Phases of anionic ordering in elpasolite structures (ordered perovskites) Текст научной статьи по специальности «Физика»

УДК 548.1:537.226.4

Phases of Anionic Ordering in Elpasolite Structures (Ordered Perovskites)

Roman G. Sevryukov* Ivan N. Safonov^

Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Maksim S. Molokeev*

Kirensky Institute of Physics SB RAS Akademgorodok, 50/38, Krasnoyarsk, 660036 Far Eastern State Transport University Serysheva, 47, Khabarovsk, 680021

Russia

Sergey V. Misyul§

Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 23.11.2015, received in revised form 09.12.2015, accepted 20.01.2016 The 108 dissymmetric phases were obtained as a result of X anions ordering in elpasolite structure A2BB'X6 (ordered perovskite) using group theoretical methods. The critical order parameters which transform according to the irreducible representations of Г and X points of Brillouin zone of Oh - Fm3m space group were considered only. Polyhedral structures showed for dissymmetric phases which are accompanied by appearing of ferroelectricity at phase transition. All results are summarized in convenient tables.

Keywords: elpasolite, structural data, a complete condensate of order parameters, irreducible representations, critical distortion, phase transition, dissymmetric phases. DOI: 10.17516/1997-1397-2016-9-1-108-118.

Introduction

A lot of compounds crystallized in space group O5 — Fm3m, part of them have important practical properties. Of particular interest are the numerous perovskite-like compounds with space group Fm3m which undergo many phase transitions (PT) during changing of external influence [1, 2]. The crystals of elpasolite with general formula A2BB'X6 (A, B, B' — cations, X — anion) are intensively studied at present and belong to this group of compounds. Such

* [email protected]

[email protected]

[email protected]

§ [email protected] © Siberian Federal University. All rights reserved

compounds can be found among halides, oxides, oxohalides, cyanides, hydrides. The cations A and B can be NH4 ammonium group also.

In elpasolites also called ordered perovskites there are two kinds of ionic groups BX6 and B'X6, which alternate along the three 4-fold axes (Fig. 1a), contrary to simple perovskite with equivalent octahedra only. Thus, elpasolite cubic cell can be considered as perovskite cell with doubled cell parameters. Cryolite A3B'X6 is a special case of elpasolite structure, where A and B ions in A2BB'X6 are the same.

(a) X anion located in 24e position (b) X is disordered, 192/ site

Fig. 1. A2BB'X6 structure. Octahedra B'X6 has gray color (ion B' is located in the center of octahedron), ions B — marked by blue color, ions A — marked by green color

Most PT, which observed in these compounds, are described as rotational distortions associated with small rotations of octahedra groups BX6 and B'X6 [1, 2]. As a rule such PT are displacive type. Relatively recent research of ammonium elpasolites (NH4)2KGaF6 [3] and (NH4)2KWO3F3 [4], solid solutions of ammonium cryolites (NH4)3Ga1_KScKF6 [5-7], oxyfluo-rides (NH4)3WO3F3, (NH4)3TiOF5 [8, 9] showed that at PT the ordering of atoms and groups including octahedral groups occur, i.e. PT is order-disorder type.

In the study of phase transitions in elpasolite compounds a well established scheme is used, on the first stage of which the group-theoretical analysis of possible distortions of structure is carried out [10]. Such analysis allows one to obtain all possible space groups of distorted phrases, correctly choose model of distorted structure of low-symmetry phase, describe the behavior of physical properties.

A lot of works are devoted to symmetric analysis of the crystals with the space group Fm3m [10-14]. Without going into details of the results of these works, it should be noted that space groups of distorted (dissymmetric) phases induced by all irreducible representations (IR) of Lifshitz points of Brillouin zone are obtained. The mechanical representation assuming rigid, linked and undistorted at PT octahedral ions BX6 and B'X6 [10, 11] was investigated to analyze experimental works about PT in halide elpasolites and cryolites. It should be mentioned the paper [13], which deals with the permutation and mechanical representations with the same space group. Let us recall that the basic functions of mechanical representation are the displacements of atoms of structure, but permutation representation — scalar values, which

in the case of PT order-disorder type are associated with the relative probabilities of atoms to occupy a certain position in the crystal.

Thus, the permutation representation of corresponding structure should be analyzed for consideration of the order-disorder PT. The analysis of representation means: 1) determination of the composition, i.e. which IR and how many times they are included in permutation representation and 2) definition of basic functions of IR, which are included in permutation representation. Such work was done in [13]. In this paper we will use the results of [13].

