Top Rated Structure Of Materials De Graef Mchenry Solution Manual
Hedda Tillmon
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In the present study, we show that time-consuming manual tuning of parameters in the Rietveld method, one of the most frequently used crystal structure analysis methods in materials science, can be automated by considering the entire trial-and-error process as a blackbox optimisation problem. The automation is successfully achieved using Bayesian optimisation, which outperforms both a human expert and an expert-system type automation despite the absence of expertise. This approach stabilises the analysis quality by eliminating human-origin variance and bias, and can be applied to various analysis methods in other areas which also suffer from similar tiresome and unsystematic manual tuning of extrinsic parameters and human-origin variance and bias.
The physical properties of materials are governed by their crystal structure; thus, crystal structure analysis is an indispensable element in materials science research1,2. Compared to the drastic improvement in material fabrication and measurement throughput, the throughput of crystal structure analysis has not been improved because the analysis heavily relies on manual time-consuming trial-and-error methods3,4,5,6,7,8. Rietveld refinement or the Rietveld method9,10, one of the most widely used crystal structure analysis methods for powder diffraction data, such as X-ray diffraction (XRD) and neutron diffraction, faces this problem as well.
Rietveld refinement involves complex curve fitting using an experimental result and a simulation based on a physical model. The physical model comprises intrinsic parameters for governing crystal structure such as lattice constants and atomic positions and extrinsic parameters such as background signals and peak shape defined by instrumental resolution. The method is designed to minimise the difference between the observed and simulated results by updating the parameters in a given physical model until the difference reaches a threshold. All parameters need initial values, including categorical ones such as background and peak shape functions. For each numerical parameter, a binary parameter (fix or refine) to control the scope of the refinement needs to be set. The number of intrinsic parameters depends on the complexity of the crystal structure and ranges from 10 to 100, while that of extrinsic parameters is typically less than 20.
BBO refers to a problem setup, in which the objective function and/or constraint function are given by blackboxes, i.e., the analytic forms of the function are unknown. BBO has recently emerged as a popular concept in the machine learning community because it performs an important task of tuning hyperparameters in machine learning models automatically, which is termed as hyperparameter optimisation (HPO)15. HPO plays a vital role in machine learning; (1) it reduces the human effort necessary for applying machine learning, (2) it improves the performance of machine learning algorithms, and (3) it facilitates fair comparisons among the results of machine learning studies by providing the same level of tuning15. HPO is formalised as the maximisation of machine learning model performance in the hyperparameter configuration space. For this case, the relationship between a hyperparameter configuration and the machine learning model performance is considered as a blackbox. At present, Bayesian optimisation16, a global search method for BBO, is the most successful in this HPO problem17,18,19,20,21. This topic is closely related to our focus, i.e., to stabilise the analysis quality and bypassing unsystematic (and often unexciting) tasks in scientific studies. Herein, we realised that the concept of HPO can be applied to Rietveld refinement.
An optimiser iteratively samples a new promising configuration and then runs a Rietveld refinement software with the sampled configuration. A refined crystal structure that achieves the best fit is obtained as a final output.
where X is the configuration space and c(x) is the set of constraint functions. In the problem, a configuration x is considered as a variable and it affects Rwp through \(y_\boldsymbolx,i^\rmCalc\). The constraints are set to ensure that the refined parameters are physically valid, or sometimes to limit the search space and reduce manual trial-and-error attempts with a plausible explanation. In this study, we limit ourselves to introduce the latter type of constraints and we only introduce an essential constraint to keep human-origin biases to a minimum (see Methods section for more details).
It should be noted that small Rwp or GOF does not guarantee that the corresponding refined crystal structure would be appropriate for the obtained experimental data. The examination of the analysis result is still an important task for human experts29.
