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Symmetry
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Crystal Symmetry and Structure
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Crystal Symmetry and Structure

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 May 2014) | Viewed by 65551

Special Issue Editor

Prof. Dr. Gervais Chapuis

E-Mail Website
Guest Editor
Federal School of Technology (EPFL), Route Cantonale, 1015 Lausanne, Switzerland
Interests: (computer aided) crystallographic teaching; aperiodic material; incommensurate crystals; superspace symmetry

Special Issue Information

Dear Colleagues,

The concept of crystal structure is intimately related to the notion of symmetry. W.L. Bragg published the description of the first structures only a year after the discovery of diffraction by crystals a century ago. By combining both diffraction and symmetry considerations he could solve the first crystal structures. The description of the symmetry properties of the 230 space groups a few years later by P. Niggli in 1919 is at the origin of the tremendous success of diffraction methods for the elucidation of structures. Space groups are systematically used nowadays to solve, describe and classify any of them.

The paradigm of the three dimensional periodicity of crystalline matter ended up in the 1970’s when aperiodic structures were discovered. The classical concept of three-dimensional symmetry had to be extended to higher dimensional symmetry considerations. Currently, it is not unusual to describe well-ordered structures but aperiodic ones in space up to six dimensions, in the frame of the so-called superspace symmetry groups.

It is the aim of this special issue of Symmetry to present a broad spectrum of modern and recent aspects of symmetry considerations, which are at the disposal of the specialists in order to improve our understanding of the fine structure details of crystalline solids.

Prof. Dr. Gervais Chapuis
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • point group symmetry
  • space group symmetry
  • superspace group symmetry
  • periodic crystal structures
  • aperiodic crystal structures
  • Bravais lattices
  • diffraction symmetry
  • local symmetry
  • scaling symmetry
  • molecular symmetry

Published Papers (10 papers)

