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Topology in the solid state sciences

Topology in the solid state sciences. José L. Mendoza-Cortés. 2011 February 17th. Materials Science. Chemistry. Physics. Why is it important? What can we learn?. Biology. What do they mean by Topology?. Main Questions.

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Topology in the solid state sciences

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  1. Topology in the solid state sciences José L. Mendoza-Cortés 2011 February 17th

  2. Materials Science Chemistry Physics Why is it important? What can we learn? Biology What do they mean by Topology?

  3. Main Questions • Fundamental question: Given an spectra (e.g. sound), can you tell the shape of the source (e.g. the instrument shape) • In other words: Is it possible that two molecules or solids can have the same properties, given the only difference is their topology? Topology is concerned with spatial properties that are preserved under continuous deformations of objects.

  4. Familiarity Voronoi-Dirichlet polyhedron Wigner-Seitz cell First Brillouin zone All are example of Voronoi-Dirichlet polyhedron but applied to an specific field

  5. Everything we are going to cover today it comes to this!

  6. And this: Zeolites

  7. rod node Different topologies could be obtained on varying the coordination geometry of the nodes... From real stuff to abstract stuff

  8. From real stuff to abstract stuff honeycomb layer

  9. “Topological” Entanglement “Euclidean” Entanglement Lets see abstract stuff

  10. Borromean links

  11. Lets see abstract stuff

  12. Models: Lattice hxl/Shubnikov plane net (3,6)Atom coordinatesC1 0.00000 0.00000 0.00000Space Group: P6/mmmCell Dimensionsa=1.0000 b=1.0000 c=10.0000 Crystallographic, not crystallochemical model

  13. Models: Net Inherently crystallochemical, but no geometrical properties are analyzed

  14. Models: Labeled quotient graph Chung, S.J., Hahn, Th. & Klee, W.E. (1984). Acta Cryst. A40, 42-50. Wrapping NaCl 3D graph NaCl labeled quotient graph

  15. Models: Embedded net Diamond (dia) net in the most symmetrical embedding

  16. Models: Polyhedral subdivision Voronoi-Dirichlet polyhedron and partition: bcu net Kd=0.5

  17. Models: Polyhedral subdivision Tilings: dia and bcu nets dia bcu ‘Normal’ crystal chemistry -> ‘dual’ crystal chemistry

  18. Abstract stuff

  19. 4

  20. 3-connected graph means that the three vertex are connected with other three vertex (therefore they have three edges)

  21. Where can we apply this? Hsieh, D. et al. A tunable topological insulator in the spin helical Dirac transport regime. Nature 460, 1101–1105 (2009).

  22. Where can we apply this?

  23. world records of Interpenetration 2002 10-fold dia MOF Ag(dodecandinitrile)2 11-fold dia H-bond [C(ROH)4][Bzq]2 Class Ia ... 18-fold srs H-bond (trimesic acid)2(bpetha)3 Class IIIb

  24. 12 interpenetrating nets TIV:[0,1,0] (13.71A) NISE: 2(1)[0,0,1] Zt=6; Zn=2 Class IIIa Z=12[6*2] dia 12f 2008

  25. ######################################### 12;RefCode:SOBTUY:C40 H42 Cd2 N12 O21 Pd1Author(s): Abrahams B.F.,Hoskins B.F.,Robson R. Journal: J.AM.CHEM.SOC. Year: 1991 Volume: 113 Number: Pages: 3606 ######################################### -------------------- Atom Pd1 links with R(A-A) Pd 1 0.5000 -0.5000 1.0000 ( 0-1 1) 19.905A Pd 1 -1.0000 0.0000 -1.5000 (-1 0-2) 17.126A Pd 1 1.0000 0.0000 1.5000 ( 1 0 1) 17.126A Pd 1 -0.5000 0.5000 -1.0000 (-1 0-1) 19.905A ------------------------- Structure consists of 3D framework with Pd(SINGLE NET) Coordination sequences ---------------------- Pd1: 1 2 3 4 5 6 7 8 9 10 Num 4 12 30 58 94 138 190 250 318 394 Cum 5 17 47 105 199 337 527 777 1095 1489 ---------------------- Vertex symbols for selected sublattice -------------------------------------- Pd1 Point/Schlafli symbol:{6^5;8} With circuits:[6.6.6.6.6(2).8(2)] With rings: [6.6.6.6.6(2).*] -------------------------------------- Total Point/Schlafli symbol: {6^5;8} 4-c net; uninodal net Classification of the topological type: cds/CdSO4 {6^5;8} - VS [6.6.6.6.6(2).*] TOPOS OUTPUT

  26. O’Keffe & Delgado-Friedrichs 3dt 2002 SyStRe 2003 Symmetry Structure Realization one can determinewithout ambiguity whether two nets are isomorphic or not

  27. SyStRe

  28. 3dt 3D Tiling

  29. Thanks to: Delgado-Friedrich, O’Keeffe, Hyde, Blatov, Proserpio.

  30. Suplementary slides Suplementary slides

  31. Self-entanglement

  32. POLYCATENATION INTERPENETRATION increase of dimensionality dimensionality unchanged

  33. ..\libro_braga\figure\asufig.jpg

  34. Borromean entanglements Polycatenation Interpenetration “Topological” “Euclidean” self-catenation Polythreading A new complexity of the solid state

  35. Data: Electronic crystallographic databases CSD ~430000 entries ICSD ~100000 entries CrystMet ~100000 entries PDB ~50000 entries

  36. Data: Electronic crystallochemical databases RCSR 1620 entries; http://rcsr.anu.edu.au TTD Collection 66833 entries; http://www.topos.ssu.samara.ru TTO Collection 3617 entries; http://www.topos.ssu.samara.ru Atlas of Zeolite Frameworks, 179 entries;http://www.iza-structure.org/databases/

  37. Data: Electronic databases of hypothetical nets EPINET 14532 entries; http://epinet.anu.edu.au/ Atlas of Prospective Zeolite Frameworks 2543772 entries; http://www.hypotheticalzeolites.net/

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