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## The Reverse Monte Carlo RMC method for modelling structural disorder

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Title: The Reverse Monte Carlo RMC method for modelling structural disorder

1
The Reverse Monte Carlo (RMC) method for
modelling structural disorder
László Pusztai Resarch Institute for Solid State
Physics and Optics, Hungarian Academy of
Sciences lp_at_szfki.hu
2
MAIN CONCERN STRUCTURE of LIQUIDS and
AMORPHOUS materials (and also, disorder in
CRYSTALLINE materials)
MAIN TOOL Structural MODELLING
3
WHY ???
4
WHY do we need (Reverse Monte Carlo) modelling?
• WHAT
• is Reverse Monte Carlo (RMC)modelling?

HOW can RMC lead us to new findings, in terms of
the structure of disordered media?
5
WHY (1) ?
• STRUCTURE the most important
• property

WE HAVE TO KNOW IT !!
DISORDERED SYSTEMS no unit cell (no
Bragg-peaks)
DIFFRACTION DATA provide highly averaged
information S(Q) only BUT still the most
important piece of information !
6
Information from a diffraction experiment
structure factor, S(Q)
Do we understand the STRUCTURE of this material,
based on its S(Q)??
7
In real space the Fourier-transform of S(Q), the
pair distribution function, g(r)
Do we understand the STRUCTURE of this material,
based on its g(r)??
8
WHY (2) ?
LIMITATIONS ERRORS of measured data
? ? ?
DIFFICULTIES in obtaining reliable -
structure factors, S(Q) - pair correlation
functions, g(r)
SUITABLE STRUCTURAL MODELS can provide more
confidence in experimental data ?
Consistent system of S(Q), g(r) and model
9
Experiment total structure factor(s)
Model (3D, thousands of atoms)
Pair correlation function(s)
10
Which measurement is more reliable? A suitable
model can tell !
11
WHY (3) ?
DIFFRACTION (particularly of neutrons) IS
EXPENSIVE
? ? ?
It is a MUST that we use EVERY SINGLE BIT of data
(And NOT to be satisfied with the usual outcome,
the one single number the coordination number.)
12
WE NEED
• Structural models that are consistent with
diffraction data the total structure factor(s)
• - within their uncertainties!

Standard simulation techniques (MC, MD) cannot
provide these in general (interatomic potential
functions!)
Inverse methods
13
What is an INVERSE method?
• DIRECT method
• Structure factor ?direct Fourier-transform
• ? pair correlation function
• Q-space ?? r-space

INVERSE method Generate g(r) ?
Fourier-transform ? ? see if SC(Q) is
consistent with SE(Q)
Only r-space ?? Q-space occurs !
14
Reverse Monte Carlo - pictorial
15
RMC algorithm (1) (McGreevyPusztai, Molec.
Simul. 1, 359 (1988)
1. Initial configuration, with periodic
boundaries Ngt4000
2. Calculate partial pair correlation functions
from the coordinates
3. Compose total pcfs, according to the
scattering powers of atoms
And generate the correspondig total structure
factors
16
RMC algorithm (2)
4. Calculate the difference between Calculated
and Experimetal S(Q)
5. Move one atom at random.
6. Calculate total structure factor(s) and ?2 for
the new position.
7. If ?2old gt ?2new then accept move. If not
then accept move with the probability
8. Repeat from step 5.
17
RMC level of consistency with experiment (1)
Liquid Ar, Yarnell et al. (1973)
18
RMC level of consistency with experiment (2)
H2O, X-rays (NartenLevy, 1971)
D2O, neutrons (Soper et al., 1997)
19
RMC details (1)
• Number of particles (system size)
• Has to be big! (say, min. 5000)

-Particle sizes (cut-offs) Not necessary but
-s-parameter Related to the assumed expt.
error, but should be considered as a control
parameter.
20
RMC details (2)
• Modelling g(r) or S(Q)?
• The process should start as close to the source
as possible, so

-Necessary Q-range?? Maybe surprising, but its
smaller than most of us would think
21
On the Q-range necessary
Amorphous Si-like model data, direct FT, with
decreasing Q(max).
Amorphous Si-like model data, RMC,
using identical Q-ranges..
22
CONSTRAINTS (1)
Even perfect data (taken over an infinitely large
Q-range, with infinitely good accuracy) are
ambiguous concerning correlations beyond
two-particle ones
and this is why constraining the configuration
space available for a given model is necessary
so that we can deal with practical problems.
In particular not even the molecular structure
is determined unequivocally by diffraction data
therefore, even for the case of simple molecules,
this question has to be addressed.
23
CONSTRAINTS (2)
• Density
• Particle sizes (packing fraction)
• Molecular structure

Diffraction data
Geometrical constraints (possibly based on
other experimental evidence)
24
UNIQUENESS (of..??) (1)
• RMC models are NOT unique!
• There are many particle arrangements
• (configurations) that are consistent with a
paricular (set of) diffraction data.

