FERROTOROIDIAL PROPERTIES OF FERROICSPECIES OF QUASI CRYSTALS

G. SIREESHA

1. Abstract

A ferroic phase of a crystal arises as a result of an actual lower-ing of the point group symmetry of the prototype. Since a pointgroup can have more than one non-trivial proper subgroup, a num-ber of distinct subgroup pairs are possible for each prototypic pointgroup. Each such pair can describe a possible ferroic species. A fer-roic crystal is said to be ferromagnetic, ferroelectric and ferroelasticif all (or) some of whose orientation states are different in spon-taneous magnetization, spontaneous polarization and spontaneousstrain tensor respectively. A fourth type of primary ferroic crystals,a ferrotoroidic crystal has been recently observed (Van Aken et al.,2007) where the domains are distinguished by a toroidial moment.This paper gives a brief account of group theoretical methods ofstudying the effect of symmetricalness on some toroidial propertiesof non-crystallographic point groups with different fold symmetries.Also the ferroic species whose ferroic point groups which are thesubgroups of order two of the prototypic point group are associatedto the alternating representations of the quasi crystals. To describethe ferrotoroidial properties of the ferroic species the number of in-dependent constants is required and it can be determined, they aretabulated

Key words: Ferrotoroidial properties; ferroic species; independentconstants; non-crystallographic point group.

2. Introduction

Landau theory provided a powerful tool for studying the stability,phase transitions and physical properties of the ordered phases. In1970s Landau theory approach was employed to look into the pos-sible crystallographic structures of the solid lower than the phase

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International Journal of Pure and Applied MathematicsVolume 118 No. 10 2018, 57-63ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: 10.12732/ijpam.v118i10.49Special Issue ijpam.eu

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transition temperature. After that number of scientific researchershas been lead to distinct crystalline and non- crystalline shapes.Levine D. and Steinhardt P.J. (1984) introduced the term quasicrystals for these unusual systems exhibiting diffraction patternswith non-crystallographic symmetry. After discovering of the quasicrystals significant progress in experimental work, theoretical stud-ies have also been developed. Penrose tiling plays a vital role andis considered as the root construction of macroscopic models ofquasi crystal structures. A theoretical explanation for quasi peri-odic structures which may have either crystallographic (or) non-crystallographic point group symmetry was given by Janssen T.(1992). Crystals in which the domain states may be distinguishedby magnetization, spontaneous polarization, (or) strain are calledprimary ferroic crystals. Crystals in which the domain states aredistinguished by the piezoelectric tensor is an example of secondaryferroic crystals treated as Ferromagnetoelectric (aev2), Ferromag-netic (aev) crystals respectively. In this context V represents a po-lar vector and e and a denotes zero rank tensors that change underspatial inversion and time inversion respectively. Wenge Yang etal., (1995) gave the group theoretical derivation of number of inde-pendent physical constants of quasi crystals. He also gave formulasfor calculating the number of independent constants of any physicalproperty tensor of quasi crystals which are derived by group repre-sentation theory. Arrangements of different quasi crystals are givenin the following figures respectively. Some ferrotoroidial propertiesof quasi crystals with 5-fold, 8-fold, 10-fold and 12-fold symmetriesare discussed. And also discussed the association of the ferroicspecies of quasi crystals which are obtained by using group the-oretical methods. The properties are tabulated in the followingtabular form.

Gopala Krishna Murthy and Uma Devi. S (1980) developed analternative method of evaluation of the number of constants ni re-quired to describe a physical property for the ferroic species GFHand can be evaluated from that of G and H is given by ni (GFH)= ni (H) ni (G) The number of components ni against the totalsymmetric representation of H should be greater than (or) equalto one is a necessary condition for the appearance of any property

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Figure1. PentagonalQuasi Crystal

Figure2. OctagonalQuasi Crystal

Figure3. DecagonalQuasi Crystal

Figure4. DodecagonalQuasi Crystal

to exist. By using the character tables of quasi crystals, we asso-ciate the alternating representation to the ferroic species of non-crystallographic point groups. From which we can find the num-ber of independent constants required to describe the ferrotoroidialproperties of the ferroic species of the quasi crystals and are tabu-lated in Table 2 and Table 3.

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Figure5. IcosahedralQuasi Crystal

Figure6. PenroseTiling

3. Conclusion

The phase transition of quasi crystals exists at different temper-atures. Quasi crystals are used for multi-purposes. They can beused as a prototype cookware with non-stick, hydrogen storage ma-terials, and thermal barrier coatings, thermo electric elements forsmall scale heating and cooling, photo thermal solar absorber coat-ings. This paper gives a brief account of group theoretical meth-ods of studying the effect of symmetry of some toroidial propertiesof quasi crystals with different fold symmetries. Also the ferroicspecies whose ferroic point groups which are the subgroups of ordertwo of the prototypic point group are associated to the alternatingrepresentations of the quasi crystals. To discuss the ferrotoroidialproperties of the above ferroic species, the number of independentconstants are required and is determined and they are tabulated inTable 1.

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References

[1] Aizu Keitsiro Possible species of ferromagnetic, ferroelectric and ferroelas-tic crystals, Phys. Rev., B, 2, No. 3 , (1970) , 754 772.

[2] Krishna Murthy T.S.G., Gopala Krishna Murthy P., Magnetic SymmetryGroups, Acta Cryst., A 25, Part 2, (1969) , 329 331. (1980), 449 451.

[3] Gopalakrishna Murthy P., Uma Devi S., An alternative approach for thedescription of physical properties in Ferroic Species, Indian Journal ofPure & Applied Physics , 18, No. 6,(1980), 449 451.

[4] Litvin D. B., Ferroic classifications extended to Ferrotoroidic crystals, ActaCryst., A64, ,(2008), 316 320.

[5] Litvin D. B., Ferroic crystals and Tensor distinction, Phase Transitions,83, ,(2010),682 693.

Department of Mathematics, VNR Vignana Jyothi Institute ofEngineering and Technology, Hyderabad, India.

E-mail address: sireesha.ganga@gmail.com

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