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TENSOR PAIRS AND DOMAIN PAIRS OF CESIUM DIHYDROGEN PHOSPHATE USING GROUP THEORETICAL TECHNIQUES

Ramesh Babu A1, Radha Madhavi M2*, Sarala S2 and Vemuri Lakshminarayana3

1Applied Sciences and Humanities Department, Sasi Institute of Technology & Engineering, Tadepalligudem, West Godavari District, Andhra Pradesh, India

2Mathematics Division, Aarupadai Veedu Institute of technology, Vinayaka Missions University, Chennai, Tamil Nadu, India

3Principal, Aarupadai Veedu Institute of Technology, Vinayaka Missions University, Chennai Tamil Nadu, India g(Received 17 June, 2017; accepted 24 November, 2017)

*Corresponding Author:
Radha Madhavi M
Mathematics Division, Aarupadai Veedu
Institute of technology, Vinayaka Missions
University, Chennai, Tamil Nadu, India
E-mail: mrmadhavi5@gmail.com

Received Date: 17 June, 2017 Accepted Date: 24 November, 2017

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Abstract

Cesium Dihydrogen Phosphate (CsH2PO4) solid single crystal, it shows different phase transitions Cubic structure is changed to Rhombohedra structure and Cubic to monoclinic structure with various temperatures. In this paper, Physical property tensor pairs and domain pairs of CsH2PO4 at different phase transitions are calculated.

Keywords

Cesium Dihydrogen Phosphate (CDP), Ferro-electric, Ferro-elastic, Tensor pairs, Domain Pairs, Magnetic, Polarization, Double coset decomposition

Introduction

The crystal CsH2PO4 (read as CDP: Cesium Dihydrogen Phosphate) under phase transition exhibit pseudo cubic direction rhombohederal, monclinic or tetragonal and this exhibit “gaint peizo – electric coefficients” and high electromechanical coupling factors.

Due to the phase transitions of the crystal CsH2PO4 (Cesium Dihydrogen Phosphate) cubic structure (m3m (Oh Group) Prototypic point group) is changed into Rhombohedra structure (2 (C2 group) ferroic point group) is around 154.2K temperature. Again, this crystal changed this structure into Rhombohedra structure (space group p2/m (C2h group) ferroic point group) is around upto ap. 505K With different temperatures. Because of its peizo electric property this crystal does not exhibit any magnetic property. At P21 it exhibits ferroelectric, ferro elastic and magneto electric polarizibility (MEP) discussed.

Cesium Dihydrogen phosphate undergoes a number of unusual properties which have so far not been found in the other ferroelectric systems. The dielectric constant of CDP shows pronounced deviation from the Curie-Weiss law over a large temperature range above θc (Aizu, 1973; Aizu, 1974; Badurski and Stroz, 1979). Also, anomalously large excess heat capacities remain over the same temperature range (Abusahmin, 2017). A detailed Raman spectroscopy investigations of CDP revealed the existence of the antiferro electric fluctuations in the vicinity of the paraelectricferroelectric phase transition (Bhagavantam and Pantulu, 1964; ). Furthermore, it has been found from pyroelectric investigations that the para electric phase exhibits fluctuations of polarization, and the pyroelectric charge does not disappear at θc but persist on heating up to ap. 230 K decreasing non-monotonically (Bradley and Cracknell, 1972; Jaffe, et al., 1971; Narayana, et al., 1990).

A ferroic crystal contains one or more domains but of the different spatial orientation, a ferroic crystal arises in a ferroic phase transition from phase of higher symmetry to a phase of lower symmetry, here grey groups G11 is the prototypic point group and H is a ferroic phase of a lower symmetry. (Aizu, 1970) has given all possible 773 species of the ferroic crystals in phase transitions (Radha, 2013; Karri, et al., 2009; Karri, 2011).

The domains in ferroic crystals can be switched by means of a magnetic field , an electric field, a mechanical stress or a combination of the and consequently ferroic crystals of technological importance for memory storage and electric and magnetic switches.

