Machine learning to predict new quasicrystals

1. Background 
Quasicrystals do not have the translational symmetry of ordinary crystals but have a high degree of order in their atomic arrangement. The first quasicrystal was discovered in 1984 by Israeli metallurgist Dr. Daniel Shechtman (Nobel Laureate in Chemistry in 2011). Since then, about 100 stable quasicrystals have been found so far. In the history of quasicrystal research, the discovery of new quasicrystals has led to the discovery of new physical phenomena such as anomalous electronic properties, insulating behavior, valence fluctuations, quantum criticality, and superconductivity. However, in recent years, the pace of discovery of new quasicrystals has slowed down significantly (Fig. 1). This trend is mainly due to the lack of clear design guidelines for the synthesis of new stable quasicrystals.

2. Prediction of chemical compositions to form quasicrystals
We tackled the problem of predicting quasicrystals using a simple workflow of supervised learning (Fig. 2). The input variable of the model is the chemical composition. The output variables represent class labels indicating "quasicrystals", "approximant crystals", and "others" including ordinary periodic crystals. As training data, we used the chemical compositions of quasicrystals, approximants, and ordinary crystals that have been discovered so far. For the data analysis, we introduced XenonPy (*2), an open-source software developed by the group at the Institute of Statistical Mathematics. We evaluated the predictability of machine learning models for the three-class classification task. The predicted quasicrystalline phases were compared with known experimental phase diagrams, and the prediction accuracy reached nearly 0.714. In particular, the phases of ordinary periodic crystals were identified to be almost perfectly predictable. We aim to use this model to narrow down the candidate compositions that form quasicrystals and approximants.

3. Rediscovery of Hume-Rothery's law by machine learning
Here, the research group noticed one interesting fact. It turns out that the machine learning model learns an empirical rule for the formation of quasicrystalline alloys called the Hume-Rothery's electron concentration law. Most quasicrystals and approximants are known to be stabilized in compositions where the average number of itinerant electrons per atom, denoted by e/a, takes on a specific value. In aluminum alloys, stable quasicrystals and approximants tend to appear in the composition that satisfies e/a = 1.8. As shown in Fig. 3, the regions of quasicrystals and approximant crystals predicted by machine learning overlap with the straight line of e/a = 1.8 for most aluminum alloys. This means that the machine learning algorithm has rediscovered this widely known rule of thumb from only the composition list of quasicrystal and approximant crystals that have been found so far.
    
4. Rules for quasicrystal formation discovered by machine learning
Furthermore, by extracting the input-output rules inherent in the black-box model of machine learning, we have uncovered rule-of-thumbs on the formation of quasicrystalline phases. These rules are expressed in five equations for the average van der Waals radius and electronegativity of constituent elements (Fig. 4). These conditions provide design guidelines for the search for new quasicrystals. There would be many other rules hidden in the model. By exhaustively examining the rule sets in the trained machine learning, we will elucidate the formation mechanism of quasicrystals, which is a central issue in solid state physics. 

5. Towards discovery of innovative quasicrystals 
With this achievement, we have taken the first step toward realizing quasicrystal discovery through data science. Using this predictive model, many researchers are now working on the synthesis of new quasicrystals. In particular, we aim to discover highly innovative quasicrystals, such as semiconductor quasicrystals, superconducting quasicrystals, and ferromagnetic quasicrystals. During nearly 35 years after the discovery of the first quasicrystal, only little is known about the formation mechanism of quasicrystals. Data science may make a significant contribution to solving this unsolved issue in quasicrystal research.