 1. Motivations for new definitions in crystallography
 2. Common confusions with cells, lattices and crystals
 3. Rigorous definitions of periodic and crystal structures
 4. Descriptors versus invariants under a given equivalence
 5. Similarities versus distance metrics and continuity
 6. Conclusions: the practical importance of definitions
 Supporting information
 References
 1. Motivations for new definitions in crystallography
 2. Common confusions with cells, lattices and crystals
 3. Rigorous definitions of periodic and crystal structures
 4. Descriptors versus invariants under a given equivalence
 5. Similarities versus distance metrics and continuity
 6. Conclusions: the practical importance of definitions
 Supporting information
 References
feature articles
The importance of definitions in crystallography
^{a}Computer Science Department and Materials Innovation Factory, University of Liverpool, Liverpool, United Kingdom, and ^{b}Clark Science Center, Smith College, USA
^{*}Correspondence email: vkurlin@liv.ac.uk
This article is part of a collection of articles from the IUCr 2023 Congress in Melbourne, Australia, and commemorates the 75th anniversary of the IUCr.
This paper was motivated by the articles `Same or different – that is the question' in CrystEngComm (July 2020) and `Change to the definition of a crystal' in the IUCr Newsletter (June 2021). Experimental approaches to crystal comparisons require rigorously defined classifications in crystallography and beyond. Since crystal structures are determined in a rigid form, their strongest equivalence in practice is rigid motion, which is a composition of translations and rotations in 3D space. Conventional representations based on reduced cells and standardizations theoretically distinguish all periodic crystals. However, all cellbased representations are inherently discontinuous under almost any atomic displacement that can arbitrarily scale up a Hence, comparison of millions of known structures in materials databases requires continuous distance metrics.
1. Motivations for new definitions in crystallography
Mathematical crystallography – including the classification of lattices, unit cells, crystal classes etc. – by symmetries has a long and rich history. But classical mathematical crystallography, grounded largely in group theory, came before the computer age and needs updating in our era of massive data.
This paper does not reinvent the wheel but extends the discrete concepts to a new continuous domain in the language of presentday crystallography for presentday crystallographers.
The entry `A crystal' appeared in the IUCr Online Dictionary of Crystallography (IUCr, 2021) in 1992 and has since been modified slightly. We propose updates to fill past gaps and meet present needs. The latest directspace definition (Chapuis, 2024a) by the Commission on Crystallographic Nomenclature (CCN) says that `a solid is a crystal if its atoms, ions and/or molecules form, on average, a longrange ordered arrangement. In most crystals, the arrangement is a periodic array that is governed by the rules of translational symmetry.'
In this paper, a crystal is a periodic crystal, so we postpone similar developments for nonperiodic materials including quasicrystals and amorphous solids to future work (see Senechal, 1996). The definition quoted above (Brock, 2021) for a periodic crystal means that the set of all atoms is preserved under all lattice translations. Since periodic crystals, lattices and unit cells are often confused (Nespolo, 2015) or used interchangeably, we provide rigorous definitions in Section 2.
The next step is to clarify which periodic crystals should be considered the same in order to reliably compare crystals. Below, we quote the paper `Same or different – that is the question' (Sacchi et al., 2020). It is correct scientific practice `to report measurable quantities with an error' because all real measurements are noisy. However, if one claims that `… two dimensions are considered the same if their values fall within the accepted error or standard deviation', quoted from section 2.1 in Sacchi et al. (2020), then an axiomatic approach logically implies that all dimensions (meaning measurements of unitcell parameters in this case) should be the `same'. This continuum paradox (Hyde, 2011) states that many small changes (indistinguishable from 0) can lead to a big overall change.
For any fixed small error , if we call any real number indistinguishable from (considered the same as) all numbers within an interval , then is the same as all numbers from , which makes x the same as any number within , similarly within if we replace ɛ with . Continuing this logical argument further, any number y becomes indistinguishable from x in steps, which is the smallest integer larger than or equal to . This argument is formalized in terms of an equivalence below.
Definition 1 (equivalence relation)
A binary relation A ∼ B between objects of any kind is called an equivalence (Raczkowski & Sadowski, 1990) if these axioms hold:
(1) reflexivity: A ∼ A, so any object A is equivalent to itself;
(2) symmetry: for any objects A, B, if A ∼ B then B ∼ A;
(3) transitivity: for any A, B, C, if A ∼ B, B ∼ C then A ∼ C.
Definition 1 is important because any well defined classification into disjoint classes requires an equivalence relation. Indeed, the equivalence class of any object is the set of all objects B equivalent to A. The transitivity axiom implies that if the classes of A, C share a common object B, these classes coincide, i.e. [A] = [C]. Hence Definition 1 guarantees that all equivalence classes are disjoint. For any fixed , the binary relation x ∼ y defined by on real numbers fails the transitivity axiom because but .
