Abstract
Crystals, the word derived from the Greek meaning “ice”, are the most stable form of all solids. They are found throughout the natural world, and have been always recognized as being distinct from other forms of matter.
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Notes
- 1.
Even before the Greek period, various properties of figures in the plane and space had been “known” by ancient people as empirical knowledge, some of which were gained from their experiences in land surveys (it is worthwhile to recall that the word “geometry” was derived from the Greek word “geometria” which is the composition of “geo” meaning land and “metria” meaning measurement).
- 2.
The Pythagorean school in Crotona, a Greek colonial town on the southeastern coast of Italy, was a religious order in which Pythagoras was a sort of cult figure.
- 3.
- 4.
More precisely, in the crystalline state, atoms oscillate around the positions of equilibrium. We are dealing with the positions of equilibrium when we talk about crystal structures.
- 5.
A primitive idea of “bond” had already been conceived as early as the twelfth century. It hypothesizes that certain types of chemical species are joined by a type of chemical affinity. By the mid-nineteenth century, it became clear that chemical compounds are formed from constituent elements, and the idea of bond had been developed as the theory of valency based on the theory of radicals. In particular, the theory of chemical structures by the German chemist August Kekulé, in which he took into account the specific combining power (or valences) of specific atoms, provided a dramatic new clarity of understanding on chemical compounds (1858).
- 6.
French chemist L. Pasteur observed that the relative spatial arrangement of atoms within molecules is vital in understanding their chemical properties (1861).
- 7.
In reality, a crystal net has finite size. But we think that the net extends to the whole space when we talk about crystals.
- 8.
\({X}_{0}^{\mathrm{ab}}\) is also called the homology universal covering graph.
- 9.
From now on we tacitly assume that deg x ≥ 2 for every vertex x.
- 10.
A base graph is also called a fundamental finite graph. In crystallography, it is called a quotient graph (of a periodic graph).
- 11.
dim X does not depend on the choice of an abstract period lattice.
- 12.
Actually one may take any norm. The norm defined here is used to describe the large deviation property of random walks on topological crystals [Notes (III) in Chap. 9].
- 13.
By abuse of language, the term “cubic lattice” is often used to express the 3-dimensional standard lattice group. In crystallography, this, as a lattice group in \({\mathbb{R}}^{3}\), is called the primitive cubic lattice. A general cubic lattice is a lattice commensurable with the primitive cubic lattice. Here two lattice groups L 1 and L 2 of \({\mathbb{R}}^{d}\) are called commensurable if \({L}_{1} \cap {L}_{2}\) are of finite index in both L 1 and L 2. See Appendix 2 for the definition of subgroups of finite index.
- 14.
The square lattice as a topological crystal is called the quadrangle lattice.
- 15.
Carbon nanotubes are allotropes of carbon discovered by Sumio Iijima in 1991 and have remarkable properties, making them potentially useful in many applications in nanotechnology.
- 16.
Zeolites are microporous, aluminosilicate minerals commonly used as commercial adsorbents.
- 17.
See Appendix 2.
- 18.
See Appendix 2.
- 19.
A d-dimensional crystallographic group is a discrete subgroup, say \(\Gamma \), of the motion group M(d) of \({\mathbb{R}}^{d}\) with compact quotient \(\mathbf{M}(d)/\Gamma \) (see Appendix 5). Crystallographic groups are virtually abelian, but not vice versa.
- 20.
The diamond crystal is a join of the same hexagonal rings (called also the chair conformation in chemical terminology).
- 21.
For a closed path c, the product c⋯c (m times) is called the m-th power of c.
- 22.
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Sunada, T. (2013). Topological Crystals. In: Topological Crystallography. Surveys and Tutorials in the Applied Mathematical Sciences, vol 6. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54177-6_6
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