RMC Method

Basic information about RMC modeling

The Reverse Monte-Carlo (RMC) method, reported by R. L. Mcgreevy and L. Putzai, is a method for probing structural model which is consistent with the observed S(Q) and G(r). It was initially commonly used as a method for probing the structure of liquids and amorphous materials¹. However, RMC methods have been widely used for local structure analysis of crystalline materials due to developing the RMCPOW simulation² by A. Mellergård and R. L. Mcgreevy and the RMCProfile simulation³ by M. G. Tucker et al.

PDFFIT⁴ and PDFgui⁵ are widely known as PDF analysis tools for crystalline materials, and they are called "Small Box Simulations" because they use a unit cell to reproduce the measured G(r). On the other hand, RMC is called a "Large Box Simulation" because it requires a large computational box to reproduce actual measurements. The RMC method provides real space information as follows: revised structure mode, partial correlation, angular histogram.

Outline of structure modelling by RMC

The flow of RMC modelling

There are two important equations to understand the RMC modelling. At first, the scattering intensity is calculated from the atomic coordination as follows:

\begin{equation}S_{RMC}(Q)= \rho_0 \displaystyle \nonumber  \int_0^{r_{max}}4πr^2(g_{RMC}(r)-1)\frac{\sin{Qr}}{Qr}dr \end{equation}

where $\rho_{0} = N/V$ is the number density (N: the number of atoms (particles) in the calculation box), $r_{\max}$ is the maximum distance in RMC modelling, and $Q$ is the scattering vector.

The estimated structure is evaluated the comparison between an observed value and calculated value using a standard χ² test as follows:

\begin{equation} x^2 = \displaystyle \nonumber \sum_{i=1}^{N} \frac{(S_{\mathrm{obs}}(Q_i)- S_{\mathrm{RMC}}(Q_i))^{2}}{\sigma^{2}(Q_i)}\end{equation}

where $S_{obs} (Q_i )$ is the observed value, $S_{RMC} (Q_i )$ is the calculated value and $σ^2 (Q_i )$ is the experimental error.

The RMC trials are generated by random numbers.

1. Select the atom and the movement trials were generated by random number.
2. If the calculated value (S(Q) or G(r)) from the atomic coordination after the move has a χ² value of the measured value,
   • Small: Accept the trial 1
   • Large: Reject the trial 1
3. Repeat steps 1 and 2 until the χ² value falls below the target value.

Actual RMC calculation

Changes in S(Q) and G(r) during RMC modelling

References：  

 [1] R.L. McGreevy, L. Pusztai, Molec. Simul. 1 (1988) 359-367. doi: 10.1080/08927028808080958