The domain pairs and tensor pairs of Ferro electric, Ferro elastic and Magneto electric polarizability for the ferro species by using coset &double coset decomposition taking grey group m3ml1 as the prototypic point group in case of magneto electric polarizability and in case of ferro electric & ferro elastic polarizibility for ordinary point group m3m were calculated and tabulated.

Representative Domain Pairs

The group G and the subgroup F one writes the left coset decomposition of G with respect to F symbolically as G = F + g1F + g2F + … + gnF

Where giF, I = 1, 2, 3, … n denotes the subset of elements of G, which is obtained by multiplying each element of the subgroup F from the left by the elements gi of G. Each subset of elements of giF, i = 1, 2, 3, …n of G are called left coset representatives of the left coset decomposition of G with respect to F (Janovec, 1989).

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 gif where f is an arbitrary element of a sub group F.Si = giS1 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 gi-1 i.e., the groups F and Fi are conjugate groups. Two domain states Si and Sj form a domain pair (Si , Si) if Sj = gijSj and Si = gijSi where gij is element of G (Karri and Babovic, 2017; Litvin and Litvin, 1990; Uma and Sireesha, 2013).

Representative Tensor Pairs

Let G be the prototypic point group, H is the ferroic point group and T is the specific form of the physical property tensor T that keeps H invariant. The number N of crystallographically equivalent ordered distinct tensor pair classes is equal to the number of double cosets decomposition of G with respect to GT.

G = GTEGT + GT g1GT + … + GT gN GT

Where GT is the stabilizer of T in G and gk k = 1, 2, …n are the double coset representatives.

Let “T” denote a spontaneous physical property tensor which arises in the low symmetry phase of the crystal. Denote by T(i), i = 1, 2, … q, the specific form of the tensor T characterizing each of the q domains, and denote T(1) = T. All ordered tensor pairs could be partitioned into classes of crystallography equivalent tensor pairs : Two tensor pairs (T(i), T(j)) and (T(i`), T(j`)) are said to be crystallographically equivalent with respect to G and to belong to the same class of ordered tensor pairs, if there is an element g of G such that (T(i), T(j)) = (gT(i), gT(j)) that is , if T(i) = gT(i`) and T(j) = gT(j`).

Let GT denote the stabilizer of T in G, this subgroup GT of G is the set of all elements g of G which leave invariant i.e., gT = T. if GT = H then T is a full physical property tensor and there are qT = q distinct forms of tensor T i.e., each of the q domains is characterized by a distinct form of the tensor T. if H is a subgroup of GT then T is a partial physical property tensor and there are qT ≤ q distinct forms of the tensor T by Ta (d), a = 1, 2, … qT and and choose Td (1) = T(1) = T.

Results and Discussion

All ordered distinct tensor pairs (Td(a), Td (b)) can be partitioned into classes of crystallographically equivalent ordered distinct tensor pairs in the same manner as T(i), T(j)). The number of classes of ordered distinct tensor pairs is same as the number of classes of tensor pairs (Litvin amd Wike, 1989).

(i) Ferro–electric domain pairs for CsH2PO4 in the state m3m F 2:

Consider the ferroic species m3m F 2, where m3m is a prototypic point group and 2 is a ferroic point group. The number of distinct domain pair classes is 12. The coset decomposition of m3m with respect to the group ‘2’ is given by

equation

The coset elements equationequation

equation Now letequationequation then we haveequation andequation henceequation forms a domain pair, instead of writing this we represent domain pair representatives of G = m3m are equationequationequationequation andequation The domain pairs for ferroic species m3m F 2 are tabulated in Table 1.

Domain pair representatives Domain Pairs
equation (0, 0, Z) (0, 0, Z)
(C2x, I) (0, 0, -Z) (0, 0, -Z)
equation (0, 0, -Z) (0, 0, -Z)
equation (Z, 0, 0) (Z, 0, 0)
equation (0, Z, 0) (0, Z, 0)
equation (-Z, 0, 0) (-Z, 0, 0)
equation (0, -Z, 0) (0, -Z, 0)
equation (0, Z, 0) (0, Z, 0)
equation (0, -Z, 0) (0, -Z, 0)
equation (Z, 0, 0) (Z, 0, 0)
equation (-Z, 0, 0) (-Z, 0, 0)
equation (0, 0, Z) (0, 0, -Z)

Table 1. Ferroelectric domain pairs for ferroic species m3m F2

(ii) Ferro-electric tensor pairs For CsH2PO4 in the state m3m F 2:

Consider the ferroic species m3m F 2 where m3m is a prototypic point group and 2 is a ferroic point group and the stabilizer GT is 4mm. The numbers of distinct tensor pair classes are 3.