If we enforce the transitivity so that x ∼ z if there is y such that x ∼ y ∼ z, this transitive extension makes all real numbers equivalent by putting them into a single equivalence class. Equality is an example of equivalence because any number can be written in many different forms: 0.5 = 1/2 = 50% = 1:2.
If the axioms of Definition 1 such as the transitivity are not satisfied, the resulting classes can overlap and become dependent on manually chosen parameters, see Zwart et al. (2008). All relations between lattices and crystals that led to 7 crystal systems, 14 Bravais classes and 230 spacegroup types are equivalences satisfying the axioms. A spacegroup type is a class of space groups under isomorphism, which is a bijection respecting the group operation, see Nespolo et al. (2018).
The most important practical motivation to agree on the main equivalences between crystals is the ongoing crisis of fake data in crystallography (Gavezzotti, 2022), which has caught attention of journalists (Chawla, 2024). Indeed, scientists could stop the `paper mills' (Bimler, 2022) that publish hundreds of articles and thousands of crystal structures, many of which are under investigation for data integrity (Francis, 2023).
In November 2023, two Nature papers described the recent `big data' attempts at generating crystal structures. The first paper (Merchant et al., 2023) reported the GNoME database of 384+ thousand `stable' predicted structures. The chemists found `scant evidence for compounds that fulfill the trifecta of novelty, credibility and utility' (Cheetham & Seshadri, 2024).
The autonomous Alab (Szymanski et al., 2023) claimed to have synthesized 43 new materials from GNoME. The review by Leeman et al. (2024) concluded that `none of the materials produced by Alab were new: the large majority were misclassified, and a smaller number were correctly identified but already known'. Section 6 will complement these conclusions by identifying thousands of duplicates in GNoME.
2. Common confusions with cells, lattices and crystals
In our papers (Widdowson et al., 2022; Widdowson & Kurlin, 2022), we introduced a lattice and periodic crystal in a single definition without explaining their logical dependencies. This approach suffices for expert mathematicians, but since many publications confuse lattices not only with crystals but also with cells, we clarify the differences here.
We are grateful to Massimo Nespolo for highlighting the differences between a periodic lattice and a ). Confusing these concepts led to the terms `lattice energy' and `lattice defects', which should be better called `structural energy' and `structural defects'. Since section 2 in Nespolo (2019) defined `the lattice of a crystal structure… as a collection of vectors expressed as a linear combination of n linearly independent vectors', we start from the more basic concepts of a basis and a lattice without requiring a whose definition needs the prerequisite concept of a lattice.
(Nespolo, 2019Definition 2 (basis and ordered basis)
(a) A basis of is an unordered set of n vectors in that are `linearly independent', i.e. if, and only if .
(b) An ordered basis of is a basis whose vectors are ordered. Equivalently, any vector can be expressed as a linear combination for unique .
For example, the vectors v_{1} = (1, 0), v_{2} = (0, 1) form a basis of because any vector is uniquely written as the linear combination . We can write coordinates of any vector in a unique order only if v_{1}, v_{2} are ordered.
Following our standards of introducing all concepts with an equivalence, these definitions imply that two bases are equivalent if they are equal as sets, while two ordered bases are equivalent if they contain the same vectors in the same order. A basis is often confused with the
defined by this basis.Definition 3 (the unit cell and lattice defined by a basis)
Any unordered basis of defines a unit cell: the parallelepiped consisting of all linear combinations with real coefficients . This basis also generates the lattice consisting of all linear combinations with integer coefficients .
Thus a (left) shows that the square cells defined by the orthonormal bases and are both unit squares, which differ only by the choice of origin and orientation. The square lattice has infinitely many bases , where A is a 2 × 2 matrix with integer coefficients and determinant ±1.
is a `box', whereas a lattice is a discrete point set. Fig. 1For the t_{i} = 1 so that all translations of by vectors tile without overlaps. The notation highlights that a is defined by a basis alone. For now, we consider unit cells (and lattices) equivalent (in the strictest possible sense) if they are equal as sets of points. The map {bases}{unit cells} is not invertible because a corner of a should be chosen for an origin. Fixing one of 2^{n} corners is equivalent to choosing n signs of ordered basis vectors . So we cannot uniquely identify an ordered basis from a without making one of 2^{n} choices.
, we excluded the valuesDefinition 3 introduced a and a lattice using only an unordered basis of vectors. Without these vectors, we cannot define their linear combinations.
However, as soon as we need to unambiguously express a point (from a motif below) using fractional coordinates in a basis, this basis should become ordered so that coordinates of any point are ordered according to the basis.
Definition 4 (motif, periodic point set, periodic crystal)
For any ordered basis of , let be a finite set M of points. We call M a motif. A periodic point set is the set of points for all and . In , if each point of M is an atom or ion with a and charge, S can be called a periodic crystal.
