The Density Fingerprint of a Periodic Point Set.

26^th March 2021, 14:00
Phil Smith
MIF

Abstract

Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction. The talk is based on the paper with H.Edelsbrunner, M.Wintraecken, T.Heiss, V.Kurlin, accepted at SoCG 2021.

MIF++ seminars in 2020 (reverse chronological order)
17 December 2020 15.15-16.45 UK time, joint with the Computational Mathematics Seminar.
Vitaliy Kurlin (Materials Innovation Factory, University of Liverpool, UK).
Title. Introduction to Periodic Geometry for applications in Crystallography and Materials Discovery.
Abstract. A periodic crystal is modeled as a periodic set of zero-sized points in 3-space. Such a periodic set is usually given by a unit cell (a parallelepiped defined by 3 edge-lengths and 3 angles) and a motif of points with fractional coordinates in this cell. Representing a crystal as a unit cell plus a motif is highly ambiguous. Hence a reliable comparison of rigid crystals should be based on isometry invariants that are preserved under any rigid motions, hence are independent of a unit cell and a motif. The talk will discuss a stable isometry classification of periodic crystals. Most past invariants of crystals such as symmetry groups are discontinuous under atomic vibrations. The new classification establishes foundations of periodic geometry. Based on joint papers with Andy Cooper, Angeles Pulido, Herbert Edelsbrunner, Mathijs Wintraecken, Teresa Heiss, Phil Smith, Marco Mosca, Dan Widdowson, e.g. arxiv:2009.02488.
22 October 2020 15.00-16.00 (UK), 7.00-8.00 (California).
Larry Andrews (Ronin Institute, US).
Title. The Rise and Fall of the V7 Metric.
Abstract. Having a measure of the difference between two lattices has several applications: databases of unit cells, clustering of images from serial crystallography, planning of epitaxial processes. Originally, unit cells were compared and cataloged using their sorted unit cell edge lengths. This technique has several problems. The first complete measure for lattices was the V7 metric (Andrews, Bernstein, Pelletier, 1980). It was complete, simple to understand, and in database retrievals it would never have false negative results. However, it was discovered to have a deficiency. The creation of V7, problems with it, and the evolution of methods for more correct distance measures will be discussed.
10 July 2020 13.30-15.00 UK time.
Marco Mosca (Materials Innovation Factory, University of Liverpool, UK).
Title. Distance-based isometry invariants of crystals.
Abstract. The talk will consider the isometry classification of periodic sets motivated by the Crystal Structure Prediction. A periodic sets of points models any solid crystalline material (a crystal) and represents any atom or a molecular centre by a zero-sized point. Since most crystals are rigid bodies, the natural equivalence on periodic sets is a rigid motion (a composition of translations and rotations) or an isometry, which preserves distances between points. Traditionally crystals were distinguished by discrete invariants such as symmetry groups that are unstable under atomic vibrations. However, large datasets of simulated crystals contain too many nearly identical crystals obtained as slightly different approximations to local minima of an energy function. The new distance-based isometry invariants are provably stable under noise. The experiments on the T2 dataset from the 2017 Nature paper show that the new invariants distinguish simulated crystals better than past tools and can automatically match experimental crystals to their closest simulated versions.
19 May 2020 10.30-12.00 UK time.
Vitaliy Kurlin (Materials Innovation Factory, University of Liverpool, UK).
Title. Isometry classification of periodic crystals.
Abstract. The Crystal Structure Prediction (CSP) tools output too many simulated crystals, including many nearly identical local minima of an energy, because there was no reliable way to quantify the similarity between crystals. Since most solid crystalline materials are rigid, the similarity problem is now rigorously stated as a classification modulo rigid motions or isometries that preserve inter-point distances. Most descriptors, especially if they are based on ambiguous unit cells, are either not isometry invariants or are unstable under atomic vibrations, hence can not reliably quantify crystal similarities. The talk introduces three approaches to build stable-under-noise and complete isometry invariants of periodic crystals.
3 February 2020 11-13 (lunch and discussion from about 12).
Uwe Grimm (Open University School of Mathematics and Statistics).
Location : the boardroom on the ground floor in the MIF.
Title. An introduction to aperiodically ordered systems and their diffraction.
Abstract. The talk will present an introductory overview on aperiodically ordered systems obtained from cut and project or inflation methods. The key methods and results will be illustrated by means of examples and pictures, leaving aside much of the mathematical details. For cut and project sets, the diffraction is well understood, and the general result and its application to examples will be discussed. In contrast, inflation systems are still only partially understood, and paradigms with different spectral type are presented. Finally, I will briefly discuss recent results on the Fourier transform of Rauzy fractals and the point spectrum of unimodular Pisot inflation tilings, which uses both the inflation and the projection structure.
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MIF++ seminars in 2019 (reverse chronological order)
13 December 2019 11-13 UK time (lunch and discussion from about 12)
Tom Hasell (Liverpool Chemistry).
