Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

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Session Overview
Session
MS-36: Generalizations of crystallographic groups and their applications
Time:
Tuesday, 17/Aug/2021:
10:20am - 12:45pm

Session Chair: Gregory L McColm
Session Chair: Eden Delight Miro
Location: 221-2

60 2nd floor

Invited:  Pavel Kalouguine (France), Martin Cramer Pedersen (Denmark)


Session Abstract

Space groups describe the global symmetry of a periodic structure. Generalizations of space groups are often necessary to investigate additional structural features, physical properties or aperiodic structures. These generalizations include superspace groups and chromatic groups (collectively called metacrystallographic groups), groups in non-Euclidean geometries and groupoids. Whereas superspace groups are now part of the daily work of the structural crystallographer, other generalizations are less well known. The microsymposium aims at reviewing these concepts and presenting new results in the field.

For all abstracts of the session as prepared for Acta Crystallographica see PDF in Introduction, or individual abstracts below.


Introduction
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Presentations
10:20am - 10:25am

Introduction to session

Gregory L McColm, Eden Delight Miro



10:25am - 10:55am

Behind the curve: Generating and analysing nets and tessellations on periodic minimalsurfaces in their universal covering space

Martin Cramer Pedersen1, Jacob J. K. Kirkensgaard1,2, Stephen T. Hyde3

1Niels Bohr Institute, University of Copenhagen; 2Department of Food Science, University of Copenhagen; 3School of Chemistry, University of Sydney

The in-surface crystallography of the family of three-periodic minimal surfaces (TPMS) consisting of the Primitive, Diamond, and Gyroid surfaces has been explored in detail in recent decades [1, 2]. We begin by reviewing the underlying group theory and geometry as well as the relationship between the TPMS and their universal covering space, hyperbolic two-space, the fundamentals of which are shown in Figure 1(a).

We describe how these methods can be used to tailor the topology and geometry of three-periodic nets realised as embeddings commensurate with the symmetries of these surfaces [3-5] as shown in Figure 1(b). Using these ideas, we demonstrate how a number of nets with pre-specified topological properties are readily produced in this manner and assess their relevance for further study in the context of reticular chemistry and soft matter materials science.

Finally, we present preliminary results on visualisation methods for understanding how these patterns and nets are realised in simulations of liquid crystals in bulk [6] and confined polymer systems. Using an array of methods from computational geometry, we visualise these simulations in hyperbolic two-space to facilitate easy comparisons and further analysis as shown in Figure 1(c).

The authors thank Anders Dahl, Myfanwy Evans, Olaf-Delgado Friedrichs, Benedikt Kolbe, Stuart Ramsden, Vanessa Robins, Gerd Schröder-Turk, and Monique Teillaud for discussions and feedback on these topics and results.

[1] Sadoc, J.-F. & Charvolin, J. (1989). Acta Crystallogr. A 45, 10-20.
[2] Robins, V., Ramsden, S., & Hyde, S. T. (2004). Eur. Phys. J. B 39(3), 365-375.
[3] Pedersen, M. C. & Hyde, S. T. (2018). Proc. Natl. Acad. Sci. U. S. A. 115(27), 6905-6910.
[4] Pedersen, M. C., Delgado-Friedrichs, O., & Hyde, S. T. (2018). Acta Crystallogr. A 74(3), 223-232.
[5] Hyde, S. T. & Pedersen, M. C. (2021). Proc. Roy. Soc. A 477, 20200372.
[6] Kirkensgaard, J. J. K., Evans, M. E., de Campo, L., & Hyde, S. T. (2014). Proc. Natl. Acad. Sci. U. S. A. 111(4), 1271-1276.

External Resource:
Video Link


10:55am - 11:25am

Flat-branched semisimplicial complexes: a versatile tool for aperiodic solids

Pavel Kalugin1, André Katz2

1Laboratoire de Physique des Solides, Université Paris-Saclay, Orsay, France; 2Directeur de recherche honoraire, CNRS, France

We introduce flat-branched semisimplicial (FBS) complexes as a universal language to describe aperiodic structures of finite local complexity. An FBS-complex naturally represents the set of local atomic arrangements occurring in the structure. It includes both metric and combinatorial data; the flexibility of the latter allows for incorporation of structural constraints on a longer range. An FBS-complex can embody "local rules" of any kind, whether or not they impose a perfect long-range order. We propose an algorithm for exploration of local rules in terms of an FBS-complex directly from the phased diffraction data. The FBS complex describing a structure entirely determines the density of atomic species, and yields experimentally verifiable constraints on their contribution to the structure factors.

External Resource:
Video Link


11:25am - 11:45am

Coordinated Colorings and their Chromatic Groups

Agatha Kristel M. Abila1, Eduard C. Taganap2, Ma. Louise Antonette N. De las Peñas3

1Southern Luzon State University - Lucena Campus, Lucena City, 4301, Philippines; 2Department of Mathematics and Physics, Central Luzon State University, Science City of Munoz, Nueva Ecija 3120, Philippines; 3Department of Mathematics, Ateneo de Manila University, Quezon City, Metro Manila 1108, Philippines

The study of chromatic symmetries to describe physical properties of crystals and mapping of individual orientations in twins has extended the field of mathematical crystallography. This paper discusses chromatic, partially chromatic or achromatic properties of symmetries or partial operations of a coordinated coloring of a symmetrical pattern, which has a potential to describe a crystalline structure. Here, we consider a symmetrical pattern P consisting of disjoint congruent copies of a symmetrical motif M.