In Definition 3, any periodic crystal has a purely geometric part, which is a periodic set of zerosized points at all atomic centers, and the physical part of atomic attributes of these points, see the history in Palgrave & Tobin (2021). Any lattice Λ can be considered a periodic point set whose motif M consists of a single point p, for example, at the origin of . More general periodic crystals, even graphite, have motifs with at least two points and thus are not lattices according to Definition 3.
Any primitive cell that is a of a minimal volume. However, Fig. 1 (right) shows that any extended cell can be made primitive by a tiny perturbation of a single atom in the initial cell. This discontinuity was reported in 1965, see page 80 in Lawton & Jacobson (1965), and emerges even in one dimension. For the integer sequence , if we shift m of every m + 1 points by a small , we obtain the periodic sequence whose every point is ɛclose to a point of (and vice versa), but the period m + 1 can be arbitrarily large after perturbation.
can be scaled by a positive integer factor along each basis vector to an extended cell. This additional ambiguity is theoretically resolved by taking aA p. The ordered vectors can be uniquely determined from their lengths and angles , , . The angles should be ordered according to their opposite vectors. A without ordered sides (ordered basis vectors) can give rise to different periodic point sets as in Fig. 2.
(CIF) contains an ordered basis of vectors in and coordinates of each point on this basis with the atomic typeOrdering basis vectors by their lengths creates another discontinuity if the vectors have equal lengths because small perturbations can change their order. Fig. 3 summarizes why an ordered basis of is more convenient for defining a periodic crystal than a When the basis is fixed, we use the shorter notations without repeating this fixed basis.
3. Rigorous definitions of periodic and crystal structures
In the past, many different equivalence relations between latices and crystals were studied. One of the simplest is by chemical composition or by equality of another property such as density. However, crystals with the same composition (say, diamond and graphite of pure carbon) or with the same density can have many different properties, so these equivalences may not suffice.
Hence we are looking for a stronger equivalence that would guarantee the same physical and chemical properties according to the structure–property hypothesis which states that the structure of a material structure should determine all of its properties (Newnham, 2012).
The IUCr online dictionary (Chapuis, 2024c) contains the following entry: `crystals are said to be isostructural if they have the same structure but not necessarily the same cell dimensions nor the same chemical composition, and with a `comparable' variability in the atomic coordinates to that of the cell dimensions and chemical composition. For instance, calcite CaCO_{3}, sodium nitrate NaNO_{3} and iron borate FeBO_{3} are isostructural'. This phrase contains a cycle of `structural' concepts (`crystals are isostructural if they have the same structure'), which should be resolved by defining a structure.
If the keyword `necessarily' was omitted above, the reflexivity axiom would fail. Any attempt to define a `comparable' variability with a threshold for deviations of cell sizes fails the transitivity axiom. Indeed, by sufficiently applying many tiny deformations, we can convert any given ) into any other cell, so the classification under this `deviation' equivalence becomes trivial.
(with an empty motif, as in Definition 3The IUCr online dictionary defines the longer term crystal structure as `a consisting of atoms'. Both Chapuis (2024b) and section 8.1.4 in Hahn (2005) defined a crystal pattern in different words but essentially as a periodic point set in Definition 4, not considered under rigid equivalence. The word pattern as in the area of often refers not to a single object but to a class of objects under an equivalence as we propose in the new Definition 6 below.
Why do we need an equivalence that distinguishes between all chemical compositions and also close polymorphs that have the same composition but different properties? Such an equivalence is important because in the past many HIV patients suffered by unknowingly taking a more stable but less soluble polymorph of ritonavir that was accidentally manufactured instead (Morissette et al., 2003).
On the other hand, the pointwise coincidence of cells, lattices and periodic sets from Section 2 is too strict. Indeed, shifting the whole motif M by a small vector within a fixed changes all fractional coordinates of atoms in a but not the actual solid material. We consider all equivalences and comparisons only for ideal periodic crystals and under the same ambient conditions such as room temperature and pressure.
Since crystal structures are determined in a rigid form, their strongest and practically important equivalence is rigid motion.
Definition 5 (rigid motion, isometry)
A rigid motion of is a composition of translations and rotations. An isometry of is any transformation that preserves all interpoint distances.
For an ordered basis of , an orientation can be defined as the sign of the n × n determinant with the columns . Any orientationpreserving of is a rigid motion. Any orientationreversing of is a composition of one (any) mirror reflection and a rigid motion.
Hence,
is a slightly weaker equivalence than rigid motion because mirror images are equivalent under but not always under rigid motion. Since mirror images can be distinguished by a (suitably chosen) sign of orientation, it almost suffices to distinguish crystals only under isometry.Definition 5 distinguishes between isometry, which makes sense for any metric space with no Euclidean structure, and more restrictive rigid motion (orientationpreserving isometry).
