phi}{2}}, S_{\phi + \frac{\Delta \phi}{2}}\). Then our the angular pair distribution function is

\[g_{\text{1d}}(\phi) = \left \langle \frac{1}{N_{g_1}} \sum_i^{N_{g_1}} \frac{\text{count} \; ({g_2}) \; \text{in} \;\Delta V_{z_i,R_i}(\phi)} {\Delta V_{z_i,R_i}(\phi)} \right \rangle\]

Similarly,

\[g_{\text{1d}}(z) = \left \langle \frac{1}{N_{g_1}} \sum_i^{N_{g_1}} \frac{\text{count} \; ({g_2}) \; \text{in} \;\Delta V_{\phi_i,R_i}(z)} {\Delta V_{\phi_i,R_i}(z)} \right \rangle\]