A coordinated coloring of P is a coloring that is perfect and transitive under the global symmetry group G of P, satisfying the condition that the coloring of M is also perfect and transitive under its symmetry group K (a local symmetry group). This means that G and K consists of elements that effect a permutation of the colors of the coloring of P and M respectively. If the coloring of P or M has two (respectively more than two) colors, then G or K is called a dichromatic (respectively polychromatic) symmetry group.

A global symmetry, a local symmetry, or a partial operation (an isometry of the plane that sends one motif to another, which may not be a global or local symmetry of P) of a coordinated coloring of P can be classified as either achromatic if it fixes all the colors; chromatic if it moves all the colors, and partially chromatic if it exchanges some colors, and fixes the rest.

As an example, consider the coordinated 4 – coloring of P given in the given figure. Each region in every motif is assigned a color from the set S={blue,yellow,red,green}. Every element of G=<2 0,0;m[10];m[01];z(1,0)>≅p2mm (a polychromatic symmetry group) permutes the colors in the 4 – coloring of P. We also note that in this coloring, every element of K=<4+ 0,0;m[10]>≅4mm permutes the colors assigned to the triangles in the motif M. The dichromatic group of M is 4'mm': we have the 90^o rotation and two reflections that change colors, while two reflections fix colors. The dichromatic group of each of the other motifs is also 4'mm'.

The local chromatic symmetries of M exchanging blue and red are 4'+,4',m'[11], and m'[11 ̅ ]. On the other hand, the local achromatic symmetries of M fixing the colors blue and red are 2,m[10] , and m[01] . The global symmetries in P can be characterized as: z(4) (1,0) (chromatic) sending blue to yellow, yellow to red, red to green, and green to blue; 2(2,2) 0,0, and m(2,2)[01] (partially chromatic) exchanging yellow and green, and fixing red and blue; and m_[10] (achromatic) fixing all the colors. The 4-coloring of the frieze pattern is described by the group [p(4) 2(2,2) m(2,2) m](4). The chromatic partial operations that map M to zM are also shown in the figure. Sending blue to yellow as well as red to green are z(2) (1,0),2(2) 1/2,0 ,m(2)[01] 1/2,0 , and g(2)[01] (1,0). Sending blue to green and red to yellow are 4(2)+ 1/2,1/2, 4(2)- 1/2,(1/2) ̅, g(2)[11](1/2,1/2) 1/2,0 and g(2)[11 ̅ ](1/2,1/2) 1/2,0.

In this talk, examples of coordinated colorings of Frieze and Plane Crystallographic patterns will be presented.

External Resource:
Video Link


11:45am - 12:05pm

Advances in the application of magnetic and non-magnetic superspace-group symmetry

Branton J Campbell, Harold T Stokes

Brigham Young University, Physics & Astronomy, Provo, Utah, USA

Superspace-group symmetry is essential to the unambiguous description of modulated structures, and a correct understanding of their physical properties. An exhaustive enumeration of superspace groups in up to 3+3 dimensions were announced in 2014. We now announce an exhaustive enumeration of magnetic superspace groups in up to 3+3 dimensions (over 250,000 groups). With these tables in hand, we have developed an algorithm and tool that detects the superspace-group (magnetic or non-magnetic) of an arbitrary modulated structure, given the amplitudes and phases of its modulations in P1 symmetry, and identifies it in the exhaustive symmetry-group table. This capability has been integrated into both the FINSYM and ISOCIF packages of the ISOTROPY software suite, and has been integrated with JANA 2000. The ISODISTORT package, which use group-representations to generate incommensurate structure models based on a given parent structure, now automatically identifies the unique magnetic superspace-group of each magnetically modulated child structure. Anyone can access these data sources and tools online to generate, symmetrize, transform, or otherwise explore magnetic or non-magnetic modulated structure models.

External Resource:
Video Link


12:05pm - 12:25pm

Magnetic modes compatible with the symmetry of crystals

Radoslaw Przenioslo, Piotr Fabrykiewicz, Izabela Sosnowska

University of Warsaw, Warsaw, Poland

We present a classification of magnetic point groups which give an answer to the question: Which magnetic groups can describe a given magnetic mode?
There are 32 categories of magnetic point groups which describe 64 unique different magnetic modes: 16 with a ferromagnetic component and 48 without. This classification focused on magnetic modes is helpful for finding the magnetic space group which can describe the magnetic symmetry of the material.

The classification selects the magnetic space groups and the magnetic site-symmetry point groups which are compatible with a number of magnetic phenomena e.g. collinear antiferromagnetism and ferromagnetism, spin roerientation, antiferromagnetism with weak ferromagnetism. The use of our classification is demonstrated on a number of well-studied materials, e.g. alpha-Fe2O3, rare earth orthoferrites, RFeO3. It is particularly useful for materials with weak ferromagnetism. Examples of use for neutron powder diffraction studies are discussed in the context of the paper by Shirane (1958).

This presentation based on the paper (ib5097) recently accepted to Acta Cryst A.

External Resource:
Video Link


 
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