Volumes and Issues
Home » Technical Physics » Year 2025, issue 1 » Article p. 51
<<<>>>
Theory of the structure of icosahedral quasicrystals: types of packings
Russian version
Madison A. E.1, Madison P. A. 1,2
1HSE University, St. Petersburg, Russia
2St. Petersburg State Electrotechnical University “LETI", St. Petersburg, Russia
Email: alex_madison@mail.ru
PDF
A unified theory of the structure of icosahedral quasicrystals is proposed. All possible variants of self-similar icosahedral packings are analyzed. These include 3 types of quasilattices (P, I, F), which are the analogues of primitive, body-centered and face-centered cubic lattices; each of them can be either centrosymmetric or non-centrosymmetric. Substitution rules for I and F-type tetrahedral tilings are fully formalized. An example of constructing a non-centrosymmetric I-type packing is presented. A method is shown for generating a zonohedral packing (P) from a tetrahedral packing (I) by joining the neighboring tetrahedra in it. For each packing type, 3 locally isomorphic patches are possible, differing in the choice of node in its center (A, B, C). When the tetrahedral packings are built up, three locally isomorphic patches cyclically transform into each other after each iteration. As a consequence, the structures of the three types of characteristic clusters are not independent. An icosahedral packing of any type can be constructed based on a unified algorithm when initialized with a single tetrahedron. Keywords: icosahedral quasicrystals, substitution rules, packings.
  1. A.E. Madison. Tech. Phys. Lett., 50 (10), 13 (2024)
  2. A.E. Madison, P.A. Madison. Tech. Phys., 69, 1967 (2024)
  3. W. Steurer, T. Haibach. In: International Tables for Crystallography Volume B: Reciprocal Space, ed. by U. Shmueli (Springer, Dordrecht, 2006), v. B, Ch. 4.6, p. 486. DOI: 10.1107/97809553602060000568
  4. W. Steurer, S. Deloudi. Crystallography of quasicrystals. Concepts, methods and structures (Springer, Berlin--Heidelberg, 2009), DOI: 10.1007/978-3-642-01899-2
  5. S. Hyde, S. Andersson, K. Larsson, Z. Blum, T. Landh, S. Lidin, B.W. Ninham. The language of shape: The role of curvature in condensed matter physics, chemistry, and biology (Elsevier, Amsterdam, 1997), DOI: 10.1016/B978-0-444-81538-5.X5000-X
  6. M. Baake, U. Grimm. Acta Cryst. A, 76, 559 (2020). DOI: 10.1107/S2053273320007421
  7. M. Senechal. Proc. Steklov Inst. Math., 288, 259 (2015). DOI: 10.1134/S0081543815010204
  8. J.E.S. Socolar, P.J. Steinhardt. Phys. Rev. B, 34, 617 (1986). DOI: 10.1103/PhysRevB.34.617
  9. L. Danzer. Discrete Math., 76, 1 (1989). DOI: 10.1016/0012-365X(89)90282-3
  10. L. Danzer, Z. Papadopolos, A. Talis. Int. J. Mod. Phys. B, 7, 1379 (1993). DOI: 10.1142/S0217979293002389
  11. P. Kramer, R. Neri. Acta Cryst. A, 40, 580 (1984). DOI: 10.1107/S0108767384001203
  12. K. Kato, T. Ninomiya. J. Alloys Compd., 342, 206 (2002). DOI: 10.1016/S0925-8388(02)00180-9
  13. A. Yamamoto, H. Takakura, A.P. Tsai. Phys. Rev. B, 68, 094201 (2003). DOI: 10.1103/PhysRevB.68.094201
  14. H. Takakura, C. Pay Gomez, A. Yamamoto, M. de Boissieu, A.P. Tsai. Nature Mater., 6, 58 (2007). DOI: 10.1038/nmat1799
  15. T. Yamada, H. Takakura, H. Euchner, C. Pay Gomez, A. Bosak, P. Fertey, M. de Boissieu. IUCrJ, 3, 247 (2016). DOI: 10.1107/S2052252516007041
  16. I. Buganski, J. Wolny, H. Takakura. Acta Cryst. A, 76, 180 (2020). DOI: 10.1107/S2053273319017339
  17. I. Buganski, R. Strzalka, J. Wolny. Acta Cryst. B, 80, 84 (2024). DOI: 10.1107/S2052520624000763
