Quasicrystals. I. Definition and structure
Abstract
In a recent paper, we introduced the concept of quasicrystals [Phys. Rev. Lett. 53, 2477 (1984)], a new class of ordered atomic structures. Quasicrystals have long-range quasiperiodic translational order and long-range orientational order. In the present paper and the following one, we discuss the details of our analysis of the mathematical and structural properties of quasicrystals. We begin with a general overview of our analysis. We then discuss our computation of the diffraction pattern of a quasilattice, using as an example the case of icosahedral orientational symmetry. We demonstrate that two quasilattices with the same orientational symmetry and quasiperiodicity which are not locally isomorphic will have diffraction patterns with different peak intensities. Finally, we describe some examples of computer modeling of atomic quasicrystals.
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It is generally believed that a quasicrystal should be ordered but not periodic and have long-range quasiperiodic translational order and long-range orientational order.
“it is generally believed that for a structure to be a quasicrystal, it should have the following properties: 1. It is ordered but not periodic.2. It has long-range quasiperiodic translational order and long-range orientational order.”
From From the Fibonacci Icosagrid to E8 (Part I): The Fibonacci Icosagrid, an H3 Quasicrystal (2024) · cites this paper as [8]
There is still no commonly agreed upon definition of a quasicrystal.
“While there is still no commonly agreed upon definition of a quasicrystal [5,8,21]”
From From the Fibonacci Icosagrid to E8 (Part I): The Fibonacci Icosagrid, an H3 Quasicrystal (2024) · cites this paper as [8]
It is generally believed that a quasicrystal should have finite types of prototiles/unit cells and a discrete diffraction pattern.
“3. It has finite types of prototiles/unit cells.4. It has a discrete diffraction pattern.”
From From the Fibonacci Icosagrid to E8 (Part I): The Fibonacci Icosagrid, an H3 Quasicrystal (2024) · cites this paper as [8]
When ρ = μ = φ (the golden ratio), x_N defines a Fibonacci chain, a 1D quasicrystal.
“In this case, x N defines a Fibonacci chain, a 1D quasicrystal [8].”
From From the Fibonacci Icosagrid to E8 (Part I): The Fibonacci Icosagrid, an H3 Quasicrystal (2024) · cites this paper as [8]
Cites
On periodic and non-periodic space fillings of<b>E</b><sup><i>m</i></sup>obtained by projection
Kramer, P., & Neri, R. (1984) · Acta Crystallographica Section A Foundations of Crystallography · doi:10.1107/s0108767384001203 · Cited by 396 · see citing works
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Quasicrystals. II. Unit-cell configurations
Socolar, J. E. S., & Steinhardt, P. J. (1986) · Physical Review B · doi:10.1103/physrevb.34.617 · Cited by 392 · see citing works
Self-citationFang, F., & Irwin, K. (2015) · arXiv · doi:10.48550/arxiv.1511.07786 · Cited by 12 · see citing works
From the Fibonacci Icosagrid to E8 (Part I): The Fibonacci Icosagrid, an H3 Quasicrystal
Fang, F., & Irwin, K. (2024) · Crystals · doi:10.3390/cryst14020152 · Cited by 1 · see citing works
- Authors
- Levine, D., & Steinhardt, P. J.
- Affiliations
- University of Pennsylvania (US)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 1986-07-15
- Volume
- 34
- Issue
- 2
- Pages
- 596-616
- ISSN
- 0163-1829 (ISSN-L)
- OpenAlex
- W1989332828
- Cited by
- 708
- References
- 1
- Citations by year
- 2026: 16 · 2025: 26 · 2024: 26 · 2023: 18 · 2022: 23 · 2021: 19 · 2020: 16 · 2019: 14 · 2018: 23 · 2017: 12 · 2016: 22 · 2015: 18 · 2014: 22 · 2013: 22 · 2012: 12
- FWCI
- 49.70
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical review. B, Condensed matter (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Materials Science › Materials Chemistry › Quasicrystal Structures and Properties
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Photonic Crystals and Applications
- Physical Sciences › Materials Science › Materials Chemistry › X-ray Diffraction in Crystallography
- Keywords
- quasicrystals, quasiperiodic order, quasilattices, diffraction patterns, icosahedral symmetry, orientational order, long-range order, local isomorphism, peak intensity
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