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Research

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3800 KiB  
Article
Crystallography and Magnetic Phenomena
by Vojtěch Kopský
Symmetry 2015, 7(1), 125-145; https://doi.org/10.3390/sym7010125 - 02 Feb 2015
Cited by 5 | Viewed by 6626
Abstract
This essay describes the development of groups used for the specification of symmetries from ordinary and magnetic point groups to Fedorov and magnetic space groups, as well as other varieties of groups useful in the study of symmetric objects. In particular, we consider [...] Read more.
This essay describes the development of groups used for the specification of symmetries from ordinary and magnetic point groups to Fedorov and magnetic space groups, as well as other varieties of groups useful in the study of symmetric objects. In particular, we consider the problem of some incorrectness in Vol. A of the International Tables for Crystallography. Some results of tensor calculus are presented in connection with magnetoelectric phenomena, where we demonstrate the use of Ascher’s trinities and Opechowski’s magic relations and their connection. Specific tensor decomposition calculations on the grounds of Clebsch Gordan products are illustrated. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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466 KiB  
Article
Polar Vector Property of the Stationary State of Condensed Molecular Matter
by Jürg Hulliger, Luigi Cannavacciuolo and Mathias Rech
Symmetry 2014, 6(4), 844-850; https://doi.org/10.3390/sym6040844 - 13 Oct 2014
Viewed by 5275
Abstract
Crystalline phases undergoing 180\(^{\circ}\) orientational disorder of dipolar entities in the seed or at growing (hkl) faces will show a polar vector property described by \(\infty\) /mm symmetry. Seeds and crystals develop a bi-polar state (\(\infty\)/mm), where domains related by a mirror plane [...] Read more.
Crystalline phases undergoing 180\(^{\circ}\) orientational disorder of dipolar entities in the seed or at growing (hkl) faces will show a polar vector property described by \(\infty\) /mm symmetry. Seeds and crystals develop a bi-polar state (\(\infty\)/mm), where domains related by a mirror plane m allow for a \(\infty\) m symmetry in each domain. The polarity of domains is due to energetic favorable interactions at the object-to-nutrient interface. Such interactions are well reproduced by an Ising Hamiltonian. Two-dimensional Monte Carlo simulations performed for real molecules with full long-range interactions allow us to calculate the spatial distribution of the electrical polarization Pel. The investigation has been extended to liquid droplets made of dipolar entities by molecular dynamics simulations. We demonstrate the development of an m\(\bar{\infty}\) quasi bi-polar state leading to a charged surface. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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961 KiB  
Article
Twinning of Polymer Crystals Suppressed by Entropy
by Nikos Ch. Karayiannis, Katerina Foteinopoulou and Manuel Laso
Symmetry 2014, 6(3), 758-780; https://doi.org/10.3390/sym6030758 - 04 Sep 2014
Cited by 10 | Viewed by 6815
Abstract
We propose an entropic argument as partial explanation of the observed scarcity of twinned structures in crystalline samples of synthetic organic polymeric materials. Polymeric molecules possess a much larger number of conformational degrees of freedom than low molecular weight substances. The preferred conformations [...] Read more.
We propose an entropic argument as partial explanation of the observed scarcity of twinned structures in crystalline samples of synthetic organic polymeric materials. Polymeric molecules possess a much larger number of conformational degrees of freedom than low molecular weight substances. The preferred conformations of polymer chains in the bulk of a single crystal are often incompatible with the conformations imposed by the symmetry of a growth twin, both at the composition surfaces and in the twin axis. We calculate the differences in conformational entropy between chains in single crystals and chains in twinned crystals, and find that the reduction in chain conformational entropy in the twin is sufficient to make the single crystal the stable thermodynamic phase. The formation of cyclic twins in molecular dynamics simulations of chains of hard spheres must thus be attributed to kinetic factors. In more realistic polymers this entropic contribution to the free energy can be canceled or dominated by nonbonded and torsional energetics. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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866 KiB  
Article
Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures
by Berthold Stöger
Symmetry 2014, 6(3), 589-621; https://doi.org/10.3390/sym6030589 - 21 Jul 2014
Cited by 4 | Viewed by 5563
Abstract
Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting [...] Read more.
Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting restrictions can be expressed in terms of linear relations of the a2, b2 and a · b scalar products of the lattice basis vectors with rational coefficients. To characterize OD families and to check their validity, these lattice restrictions are expressed in the bases of different layers and combined. For a more familiar notation, they can be expressed in terms of the lattice parameters a, b and . Alternatively, the description of the lattice restrictions may be simplified by using centered lattices. The representation of the lattice restrictions in terms of scalar products is dependent on the chosen basis. A basis-independent classification of the lattice restrictions is outlined. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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493 KiB  
Article
Development of Symmetry Concepts for Aperiodic Crystals
by Ted Janssen
Symmetry 2014, 6(2), 171-188; https://doi.org/10.3390/sym6020171 - 31 Mar 2014
Cited by 1 | Viewed by 5464
Abstract
An overview is given of the use of symmetry considerations for aperiodic crystals. Superspace groups were introduced in the seventies for the description of incommensurate modulated phases with one modulation vector. Later, these groups were also used for quasi-periodic crystals of arbitrary rank. [...] Read more.
An overview is given of the use of symmetry considerations for aperiodic crystals. Superspace groups were introduced in the seventies for the description of incommensurate modulated phases with one modulation vector. Later, these groups were also used for quasi-periodic crystals of arbitrary rank. Further extensions use time reversal and time translation operations on magnetic and electrodynamic systems. An alternative description of magnetic structures to that with symmetry groups, the Shubnikov groups, is using representations of space groups. The same can be done for aperiodic crystals. A discussion of the relation between the two approaches is given. Representations of space groups and superspace groups play a role in the study of physical properties. These, and generalizations of them, are discussed for aperiodic crystals. They are used, in particular, for the characterization of phase transitions between aperiodic crystal phases. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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244 KiB  
Article
On the Notions of Symmetry and Aperiodicity for Delone Sets
by Michael Baake and Uwe Grimm
Symmetry 2012, 4(4), 566-580; https://doi.org/10.3390/sym4040566 - 10 Oct 2012
Cited by 5 | Viewed by 7176
Abstract
Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept [...] Read more.
Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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306 KiB  
Article
A Higher Dimensional Description of the Structure of β-Mn
by Sven Lidin and Daniel Fredrickson
Symmetry 2012, 4(3), 537-544; https://doi.org/10.3390/sym4030537 - 27 Aug 2012
Cited by 9 | Viewed by 6309
Abstract
The structure of β-Mn crystallizes in space group P4132. The pseudo 8-fold nature of the 41 axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a five-dimensional super space group containing mutually perpendicular [...] Read more.
The structure of β-Mn crystallizes in space group P4132. The pseudo 8-fold nature of the 41 axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a five-dimensional super space group containing mutually perpendicular 8-fold axes cannot generate P4132 on projection to 3-d space and how this may instead be accomplished from a six-dimensional model. A procedure for generating the actual structure of β-Mn lifted to six-dimensional space is given. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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20907 KiB  
Article
Symmetry-Adapted Fourier Series for the Wallpaper Groups
by Bart Verberck
Symmetry 2012, 4(3), 379-426; https://doi.org/10.3390/sym4030379 - 17 Jul 2012
Cited by 7 | Viewed by 7452
Abstract
Two-dimensional (2D) functions with wallpaper group symmetry can be written as Fourier series displaying both translational and point-group symmetry. We elaborate the symmetry-adapted Fourier series for each of the 17 wallpaper groups. The symmetry manifests itself through constraints on and relations between the [...] Read more.
Two-dimensional (2D) functions with wallpaper group symmetry can be written as Fourier series displaying both translational and point-group symmetry. We elaborate the symmetry-adapted Fourier series for each of the 17 wallpaper groups. The symmetry manifests itself through constraints on and relations between the Fourier coefficients. Visualising the equivalencies of Fourier coefficients by means of discrete 2D maps reveals how direct-space symmetry is transformed into coefficient-space symmetry. Explicit expressions are given for the Fourier series and Fourier coefficient maps of both real and complex functions, readily applicable to the description of the properties of 2D materials like graphene or boron-nitride. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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Review