The double coset decomposition of m3m with respect to the stabilizer 4 mm is given by

G = m3m = (4 mm) E (4 mm) + (4 mm) C2x (4 mm) + (4 mm) C31+ (4 mm)

Here m3m is a prototypic point group and stabilizer 4 mm is a ferroic point group. The ferro-electro tensor pairs for ferroic species m3m F 2 are tabulated in Table 2.

Double coset representatives Stabilizer
GT
Tensor Pairs
(a) (b) (c) (d)
E 4 mm (0, 0, Z) (0, 0, Z)
C2x 4 mm (0, 0, Z) (0, 0, Z)
C31+ 4 mm (0, 0, Z) (Z, 0, 0)

Table 2. Ferro-electro tensor pairs for ferroic species m3m F2

(iii) Ferro-Elastic domain pairs for CsH2PO4 in the state m3m F 2:

Consider the ferroic species m3m F 2 where m3m is a prototypic point group and 2 is a ferroic point group. The number of distinct domain pair classes is 12. The coset decomposition of m3m with respect to the group 2 is given by

equation

The coset elements equationequationequation

The Ferro Elastic Domain pairs representatives of m3m F 2 are

equationequationequation

The Ferro Elastic Domain pairs of m3m F 2 are tabulated in Table 3.

Domain Pair Representatives Domain pairs
equation (xx, yy, zz, xy) equation
(C2x, I) equation (xx, yy, zz, xy)
equation (yy, xx, zz, yx) equation
equation (zz, yy, xx, zy) (zz, yy, xx, zy)
equation equation (xx, zz, yy, xz)
equation equation equation
equation equation
equation (yy, zz, xx, yz) equation
equation equation equation
equation equation equation
equation equation equation
equation equation equation

Table 3. Ferro-elastic domain pairs for ferroic species m3m F 2

(iv) Ferro-Elastic Tensor pairs for CsH2PO4 in the state m3m F 2:

Consider the ferroic species m3m F 2 where m3m is a prototypic point group and 2 is a ferroic point group and the stabilizer GT is 2/m. The number of distinct tensor pair classes is 8. The double coset decomposition of m3m with respect to the stabilizer 2/m is given by

G = m3m = (2/m) E (2/m) + (2/m) C2a (2/m) + (2/m) C2c (2/m) + (2/m) C2d (2/m) + (2/m) C31 (2/m) + (2/m) C2x (2/m) + (2/m) C31 (2/m) + (2/m) C31+ (2/m) . The Ferro Elastic tensor pairs of m3m F 2 are tabulated in Table 4.

Double Coset Representations Stabilizer Tensor Pairs
(a) GT (b) (c)
E 2/m equation equation
C2a 2/m equation equation
C2c 2/m equation equation
C2d 2/m equation equation
C4z- 2/m equation equation
C2x 2/m equation equation
C31+ 2/m equation equation
C31+ 2/m equation equation

Table 4. Ferro-elastic tensor pairs for ferroic species m3m F2

(v) The Magneto-Electric polarizability (MEP) tensor pairs for CsH2PO4 in the state of m3ml1 F 2:

Consider the ferroic species m3ml1 F 2, where m3m is a prototypic point group and 2 is a ferroic point group and the stabilizer GT is 2/ml. The number of distinct tensor pair classes is 22. The double coset decomposition of m3m with respect to the stabilizer 2/ml is given by

G = m3ml1 = (2/m1) E (2/m1) + (2/m1) R2 (2/m1) + (2/m1) C2x (2/m1) + (2/m1) C31+ (2/m1) + (2/m1) C31 (2/m1) + (2/m1) C31 (2/m1) + (2/m1) C32 (2/m1) + (2/m1) C2a (2/m1) + (2/m1) C2c (2/m1) + (2/m1) C2d (2/ ml) + (2/m1) C2e (2/m1) + (2/m1) C2f (2/m1) + (2/m1) R2 C2x (2/m1) + (2/m1) R2 C31+ (2/m1) + (2/m1)