  18. L.S. Levitov, J. Rhyner. JETP Lett., 47 (12), 760 (1988). http://jetpletters.ru/ps/1099/article_16620.pdf
  19. L.S. Levitov, J. Rhyner. J. Phys. France, 49, 1835 (1988). DOI: 10.1051/jphys:0198800490110183500
  20. P. Kramer, Z. Papadopolos, D. Zeidler. In: Symmetries in science V, ed. by B. Gruber, L.C. Biedenharn, H.D. Doebner (Springer, Boston, 1991), p. 395. DOI: 10.1007/978-1-4615-3696-3_19
  21. P. Kramer, Z. Papadopolos, M. Schlottmann, D. Zeidler. J. Phys. A: Math. Gen., 27, 4505 (1994). DOI: 10.1088/0305-4470/27/13/024
  22. A.E. Madison. RSC Adv., 5, 5745 (2015). DOI: 10.1039/C4RA09524C
  23. A.E. Madison. RSC Adv., 5, 79279 (2015). DOI: 10.1039/C5RA13874D
  24. A.E. Madison, P.A. Madison, V.A. Moshnikov. Tech. Phys., 69, 528 (2024)
  25. D.A. Rabson, N.D. Mermin, D.S. Rokhsar, D.C. Wright. Rev. Mod. Phys., 63, 699 (1991). DOI: 10.1103/RevModPhys.63.699
  26. D.B. Litvin. Acta Cryst. A, 47, 70 (1991). DOI: 10.1107/S0108767390010054
  27. A.E. Madison, P.A. Madison. Struct. Chem., 31, 485 (2020). DOI: 10.1007/s11224-019-01430-w
  28. J.H. Conway, N.J.A. Sloane. Sphere packings, lattices and groups, 3rd ed. (Springer, NY., 1999), DOI: 10.1007/978-1-4757-6568-7
  29. M. Baake, U. Grimm. Aperiodic order. V. 1: A mathematical invitation (Cambridge Univ. Press, Cambridge, 2013), DOI: 10.1017/CBO9781139025256
  30. A. Al-Siyabi, N. Ozdes Koca, M. Koca. Symmetry, 12, 1983 (2020). DOI: 10.3390/sym12121983
  31. L.S. Levitov. EPL, 6, 517 (1988). DOI: 10.1209/0295-5075/6/6/008
  32. A.E. Madison. Struct. Chem., 26, 923 (2015). DOI: /10.1007/s11224-014-0559-3
  33. A.E. Madison. Struct. Chem., 29, 645 (2018). DOI: 10.1007/s11224-018-1083-7
  34. S. Pautze. Symmetry, 9, 19 (2017). DOI: 10.3390/sym9020019
  35. J.H. Conway, C. Radin. Invent. Math., 132, 179 (1998). DOI: 10.1007/s002220050221
  36. D. Frettlöh, A.L.D. Say-awen, M.L.A.N. De Las Penas. Indag. Math., 28, 120 (2017). DOI: 10.1016/j.indag.2016.11.009
  37. K. Shea. In: Design Computing and Cognition '04, ed. by J.S. Gero (Springer, Dordrecht, 2004), p. 137. DOI: 10.1007/978-1-4020-2393-4_8
  38. J. Roth. J. Phys. A: Math. Gen., 26, 1455 (1993). DOI: 10.1088/0305-4470/26/7/008
  39. M. Baake, M. Schlottmann, P.D. Jarvis. J. Phys. A: Math. Gen., 24, 4637 (1991). DOI: 10.1088/0305-4470/24/19/025
  40. A.E. Madison, P.A. Madison. Proc. Roy. Soc. A, 475, 20180667 (2019). DOI: 10.1098/rspa.2018.0667
  41. S.-Y. Jeon, H. Kwon, K. Hur. Nat. Phys., 13, 363 (2017). DOI: 10.1038/nphys4002
  42. L. Casas. J. Appl. Cryst., 53, 1583 (2020). DOI: 10.1107/S1600576720011772
  43. T. Yamada, N. Fujita, F. Labib. Acta Cryst. B, 77, 638 (2021). DOI: 10.1107/S2052520621006715
  44. V.E. Dmitrienko. Acta Cryst. A, 50, 515 (1994). DOI: 10.1107/S0108767393013960
  45. V.E. Dmitrienko, V.A. Chizhikov. Crystallogr. Rep., 51, 552 (2006). DOI: 10.1134/S106377450604002X
  46. S. Lidin. In: Handbook of solid state chemistry, ed. by R. Dronskowski, S. Kikkawa, A. Stein (Wiley-VCH, Weinheim, 2017), p. 73. DOI: 10.1002/9783527691036.hsscvol1002
  47. T. Matsubara, A. Koga, A. Takano, Y. Matsushita, T. Dotera. Nat. Commun., 15, 5742 (2024). DOI: 10.1038/s41467-024-49843-4
  48. C.T. Hann, J.E.S. Socolar, P.J. Steinhardt. Phys. Rev. B, 94, 014113 (2016). DOI: 10.1103/PhysRevB.94.014113
  49. M. de Boissieu. Chem. Soc. Rev., 41, 6778 (2012). DOI: 10.1039/C2CS35212E
Publisher:

Ioffe Institute

Institute Officers:

Director: Sergei V. Ivanov

Contact us:

26 Polytekhnicheskaya, Saint Petersburg 194021, Russian Federation
Fax: +7 (812) 297 1017
Phone: +7 (812) 297 2245
E-mail: post@mail.ioffe.ru