RMC
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RMC++ in short |
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The spatial distribution of atoms and
molecules in a disordered material is described by the so-called partial pair
correlation functions ( Fig. 1 ).
Fig 1: Def. of partial pair correlation functions
The structure can be probed by the
diffraction (elastic scattering) of waves (neutrons or X-ray) of adequate wavelength
. The structure factors are obtained by sine-Fourier transform from the partial
pair correlation functions:
and the diffracted intensity
is a linear combination of the structure factors, weighted by the
concentrations c and scattering lengths b:
Direct inversion of this inverse problem needs several measurements
with isotopic substitution to disentangle the different partials. Furthermore,
it is plagued with the usual problems of instability, over- and
under-determination, and the Fourier ripples due to truncated Q-range (Fig 2).
Fig 2: forward and inverse problems
One alternative solution to direct
inversion, is to explore the parameter space and solve the direct problem as
often as required (Fig.3 ).
Fig 3: repeated computation of the forward problem
In MCGR [Monte Carlomodelling
of G(r)], the parameter space consists in the discretised pair correlation
functions. This technique allows to model partials while avoiding some of the
usual pitfalls of direct inversion (still, it does not guarantee that the
partials are physically sound).
Reverse Monte Carlo (RMC) goes one step further and considers the positions
of a set of N atoms (the configuration) as the parameter space, i.e.
it will derive a set of coordinates of the N atoms of the configuration that is
in agreement with the data ( Fig 4).
Systematic or exhaustive exploration of the
parameter space is ruled out by its mere dimension, and therefore a more
efficient way of sampling is needed in order to converge towards one
solution.
Both MCGR and RMC are based on the Metropolis algorithm , i.e. the
parameter space is explored by a `guided' random-walk. In short, the acceptance
of generated random moves in the parameter space is defined by the agreement of
the corresponding calculated data with the experimental data ( Fig 5).
Fig 5: Flowchart of the metropolis algorithm
In `standard' RMC++ (as in the existing
RMCA) the random move in the parameter space consists in the displacement of one
atom of the configuration. The atom moved, as well as the amplitude and the
direction of the move are defined randomly (within some predefined range).
One essential additional component in RMC is the introduction of constraints.
The Metropolis algorithm indeed produces solutions with the largest amount of
disorder, for a given set of constraints (including the Chi Squared).
Constraints can be introduced at 2 stages in the algorithm:
- at the definition of the move i.e. reject moves that bring
atoms to close to one another (distances of closest approach).
- at the definition of the Chi squared, by adding terms corresponding
to coordination constraints or potential.
Individual molecules can
also be mimicked by imposing a distance range between defined atoms in the
configuration (the so-called fixed neighbour constraints - FNC -).
The flowchart of the `standard' RMC algorithm appears in Fig 6.
Fig 6: Flowchart of the RMC algorithm
For more details, look at the references.
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RMC implementations |
- Original RMC: RMCA
The original RMC code was written in Fortran by Malcolm Howe and Jim Wicks in Oxford in 1992.
A series of RMC codes for different applications can be found on the ISIS website, including the RMCA code for structure modelling of disordered materials. - RMCA+FNC=rmc_fi
László Pusztai introduced the fixed neighbours constraints (FNC) to the code in Studsvik in 1996 (n.b. the RMCA code available from ISIS is the original one, without FNC's). Unfortunately, the FNC option of the rmc_fi code is not documented. - RMC++:
The RMC++ implementation has been designed to get rid of some outdated features of the Fortran code (noticeably the creation of a large array designed for minimizing memory usage). Special care has been taken in order to optimize the speed of execution. Additional priorities in the programme design were the upgradability and the documentation. For comparison purposes, RMCA has been used as a standard. - RMC++_new:
This is the upgraded version of RMC++, where structural changes was made to increase speed, new features (for example additional constraints) are implemented. - RMC++_multi:
This is the same in functionality as RMC++_new, but designed for multi-processor computers using multi-threading implemented according to POSIX standard. - RMC_POT
All the features of RMC_new and multi combined in one source code, compiler option decided, whether it is compiled for consecutive or parallel implementation. Increased functionality, such as bonded and non-bonded potential, and local invariance calulation.
RMC++ should look very familiar to RMCA users, since it uses the same
input and similar output format.
Look at the examples to see what
can be done with RMC++, and for the performance comparison with RMCA.
Look at the manual
for details concerning input and output.
Last modified 21/06/2010) by Orsolya Gereben
(comments welcome!)