The word motion is justified by the fact that any rigid motion f, which excludes mirror reflections by definition, can be realized through a continuous (motion) family of isometries , where , f_{1} = f and is the identity map. was called a in section 8.1.3 of Hahn (2005). Since symmetry has a wider meaning in science, we use the more specific concepts of rigid motion and isometry. The comprehensive books by Engel et al. (2004) and Zhilinskii (2016) studied lattices through group actions. In this language, any periodic structure from Definition 6 is a class in the quotient of all periodic point sets under the action of the special Euclidean group of all rigid motions in .
Definition 6 (periodic and crystal structures)
A periodic structure is an equivalence class of periodic point sets under rigid motion. A crystal structure is an equivalence class of periodic crystals with atomic attributes under rigid motion in .
Section 2 in Nespolo et al. (2018) defined a as `an idealized periodic pattern of atoms in 3D space using the corresponding coordinates with respect to the chosen coordinate system'. This pattern coincides with a (Chapuis, 2024b) in section 8.1.4 of Hahn (2005) and is a single representative of a periodic structure, introduced as a class of all rigidly equivalent crystals in Definition 6.
Any explicit use of coordinates for a crystal representation, like in a called such objects periodic point sets and periodic crystals. Shifting a motif by a fixed vector changes the description in a but not the real structure which is considered to be a class of equivalent representations
requires choosing an ordered basis and a motif of points with fractional coordinates in this basis. Definition 4Then a periodic crystal in the sense of the classical cellbased Definition 4 becomes one of infinitely many coordinatebased representations of a in the sense of the new Definition 6. Hence crystals are defined as the same if all their atoms can be matched by rigid motion. If there is no ideal match, any slightly different structures can be called close rather than `the same' because any tolerance makes the classification trivial.
Ignoring atomic attributes maps any periodic crystal to a periodic set of points (atomic centers). Though this projection might seem to lose all chemistry, Richard Feynman gave us a visual hint in his first lecture on atomic theory (Fig. 4) to compare crystals only by atomic centers without chemical elements.
Despite the apparent simplicity, Definition 6 brings up a hard problem of efficiently distinguishing periodic structures, which will be stated in Section 6 when defining a few more concepts. A recent and almost complete solution to this problem has made Definition 6 practically important, especially for detecting thousands of previously unknown nearduplicates in major databases. Sections 4 and 5 will discuss how to distinguish crystal structures and continuously quantify their differences.
4. Descriptors versus invariants under a given equivalence
Distinguishing objects under any equivalence relation from Definition 1 necessarily requires the concept of an invariant. Such a numerical property is often called a feature or descriptor without specifying an equivalence. In the sequel, for simplicity, we use as our main equivalence, denoted by S ≃ Q. Extensions to rigid motion will need a sign of orientation.
Definition 7 (invariant, complete invariant)
A function I on periodic point sets is called an isometry invariant if any isometric sets S ≃ Q have I(S) = I(Q) or, equivalently if then . An invariant I is called complete (injective or separating) if the converse also holds: if then .
Though it is very tempting to reduce a periodic point set to a finite subset such as an extended motif, this reduction can lead only to many nonisometric subsets as in Fig. 5. Hence, there is no simple way to reduce a periodic point set to a single finite subset. Taking finite clouds around every atom in a motif can lead to a complete invariant of periodic point sets under (Anosova & Kurlin, 2021), but the continuity under perturbations needs careful justifications (Anosova & Kurlin, 2022).
A simple S is the number m of points within a primitive U of S. This invariant is weak and cannot distinguish any lattices. A complete (injective or separating) invariant I is the strongest possible in the sense that I distinguishes all nonisometric sets.
invariant of a periodic point setThe sidesideside (SSS) theorem from school geometry can be rephrased in terms of invariants by stating that a complete invariant of three unordered points under m given points are ordered, the m × m matrix of their pairwise distances is complete under by Schoenberg (1935).
of consists of three interpoint distances up to permutations. If allThe case of unordered points is more practical for molecules whose many atoms can be indistinguishable as in the benzene ring. The naive extension of distance matrices to m unordered points requires m! permutations, which is impractical even for small m. Hence the important requirement for invariants is their computability, e.g. in polynomial time of the input size.
The invariance condition is the minimal requirement for a descriptor to be practically useful. A noninvariant such as the list of fractional coordinates of all motif points cannot distinguish between any periodic structures even under translation because all points of a motif M can be slightly moved along the same vector within a fixed without changing the underlying periodic structure in the sense of Definition 6.
The related concept of an equivariant means a function E(S) such that any rigid motion f affects E(S) in a way controlled by f so that , where T_{f} depends only on f but not on S. The invariance means that T_{f} is the identity.
