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3800 KiB

Open AccessArticle

Crystallography and Magnetic Phenomena

by Vojtěch Kopský

Symmetry 2015, 7(1), 125-145; https://doi.org/10.3390/sym7010125 - 2 Feb 2015

Cited by 6 | Viewed by 7982

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

Open AccessArticle

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 5549

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

Open AccessArticle

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 - 4 Sep 2014

Cited by 12 | Viewed by 7261

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

Open AccessArticle

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 5891

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

Open AccessArticle

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 5788

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

Open AccessArticle

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 7601

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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Figure 1

Figure 1
Sketch of the incompatibility between points of fivefold rotational symmetry at minimal distance (black line) and periodicity in the Euclidean plane. Full article ">Figure 2
Inflation rule for the (essentially undecorated) Ammann–Beenker tiling (left), and a patch (right) obtained by three inflation steps from a square-shaped patch (consisting of two triangles) in the centre; see text for details. The patch has no rotation symmetry, but is reflection symmetric in the diagonal. Full article ">Figure 3
Central patch of an Ammann–Beenker tiling with exact <math display="inline"> <msub> <mi>D</mi> <mn>8</mn> </msub> </math> symmetry. Full article ">Figure 4
Central patch of a rhombic Penrose tiling with exact <math display="inline"> <msub> <mi>D</mi> <mn>5</mn> </msub> </math> symmetry. Full article ">Figure 5
The SCD monotile (left) and the layer structure of the corresponding tiling (right). Full article ">Figure 6
Aperiodic prototile set for the rhombic Penrose tiling. Full article ">Figure 7
Inflation rule (left) and patch (right) of the pinwheel tiling. The prototiles are right triangles with side length 1, 2 and <math display="inline"> <msqrt> <mn>5</mn> </msqrt> </math>. The dots represent control points of an equivalent Delone set. Starting from a single triangle with the control point in the origin leads to a fixed point tiling with circular symmetry. Full article ">

306 KiB

Open AccessArticle

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 6692

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

Open AccessArticle

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 7940

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

Jump to: Research

616 KiB

Open AccessReview

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 - 5 May 2015

Cited by 11 | Viewed by 6745

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

Open AccessReview

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 4 | Viewed by 7919

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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