To evaluate the proposed BBO-Rietveld approach, we optimised parameter configurations of Rietveld refinement for XRD patterns of Y2O3, Dy0.5Sr0.5MnO3 (DSMO) and LiCoO2 which are chosen as benchmark materials. The tree-structured Parzen estimator (TPE)17,20, one of the variants of Bayesian optimisation, is used as the optimiser. The details of our optimisation system are provided in the Methods section. Here we discuss the results of Y2O3 and DSMO. Readers who are interested in the result of LiCoO2 should refer to the supplementary resources provided in the BBO-Rietveld repository (see Code availability). For each optimisation run, the optimiser iteratively evaluated 200 configurations including the initial 20 random configurations and returned the best one with the minimum Rwp. The search spaces for each parameter are shown in Table 1. The initial crystal structure models were taken from crystallographic information files (CIFs) on the GitHub repository of AutoFP47, presumably those used in ref. 43. To evaluate statistical property and reproducibility, optimisations have been performed 100 times with different random seeds each for Y2O3 and DSMO. Histograms of 100 Rwp values for both materials are shown in Fig. 2. Reference values for a human expert and AutoFP are also shown for comparison. The latter ones are taken from the previous study43.
Surprisingly, 90% and 99% of our optimisation runs for both test datasets, Y2O3 and DSMO, achieve better Rwp than the human expert and AutoFP, although no additional expertise in Rietveld refinement is used other than the conventional refinement procedure (see Methods section for details). Fig. 3 shows the refinement results of the best configuration derived by BBO for each benchmark material. We cannot find apparent flaws in the results refined with the best configuration. The changes of Rwp and goodness-of-fit (GOF) are shown in Fig. 4 (Y2O3) and 5 (DSMO). Both Rwp and GOF were improved as optimisation progresses, and in both metrics, our proposed method exceeded human expertise at around 100 evaluations. Rwp and GOF will be further improved if we continue the optimisation.
To visualise the refined structure features (x, y, z positions and Uiso of each atom) in two-dimensional space, we utilise the multidimensional scaling (MDS)48,49, one of the dimensionality reduction algorithms. MDS can represent high-dimensional data in a low-dimensional space by approximating the distance in the original space. The number of original structure feature dimension is 16 for Y2O3 and 20 for DSMO. The MDS visualisation for DSMO is shown in Fig. 6. Each point in this figure represents a refined crystal structure, and the distance between points indicates their similarity. The majority of the refined structures form a loose cluster at the lower-right side of the figure, which means they are similar to each other. In the cluster, there are three special points; the best one from BBO, the one from the human expert, and the one from BBO and very close to the human expert result as shown in Fig. 6. While the best structure obtained by BBO (Table 2) scores smaller Rwp than that of the human expert result (Table 2), Uiso of O1 atom in the BBO result is noticeably smaller than that of O2 atom, which violates a conventional criterion that Uiso of atoms with similar mass should be comparable. On the other hand, the third special point, obtained by BBO and close to the human expert result, passes the criterion (Table 2). This demonstrates that BBO efficiently obtains candidates with small Rwp values but additional examinations are still required to choose the most preferable one.
Each point in this figure represents a crystal structure refined with the 100 configurations optimised by BBO (i.e., 100 crystal structures with the best Rwp among 200 configurations from each of 100 runs) or the crystal structure refined with the best configuration by the human expert.
Next, we discuss the interesting outlier at the upper left of Fig. 6. The outlier (Table 2) has good converged Rwp, but the x position of O1 atom in the refined structure is considerably different from others. The difference exceeds the uncertainty calculated by the software, and, from the structural point of view, the positional shift corresponding to 5% of the lattice parameter would be enough to affect physical properties. This implies that the outlier corresponds to a local minimum not belonging to the cluster discussed above. In such a situation, multiple crystal structure candidates can explain the experimental data sufficiently with a similar level, which may affect the conclusion of a study or evoke a new discussion. Despite that it meets the constraint in the optimisation (positive Uiso), the outlier can be rejected due to the violation of the conventional criterion for Uiso as well as the best result by BBO. We believe that imposing constraints other than universal physical requirements in BBO can result in biased optimisation of results toward human expectations. Experts can examine the list of candidates using conventional criteria and other knowledge, and can eliminate inappropriate solutions at any time. This ability to propose multiple candidates without human effort and bias is a great advantage of automation and may lead to the discovery of hitherto unnoticed new knowledge beyond the standard practices of conventional manual tuning and the expert system simulating the practices. The use of empirical constraints in a manual analysis by experts also has the purpose of bounding the search space to reach a valid solution as soon as possible. However, Rietveld refinement using BBO can evaluate a large number of parameter combinations and suggest candidate structures much more efficiently than manual approach, that is, the constraints imposed to save time are no longer essential.
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