RMC
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RMC++ topics |
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This page describes specific topics relevant to the use of the RMC algorithm in general, and of RMC++ in particular.
The problems discussed below include:
· the requirements on r-space discretization due to the Q-range of the data
· the definition of the configuration size
· the renormalisation of the histograms
Link to topics discussed separately are the following:
· quadratic background correction
The first thing to do in a RMC run is to define the configuration that will
be simulated.
The choice of the configuration size, the Q-range of the data and
the histogram bin width are linked by mathematical requirements
intrinsic to the method, by finite computer time, and by what kind of
information is expected from the RMC simulation (this latter being actually the
most important).
The histograms bin width:
- The bin width dr of the histograms interval must be small enough to allow the approximation of the sine-Fourier transform for the data points up to the highest Q-value. This requirement reads:
- The bin width dr defines the resolution of the final g(r) partials. But logically, fine details in the g(r)'s can only be constrained by high Q-values (i.e. the relevance on the high resolution details in the g(r) should be estimated from the Q-range)
- What is more, the finer the r-bins, the fewer distance in the bins, and therefore, the greater the statistical uncertainty on the derived g(r) values. So for smaller bins only makes sense for large system.
- The usual bin size is 0.1 Å for a system containing <10000 particles. If there is more than one component, than the histogram counts will be divided among the partial histograms, so the statistic will be worse than for a mono-atomic system.
In short, the bin width dr should be chosen as large as possible, taking into account the maximum Q-values in the data and the desired degree of detail in the g(r) partials.
The configuration size:
- The configuration size (i.e.
the largest distance binned in the histograms) must be greater than the
minimal range after which there is complete disorder (i.e.
positions of atoms are not correlated and the partial pair correlation
functions take the value 1).
This is a mathematical requirement for the approximation of the sine-Fourier transform - The number of atoms must be as large as possible for the sake of statistical accuracy. The density being fixed, that means that the larger the configuration the better. From a `hand waving' argument, it can be seen that there is a relation between the statistical uncertainties on the derived g(r) values at small r-range, the bin width dr, and the total number of atoms N which reads:
Note that at the moment, there is no tool for
assessing the uncertainties on RMC results.
- On the other hand, the number of distances calculated for each move grows as N, and this is the most time-consuming part of the algorithm.
In short, the configuration size (and the optional xmax
value extending the histogram range beyond the half-size of the cubic box in
RMC++) must be chosen as large as possible, for the computing time available.
Note that in general it is not the long range order that fixes the
configuration size, but rather the statistical uncertainty due to small bin
counts at small distances. In other words, the configuration size is dictated
by the r resolution that one wants rather than from the long range order
of the material.
The renormalisation of the histograms:In RMC, the g(r) partials are estimated by counting and binning distances between atoms in the configuration. This operation requires the renormalization of the histograms defined by
where i is the bin index, r the corresponding radius (distance), ρ the number
density, dr the histogram bin width and S a surface factor. If the sphere of
radius r is contained in the configuration box, the the factor S is just the
surface of this sphere.
In RMCA only distances in this case are considered: if L is the (half) size of
the configuration box, histograms (and partials) are computed up to L. In other
words, for a central atom, only neighbours up to a distance L are used for the
computation of the histogram. This means that (4/3 πL³)/(8 πL³)=52.3 % of all
available (and computed) distances are effectively used for the g(r)
computation.
In RMC++, this range can be extended by using the appropriate surface factor.
The maximum distance range is √3 L, and there is an analytical formula for S up
to √2 L (see RMC++ manual).
This allows to use a smaller box with systems with long range order. But as
noted above, the limiting factor for the configuration size is usually the
number of centers.
Uncertainty relation for g(r) partials (handwaving argument)
For disordered materials, one focuses on
"local" order, i.e. how neighbouring atoms are arranged.
In RMC, the g(r) partials are estimated via the histograms of distances. The
number of distances in the spherical shell [r, r+dr] grows as r
squared, but 'locally' (i.e. at very short range) it is proportional to
the number of centers (i.e. number of atoms N).
This number of 'local' distances is shared between the histogram bins, whose
number is proportional to 1/dr.
The average number of local distances per bin is therefore proportional to N
dr, and the uncertainty on this number (standard deviation) is therefore
proportional to (N dr)^(1/2).
The partial g(r) is obtained by normalising the histograms, and the normalising
factor is proportional to dr. Consequently,the derived (absolute)
uncertainty (standard deviation) on the g(r) partial reads
However, for the relative statistical uncertainty on g(r) one has:
which indicates that for maximum precision, the number of atoms in the
configurations must be chosen as large as possible, and that any gain in r-resolution
is paid by a loss in the g(r) precision.
Last change by Orsolya Gereben 23/06/2010
(comments welcome!)