For example, the center of mass of a finite molecule M is equivariant (rigidly moves together with M). But the center of mass of a motif M is not equivariant for a periodic point set S because a translation can push one point through a side face of a U, so the new periodic translate of p in the cell U nonequivariantly changes M and its center of mass.
Any linear combination of given point coordinates is equivariant under linear transformations, while invariants are much more restrictive and hence valuable. Equivariants are often used for representing interatomic forces by vectors that should be rigidly moved with the whole structure. Any collection of forces (one vector at every atom) can be interpreted as an ordered pair (initial structures, final structure moved by these forces).
Hence, complete invariants suffice to describe not only static structures but also any dynamics in the space of structures. Mathematical crystallography has developed many approaches to unambiguously identify a periodic structure under rigid motion, for example using a theoretically unique ). Then any periodic structure can have standard settings in the (Parthé et al., 2013). In theory, this conventional representation is complete under rigid motion.
(Niggli, 1928Fig. 1 (right) shows that almost any noise can arbitrarily scale up any Theorem 15 in Widdowson et al. (2022) states that this discontinuity under tiny perturbations holds even for lattices, which have motifs consisting of only one point.
The discontinuity of cellbased representations allows anyone to disguise a nearduplicate as a new material by making any extended cell primitive due to a slight displacement of atoms and by replacing some atoms with similar ones. To stop potential duplicates, we need continuous invariants that can quantify any (near)duplicates in terms of a distance metric. The more practically important requirements of continuity and reconstructability in Fig. 6 will be formalized in Section 5.
5. Similarities versus distance metrics and continuity
Section 4 justified the importance of invariants for distinguishing periodic structures. This section formalizes the concept of continuity with respect to a distance metric. We start from the simplest nontrivial case of 2D lattices.
De Lagrange (1773) classified all lattices under by using the quadratic form q_{11}x^{2}+2q_{12}xy+q_{22}y^{2}, whose coefficients are expressed via a basis v_{1}, v_{2} of a lattice Λ by the formulae , and . The extra conditions and guarantee the uniqueness of the form Q. The corresponding basis of Λ is called reduced and is unique under the of but not under rigid motion because the bases v_{1} = (1, 0), for have the same reduced form and generate lattices that are mirror images and not related by the rigid motion of .
In a more geometric approach, Selling (1874) and later Delone et al. (1934) added to any basis v_{1}, v_{2} of , the extra vector and the restriction that all pairwise angles between these vectors are nonacute, which means 90° or more. More recently, Conway & Sloane (1992) called such a collection an obtuse superbase. This name is justified by the fact that any vector can be written as for unique in a basis v_{1}, v_{2} and also as for unique , and so that .
While any lattice in has infinite nonisometric bases [see Fig. 1 (left)], its obtuse superbase is unique up to Indeed, any nonrectangular lattice has only two opposite superbases , which are related by the twofold rotation around , and whose all six vectors are orthogonal to the boundary of the hexagonal in Fig. 7 (left) (see Voronoi, 1908). All obtuse superbases of a rectangular lattice are related by reflections and are not unique under rigid motion. Fig. 7 (right) shows two obtuse superbases (mirror images) for v_{1} = (a, 0), v_{2} = (0, b) and v_{0} = (−a, −b).
 Figure 7 but not under rigid motion (for the rectangular lattice on the right). 
Definition 8 (root invariant )
Let a lattice have an obtuse superbase v_{0}, v_{1}, v_{2}, so that v_{1}, v_{2} generate Λ, and for all distinct . Write the root products in increasing order , which might reorder the vectors without changing Λ. The root invariant is the ordered triple , where only r_{12} can be 0.
Theorem 4.2 in Kurlin (2022b) proved that is a complete invariant of all lattices under also under rigid motion after enriching with a sign of orientation. The key advantage of in comparison with a reduced basis is the continuity under perturbations. In Kurlin (2022b), figure 4 explains the discontinuity of reduced bases, while theorems 7.5 and 7.7 prove the bicontinuity of the root invariant .
Fig. 8 visualizes the continuous space of all 2D lattices under composed (for simplicity) with uniform scaling, which maps each root product to . Since , we can use only two independent coordinates and which define the quotient triangle . Any rectangular lattice with an obtuse superbase v_{1} = (a, 0), v_{2} = (0, b), v_{0} = (−a, −b) for has and . All square lattices with a = b are represented by the origin (x, y) = (0, 0). The point (1, 0) is excluded as a limit case of lattices with infinitely thin and long cells.