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616 KiB  
Review
Group Theory of Wannier Functions Providing the Basis for a Deeper Understanding of Magnetism and Superconductivity
by Ekkehard Krüger and Horst P. Strunk
Symmetry 2015, 7(2), 561-598; https://doi.org/10.3390/sym7020561 - 05 May 2015
Cited by 11 | Viewed by 6308
Abstract
The paper presents the group theory of optimally-localized and symmetry-adapted Wannier functions in a crystal of any given space group G or magnetic group M. Provided that the calculated band structure of the considered material is given and that the symmetry of the [...] Read more.
The paper presents the group theory of optimally-localized and symmetry-adapted Wannier functions in a crystal of any given space group G or magnetic group M. Provided that the calculated band structure of the considered material is given and that the symmetry of the Bloch functions at all of the points of symmetry in the Brillouin zone is known, the paper details whether or not the Bloch functions of particular energy bands can be unitarily transformed into optimally-localized Wannier functions symmetry-adapted to the space group G, to the magnetic group M or to a subgroup of G or M. In this context, the paper considers usual, as well as spin-dependent Wannier functions, the latter representing the most general definition of Wannier functions. The presented group theory is a review of the theory published by one of the authors (Ekkehard Krüger) in several former papers and is independent of any physical model of magnetism or superconductivity. However, it is suggested to interpret the special symmetry of the optimally-localized Wannier functions in the framework of a nonadiabatic extension of the Heisenberg model, the nonadiabatic Heisenberg model. On the basis of the symmetry of the Wannier functions, this model of strongly-correlated localized electrons makes clear predictions of whether or not the system can possess superconducting or magnetic eigenstates. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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1355 KiB  
Review
Symmetry Aspects of Dislocation-Effected Crystal Properties: Material Strength Levels and X-ray Topographic Imaging
by Ronald W. Armstrong
Symmetry 2014, 6(1), 148-163; https://doi.org/10.3390/sym6010148 - 20 Mar 2014
Cited by 3 | Viewed by 7569
Abstract
Several materials science type research topics are described in which advantageous use of crystal symmetry considerations has been helpful in ferreting the essential elements of dislocation behavior in determining material properties or for characterizing crystal/polycrystalline structural relationships; for example: (1) the mechanical strengthening [...] Read more.
Several materials science type research topics are described in which advantageous use of crystal symmetry considerations has been helpful in ferreting the essential elements of dislocation behavior in determining material properties or for characterizing crystal/polycrystalline structural relationships; for example: (1) the mechanical strengthening produced by a symmetrical bicrystal grain boundary; (2) cleavage crack formation at the intersection within a crystal of symmetrical dislocation pile-ups; (3) symmetry aspects of anisotropic crystal indentation hardness measurements; (4) X-ray diffraction topography imaging of dislocation strains and subgrain boundary misorientations; and (5) point and space group aspects of twinning. Several applications are described in relation to the strengthening of grain boundaries in nanopolycrystals and of multiply-oriented crystal grains in polysilicon photovoltaic solar cell materials. A number of crystallographic aspects of the different topics are illustrated with a stereographic method of presentation. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
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