Experimental [3-9] and theoretical investigations [9-11] of A2BB'X6 crystals prove that order parameters (OP) at PT, which are associated to ordering of X anions, are transformed by IR of center, point r, and boundary, point X, of Brillouin zone (Tab. 1) of space group Fm3m (designation of IR are in accordance with reference books [15, 16]). IR and OP, which induce symmetry changes, are commonly known as critical or primary.

Table 1. Lifshitz points of Brillouin zone of space group Fm3m. The main periods of primitive lattice: a1 = t (0,1,1), a2 = t (1,0,1), aa = t (1,1,0), edge of face-centered cube a = 2t. The main periods of reciprocal lattice: b1 = n/T ( — 1,1,1), b2 = n/T (1, —1,1), ba = n/T (1,1, —1)

Type of star from [15, 16]

Rays of star

K

10

X

1 n

ki = -(bi + b2) = - (0,0,1),

2 t

1 n

k2 = -(bi + ba) = - (0,1,0),

2 t

1 n

ka = 1(b2 + ba) = - (1,0,0)

2 t

Kii — r

ki

0

However, the distortion of the structure of the parent phase in some cases is impossible to describe using only critical OP. In the distorted (dissymmetric) phase the appeared displacements or ordering of ions can be consistent with the symmetry of the phase and they are controlled by non-critical (secondary) OP and IR. All OP, critical and non-critical, which are appeared at PT, form a complete condensate of OP [17]. Symmetry analysis indicates the presence and type of non-critical OP only. Numerical values of both critical and non-critical distortions and OP, which participate in complete condensate, are determined from experimental, in the best case structural, data.

The purpose of the current work is to obtain and visualize structures of dissymmetric phases which appeared in result of X anion ordering in elpasolite crystals A2BB' X6. For the reasons stated above, the dissymmetric structures were considered which are associated with critical OP transformed by IR of r and X points of Brillouin zone (Tab. 1) of space group Fm3m. Herewith non-critical OP and IR are not considered. The work about non-critical deformations will be presented in further publications.

The results of [13] and program ISOTROPY [18] were used to achieve this goal. Crystal structures of dissymmetric phases were presented using program Diamond [19].

1. Parent disordered phase A2BB/X6

The results of works [7-9], allow us to suppose that in the parent phase Go of elpasolite A2BB'X6 there are 24 anions X, which can be disordered among 192/ site of cubic face-centered cell of space group Fm3m [20]. The probability for X anion to occupy some site is equal to 0.125. Polyhedral representation of such disordered structure is showed in Fig. 1b, coordinates of ions of A2BB'X6 are in Tab. 2.

According to the work [13], one can argue that all IR of Lifshitz points of Brilloiun zone of Fm3m group are included in permutation representation on 192/ site at which the X anion is disordered. The IR matrixes for r and X points of Brillouin zone of Fm3m space group are in Tab. 3. The IR matrixes are showed for generators of group only in accordance with reference books [15]: h2 — rotation on 180° around axis [1, 0,0]; h3 — rotation on 180° around axis [0,1,0]; h5 — rotation on 240° around axis [1,1,1]; h9 — rotation on 120° around axis [1,1,1]; h13 — rotation on 180° around axis [—1,1,0]; h25 — inversion.

Table 2. Coordinates of atoms of cubic elpasolite phase A2BB'X6 (sp.gr. Fm3m). For X anion there is coordinates in 192/ disordered general position (upper string) and average value in 24e position (lower string)

Atom Site x/a y/b z/c Occupancy

B' 4a 0 0 0 1

B 4b 0.5 0.5 0.5 1

A 8c 0.25 0.25 0.25 1

X 192/ 0.015 0.05 0.20 0.125

24e 0 0 0.20 1

Notes. The type of IR is defined by two numbers divided by dash: first value is number of star of hi vector, second value defines sequence number of IR of this vector. In addition to notation of IR, corresponding to [15], there are notations which is accepted in foreign literature from [16]. The IR matrixes are showed for generators of group only in accordance with reference books [15]. The diagonal matrixes are presented in form of column with elements corresponding to elements on the main diagonal of matrixes. The sign «±» defines two IR, the matrix presents in first IR with sign «+», and in second IR with sign «—». The letters were used for the matrixes in form:

2. Ordered phases in elpasolites A2BB/X6

The following Tab. 4 and Fig. 2 shows the results of a symmetry analysis, which includes space groups of ordered phases of A2BB'X6 structure. As a critical OP were chosen only those, which are transformed by IR related to the r and X stars of Brillouin zone of face-centered cubic cell of Fm3m group.