In summary, all classes of 2D lattices under x + y = 1 and vertical side x = 0, .
and uniform scaling are in a 1–1 bicontinuous correspondence with all points in the quotient triangle QT. The Bravais classes of square and hexagonal lattices are the points (0, 0) and (0, 1), respectively. The of centered rectangular lattices consists of two boundary edges (without endpoints): the hypotenuseAny continuous path in QT is realized as a continuous deformation of lattices. For example, the unit square lattice Λ_{0} with the obtuse superbase (3, 0), (0, 3), (−3, −3) and can be continuously deformed into the hexagonal lattice Λ_{1} with the obtuse superbase , and along the vertical side x = 0, through the lattices Λ_{y} with and the bases and , where and y continuously moves from 0 to 1.
Fig. 8 contrasts the discrete tree of five Bravais classes of 2D lattices with the continuous map on the quotient triangle QT. Although every orthorhombic crystal from the CSD is represented by three rectangular lattices (on three pairs of reduced basis vectors), about 45% of all resulting lattices are oblique and continuously fill the interior of QT apart from the sparse corner close to (1, 0), where lattices have very thin and long primitive unit cells. All nongeneric lattices occupy lowerdimensional subspaces in the continuous space of lattices.
One can define many continuous distances between points in the quotient triangle QT in Fig. 8, hence between classes of 2D lattices under and uniform scaling. Section 5 in Kurlin (2022b) gave closedform expressions for metrics between root invariants and section 6 quantified deviations from symmetry by continuous chiral distances (see Bright et al., 2023a).
Any lattice in has an obtuse superbase, which is unique under a) explicitly described all nonisometric obtuse superbases for the five Voronoi types of 3D lattices. These results led to a complete root invariant of lattices under in in Kurlin (2022a). The root invariant of a 3D lattice requires complicated continuous distances satisfying the metric axioms in Definition 9 below and will appear in a forthcoming work.
only for generic lattices whose is a truncated octahedron. Lemmas 4.1–4.5 in Kurlin (2022The even more general case of periodic point sets needs a metric satisfying the axioms below. This metric is a distance between two objects, not a numerical property of a single object.
Definition 9 (distance metric)
For any objects under an equivalence relation from Definition 1, a distance metric d(A,B) is a function satisfying these axioms:
(1) coincidence: if and only if ;
(2) symmetry: for any objects A, B;
(3) inequality: for any A, B, C.
The positivity property follows the axioms above. A metric is needed to formalize the continuity of invariants in Problem 10 below. Though classical crystallography theoretically achieved the completeness of cellbased invariants, Problem 10 asks for more practically important invariants that have no discontinuities at boundaries of 230 (or any other number of) classes in the fully connected crystal universe.
Problem 10 (isometry classification of periodic structures)
Find a function I on all periodic point sets satisfying the following practically important conditions:
(a) invariance: if are isometric, then ;
(b) completeness: if , then are isometric;
(c) continuity: there is a metric d satisfying the axioms of Definition 9 under and the continuity below: for any and a periodic point set S, there exist C and such that if Q is obtained by perturbing any point of S up to δ in Euclidean distance, then ;
(d) reconstructability: any periodic point set can be reconstructed (uniquely up to isometry) from its invariant I(S);
(e) computability: the invariant I, metric d and reconstruction of can be obtained in polynomial time of the motif size from a suitably reduced basis of S and motif points in this basis.
Due to the coincidence axiom of a metric in Definition 9, the equality I(S) = I(Q) in the completeness condition (b) of Problem 10 is best checked as . If computability condition (e) of Problem 10 is missed, one impractical invariant I(S) satisfying all other conditions can be defined as the class of all (infinitely many) periodic point sets isometric to S. We assume that a periodic point set S is given with a reduced basis such as the Niggli basis in or the Minkowski basis in a higher dimension n since lattice reductions can be slow for (see Nguyen & Stehlé, 2009).
The continuity condition (c) of Problem 10 is a classical but weak version of continuity. The stronger Lipschitz continuity states that C and δ are independent of S and ɛ, so if Q is ɛclose to S, then , where a constant δ was absorbed by .
For 2D lattices Λ, theorem 7.5 in Kurlin (2022b) proved the intermediate Hölder continuity, stating that if the coordinates of the basis vectors of Λ are perturbed up to ɛ, the root invariant changes up to in the Euclidean metric, where l is the maximum length of given basis vectors of Λ.
The stronger Lipschitz continuity (without the factor ) seems unrealistic for lattices because the rectangular lattices with the ɛclose bases and can substantially differ even by unitcell areas and whose difference can be arbitrarily large if l has no upper bound.
Fig. 9 visualizes the advantages of invariants that satisfy all the conditions of Problem 10. In the past, incomplete, discontinuous or noninvariant descriptors mapped periodic crystals to latent spaces (image spaces of descriptor functions).
