Symmetries of coincidence site lattices of cubic lattices

Zeiner, Peter

doi:10.1524/zkri.2005.220.11_2005.915

Cited by 16 publications

(30 citation statements)

References 13 publications

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“…Nevertheless, quite a bit is known for simple coincidences [2,24,25], and first steps are in sight for multiple ones [26]. Also, in the planar case, one would like to get rid of the assumption (CN1).…”

Section: Discussionmentioning

confidence: 99%

Multiple planar coincidences with N-fold symmetry

Baake¹,

Grimm²

2006

Zeitschrift Für Kristallographie - Crystalline Materials

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Abstract. Planar coincidence site lattices and modules with N -fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions. Key Words:Lattices, Coincidence Ideals, Planar Modules, Cyclotomic Fields, Dirichlet Series, Asymptotic Properties IntroductionGiven a lattice Γ ⊂ R d , with d ≥ 2, it is interesting to know its coincidence site lattices (CSLs), which originate from intersections of Γ with a rotated copy. In fact,is a group, whose structure in general (i.e., for d > 2) is not well understood. The coincidence indexof a rotation R is defined as the number of cosets of Γ ∩ RΓ in Γ (which can be ∞), and the spectrum of finite Σ-values, Σ SOC(Γ ) , is often the first quantity considered, followed by counting all CSLs of a given index. Note that one can also consider the obvious extension to general orthogonal transformations R ∈ O(d) and the corresponding group OC(Γ ).Coincidence lattices play an important role in the theory of grain boundaries [9,13], and small indices can be determined in suitable experiments. The classification of these elementary or simple coincidences is partly done, in particular for low-dimensional lattices with irreducible symmetries, compare [7,2] and references given there, but also for certain generalizations to quasicrystals [17,2]. The situation for various related problems More recent is the problem of optimal lattice quantizers [19], and connected with it is the question for multiple coincidences. Here, one would like to classify all lattices that can be obtained as multiple intersections of the formwith R ℓ ∈ SO(d). One defines the corresponding index aswhich is finite if and only if each R ℓ ∈ SOC(Γ ), due to the mutual commensurability of the lattices Γ ∩ R ℓ Γ (we shall explain this in more detail below). Consequently, one attaches the group SOC(Γ ) m to the setting of m-fold coincidences. The corresponding spectrum is its image under Σ, while the full (or complete) coincidence spectrum of Γ isThis is an inductive limit, withwhich often stabilizes: from a certain m on, the spectra are stable, i.e., they do not grow any more [26]. In our examples below, this actually happens at m = 1, so that the spectrum is Σ Γ = Σ SOC(Γ ) . Clearly, multiple coincidences are also relevant in crystallography, as they are the basis for triple or multiple junctions, in obvious generalization of twinning, compare [11].The problem of multiple coincidences is considerably more involved than that of the simple ones, in particular in dimensions d ≥ 3. However, for d = 2, one can rather easily extend the treatment of elementary coincidences to multiple ones, building on the powerful and well understood connection to the ...

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Section: Discussionmentioning

confidence: 99%

Multiple planar coincidences with N-fold symmetry

Baake¹,

Grimm²

2006

Zeitschrift Für Kristallographie - Crystalline Materials

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“…[9,10] for other applications of quaternion algebra to the study of grain boundaries; an overview of quaternion algebra is given in Ref. [11]).…”

Section: Generating Grain Boundary Structuresmentioning

confidence: 99%

“…[12,13]) that is, a periodic sublattice of space points where red and blue atoms coincide. In this case, given q, closed expressions exist that readily provide a set of basis vectors for the coincident site sublattice [10]. Besides making it easy to perform rotations with no need for using rotation matrices, there is an additional favourable aspect of the quaternion scheme for generating symmetric tilt grain boundaries.…”

Section: Generating Grain Boundary Structuresmentioning

confidence: 99%

Imeall: A computational framework for the calculation of the atomistic properties of grain boundaries

Lambert

Fekete

Kermode

et al. 2018

Computer Physics Communications

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a b s t r a c tWe describe the Imeall package for the calculation and indexing of atomistic properties of grain boundaries in materials. The package provides a structured database for the storage of atomistic structures and their associated properties, equipped with a programmable application interface to interatomic potential calculators. The database adopts a general indexing system that allows storing arbitrary grain boundary structures for any crystalline material. The usefulness of the Imeall package is demonstrated by computing, storing, and analysing relaxed grain boundary structures for a dense range of low index orientation axis symmetric tilt and twist boundaries in α-iron for various interatomic potentials. The package's capabilities are further demonstrated by carrying out automated structure generation, dislocation analysis, interstitial site detection, and impurity segregation energies across the grain boundary range. All computed atomistic properties are exposed via a web framework, providing open access to the grain boundary repository and the analytic tools suite. Program summaryProgram Title: Imeall Program Files doi: http://dx.doi.org/10.17632/nj77stc62b.1 Licensing provisions: Apache-2.0 Programming languages: python, fortran, javascript Nature of problem: Determining the minimum energy structure for a specific grain boundary and interatomic force field involves extensive searches in configuration space. Duplication of this effort should be avoided, and providing a unique database to gather the resulting structures is needed for this. Accurately cataloguing the chemical and electronic environments associated with the interface atoms in the grain boundary furthermore requires a useable interface between a database of grain boundary structures and an expandible set of interatomic potential calculators. Solution method: We introduce a standard indexing convention that allows the integration of grain boundary structures for arbitrary materials, generated by different users/research groups, into a normalised database that can be easily queried and used as a starting point for further research projects. The Imeall package accomplishes this by specifying a standard naming convention for the grain boundary database and by providing the software routines necessary to populate and query such a database, as well as an interface to interatomic calculators and analysis tools.

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“…First of all, having counted the CSMs, one would like to have a finer classification into Bravais types. Some results in this direction, for the cubic lattices, are presented in [25]. Then, there is no compelling reason to stop at single intersections, and recent developments make an extension to multiple coincidence site modules desirable.…”

Section: Extensions and Outlookmentioning

confidence: 99%