Full length articleA new approach to the analysis of short-range order in alloys using total scattering
Graphical abstract
Introduction
The effects of short-range order on physical properties have long been noted in a variety of alloys systems. Thermodynamic discontinuities [1] in nickel chromium alloys, along with observed anomalies in electrical resistivity [2], [3] have led to the description of a Komplex (K-) state containing a different order to that expected for a random solid solution. Similarly, changes in magnetic properties, such as an increase in the observed spin wave stiffness of Ni3Mn [4] provide evidence for short-range ordering. In addition, short-range order has been shown to influence dislocation motion [5], [6] and has been observed in a number of systems used for structural applications, including nickel superalloys [7]. In spite of this influence on numerous materials properties, the study of short-range ordering in alloy systems has been largely abandoned over the last 20 years or so. This is due mainly to the lack of experimental evidence available from conventional diffraction experiments and the difficulties associated with the calculation of the ordering parameters from the diffuse scattering observed in single crystal experiments.
In this paper, a methodology is presented by which total scattering techniques may be used for the direct observation of short-range order in crystalline alloy systems. Simulated supercells based on the face-centred cubic (fcc) structure, containing various types and degrees of short-range order, are used to explore the information content of pair distribution functions, which are experimentally accessible through Fourier transformation of total scattering data. The atomistic models that may be obtained from the analysis of such experimental data can be used to provide quantitative information about the types of short-range order present, and approaches by which this can be achieved are discussed.
Standard diffraction experiments, particularly of powder samples, derive structural information from the characteristic Bragg pattern [9]. Inherently, this analysis is based on the long-range average structure of the material and information on the short-range ordering that may be present in the system is lost in the averaging. The study of amorphous materials, which by definition lack the long-range order that leads to Bragg peaks, motivated the development of the total scattering technique, in which the Bragg and diffuse scattering from a sample are measured and analysed simultaneously. The technique has, more recently, been applied with great success to study many crystalline and disordered-crystalline systems [10] providing insight into the local information through analysis of deviations from the average structure. The basic scattering function for neutrons [11] is given as:where i and j are atomic labels, r the instantaneous position of an atom, b the atomic scattering length, N the number of atoms and Q the magnitude of the scattering vector Q i.e. the momentum transfer of the incident radiation. An equivalent expression for X-rays can be derived, replacing the neutron scattering length bwith the Q-dependent atomic form factor. For a periodic crystal this equation simplifies to the Bragg condition. In the case of an amorphous glass this may be written as:Separating the self-scattering components:which can be recast in the form:where ci is the concentration of species i and the total scattering function, F(Q), is given by:in which ρ0 is the average density of the structure and:where the gij(r) terms are the partial pair distribution functions. The function G(r) is often known as the pair distribution function (PDF). The inverse Fourier transform of Eq. (5), used for the calculation of a PDF, is given by:
Alternative normalisations of the PDF are sometimes used for convenience, to emphasise specific features of the function related to the properties and length-scales of interest. A full description of the other formulations of the PDF has been provided by Keen [12]. Unless otherwise stated, the terms G(r) and PDF are used interchangeably here.
The partial PDFs are described mathematically as:where nij(r) is the number of atoms lying within r and r+dr, and ρi = ci ρ0, where ρ0 is the density of the substance and ci the concentration of species i. Critically, this description is as applicable to periodic crystals as to amorphous materials.
The PDF, by definition, is a weighted histogram of the distribution of distances between atoms in the structure. A peak in the PDF will indicate the average distance of one atom relative to another; its width being dependent on the distribution of the inter-atomic distances (thermal vibration or static displacements) and its area being governed by the number and scattering length of the correlating atoms.
As is apparent from Eqs. (5), (7)), the PDF and F(Q) are effectively different representations of the same information (both contain the local structural information lacking in the Bragg peaks alone). Here the PDF is considered, as it is intuitively understandable as a visual representation of the local structure, but an equivalent analysis could be achieved through consideration of the F(Q).
