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Igor A. Baburin

Publications and source records attributed to Igor A. Baburin.

8 recordsLinked to original sources
arXiv2026

Subperiodic groups and bounded automorphisms of periodic graphs

A subperiodic group is a group of motions of $d$-dimensional Euclidean space $\R^d$ which contains a translation lattice $\Z^r$ of rank $r < d$ as a subgroup of finite index. A classification into abstract group isomorphism classes is performed for subperiodic groups in dimension~3: 75 \emph{crystallographic} rod groups ($r=1$) and 80 layer groups ($r=2$) are shown to belong to 32 and 34 isomorphism classes, respectively. An easy-to-compute set of invariants is developed for recognizing these isomorphism classes from finite presentations which makes use only of the number of subgroups up to a given finite index~$n$ ($n \leq 12$ for rod groups and $n \leq 8$ for layer groups) and how many of them are normal. Cayley graphs of rod and layer groups are used to illustrate the concept of bounded automorphisms of finite order, \emph{i.e.} those when the distance between a graph vertex and its image has an upper bound. It is proven that a Cayley graph of a crystallographic space group $G$ (in which case $r=d$) possesses bounded automorphisms of finite order, if and only if the respective inverse-closed generating set is stabilized by conjugation by an element of finite order in $G$. As an application, subperiodic groups in $\R^4$ with a three-dimensional translation lattice are used to systematically derive embeddings of three-periodic \emph{ladder graphs} in~$\R^3$.

math.GR↗
arXiv2025

On colourings of cubic lattices

Given the integral lattice $Λ^d$ in $d$-dimensional Euclidean space, partitions of the lattice nodes into orbits of finite-index subgroups of $Aut(Λ^d)$ have been computed for $d \leq 4$. These partitions can be interpreted as colourings of orbits defined up to permutation of colours. Complete results are obtained for $d=2$ up to 64 orbits, for $d=3$ up to 8 orbits, and for 2 orbits in dimension 4. The automorphism groups of the partitions are also determined. Our results for two orbits in dimension 3 correct the old result of H. Heesch [Z. Kristallogr., (1933), 85, 335--344] who overlooked one partition.

math.CO↗
arXiv2025

Short presentations for crystallographic groups

A practical approach is proposed to construct short presentations for Euclidean crystallographic groups in terms of generators and relations. For our purposes a short presentation is the one with a small number of short relators for a given generating set. The connection is emphasized between relators of a group presentation and cycles in the associated Cayley graph. It is shown by examples that a short presentation is usually the one where relators correspond to strong rings in the Cayley graph and therefore provide a natural upper bound for their size. Presentations are computed for vertex-transitive groups which act with trivial vertex stabilizers on a number of high-symmetry 2-, 3- and 4-periodic graphs. Higher-dimensional as well as subperiodic examples are also considered. Relations are explored between geodesics in periodic graphs and corresponding cycles in their quotients.

math.GR↗
arXiv2022

Ge$_{136}$ type-II clathrate as precursor for the synthesis of metastable germanium polymorphs: a computational study

The response to compression of the clathrate type-II structure Ge(cF136) is investigated by means of \textit{ab initio} small-cell metadynamics at different temperatures and pressures. At lower pressure $p$=2.5 GPa the metastable metallic bct-5 phase competes against $β$-Sn Ge(tI4), which forms at higher pressures. On lowering temperature, the presence of amorphous intermediates obtained from Ge$_{136}$ is more pronounced and is instrumental to the formation of denser structural motifs, from which metallic bct-5 can form. Therein, anisotropic box fluctuations promote phase formation. The metadynamics runs are analysed in depth using a set of topological descriptors, including coordination sequence and ring statistics. Differences in the structural landscape history and amorphous intermediates are critical for the selective formation of particular metastable polymorphs, towards turning crystal structure predictions into actual materials.

cond-mat.mtrl-sci↗
arXiv2018

On the Origin of Crystallinity: a Lower Bound for the Regularity Radius of Delone Sets

The local theory of regular or multi-regular systems aims at finding sufficient local conditions for a Delone set $X$ to be a regular or multi-regular system. One of the main goals is to estimate the regularity radius $\hatρ_d$ for Delone sets $X$ in terms of the radius $R$ of the largest "empty ball" for $X$. The present paper establishes the lower bound $\hat{ρ_d}\geq 2dR$ for all $d$, which is linear in $d$. The best previously known lower bound had been $\hatρ_d\geq 4R$ for $d\geq 2$. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent $(2dR-\varepsilon)$-clusters, which are not regular systems.

math.MG↗
arXiv2012

Novel metastable metallic and semiconducting germaniums

By means of ab initio metadynamics runs we explored the lower-pressure region of the phase diagram of germanium. A monoclinic germanium phase with four-membered rings, less dense than diamond and compressible into β-tin phase (tI4) was found. A metallic bct-5 phase, mechanically stable down to room conditions appeared between diamond and tI4. mC16 is a narrow-gap semiconductor, while bct-5 is metallic and potentially still superconducting in the very low pressure range. This finding may help resolving outstanding experimental issues.

cond-mat.mtrl-sci↗
arXiv2012

Superhard sp3 carbon allotropes with odd and even ring topologies

Four sp3 carbon allotropes with six, eight, and 16 atoms per primitive cell have been derived using a combination of metadynamics simulations and topological scan. A chiral orthorhombic phase oC16 (C2221) was found to be harder than monoclinic M-carbon and shows excellent stability in the high-pressure range. A second orthorhombic phase of Cmmm symmetry, by \sim 0.028 eV/atom energetically lower than W-carbon, can be formed from graphite at \sim 9 GPa. In general, the mechanical response under pressure was found to depend on the structure topology, which reflects the way rings are formed from an initial graphene layer stacking.

cond-mat.mtrl-sci↗