Table of contents

This special issue contains a collection of papers reflecting the content of the second workshop on reverse Monte Carlo (RMC) methods, held in a hotel on hills overlooking Budapest in October 2003. Around forty participants gathered to hear talks and discuss a broad range of science based on the RMC technique in very convivial surroundings.

Reverse Monte Carlo modelling is a method for producing three-dimensional disordered structural models in quantitative agreement with experimental data. The method was developed in the late 1980's and has since achieved wide acceptance within the scientific community. It is particularly suitable for studies of the structures of liquid and amorphous materials, although it has also been used for a number of disordered crystalline systems. There is currently a great interest in the properties of disordered materials and this has produced a resurgence in methods for investigating their structures, with an increased number of high-quality instruments at central facilities for neutron and x-ray scattering from disordered materials. Methods such as RMC are currently in great demand for analysing the resulting total scattering and XAS data and the RMC methodology is actively being developed by a number of groups worldwide. Within this context, the RMC workshop was particularly timely, providing a forum for those workers in the field to take stock of past achievements and to look forward to future developments. It is our hope that the collection of papers within this special issue will also communicate this to the wider scientific community, by providing a balance between papers that have more of an introductory review flavour and those that concentrate on current state of the art research opportunities using the RMC method.

The order of the papers within this special issue reflects this balance. The first two papers are introductory reviews of the RMC method in general and as applied specifically to crystalline systems, respectively. The next, largest group describes a wide range of scientific applications of the RMC method within liquid, amorphous, crystalline and nanocrystalline materials. The final group of papers have a bias towards method development and testing, particularly with regard to the use of XAS data in RMC modelling.

We are very grateful to Institute of Physics Publishing for their willingness to publish the proceedings of this meeting in a special issue of Journal of Physics: Condensed Matter.

The basic reverse Monte Carlo algorithm, as applied primarily for the study of disordered systems, is introduced, using an example of a new reverse Monte Carlo computer code. RMC++ is a new implementation of the RMC algorithm in C++. Its main purpose is to provide the community with a fast, flexible and documented code for RMC simulations, compatible with the rmca distribution. The source code, the documentation and the executable files are made available through the Internet. The flexibility of the code is exemplified by the implementation of a 'molecular move' step in the Metropolis algorithm. This feature, as well as a performance comparison, is illustrated with simulations performed for molecular liquids such as CCl₄ and C₂Cl₄.

The reverse Monte Carlo (RMC) modelling method, although initially developed for interpreting structural data from liquids and amorphous materials, has been extensively applied to similar data from crystalline systems. This has been especially beneficial for materials which display a large amount of disorder. The work in this area will be briefly reviewed here, including a summary of the range of crystalline materials which have been studied using RMC modelling. Recent developments made specifically to improve the RMC modelling method for crystalline systems will also be described.

The structures of liquid Na_xCs_1−x alloys have been modelled using the reverse Monte Carlo method on the basis of x-ray and neutron diffraction data. The partial structure factors and partial radial distribution functions obtained were consistent with previous work. The average coordination numbers and coordination number distributions for the four distinct kinds of atom pairs have been calculated over a wide range of composition. Cs forms complex clusters near 85 at.% Na, near to the maximum of the density fluctuation in these alloys.

A model for the structure of copper–tin liquid alloy has been developed using the standard reverse Monte Carlo method. The interstitial void structure (size distribution) was analysed. The effects of various kinds of voids (small size and large size) on the interference function and radial distribution function were investigated. Predictions related to the formation of some ternary alloys by filling the interstices of the basic alloy were advanced.

In reverse Monte Carlo modelling, experimental information (i.e. diffraction data) and a priori information (i.e. constraints introduced in the algorithm) are partly redundant. The extent of this redundancy for ('fixed neighbour') constraints determining the molecular geometry is studied systematically in the typical case of liquid CCl₄. Results indicate that data with a very limited momentum transfer range are sufficient for deriving the intermolecular structure of such disordered systems when (intra)molecular geometry and intermolecular distances of closest approach are introduced by the appropriate algorithmic constraints.

Neutron diffraction results obtained for liquid CO and NO have been modelled by means of the reverse Monte Carlo method. Partial pair correlation functions, centre–centre pair correlation functions and the relative orientations of molecular axes as a function of distances between molecular centres have been calculated from the models. It was found that, contrary to earlier suggestions, well defined orientational correlations exist in liquid carbon monoxide, even at lower densities. For liquid nitric oxide, the existence of (NO)₂ dimers has been confirmed, but the 'cis-planar' structure of the dimers, suggested earlier, could not be made consistent with available diffraction data.

Neutron and x-ray diffraction data taken on aqueous HCl/DCl solutions at different concentrations (Triolo and Narten 1975 J. Chem. Phys.63 3624) have been analysed by means of the reverse Monte Carlo modelling technique. Partial radial distribution functions, numbers of neighbours and cosine distributions of bond angles have been calculated from the resulting particle configurations. Based on these results, the original interpretation of experimental data must be altered; Cl–H/D correlations, for instance, seem to have a more important role than had been thought previously.

A model of silica glass consisting of a fully connected corner-sharing network of SiO₄ tetrahedra is refined using neutron diffraction data and reverse Monte Carlo modelling. This model is then used to investigate optimal inter-tetrahedral Si–O–Si bond angle distributions. The distribution which is most consistent with the data is found to be centred around θ_Si−O−Si = 151.0° with a standard deviation of between 9° and 12°. Other recent determinations of the Si–O–Si bond angle distribution are in good agreement with this result.