Whilst other local structural probes exist, such as EXAFS and NMR spectroscopy, they are of limited use for the analysis of short-range order in alloys systems, owing to the fact that ordering information for a large number of coordination shells can be extracted from a PDF, whilst EXAFS and NMR provide information for only the first couple of shells.
It is straightforward to achieve a qualitative understanding of the information provided by a PDF, however producing structural models is more involved. There are two popular approaches. The first, often known as ‘small-box’ modelling, utilises tools such as PDFgui [13], and is analogous to Rietveld refinement though the models are constrained by the PDF instead of the Bragg diffraction pattern. This technique is not well suited to the characterisation of short-range order in alloys since it produces a crystallographic description of a structure (i.e. one that comprises cell parameters, thermal parameters, fractional site occupancies), albeit one that is biased towards the local, rather than the average, structure. The second, a ‘large-box’ modelling technique using the reverse Monte Carlo [14] algorithm, is arguably better suited for the analysis of ordering in alloy systems as it is unconstrained by symmetry and produces large (>10,000 atom) models that provide the ability to probe the ordering across appropriate length scales.
Traditionally, analysis of total scattering data involves visual inspection of the scattering functions and the use of peak fitting to obtain information about the arrangement of the first one or two coordination shells. It is also common to use molecular dynamics or hard-sphere Monte-Carlo to calculate structural models, from which theoretical total scattering functions can be calculated and compared with the observed data. Analysis of the system is then possible only if the two are found to be in good agreement. The tuning of initial potentials to create a structural model that more accurately reflects the data is theoretically possible, but the direct effect of the potentials on the diffuse scattering is often difficult to quantify and the process of fine-tuning is laborious. To this end, the empirical potential structure refinement (EPSR) technique [15] combines direct Monte-Carlo simulations with the refinement of an empirical potential in order to obtain calculated scattering functions that are in good agreement with the observed data. In contrast, the reverse Monte Carlo (RMC) technique [14] is an iterative process in which goodness-of-fit parameters, which describe the statistical quality of the agreement between the model and the experiment, are minimised.
In the RMC method, an initial arrangement of atoms is created that reflects the average structure of the system, as determined by analysis of the Bragg diffraction data. The PDF calculated from this will consist of a series of delta functions, as every atom will be on its ideal position, with no allowance made for positional variations associated with thermal vibrations. However, a more realistic initial model can be obtained by applying small off-site displacements to each atom according to a pseudo-random Gaussian distribution generated using the Box-Muller method [16]. From this, a number of scattering functions may then be calculated and compared with the experimental data.
In general, the quality of the fit is defined by an agreement factor, χ2, which is calculated as follows:where j indicates a data point, the experimental value at that point, the value calculated from the refined box and σj the weighting factor for an individual data set. The χ2 parameters for different data sets are then summed as follows:
Any number of data sets can be summed, allowing the independent fitting of multiple different data sets simultaneously. The implementation of physical constraints (e.g. ) helps to ensure physically realistic results are obtained from the refinement.
The refinement makes random atomic translations and swaps and calculates the change in the scattering functions that result. Changes that lower will be automatically accepted, whilst those that increase by will be accepted with a probability of:
By accepting a certain number of ‘bad’ moves, the system may be prevented from becoming stuck in false minima. The result of the RMC refinement is an atomistic model that adequately fits all datasets and obeys all applied constraints.
The RMC algorithm has been implemented in a number of software packages (for example [8], [15], [17], [18]) for the analysis of local structure and disorder. However, for accurate analysis of the structure of crystalline alloy systems, it is necessary to explicitly fit the Bragg pattern as well as the total scattering data. The RMCProfile [8] code, developed and maintained by some of the authors of this work, which incorporates this functionality, is therefore optimised for studies of this nature.
