Analyse der vierperiodischen Minimalnetze

Beukemann, Alexander

dc.contributor.advisor	Kuhs, Werner F. Prof. Dr.
dc.contributor.author	Beukemann, Alexander
dc.date.accessioned	2015-03-06T13:03:34Z
dc.date.available	2015-03-06T13:03:34Z
dc.date.issued	2015-03-06
dc.identifier.uri	http://hdl.handle.net/11858/00-1735-0000-0022-5DE7-F
dc.identifier.uri	http://dx.doi.org/10.53846/goediss-4969
dc.language.iso	deu	de
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/3.0/
dc.subject.ddc	910	de
dc.subject.ddc	550	de
dc.title	Analyse der vierperiodischen Minimalnetze	de
dc.type	doctoralThesis	de
dc.title.translated	Analysis of 4-periodic minimal nets	de
dc.contributor.referee	Scheitler, Gerhard Prof. Dr.
dc.date.examination	2015-02-11
dc.description.abstracteng	Periodic structures – e. g. crystal structures – are analyzed by means of group theory. The concept of a crystal structure is extendable to the n-dimensional space and the 4-dimensional space groups are known [Brown, Bülow, Neubüser, Wondratscheck, Zassenhaus, 1978]. As quasi-periodic and incommensurate structures can be the result of a projection of a higher - periodic structure, it is also worth to enhance the knowledge about higher-periodic structures by graph theoretical means. Thereby the atoms of a crystal can be assigned to the vertices, and the chemical bonds to the edges of a graph being a periodic net. Minimal nets [Beukemann, Klee, 1992] are nets, where the deletion of any edge together with and its translational equivalent edges results in a structure that is no longer connected. It’s special about a minimal net to have a 1:1 relation to its quotient graph [Chung, Hahn, Klee, 1984] that represents the topology of the net by a finite graph, presenting the adjacency relations between the classes of translational equivalent vertices and edges of the periodic net. 3-periodic minimal nets are analyzed by several authors such as Bonneau, Delgado-Friedrichs, O’Keeffe, Yaghi, Eon. Now the complete class of 4-periodic minimal nets (111 nets) is analyzed by comparing their substructures such as trees, chains, cycles and sub-networks. Net to subnet relations are identified and methods for net conversions are proposed. Different principles for the construction of those structures are applied. The embedding of two nets is discussed in detail. Single cycles of the minimal net 6(3)1 – number 1 of minimal nets with 6 classes of vertices and vertex degree 3 (quotient graph K3,3) – where mapped to polygons of a Penrose pattern. The minimal net 3(4)1 could be embedded into the 4-dimensional space by a “symmetric labelling” of the edges of its quotient graph. A reference work with graphics and cycle class sequences of all 2-, 3- and 4-periodic minimal nets is attached.	de
dc.contributor.coReferee	Sowa, Heidrun Prof. Dr.
dc.subject.ger	Periodische Strukturen	de
dc.subject.ger	Quotientengraph	de
dc.subject.ger	Minimalnetz	de
dc.subject.ger	Kreisklassen	de
dc.subject.ger	Periodische Netze	de
dc.subject.ger	Referenz	de
dc.subject.ger	2-periodische Minimalnetze	de
dc.subject.ger	3-periodische Minimalnetze	de
dc.subject.ger	4-periodische Minimalnetze	de
dc.subject.eng	Periodic structures	de
dc.subject.eng	quotient graph	de
dc.subject.eng	minimal nets	de
dc.subject.eng	cycle classes	de
dc.subject.eng	periodic nets	de
dc.subject.eng	reference	de
dc.subject.eng	2-periodic minimal nets	de
dc.subject.eng	3-periodic minimal nets	de
dc.subject.eng	4-periodic minimal nets	de
dc.identifier.urn	urn:nbn:de:gbv:7-11858/00-1735-0000-0022-5DE7-F-9
dc.affiliation.institute	Fakultät für Geowissenschaften und Geographie	de
dc.subject.gokfull	Geologische Wissenschaften (PPN62504584X)	de
dc.identifier.ppn	819684147

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