-Is this a particular feature of the
technique?? Or is it something to do with THE
DATA THEMSELVES ???
(Who is to blame if identically perfect fits hide
totally different atomic level structure?)
IS THIS A PROBLEM ????
25
UNIQUENESS (of..??) (2)
An example Ni81B19 metallic glass
However the two models ARE identical at
the level of 2- and 3-particle correlations !!
26
UNIQUENESS (conclusions)
• RMC models are NOT unique!
• But this can be viewed as an ADVANTAGE, instead
of as a disadvantage
• RMC is a method which is capable of tackling the
non-uniquess of the DATA.

NOT A BLACK BOX!!
27
Are partial g(r)s unique ?
NO! In general (see molten CuBr, water, etc),
even if one has the sufficient number of
independent measurements, there are several
different sets of ppcfs. (Finite data of finite
accuracy - not a magic.)
And this is yet another reason why total S(Q)
should be modelled.
28
WHY RMC ?
• The full diffraction data is exploited in a
quantitative manner.
• Interatomic potentials are not necessary
(although, can be included as constraints).
• Different types of data (ND, XRD, EXAFS,) can be
combined naturally.
• RMC models are always self-consistent and they
always correspond to physically possible particle
arrangements.
• Some hope even when the number of available data
sets is less than the number of partials.

29
Experiment total structure factor(s)
RMC
Model (3D, thousands of atoms)
Pair correlation function(s)
30
Some applications of RMC
• Liquid noble gases (liquid Ar was first)
• Liquid elements
• Molten salts (and solid superionic conductors)
• Amorphous alloys (metallic glasses)
• Covalent glasses
• Amorphous tetrahedral semiconductors (and
amorphous carbons)
• Molecular liquids simple and H-bonded
•  etc.

31
Metallic glasses Ni62Nb38 (1)
32
Metallic glasses Ni62Nb38 (2)
33
Metallic glasses Pd52Ni32P16 (structural
relaxation)
34
MOLECULAR STRUCTURE via fixed neighbours
constraints (FNC)
• Rigid vs. FLEXIBLE molecules
• For well-defined intramolecular distances, one
should measure
• (and use RMC) up to Q? (with high accuracy)
• Since this is impossible, the current practice is
the subtraction
• of the intramolecular part of the structure
factor
• However, this is an avoidable source of errors,
IF flexible
• molecules can be applied in RMC.

FNCs are special neighbour list which fix the
NUMBER and IDENTITY of neighbours of a given
TYPE that must be kept within SPECIFIED DISTANCE
LIMITS from a given centre.
35
Worked example liquid water (1)
1500 to 2000 FLEXIBLE molecules
Molecular structure (by FNCs) O-H distance
0.9 to 1.1 Å H-H distance 1.5 to 1.65 Å
Modelling 1, 2, 3 total structure factors (and,
for cross-checks, sets of partial g(r)s).
36
Worked example liquid water (2)
? ? ?
37
Worked example liquid water (3)
(Some data may be even harder to take/handle than
the most pessimistic person of us would imagine!)
38
Amorphous silicon, a-Si (1)
39
Amorphous silicon, a-Si (2)
??
? ?
40
Liquid (?) phosphorus at high P and T
Synchrotrone X-rays, Katayama et al, Nature, 2000
41
Liquid tungsten-hexachloride, WCl6 ionic or
molecular?
A NICE MOLECULAR LIQUID!
42
RMC modelling of powder diffraction data (RMCPOW)
(1)
There is a lot of information under the
Bragg-peaks!
(Which is taken as background by
standard methods like Rietveld.)
RMC may be able to help, BUT
43
RMC modelling of powder diffraction data (RMCPOW)
(2)
but one has to do both powder diffraction and
RMC differently!
POWDER DIFFRACTION has to be a total
scattering type experiment (to be carried out
like a liquid/amorphous measurement).
Reverse Monte Carlo has to be able to calculate
S(Q) DIRECTLY FROM THE COORDINATES.
RMC for POWder patterns RMCPOW MellergårdMcGre
evy (1999), Acta. Cryst. A55, 783
44
RMC modelling of powder diffraction data (RMCPOW)
(3)
RMCPOW models nuclear and magnetic, Bragg- and
diffuse scattering SIMULTANEOUSLY.
• Some applications
• MnO
• Perovskite structures
• Frustrated antiferromagnets
• Diffusion pathways in CdHSO4 (and similar
crystals)

http//www.studsvik.uu.se (software)
45
The different levels of applying Reverse Monte
Carlo modelling
Interpretation of diffraction (and/or related)
data, in terms of 3D models.
Evaluation of partial pair distribution functions.
Integral part of data evaluation (assessing data
quality helping difficult stages like density
determination )