Location: the 3rd floor meeting room in the MIF.
Title. Making polymers from elemental sulfur
Abstract. Sulfur is an industrial by-product, removed as an impurity in oil-refining. This has led to vast unwanted stockpiles of sulfur, and resulted in low bulk prices. Sulfur is therefore a promising alternative feedstock to carbon for polymeric materials. Sulfur normally exists as S8 rings – a small molecule with poor physical properties. On heating, these sulfur rings can open and polymerise to form long chains. However, because of the reversibility of sulfur bonds, these polymers are not stable, and decompose back to S8 over time, even at room temperature. Inverse vulcanisation has made possible the production of high sulfur polymers, stabilised against depolymerisation by crosslinking [1]. These high sulfur-materials show excellent potential as low cost water filters to remove mercury [2-4]. Heavy metal contamination exists in the waste streams of many industries, and mercury is of particular concern for human health. Alternative crosslinkers for inverse vulcanisation, from industrial by-products or bio-renewable sources, can be used to reduce the cost and improve the properties of the resultant polymers [4,5]. Polymers made from sulfur also have many other intriguing properties and applications in optics, electronics, insulation, and antimicrobial materials. We recently reported a catalytic route that reduces the required reaction time, temperature, and by-products – and allows otherwise unreactive crosslinkers to be used [6].
[1] Nat. Chem. 2013, 5, 518–524. [2] Chem. Commun., 2016, 52, 5383 - 5386. [3] J. Mater. Chem. A. 2017,5, 11682-11692 [4] J. Mater. Chem. A., 2017, 5, 18603 [5] Chem.-Eur. J. 2019, 25, 10433-10440 [6] Nat. Commun. 2019, 10, 647
4 November 2019 11-13 UK time nd discussion from about 12)
Linjiang Chen (Liverpool Chemistry).
Location : the 3rd floor meeting room in the MIF.
Title. Functional materials discovery: ‘Seeing’ structures and functions?
Abstract. We have recently demonstrated a strategy for how new, functional molecular materials can be discovered. By carrying out a priori prediction of both the crystal structure and its functional properties, energy–structure–function (ESF) maps are created to aid researchers, without computational expertise, in realizing several remarkable porous materials promising for different possible applications.
As large-scale computational screening studies become routinely carried out, new opportunities have arisen for accelerating materials discovery by taking advantage of the availability of big data. It, however, remains a challenge to properly understand and use the vast amount of data generated by simulations. In my talk, I will show how we use computation to accelerate serendipitous discovery of new functional materials, as well as the prowess of data-driven machineries becoming available. I will outline our current focuses on and approaches to the following topics:
1. ESF mapping to identify synthetic targets with desired functional properties;
2. Interactive visualization of high-dimensional structure–property relationships;
3. Smart navigation of the ESF space with machine learning.
21 October 2019 11-13 UK time (lunch and discussion from about 12). Georg Osang (IST Austria Computational Geometry).
Location: the boardroom on the ground floor in the MIF.
Title. The Multi-cover Persistence of Euclidean Balls
Abstract. Persistent homology has become a popular tool to analyse various kinds of data, in particular in material sciences. Specifically, persistence of discrete point sets has recently been used to analyse sphere packing data, to shed light on structures arising in sphere packings at different packing densities. We generalize this notion and introduce higher-order persistence of discrete point sets. We address computational challenges and show how this notion can deal with noisy point samples. In the setting of sphere packings we show that this notion can also capture a wider variety of local structures, and in particular can distinguish between the hexagonal close packing and the face centered cubic lattice packing, two structures know to have optimal packing density in 3 dimensions.
1 July 2019 11-13 UK time (lunch and discussion from about 12).
John Claridge (Liverpool Chemistry).
Location: the 3rd floor meeting room in the MIF.
Title : Challenges and opportunities in condensed matter science.
Abstract. Solid state chemistry, is the study of the synthesis, structure and properties of solid state materials and particularly how these are related to each other. It is obviously closely linked to condensed matter physics, mineralogy, crystallography, ceramics, metallurgy, materials science and electronics. We'll begin with some simple illustrations of these relationships, before looking at how structural/crystallographic methods have advanced in order to both characterise and describe the materials we can now make, and where I believe there are limitations in our current tools and understanding, with examples taken from my own work and more generally, and why they are important.
13 May 2019 11-12.15 UK time (with lunch at VGM from 12.30).
Vanessa Robins (ANU Applied Mathematics).
Location: the boardroom on the ground floor in the MIF.
Title. An introduction to the mathematical description and computational enumeration of periodic tilings and nets.
Abstract. Periodic tilings and nets are simplified models of atomic arrangements in crystals. Systematic methods for describing and enumerating possible crystal structures are based on factoring out either the translational periodicity or the full crystallographic symmetry. The mathematical version of “factoring out” gives us an object called an orbifold. I will explain how orbifolds encapsulate the essential elements of a crystallographic group and provide an effective basis for the enumeration of hypothetical crystalline structures. The presentation will be informal and largely based on pictures and examples, hopefully accessible to non-specialists.
