IJCRR - 5(21), November, 2013
Pages: 92-95
Date of Publication: 21-Nov-2013
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DOMAIN PAIRS OF THE FERROIC SPECIES OF LANGBEINITE FAMILY OF CRYSTALS
Author: G. Sireesha, S. Uma Devi
Category: Technology
Abstract:In this paper we have taken the Langbeinite family ? ? + 2 3 2 2 4 3 A M so ? ? of crystals with
A = NH4, K, Rb, Cs, Ti and M = Mg, Ca, Mn, Fe, Co, Ni, Zn and Cd. The Langbeinite family of
crystals which is exhibiting different higher order physical properties at various phase transition
temperatures. Here we calculated ferromagnetoelectric (aev2) domain pairs of the crystal at different phase transition temperatures with the help of group theoretical methods of studying the effect of symmetry on some physical properties.
Keywords: Prototypic point group, ferroic point group, coset decomposition, Domain pairs, ferromagnetoelectric property.
Full Text:
INTRODUCTION
Physical properties of crystals generally express the relation between the two quantities. These may be scalars and vectors. These properties differ from one another by their transformation properties. A ferroic crystal contains two (or) more equally stable domains, volumes of the same homogeneous bulk structure of these domains in a polydomain sample are referred to as domain states. The analysis of domains of a ferroic crystals can be found using coset decomposition of point group (Aizu, 1970; Janovec,1972). Here the number of distinct right or left cosets and coset representatives are F- operations that change one orientation state to another. The ferroic crystal with A = NH4, K, Rb, Cs, Ti and M = Mg, Ca, Mn, Fe, Co, Ni, Zn and Cd which is exhibiting different physical properties at various phase transition temperatures. In this chapter we have taken this crystal and found its physical property like ferromagnetoelectric domain pairs at different phase transition temperatures with the help of group theoretical methods of studying the effect of symmetry on the physical properties.
Coset Decomposition
Let H be a subgroup then the left (right) cosets of H in G provide a decomposition of G as a sum of mutually disjoint left (right) cosets. i.e. G = This is known as coset decomposition of G with respect to H. The arbitrary element ai of each left (right) coset is called a representative of coset.
The concept of coset decomposition of a group with respect to one of its subgroup has wide application in crystallography and solid-state physics. The points of any crystallographic orbit are in a one-to-one correspondence with the cosets of the coset decomposition of the crystallographic group with respect to the site symmetry group of one of its points (wondratschek, 1983). Coset decomposition has been applied in the analysis of domains of ferroic crystals using coset decomposition of point groups and space groups (Aizu, 1970 ; Janovec, 1972). Here the number of distinct domains of ferroic crystals are equal to the
number of distinct right (or) left cosets and the coset representatives are F - operations that change one orientation state to another.
Domain
A phase transition of a crystalline structure from a phase of higher symmetry G to a phase of lower symmetry F of ferroic materials consists of homogeneous regions called domain.
Representative Domain Pairs
Consider a ferroic phase transition of a crystalline structure from a phase of higher symmetry G to a phase of lower symmetry F of ferroic materials consists of homogeneous regions called domains, the interior bulk structures of the domains are called domain states and in the continum description differ only in orientation. The properties of domain states are described by the property of tensors. By using the coordinate system components, in which the properties of the tensors that distinguish between the domains of magnetic domains which differ solely in the two domains by change in the sign.
Two domain states that have different spontaneous magnetization vector are denoted as a ferromagnetic domain pairs. Consider a phase transition between phases of symmetry G & F. The crystal splits into n = [G/F] single domain states denoted by S1, S2 … Sn. Let Si & Sj be two arbitrary orientation states of ferroic crystals. They are identical (or) an antimorphism in structure. This means that mathematically Sj is said to be obtained by performing a certain operation of G upon Si. We refer to this operation as an F – operation from Si to Sj of the ferroic crystal. In particular, when Si and Sj (or) Si itself is just an element of the point group of Si. In general there is more than one F operation from Si to Sj. Let a ferroic crystal have q orientation states in all and let S be one of those operation states. We refer to a set of q – F operations each from S to each orientation state as a “Set of representative F operation on S” of the ferroic crystals. In general, there are more than one set of representative F operation on S. These can be obtained by the coset decomposition of G with respect to F. For the given group G and subgroup F one writes the left coset decomposition of G with respect to F symbolically written as G = F + g1F + g2F + …. + gnF. Where gi F denote the subset elements of G obtained by multiplying each element of the subgroup F from the left side of the elements gi of G. Each subset of elements of gi F. i = 1, 2 ….. n of G are called left coset representatives of the left coset decomposition of G with respect to F.
The subset of elements of G in each coset of the left coset decomposition of G with respect to F is unique but the coset representatives are not unique. A coset representative gi can be replaced by the element gi f where f is an arbitrary element of a subgroup F. Si = gi S1 i.e the orientation of the ith domain Si is related to the orientation of the domain S1 by the element gi of this coset decomposition for i = 1, 2 …. N. The symmetry group Fi = gi F i.e the groups F and Fi are conjugate groups. Two domain states Si and Sj form a domain pair (Si, Sj) if Si = gij Sj where gij is element of G. Here we calculated ferromagnetoelectric domain pairs for the ferroic species by using coset decomposition.
Ferromagnetoelctric domain pairs for Langbeinite family in the state 23F1 :
Consider the ferroic species 23F1. Where 23 is a prototypic point group and 1 is a ferroic point group. The number of distinct domain pair classes are 6.
The coset decomposition of 23 with respect to the group 1 is given by-
Ferromagnetoelctric domain pairs for the ferroic species 23F2 :
Consider the ferroic species 23F2. Where 23 is a prototypic point group and 2 is a ferroic point group. It has 3 domain pairs. The coset decomposition of 23 with respect to the group 2 is given by . The coset elements .It has six coset elements and hence they form three domain pairs.
Ferromagnetoelctric domain pairs for the Ferroic species 23F3 :
Consider the ferroic species 23F3. Where 23 is a prototypic point group and 3 is a ferroic point group. The number of distinct domain pair classes are 2. The coset decomposition of 23 with respect to the group 3 is given by .
The coset elements gi’s are E, . Domain pairs of 23F3 are
Consider the ferroic species 23F222. Where 23 is a prototypic point group and 222 is a ferroic point group. It has 2 domain pairs. The coset decomposition of 23 with respect to the group 222 is given by . The coset elements are.
CONCLUSION
Due to the phase transitions of the crystals A2+ M22+ (so43-)3 with A = NH4, K and M = Mg, Ca, Mn & Cd, cubic structure P213(T group) which is the prototypic point group is changed at high temperature as trigonal structure R3 (C3 group), orthorhombic P212121 (D2 group),mono clinic structure P21 (C2 group) and triclinic structure P1 (C1 group) with different temperatures. Consider P213(T group) as prototypic point group and R3, P212121, P21 and P1 are taken as ferroic point groups. The first three involve improper ferroelectric transitions. The change of phase from P213 to P212121 involves a ferroelastic transition. While considering ferromagnetoelectric property, ordinary point group 23 is taken as prototypic point group and 222, 2,3 and 1 are taken as ferroic point groups. In this paper ferromagnetoelectric domain pairs of Langbeinite family of crystals at different temperatures are calculated by using group theory techniques and the results are tabulated.
References:
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- Wadhawan, V.K – 1982. “Phase transitions, 3 : 3, 1982
- Wooster, W.A, “Tensors and Group theory for the physical property tensors of crystals”, Clarendon press, 1973.
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