Reverse Monte Carlo: Difference between revisions
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#Start with a configuration of atoms with periodic boundary conditions. This can be a random or a crystalline configuration from a different simulation or model. | #Start with a configuration of atoms with periodic boundary conditions. This can be a random or a crystalline configuration from a different simulation or model. | ||
#Calculate the total radial distribution function <math>g_o^C(r)</math> for this old configuration. | #Calculate the total radial distribution function <math>g_o^C(r)</math> for this old configuration (''C''=Calculated, ''o''=Old). | ||
#Transform to the total structure factor: | #Transform to the total structure factor: | ||
#:<math>S_o^ | #:<math>S_o^C (Q)-1=\frac{4\pi\rho}{Q}\int\limits_{0}^{\infty} r(g_o^C(r)-1)\sin(Qr)\, dr</math> | ||
#:where ''Q'' is the momentum transfer <math>\rho</math> | #:where ''Q'' is the momentum transfer and <math>\rho</math> the number density. | ||
#Calculate the difference between the measured structure factor <math>S^E(Q)</math> and the one calculated from the configuration <math>S_o^C(Q)</math>: | #Calculate the difference between the measured structure factor <math>S^E(Q)</math> and the one calculated from the configuration <math>S_o^C(Q)</math>: | ||
#:<math>\chi_o^2=\sum_i(S_o^C(Q_i)-S^E(Q_i))^2/\sigma(Q_i)^2</math> | #:<math>\chi_o^2=\sum_i(S_o^C(Q_i)-S^E(Q_i))^2/\sigma(Q_i)^2</math> | ||
#:this sum is taken over all experimental points <math>\sigma</math> is the experimental error. | #:this sum is taken over all experimental points <math>\sigma</math> is the experimental error. | ||
#Select and move one atom at random and calculate the new distribution function, structure factor and: | #Select and move one atom at random and calculate the new (''n''=New) distribution function, structure factor and: | ||
#:<math>\chi_n^2=\sum_i(S_n^C(Q_i)-S^E(Q_i))^2/\sigma(Q_i)^2</math> | #:<math>\chi_n^2=\sum_i(S_n^C(Q_i)-S^E(Q_i))^2/\sigma(Q_i)^2</math> | ||
#If <math>\chi_n^2<\chi_o^2</math> accept the move and let the new configuration become the old. If <math>\chi_n^2>\chi_o^2</math> then the move is accepted with probability <math>\exp(-(\chi_n^2-\chi_0^2)/2)</math> otherwise it is rejected. | #If <math>\chi_n^2<\chi_o^2</math> accept the move and let the new configuration become the old. If <math>\chi_n^2>\chi_o^2</math> then the move is accepted with probability <math>\exp(-(\chi_n^2-\chi_0^2)/2)</math> otherwise it is rejected. |
Revision as of 12:56, 21 February 2007
Reverse Monte Carlo (RMC) [1-4] is a variation of the standard Metropolis Monte Carlo (MMC) method. It is used to produce a 3 dimensional atomic model that fits a set of measurements (Neutron-, X-ray-diffraction, EXAFS etc.). In addition to measured data a number of constraints based on prior knowledge of the system (like chemical bonds etc.) can be applied. Some examples are:
- Closest approach between atoms (hard sphere potential)
- Coordination numbers.
- Angles in triplets of atoms.
The 3 dimensional structure that is produced by RMC is not unique, it is a model consistent with the data and constraints provided.
The algorithm for RMC can be written:
- Start with a configuration of atoms with periodic boundary conditions. This can be a random or a crystalline configuration from a different simulation or model.
- Calculate the total radial distribution function for this old configuration (C=Calculated, o=Old).
- Transform to the total structure factor:
- where Q is the momentum transfer and the number density.
- Calculate the difference between the measured structure factor and the one calculated from the configuration :
- this sum is taken over all experimental points is the experimental error.
- Select and move one atom at random and calculate the new (n=New) distribution function, structure factor and:
- If accept the move and let the new configuration become the old. If then the move is accepted with probability otherwise it is rejected.
- repeat from step 5.
When have reached an equilibrium the configuration is saved and can be analysed.
References
- R. L. McGreevy and L. Pusztai, "Reverse Monte Carlo Simulation: A New Technique for the Determination of Disordered Structures", Molecular Simulation, 1 pp. 359-367 (1988)
- [R. L. McGreevy, "Reverse Monte Carlo modelling", J.Phys.:Cond. Matter 13 pp. R877-R913 (2001)]
- [R. L. McGreevy and P. Zetterström, "To RMC or not to RMC? The use of reverse Monte Carlo modelling", Current Opinion in Solid State and Materials Science. 7 no. 1 (2003) pp. 41-47 Elsevier Science]
- [G. Evrard, L. Pusztai, "Reverse Monte Carlo modelling of the structure of disordered materials with RMC++: a new implementation of the algorithm in C++", J.Phys.:Cond. Matter 17 pp. S1-S13 (2005)]