Section snippets
Super-lattice structures in binary alloys
The phase diagrams of many alloy systems show regions where long-range ordered structures form, that differ from the disordered parent phases only in that the atoms are not randomly distributed but occupy specific lattice sites, either within a single cell or a super-cell of the parent phase. This results in the breaking of the symmetry of the parent structure and the appearance of super-lattice peaks in the diffraction pattern. To demonstrate the efficacy of analysing short-range order though
The measurement of high-quality PDFs
Owing to the low intensity of the diffuse scattering signal, compared with the much stronger Bragg peaks, high quality data are required in order to obtain a statistically significant measurement of the diffuse scattering in a total scattering experiment. Care must be taken in both the acquisition of the data and the subsequent correction, processing and analysis.
It can be seen from Eq. (7) above, that the integral should be calculated over an infinite Q-range, and while this is obviously not
Discussion and conclusions
It is clear from the preceding sections that the use of total scattering for the analysis of short-range order in crystalline alloys could provide valuable insight into the structure and properties of many complex alloy systems. To increase confidence in the results obtained in such a manner, it is essential to establish a systematic methodology for the collection and analysis of data.
Acknowledgements
This work was supported by the STFC ISIS Facility and the Rolls-Royce plc/EPSRC Strategic Partnership under EP/H022309/1 and EP/M005607/1. The authors gratefully acknowledge STFC for the provision of beamtime at Diamond Light Source Ltd (EE10354, EE11665) and the ISIS Facility (RB1510579, RB1520332), and thank Dr Stephen Hull for useful discussions.
References (33)
- et al.
Spin-wave dispersion relation in partially ordered Ni3Mn
J. Magn. Magn. Mater.
(1979) - et al.
Kinetic study of short range order in alpha-CuAu alloys
Acta Metall.
(1978) - et al.
Specific-heat anomalies in solid solutions of chromium and molybednum in nickel - evidence for short-range order
J. Inst. Metals
(1966) Uberstruktur und K-Zustand im system nickel-chrom
Z. Fur Met.
(1958)- et al.
The dependence of the electrical resistivity on short-range order
J. Phys. C. Solid. St. Phys.
(1971) On the strength of solid solution alloys
Acta Metall.
(1954)Solid solution strengthening
- et al.
Characterisation of short-range order using dislocations
Zeitscrift fur Met.
(2006) - et al.
RMCProfile: reverse Monte Carlo for polycrystalline materials
J. Phys. - Condens. Matter
(2007) The diffraction of short electromagnetic waves by a crystal
Proc. Camb. Philos. Soc.
(1913)
New insights into complex materials using Reverse Monte Carlo modeling
Annu. Rev. Mater. Res.
Dispersion of Rontegen rays
Ann. Phys.
A comparison of various commonly used correlation functions for describing total scattering
J. Appl. Crystallogr.
PDFfit2 and PDFGui: computer programs for studying nanostructure in crystals
J. Phys. Condens. Matter
Reverse Monte Carlo simulation: a new technique for the determination of disordered structures
Mol. Simul.
Tests of the empirical potential structure refinement method and a new method of application to neutron diffraction data on water
Mol. Phys. Int. J. A. T. Interface Between Chem. Phys.
Cited by (49)
Graphene enables equiatomic FeNiCrCoCu high-entropy alloy with improved TWIP and TRIP effects under shock compression
2024, Journal of Materials Science and TechnologySolute-strengthening in metal alloys with short-range order
2024, Acta MaterialiaAtomistic simulations reveal strength reductions due to short-range order in alloys
2024, Acta MaterialiaOrientation dependence of the effect of short-range ordering on the plastic deformation of a medium entropy alloy
2023, Materials Science and Engineering: AUncertainty quantification of predicting stable structures for high-entropy alloys using Bayesian neural networks
2023, Journal of Energy Chemistry
- 1
Now at Spallation Neutron Source, One Bethel Valley Road, Oak Ridge, Tennessee, USA.