The 80 Diperiodic Groups in Three Dimensions

Wood, Elizabeth A.

doi:10.1002/j.1538-7305.1964.tb04077.x

Cited by 114 publications

(49 citation statements)

References 7 publications

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“…At the same time, the spatial symmetry group of the crystalline lattice of single-layer graphene, 6/ ( 80) [5] or the triperiodic group 1 6ℎ , whose diagram coincides with that of the biperiodic spatial group of single-layer graphene, is symmorphic, and all its "rotational" elements -the symmetry elements of the point group 6/ ( 6ℎ ) -do not contain nontrivial (partial) translations.…”

Section: Crystal Structures Brillouinmentioning

confidence: 99%

“…The corresponding result is presented in Table 6. After calculating the values of the coefficients , , and , one can easily see that the constructed factor-system belongs to the projective class 5 . The values of the function on the group 1, ( ), which bring the factorsystem 1, ( 2 , 1 ) to the standard form ′ 1, ( 2 , 1 ), can be easily found by the method described in work [6, formulas (14.33)] and in works [10][11][12].…”

Section: Characters Of Irreduciblementioning

confidence: 99%

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Energy Spectra Correlation of Vibrational and Electronic Excitations and Their Dispersion in Graphite and Graphene

et al. 2018

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The correlation between the vibrational and electron excitation modes in the energy spectra of single-layer graphene and crystalline graphite, as well as the dispersion dependences of those modes, has been studied. The methods of the theory of projective representations of the point and spatial symmetry groups are used for the first time in order to interpret those correlations. The correlations of vibrational and electron excitation spectra and the compatibility conditions for irreducible projective representations in the descriptions of quantum states of graphene and crystalline graphite at various points of their Brillouin zones are determined. For the projective representations of all projective classes belonging to the hexagonal system, standard factor-systems are constructed for the first time. In particular, the factor-systems for electron states are first determined. The results obtained are used to calculate, also for the first time, the correct spinor multiplication tables, i.e. the multiplication tables for elements in double symmetry groups. The developed method is applied to classify all high-symmetry points in the Brillouin zones of single-layer graphene and crystalline graphite with respect to the symmetry type of vibrational excitations.K e y w o r d s: spinor representation of symmetry groups, factor-system, dispersion of elementary excitations.

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Section: Crystal Structures Brillouinmentioning

confidence: 99%

Section: Characters Of Irreduciblementioning

confidence: 99%

Energy Spectra Correlation of Vibrational and Electronic Excitations and Their Dispersion in Graphite and Graphene

et al. 2018

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“…The correspondence between layer groups and three-dimensional three-periodic space groups is provided in Ref. [14]. The two-dimensional Brillouin Zone (BZ) of the layer group coincides with the (k x , k y ) restriction of the BZ of the corresponding threedimensional three-periodic space group and presents the same symmetry properties as the restriction.…”

Section: Symmetry Analysismentioning

confidence: 99%

Symmetry and optical properties of wurtzite nanostructures with the c axis in the layer plane

Tronc¹,

Vennéguès²

2008

Phys. Solid State

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In wurtzite-based quantum wells and superlattices with the c-axis parallel to the layer plane, the plane is parallel to either a summetry plane of the wurtzite lattice (type I structures, the 1120 growth direction) or a glide plane containing the c-axis (type II structures, the 1010 growth direction). In both cases, the space symmetry of the structure depends on the parity of the number of monolayers within the slab(s). The point symmetry is C 2v except for the type II structures with odd monolayer number(s). The latter structures have the σv point symmetry and can present a built-in electric field. Quite different selection rules, depending on the structure symmetry, govern electron optical transitions and exciton radiative recombination.

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“…In addition, the program has been deposited at the University of Illinois NCSA facility and can be accessed via standard file-transfer protocol (ftp.ncsa.uiuc.edu). Crystallography, 1989) and for the layer groups (Weber, 1929;Wood, 1964;Chapuis, 1966; International Tables for Crystallography, 1993). A second twodimensional perspective general position diagram for some layer groups has been given by Grell, Krause & Grell (1988).…”

mentioning

confidence: 99%

“…and for the layer groups (Weber, 1929;Wood, 1964;Chapuis, 1966;International Tables for Crystallography, 1993). A second twodimensional perspective general position diagram for some layer groups has been given by Grell, Krause & Grell (1988).…”

mentioning

confidence: 99%

LAYERGPD– a program to visualize the three-dimensional layer group general position diagrams

Litvin¹,

Литвин²

1993

J Appl Cryst

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Documentation:The write up for MACINPLOTII is approximately 800 lines and can be supplied as an ASCII file or a MS Word 5.0 document.* Availability: Copies of the program can be obtained by writing to the author. The program and documentation will be supplied on a 3.5 in double-sided double-density floppy disk formatted for the Macintosh. A nominal fee will be charged to cover the cost of the disk, or a blank disk can be sent with the request. In addition, the program has been deposited at the University of Illinois NCSA facility and can be accessed via standard file-transfer protocol (ftp.ncsa.uiuc.edu). Crystallography, 1989) and for the layer groups (Weber, 1929;Wood, 1964;Chapuis, 1966; International Tables for Crystallography, 1993). A second twodimensional perspective general position diagram for some layer groups has been given by Grell, Krause & Grell (1988). The goal in developing LAYERGPD was to overcome, for the layer groups, the constraint of a single parallel view of the general positions. We provide the user with options to choose the orientation and type of view, parallel or projective, of the threedimensional general positions. Keywords Method of solution:A PHIGS (programmer's hierarchical interactive graphics system)-based development product (Z-PHIGS by Weiss Software, LL~beck, Germany) was used to create the three-dimensional general position diagrams. At each position, an atom is represented by a multicolored asymmetric triangular pyramid to aid visualization of the symmetry relationships among the general positions.Software environment: LAYERGPD was written in TurboPascal6.0 and is compiled. The PHIGS system necessary to view the general positions is included. There are no overlays.Hardware environment: The program runs under MS-DOS 3.1 or higher on an IBM-compatible PC with a math coprocessor, 2 Mbytes of expanded memory and at least a VGA color monitor. As this is a graphics-intensive program, a 486/25 or better microprocessor is recommended. If stored on hard disk, the program takes up 680 kbytes of disk space.The single view can be rotated independently about all three coordinate axes, zoomed and set to either a parallel or projective view. Documentation:A 'readme' file is provided with initial instructions on running the program from disk or installing it onto and running it from a hard drive. International Tables for Crystallography (1989). Vol. A, 2nd ed., edited by T. Hahn. Dordrecht: Kluwer Academic Publishers. J. Appl. Cryst. (1993). 26, 498 New GK 311/M All-Round Goniospectrometer for Colour Availability: International Tables for Crystallography New Commercial Products Measurement of Special-Effect PaintsSurfaces with pronounced anisotropic scattering, such as metallic and pearl-

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The 80 Diperiodic Groups in Three Dimensions

Cited by 114 publications

References 7 publications

Energy Spectra Correlation of Vibrational and Electronic Excitations and Their Dispersion in Graphite and Graphene

Energy Spectra Correlation of Vibrational and Electronic Excitations and Their Dispersion in Graphite and Graphene

Symmetry and optical properties of wurtzite nanostructures with the c axis in the layer plane

LAYERGPD– a program to visualize the three-dimensional layer group general position diagrams

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