The noninvariance (existence of false negatives) means that the same maps to different points, which makes the problem of distinguishing structures even harder. The incompleteness (existence of false positives) means that nonisometric structures map to the same point, which leaves no chance to reconstruct a correct crystal. The discontinuity under tiny atomic displacements means that nearduplicates can appear very distant in the latent space.
All the conditions of Problem 10 guarantee that a required invariant I is a bijective and continuous map from the space of crystal structures to the space of invariant values. The inverse map I^{−1} reconstructs any periodic point set S from I(S).
6. Conclusions: the practical importance of definitions
This section summarizes the progress in developing invariants that satisfy the conditions of Problem 10. The root invariant from Definition 8 satisfies all conditions of Problem 10 for all 2D lattices even with the stronger Hölder continuity (instead of the weaker continuity) under rigid motion, which is stronger than For 3D lattices, Kurlin (2022a) defined a complete invariant whose continuity under perturbations is being finalized.
Past approaches defined metrics between lattices that allowed only slow or approximate computations. Some of these theoretical metrics were proved to be continuous for ).
classes of lattices in any dimension (Mosca & Kurlin, 2020In Widdowson & Kurlin (2022), for arbitrary periodic point sets S in , definition 3.3 defined the Pointwise Distance Distribution , where k is the number of neighbors taken into account for any point in a motif. Theorem 4.3 proved the Lipschitz continuity, stating that perturbing any atom up to ɛ changes only up to in a suitable metric. Theorem 4.4 proved that is generically complete in the sense that almost any periodic structure (outside singular subspaces of measure 0) can be reconstructed from a lattice of S and with an explicit upper bound on k depending on a given and motif of S. Hence PDD can be considered a DNAstyle code that uniquely identifies almost any real periodic crystal. PDD is stronger for periodic crystals than DNA, which allows identical twins (about 0.3% among humans) with indistinguishable DNA (see Osterman et al., 2022).
In practice, distinguished all (more than 660 000) different periodic crystals in the Cambridge Structural Database (CSD) through more than 200 billion pairwise comparisons, which were completed within two days on a modest desktop. Section 6 in Widdowson & Kurlin (2022) lists several pairs that turned out to be nearduplicate CIFs, where all numbers (unitcell parameters and fractional coordinates) were identical almost to the last decimal place, but one atom was replaced with a different one, e.g. Cd with Mn in the pair JEPLIA versus HIFCAB. The integrity office of the Cambridge Crystallographic Data Centre and all other crystallographers who looked at these previously unknown nearduplicates agreed that such an atomic replacement should more substantially perturb the geometry of atomic centers, so five journals are investigating the data integrity of the underlying publications.
A forthcoming paper will extend PDD invariants to distinguish all known pairs of homometric crystals that (by definition) have the same (infinite) list of all interatomic distances. We conjecture that the extended invariants are theoretically complete for all periodic point sets under in any Euclidean .
The comparisons above use only geometry of atomic centers without chemical elements. After excluding the unrealistic duplicates found in the CSD, the PDD invariants mapped all nonisometric crystal structures to nonisometric periodic structures, where each atom is replaced with a zerosized point.
Since this map is injective, the more important conclusion is the crystal isometry principle (CRISP) which states that any real periodic structure has a unique location in a common Crystal Isometry Space of all periodic structures (isometry classes of periodic point sets) independent of symmetry, see Fig. 10.
Hence, in principle, all atomic types in a real periodic crystal can be reconstructed from a sufficiently precise geometry of their atomic centers. The Eureka moment for this insight happened in May 2021 when the second author was reading Richard Feynman's first lecture `Atoms and motion' (see Fig. 4) with the table of seven cubic crystals whose chemistry can be reconstructed from the only geometric parameter d (smallest interatomic distance) known to two decimal places.
The crystal showed a map of 2D lattices under and uniform scaling. Continuous maps of the CSD and other databases in invariant coordinates were presented at the IUCr congress (see Kurlin, 2023) and will be discussed in future work.
principle does not claim that any periodic point set gives rise to a real periodic crystal because interatomic distances cannot be arbitrary. However, any newly discovered periodic crystal will appear in the same continuous universe, where all known crystals are already visible. Fig. 8While the realizability of root invariants by lattices in two and three dimensions has been established in Kurlin (2022b,a), we keep working on the harder problem of realizability of PDD invariants. The implemented application of PDD is the ultrafast detection of (near)duplicates in structural databases. The final sections in Widdowson et al. (2022) and Widdowson & Kurlin (2022) reported over a dozen such pairs in the CSD. Another forthcoming work will report less obvious (near)duplicates in the CSD and many more duplicates in the Crystallography Open Database (COD), Inorganic Database (ICSD), Materials Project and others.