R2C32+ (2/m1) + (2/m1)R2 C31 (2/m1) + (2/m1)R2 C32 (2/m1) + (2/m1) R2C2a (2/m1) + (2/m1) R2C2c (2/m1) + (2/m1) R2C2d (2/m1) + (2/m1) R2C2e (2/m1) + (2/m1) R2C2f (2/m1)

The Magneto-Electric polarizability (MEP) tensor pairs of m3ml1 F 2 are tabulated in Table 5.

Double
cosset representatives
Stabilizer (GT) Tensor Pairs
( a ) ( b ) ( c ) ( d )
E 2/ml equation equation
R2 2/ml equation equation
C2x 2/ml equation equation
C31+ 2/ml equation equation
C32+ 2/ml equation equation
C31- 2/ml equation equation
C32- 2/ml equation equation
C2a 2/ml equation equation
C2c 2/ml equation equation
C2d 2/ml equation equation
C2e 2/ml equation equation
C2f 2/ml equation equation
R2C2x 2/ml equation equation
R2C31+ 2/ml equation equation
R2C32+ 2/ml equation equation
R2C31- 2/ml equation equation
R2C32- 2/ml equation equation
R2C2a 2/ml equation equation
R2C2c 2/ml equation equation
R2C2d 2/ml equation equation
R2C2e 2/ml equation equation
R2C2f 2/ml equation equation

Table 5. The MEP tensor pairs for ferroic species m3ml1 F2

(vi) Ferro-elastic domain pairs for CsH2PO4 in the state m3m F 2/m:

Consider the ferroic species m3m F 2/m where m3m is a prototypic point group and 2/m is a ferroic point group. The number of distinct domain pair classes is 6. The coset decomposition of m3m with respect to the group 2 is given by

equation

The coset element gi`s are equation

The Ferro Elastic Domain pair representatives of m3m F 2/m are equationequation and The Ferro Elastic Domain pairs of m3m F 2/m are tabulated in Table 6.

Domain Pair Representatives Domain pairs
(E, C2x) (xx, yy, zz, xy) equation
equation (yy, xx, zz, yx) equation
(C2c ,C2e) equation equation
(C2d , C2f ) equation equation
equation (zz, xx, yy, zx) equation
equation (yy, zz, xx, yz) equation

Table 6. Ferro-elastic domain pairs for ferroic species m3m F 2

(vii) Ferro-Elastic tensor pairs for CsH2PO4 in the state m3m F 2/m:

Consider the ferroic species m3m F 2/m where m3m is a prototypic point group and 2 is a ferroic point group and the stabilizer GT is also same ferroic point group 2/m. The number of distinct tensor pair classes is 8. The double coset decomposition of m3m with respect to the stabilizer 2/m is given by

G = m3m = (2/m) E (2/m) + (2/m) C2a (2/m) + (2/m) C2c (2/m) + (2/m) C2d (2/m) + (2/m) C4z (2/m) + (2/m) C2x (2/m) + (2/m) C31 (2/m) + (2/m) C31+ (2/m)

Conclusion

In this paper the ferroelectric, ferroelastic, magneto electric polarizibility of the crystal CDP (Cesium Dihydrozen Phosphate with structural formula CsH2PO4 ) domain pairs & tensor pairs are calculated by group theoretical techniques. While considering ferroelectric and ferro elastic properties only ordinary point group P-m3m is considered as prototypic point group and 2, 2/m are ferroic point subgroups. Since they are non-magnetic properties. Ferroelectric and ferroelastic tensor pairs of m3m F2 is calculated by using the stabilizer 2 m, but in the case of magneto-electric polarizibility (MEP) grey group m3ml1 is taken as prototypic point group and 2/m1 is taken as stabilizer. Similarly Ferroelastic tensor pairs of m3m F2/m is calculated by same stabilizer 2/m1.

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