29 April 2019 11-13 UK time (lunch and discussion from about 12).
Vladimir Gusev (Liverpool Chemistry).
Location: the boardroom on the ground floor in the MIF.
Title. New approaches to the crystal structure prediction
1 April 2019 11-13 UK time (lunch and discussion from about 12).
Chris Collins (Liverpool Chemistry).
Location: the boardroom on the ground floor in the MIF.
Title. Computationally driven materials discovery
Abstract. In the previous seminar by Dr D. Antypov, we discussed the general concepts of energy calculations and crystal structure prediction (CSP). In this session are to discuss how CSP is used in practice in the search for new inorganic materials. We present how the use of ‘probe’ structures accelerates the experimental exploration of composition space. A probe structure is a hypothetical crystal structure for a given composition, it need not be the ground state. However, it needs to be complex enough that it can contain enough co-ordination chemistry to produce an energy which is representative of potential single phases. When probe structures are generated across a range of compositions we can then construct the convex hull, and look for minima – regions of compositional space in which new compounds are likely to be found. We will discuss the methods for creating probe structures. The development of a method is split into two streams: how can we best represent crystal structures in a computer? And how can we search through the possible configuration space? We will then present our latest codes developed for probe structure prediction (Monte Carlo – Extended Module Materials Assembly (MC-EMMA[1]) and the Flexible Unit Structure Engine (FUSE[2]), along with example phase fields computed using them and show how probe structure generation for composition prediction is a powerful, predictive tool to accelerate the discovery of materials.
1. C. Collins, M. S. Dyer, M. J. Pitcher, G. F. S. Whitehead, M. Zanella, P. Mandal, J. B. Claridge, G. R. Darling and M. J. Rosseinsky, Nature, 2017, 546, 280-284.
2. C. Collins, G. R. Darling and M. J. Rosseinsky, Faraday Discussions, 2018, 211, 117-131.
11 March 2019 11-13 UK time (lunch and discussion from about 12).
Dmytro Antypov (Liverpool Chemistry).
Location: the 3rd floor meeting room in the MIF.
Title. Introduction to crystal structure prediction
Abstract. Crystalline solids such as metal oxides and perovskites are functional materials used in solar cells, batteries and many other devices. These materials are often composed from 3, 4 or 5 different chemical elements arranged in a periodic three dimensional structure. Such chemical diversity on the one hand allows for fine-tuning of material properties but on the other hand makes the identification and synthesis of stable compounds difficult. To accelerate the design of such materials we use computation to predict the combinations of the constituent chemical elements that will lead to stable crystalline structures. In this talk I will explain how this chemistry problem is formalised to have a tractable computational solution and show some examples.
25 February 2019 11-13 UK time (lunch and discussion from about 12).
Phillip Maffettone (Liverpool Chemistry).
Location: the 3rd floor meeting room in the MIF.
Title. Describing solids, and challenges for high-throughput analysis.
Abstract. The talk will introduce reciprocal space and the reciprocal lattice as the inverse of the direct lattice, building on the previous introduction describing periodic structures. The reciprocal lattice plays a fundamental role in understanding wave mechanics in crystalline materials, and is implemented in most analytical studies of materials through the theory of diffraction. These concepts will be presented with a focus on their experimental utility, necessary limitations, and potential implications for high-throughput automation. In particular, we will explore the question: When can we meaningfully invert the diffraction pattern and the pair-wise information it contains, and how can we automate this inversion?
11 February 2019 11-13 UK time (lunch and discussion from about 12).
Mike Gaultois (Liverpool Chemistry).
Location: the 3rd floor meeting room in the MIF.
Title. Searching for new magnetic materials.
Abstract. Magnetic materials have fascinated humans since the antiquity, but only recently since the 1800's with the work of Oersted and Maxwell did we begin to develop a deeper understanding of magnetism. Magnetic materials are now involved in many technological applications, many of which are limited by the material performance. One area of enquiry is the search for materials that can make hard permanent magnets, and another area of enquiry is the search for materials that conduct electricity without resistance (i.e., superconductors). This discussion is interested in exploring how we can best computationally predict (through first principles, or otherwise) the magnetic properties of a given material. We will describe what others have done to search for new materials using machine learning techniques, and discuss potential areas of interaction between MIF++ where advances can be made.
28 January 2019 11-13 UK time (lunch and discussion from about 12).
Vitaliy Kurlin (Materials Innovation Factory, University of Liverpool, UK).
Location: the boardroom on the ground floor in the MIF.
Title. Mathematical challenges of solid crystalline materials.
Abstract. The talk will introduce periodic structures that model all solid crystalline materials, for example molecular crystals, covalent organic frameworks and inorganic crystals based on isolated atoms or ions. The key problems of materials discovery will be stated in mathematical terms: what structures should be equivalent and how to quantify a similarity between crystals. We will discuss several requirements for potentials solutions that should work for real crystals.
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