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Reverse Monte Carlo

2007-02-26T17:44:03Z

Per:

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 <math>g_o^C(r)</math> for this old configuration (''C''=Calculated, ''o''=Old).
#Transform to the total [[Structure factor | structure factor]]:
#:<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 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>:
#:<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.
#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>
#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 \geq \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.
#repeat from step 5.
When <math>\chi^2</math> have reached an equilibrium the configuration is saved and can be analysed.

== References ==
#[http://dx.doi.org/10.1080/08927028808080958 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)]
#[http://dx.doi.org/10.1088/0953-8984/13/46/201 R. L. McGreevy, "Reverse Monte Carlo modelling", J.Phys.:Cond. Matter '''13''' pp. R877-R913 (2001)]
#[http://dx.doi.org/10.1016/S1359-0286(03)00015-9 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]
#[http://dx.doi.org/10.1088/0953-8984/17/5/001 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)]

Reverse Monte Carlo

2007-02-21T10:56:37Z

Per:

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 <math>g_o^C(r)</math> for this old configuration (''C''=Calculated, ''o''=Old).
#Transform to the total structure factor:
#:<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 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>:
#:<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.
#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>
#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.
#repeat from step 5.
When <math>\chi^2</math> have reached an equilibrium the configuration is saved and can be analysed.

----

== References ==
#[http://dx.doi.org/10.1080/08927028808080958 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)]

Reverse Monte Carlo

2007-02-21T10:39:57Z

Per:

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 <math>g_o^C(r)</math> for this old configuration.
#Transform to the total structure factor:
#:<math>S_o^2 (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> and 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>:
#:<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.
#Select and move one atom at random and calculate the 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>
#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.
#repeat from step 5.
When <math>\chi^2</math> have reached an equilibrium the configuration is saved and can be analysed.

----

== References ==
#[http://dx.doi.org/10.1080/08927028808080958 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)]

Reverse Monte Carlo

2007-02-21T10:39:25Z

Per:

Reverse Monte Carlo (RMC) [1] 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 <math>g_o^C(r)</math> for this old configuration.
#Transform to the total structure factor:
#:<math>S_o^2 (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> and 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>:
#:<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.
#Select and move one atom at random and calculate the 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>
#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.
#repeat from step 5.
When <math>\chi^2</math> have reached an equilibrium the configuration is saved and can be analysed.

----

== References ==
#[http://dx.doi.org/10.1080/08927028808080958 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)]

Reverse Monte Carlo

2007-02-19T17:46:27Z

Per:

Reverse Monte Carlo (RMC) [1] 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 chemocal bonds etc.) can be applied. Some examples are:

#Closest approach between atoms (hard sphere potential)
#Coordination numbers.
#Angels in triplets of atoms.

The algorithm for RMC can be written:

1. 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.
2. Calculate the total radial distribution function <math>g_o^C(r)</math> for this old configuration.
3. Transform to the total structure factor:
<math>S_o^2 (Q)-1=\frac{4\pi\rho}{Q}\int\limits_{0}^{\inf} r(g_o^C(r)-1)sin(Qr)\, dr</math>

where ''Q'' is the momentum transfer <math>\rho</math> and the number density.
4. 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>

this sum is taken over all experimental points <math>\sigma</math> is the experimental error.
5. Select and move one atom at random and calculate the 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>
6. 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> otherwiase rejected.
7. repeat from step 5.
When <math>\chi^2</math> have reached an equilibrium the configuration is saved and can be analysed.

----

== References ==
#R.L.McGreevy and L. Pusztai, ''Mol. Simulation,'' '''1''' 359-367 (1988)

Reverse Monte Carlo

2007-02-19T17:31:47Z

Per:

Reverse Monte Carlo (RMC) [1] 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 chemocal bonds etc.) can be applied. Some examples are:

#Closest approach between atoms (hard sphere potential)
#Coordination numbers.
#Angels in triplets of atoms.

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 <math>g_o^C(r)</math> for this old configuration.
#Transform to the total structure factor:
<math>S_o^2 (Q)-1=\frac{4\pi\rho}{Q}\int\limits_{0}^{\inf} r(g_o^C(r)-1)sin(Qr)\, dr</math>

where ''Q'' is the momentum transfer <math>\rho</math> and 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>:
<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.
#Select and move one atom at random and calculate the 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>
#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> otherwiase rejected.
#repeat from step 5
When <math>\chi^2</math> have reached an equilibrium the configuration is saved and can be analysed.

----

== References ==
#R.L.McGreevy and L. Pusztai, ''Mol. Simulation,'' '''1''' 359-367 (1988)

Reverse Monte Carlo

2007-02-19T17:24:45Z

Per:

Reverse Monte Carlo (RMC) [1] 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 chemocal bonds etc.) can be applied. Some examples are:

#Closest approach between atoms (hard sphere potential)
#Coordination numbers.
#Angels in triplets of atoms.

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 <math>g_o^C(r)</math> for this old configuration.
#Transform to the total structure factor:
<math>S_o^2 (Q)-1=\frac{4\pi\rho}{Q}\int\limits_{0}^{\inf} r(g_o^C(r)-1)sin(Qr)\, dr</math>

where ''Q'' is the momentum transfer <math>\rho</math> and 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>:
<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.
#Select and move one atom at random and calculate the 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>

----

== References ==
#R.L.McGreevy and L. Pusztai, ''Mol. Simulation,'' '''1''' 359-367 (1988)

Reverse Monte Carlo

2007-02-19T16:55:07Z

Per: