Determining the crystal structure of a synthesized material is crucial, and X-ray diffraction (XRD) analysis is a widely used technique for this purpose. Traditionally, identifying a crystal structure from an XRD pattern involves searching material databases like the Powder Diffraction File (PDF) and Inorganic Crystal Structure Database (ICSD) for matches. However, this approach fails when the XRD pattern corresponds to an unknown or novel material. While advanced techniques like machine learning models and Rietveld refinement have been developed to improve crystal structure identification, they still heavily rely on existing databases and often struggle with unknown structures. Rietveld refinement, in particular, requires a good initial structure, and if the initial structure is significantly different from the actual structure, refinement can fail. This paper addresses this limitation by proposing a database-independent inverse design method for directly creating a crystal structure that reproduces a target XRD pattern.

Existing methods for crystal structure determination from XRD patterns primarily rely on database searches. Machine learning approaches have been proposed to predict crystal systems and space groups from XRD patterns, but these methods are limited by the available data in the database. Other methods, such as those proposed by Griesemer et al. and Dong et al., attempt to improve the efficiency and accuracy of database searches or predict XRD patterns from chemical composition, respectively. However, these approaches still depend heavily on the completeness and accuracy of existing databases, making them unsuitable for determining the structures of novel materials. Rietveld refinement offers a way to refine a candidate structure to match the measured XRD pattern, but it requires a good initial guess and expertise in parameter optimization. The Black-Box Optimization Rietveld (BBO-Rietveld) method automates this process, but still depends on an initial structure from a database. This study aims to overcome these limitations by creating a database-independent method.

The proposed method, Evolv&Morph, consists of two main stages: an evolutionary algorithm and crystal morphing, both supported by Bayesian optimization. The evolutionary algorithm generates a large number of candidate crystal structures using genetic operators like crossover and mutation. The crossover operator combines parts of two parent structures to create offspring, while the mutation operator introduces variations in a single parent structure. These genetic operators explore a wide range of structural possibilities. The fitness of each candidate structure is evaluated based on the cosine similarity (S_cos) between its simulated XRD pattern and the target XRD pattern. Structures with higher S_cos scores are favored in subsequent generations. First-principles calculations using the VASP code with the PBEsol functional are employed to refine the structures generated by the evolutionary algorithm. This step helps to ensure the generated structures are physically reasonable and stable. Following the evolutionary algorithm, crystal morphing is used to further refine the best-performing structures from the evolutionary algorithm. Crystal morphing uses the SOAP (Smooth Overlap of Atomic Positions) distance as a metric to interpolate between two input structures, creating intermediate structures. Bayesian optimization, implemented using the GPyOpt code, guides this process by selecting the morphing parameters that maximize S_cos. Two strategies are used for Bayesian optimization: greedy optimization and all-pairs investigation. Greedy optimization iteratively selects the two best structures and searches for intermediate structures with higher S_cos. All-pairs investigation explores intermediate structures between all pairs of structures in the current set. Finally, Rietveld refinement and symmetrization are used for additional fine-tuning of the structures.

Evolv&Morph successfully created crystal structures that reproduced the target XRD patterns for sixteen different material systems. For twelve systems with simulated target XRD patterns, the method achieved a cosine similarity of approximately 99%. For four systems with experimentally measured powder XRD patterns (after background removal), the cosine similarity exceeded 96%. These results demonstrate the effectiveness of Evolv&Morph in reproducing XRD patterns and identifying unknown crystal structures without relying on databases. The study also highlights the potential of Evolv&Morph as an inverse design tool, where material properties can be specified as the optimization target. The success rates obtained across diverse crystal systems, including binary and ternary compounds with various structures, underscore the robustness and general applicability of the proposed method. The use of first-principles calculations, while adding computational cost, helps ensure the stability and physical validity of the predicted structures. The combination of evolutionary algorithms, crystal morphing, and Bayesian optimization provides a powerful approach for searching the large and complex space of possible crystal structures. The high cosine similarity achieved indicates excellent agreement between the predicted and experimental XRD patterns.

The success of Evolv&Morph in reproducing target XRD patterns for a diverse range of materials demonstrates its potential as a powerful tool for crystal structure determination and material design. The database-independent nature of the method is particularly significant, as it allows the identification of novel materials with previously unknown structures. The ability to define specific material properties as the optimization target opens up new avenues for inverse design, where the desired properties are specified first, and then materials with the target properties are systematically searched for. The method's ability to handle both simulated and experimental XRD data shows its applicability to both theoretical and experimental studies. While the inclusion of first-principles calculations is computationally expensive, the high accuracy and reliability of the results justify the additional computational cost. Future research could focus on optimizing the computational efficiency of the algorithm to handle larger and more complex systems, exploring alternative optimization algorithms and fitness functions, and integrating machine learning potentials to further speed up calculations.

This study presented Evolv&Morph, a novel database-independent method for creating crystal structures that reproduce target XRD patterns. The method effectively combines evolutionary algorithms, crystal morphing, and Bayesian optimization to efficiently search the vast space of possible structures. High accuracy in reproducing XRD patterns was demonstrated for diverse material systems. Evolv&Morph presents a significant advancement in crystal structure determination, offering a powerful tool for both experimental and theoretical material science and opening opportunities for inverse material design.

The computational cost associated with first-principles calculations can be significant, particularly for large and complex systems. The accuracy of the predicted structures depends on the accuracy of the first-principles calculations and the employed interatomic potentials. While the method has demonstrated success for a range of materials, its performance on highly complex systems with numerous atoms and long-range interactions remains to be fully explored. Further optimization and development of the algorithm could potentially improve its efficiency and applicability to a wider range of materials.