The total structure factors, S(Q), obtained from high-energy x-ray and neutron diffraction measurements on vitreous SiO₂ (v-SiO₂) and vitreous GeO₂ (v-GeO₂) have been analysed by the reverse Monte Carlo (RMC) modelling technique to generate a three-dimensional structural model. The bond angle distributions and the ring size distributions from the model indicated that the sixfold ring and six- and sevenfold rings are dominant in v-SiO₂ and v-GeO₂, respectively. However, the fraction of threefold rings of Ge in v-GeO₂ is larger than that of Si in v-SiO₂ glass. These features are consistent with the published neutron diffraction and Raman scattering studies.

Reverse Monte Carlo (RMC) modelling is now widely used for obtaining structural models of amorphous materials that are in quantitative agreement with available experimental structure data. In the case of ion conducting glasses it has been possible to determine the distribution and local environment of the mobile ions. However, only little information has been gained concerning the conduction process in the glasses. In an attempt to at least partly overcome this problem we have applied the bond valence method to reverse Monte Carlo produced structural models of ion conducting glasses. Although we are still not able to directly elucidate the dynamical process, it has been possible to understand experimental conductivity data on a structural basis. In fact, the experimental conductivity and its associated activation energy can be predicted directly from the structural models and conduction anomalies, such as the well-known mixed alkali effect, can be understood using the bond valence approach. Here, we also discuss the possibility of exploring the conduction process in more detail by elucidating the number of available sites for the mobile ions and the dimensionality of the conduction pathways as a function of temperature.

We measured structure factors of hydrogenated amorphous Si by x-ray diffraction and analysed the obtained structures using a reverse Monte Carlo (RMC) technique. A small shoulder in the measured structure factor S(Q) was observed on the larger Q side of the first peak. The RMC results with an unconstrained model did not clearly show the small shoulder. Adding constraints for coordination numbers 2 and 3, the small shoulder was reproduced and the agreement with the experimental data became better. The ratio of the constrained coordination numbers was consistent with the ratio of Si–H and Si–H₂ bonds which was estimated by the Fourier transformed infrared spectra of the same sample. This shoulder and the oscillation of the corresponding pair distribution function g(r) at large r seem to be related to the low randomness of cat-CVD deposited a-Si:H.

We report a study of the displacive phase transition in SrTiO₃ using neutron total scattering with analysis by the reverse Monte Carlo method. The resultant configurations have been analysed in terms of bond distance and bond angle distribution functions, and using recently developed methods based on geometric algebra (GA). The resultant picture is that the short-range order in SrTiO₃ closely follows the long-range order, with very little disorder in the high-temperature phase. This result is in contrast to similar work carried out on the displacive phase transition in quartz (Tucker et al 2000 J. Phys.: Condens. Matter.12 L723–30). The differences between these two systems arises from the fact that there are many more Rigid Unit Modes in quartz, a point that is quantified by the GA analysis.

The local structure of nanoparticles can be determined by a direct interpretation of the first peaks observed in the radial distribution function obtained from powder diffraction data measured to high Q values. Single peak fits to these maxima yield the mean distances and width of the distribution in the first few neighbouring shells. An algorithm is presented to calculate the radial distribution function based on the simulation of a single nanoparticle. This algorithm allows a flexible description of nanoparticle size and shape. Based on this algorithm the structure of CdS:glutathione nanoparticles of less than 2 nm diameter is presented. This structure consists of a CdS core with tetrahedrally coordinated first neighbours and a shell of glutathione molecules. Within the core several stacking faults lie in different orientations with a stacking fault probability of 40%.

We have developed a successful method for structural refinement, based on the reverse Monte Carlo (RMC) algorithm, that can be applied simultaneously to diffraction and x-ray absorption (XAS) data. The method is designed to study molecular and condensed systems incorporating all of the advances related to the application of modern multiple-scattering (MS) codes and the n-body expansion for XAS data-analysis (GNXAS). Convergence properties and the dependence of the structural model upon specific features of the XAS spectra and calculation parameters are discussed. Limitations and advances of the method are elucidated in the light of the structural sensitivity of the XAS technique.

Over the past two decades x-ray absorption spectroscopy has proven to be a valuable tool for the study of the short-range order in a wide variety of materials, including disordered systems such as superionic conductors as well as glasses, amorphous and liquid systems in general. A number of methods have been proposed to analyse EXAFS data. However, in the case of disordered systems, only the ones taking the distribution of atomic environments into account should be retained. Molecular dynamics (MD) simulations are a valuable tool in this respect, as will be shown from results obtained in a supercritical aqueous solution.

We studied and evaluated the two simulation-assisted inverse methods: potential determination and the reverse Monte Carlo method. First, we proved that one step of the iterative scheme of Lyubartsev and Laaksonen corresponds to a perturbative potential determination of a dilute gas. Second, we performed reverse molecular dynamics simulations to study the dynamics of the system governed by the formal potential of χ², the acceptance-criterion function in the stochastic scheme. We found that the dynamics is unrealistic. These results indicate the need for caution when one attempts to calculate dynamical properties with reverse Monte Carlo-like methods.

A description of how the Monte Carlo simulation of a neutron scattering instrument can be combined with an optimization routine in order to design an instrument with the best performance is given. The parameters optimized here describe the converging guide exit of an instrument. A least-square fitting routine and a Metropolis algorithm (as used in the reverse Monte Carlo method) were used to reach this goal. The advantages and disadvantages of the use of these routines are discussed. A combination of both methods is proposed to find the optimal values with high accuracy. In general, the numerical optimization yields results that can sometimes not be obtained by varying the parameters by hand.