The most important practical impact of CRISP is the scientific barrier for `paper mills' and `duplicate generators' that can output thousands and even millions of `predicted' and sometimes `synthesized' materials by disguising known structures as new by tiny perturbations of cell parameters and atomic coordinates (structure factors or other experimental data if needed) to scale up a
and finally by changing some nonstandard chemical elements to their suitable neighbors in the periodic table. Google's example below shows that even big numbers cannot mask (near)duplicates that we can filter out by numbers in given CIFs even before computing invariants.The paper finishes by describing embarrassing coincidences in Google's GNoME database of 384 398 `stable' structures in Google (2023) predicted by expensive DFT optimization (Mardirossian & HeadGordon, 2017). The following crystal bug test can substantially reduce further invariant computations for such a large database. Ordering all CIFs by the unitcell volume detected many thousands of pairs of CIFs in GNoME that have identical volumes to all (eight) decimal places (digits).
Other colleagues found some duplicates after ordering all CIFs by file sizes in bytes, but filtering by the unitcell volume is more justified. Further filtering by six parameters (three lengths and three angles) of a
found 30 000+ CIFs with identical unit cells, again with all given digits.Table 1 summarizes more hardtoexplain coincidences. The CIFs with GNoME IDs 4135ff7bc7, 6370e8cf86, c6afea2d8e and e1ea534c2c are identical texts (symbolbysymbol). The supporting information contains an Excel table listing more than a thousand pairs of identical CIFs. If chemical elements are ignored, GNoME has 1481 pairs of CIFs with all equal numbers (unitcell parameters and fractional coordinates). If we round all numbers to four and two decimal places for the precision of 10^{−4} Å and 10^{−2} Å, respectively, the last two columns in Table 1 show many more groups of CIFs that become numerically identical to each other. Table 2 shows chemical compositions for the three largest groups of CIFs.

The first part of Table 2 shows that GNoME contains a group of nine CIFs, where all numbers are equal (with all decimal places) but chemical compositions differ by one or two atoms. For example, Dy, Y, Ho and Tb are often swapped. If all numbers are rounded to two digits, one more (a18d30a9fc) joins the group of duplicates, where Ru is replaced with Re. So comparisons by unitcell parameters and fractional coordinates can help to filter out obvious (near)duplicates even in big data.
This paper clarifies the concept of a periodic crystal in terms of an ordered basis whose reordering creates ambiguity or discontinuity in Fig. 2. Definitions 2, 3 and 4 are visually summarized in Fig. 3. Rigid motion (or slightly weaker isometry) is motivated as the strongest equivalence between crystals whose structures are determined in a rigid form. The practical importance of distinguishing nearduplicates in major structural databases requires us to define a periodic (crystal) structure as an equivalence class under rigid motion. Any deviations from an ideal rigid matching should be continuously quantified in terms of a distance metric satisfying all axioms and at least the classical form of continuity.
As a visual summary, Fig. 6 highlights the importance of invariants versus noninvariant descriptors. Fig. 5 explains that similarities based on single (hence noninvariant) finite subsets are hard to justify for periodic structures. In the past, crystallography developed conventional representations based on reduced that can be considered complete invariants in theory.
Now computational resources are used for generating millions of structures, many of which turn out to be nearduplicates. Hence Problem 10 has become the important scientific barrier for paper `milling' by validating any newly discovered crystals versus all known ones. Future work will extend PDD to a full solution of Problem 10. The crystal principle and underlying invariants were used for property predictions in the literature (Ropers et al., 2022; Balasingham et al. 2024a,b) and were presented at the IUCr congresses in 2021 and 2023, the European Crystallographic Meeting 2022, the BCA annual meetings 2022–2024, and MACSMIN 2021–2023 (Mathematics and Computer Science for Materials Innovation).
Supporting information
Zipped tables listing more than a thousand pairs of identical CIFs from GNoME. DOI: https://doi.org/10.1107/S2052252524004056/gq5017sup1.zip
Acknowledgements
VK thanks Matt McDermott for a comprehensive tour of the Alab in Berkeley, where 43 materials were synthesized (Szymanski et al., 2023); and Ram Seshadri (UCSB), Leslie Schoop (Princeton) and Robert Palgrave (UCL) for fruitful discussions. We thank Andy Cooper (Liverpool), Simon Billinge (Columbia) and all members of the Data Science Theory and Applications group in the Materials Innovation Factory at Liverpool, especially Daniel Widdowson and Yury Elkin. We thank all reviewers for their valuable time and helpful advice.
Funding information
This work was supported by the VK's Royal Academy of Engineering Fellowship `Data Science for Next Generation Engineering of Solid Crystalline Materials' (grant No. IF2122/186), the Engineering and Physical Sciences Research Council New Horizons grant `Inverse design of periodic crystals' (grant No. EP/X018474/1 awarded to VK), and the Royal Society APEX fellowship `New geometric methods for mapping the space of periodic crystals' (grant No. APX/R1/231152 awarded to VK).
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