Table 3. Complete IR of Fm3m group for Lifshitz points r and X of Brillouin zone

IR

ai

a-2

as

(h2/0) (hs/0) (h5/0) (his/0) (W0)

10-1, X+ 10-2, X-

10-3, X+ 10-4, X-

10-5, X+ 10-6, X-

10-7, X+ 10-8, X-

10-9, X+ 10-10, X-

11-1, r+

11-2, r-

+

11-3, r+ 11-4, r-11-5, r+ 11-6, r-

11-7, r+ 11-8, r-

11-9, r+

11-10, r-

/ — /—1\

1 _1 1 I i )

V i /

n1)

V i / / i \ / 1 \

n1)

V1/ / 1 \ / 1 \

i-1)

V1/

W w

1 1

1 1

I I

1

'-V

T

'-V

'-r 1

I

'-r

fv l

1 1

I 1

1

A A A A

1 1

W A

A

B

—B

-B

B 1

Q

B

B

±

±

±

±

l

±

±

I

±

±

l

l

l

l

1

1

1

Table 4. Probable dissymmetric phases with X ions ordering in elpasolite A2BB'X6 structure. The cell parameters of dissymmetric phases are presented in terms of translation vectors of cubic face-centered unit cell of parent phase: a'x = 2t (1,0, 0), a'2 = 2t (0,1,0), a' = 2t (0,0,1)

Critical IR Critical values Subgroup Basis

of order parameters Schoenflies International

11-2, r- (a) Os F 432 (1, 0,0), (0,1, 0), (0, 0,1)

11-3, r+ (a) T s Fm3 (1, 0,0), (0,1, 0), (0, 0,1)

11-4, r- (a) T 2 F 43m (1, 0,0), (0,1, 0), (0, 0,1)

11-5, r+ (a, 0) D4h I4/mmm (2,2, 0), (1,2, 0), (0,0,1)

Critical IR

Critical values of order parameters

Subgroup Schoenflies International

Basis

11-6, rj

11-7, r+

11-8, T5-

11-9, r+

11-10, rj

10-1, X+

(a, b)

(a, 0) (0, a) (a, b)

10-2, Xj

( a, 0, 0)

( a, a, a)

(a, a, b)

( a, b, c)

( a, 0, 0)

(a, a, 0)

( a, a, a)

(a, b, 0)

(a, a, b)

( a, b, c)

( a, 0, 0)

(a, a, 0)

( a, a, a)

( a, b, c)

( a, 0, 0)

(a, a, 0)

( a, a, a)

(a, b, 0)

(a, a, b)

( a, b, c)

( a, 0, 0)

( a, a, a)

( a, b, a)

( a, b, c)

( a, 0, 0)

(a, 0, a)

D23 n2h

D9

n2d

D9

D7

D25 n2h

n3d C2h C1

nu

n2d

C2v

D cS

C3

ci

C4h

C 3

C2h

Ci Ci

9

C4v

r'20 C2v

C3v

Cs3 Cs3

Cii

n4ih

Oh

n4ih

Dih

n4 n4h

n4h

Fmmm

14m2 I422 F222

Immm

R3m C 2/m P1

I42m

Imm2

R32

Cm

C2

P1

I4/m C2/m R3 P1

I4mm

Imm2

R3m

Cm

Cm

P1

P4/mmm Pm3m P4/mmm Pmmm

P 4/nnc P 4/nbm

1, 0, 0), (0,1, 0), (0,0,1) , 2, 0), (1,2, 0), (0,0,1)

2 ' 2

1 1 0)

2 , 2 , 0)

(1, 0, 0),

2,2, °) 1, 2, 0)

[ 1 1 )

2 , 1, 2 )

0 1 1 )

0 , 2,2 )

2^0) 0,2 ) 1,2, 0) (0, 0, 1),

111 )

2 , 1, 2 )

0 11 )

0, 2 , 2 )

0 11 )

0, 2 , 2 )

1, 2, 1) 1, 2, 0)

011 )

0 , 2,2)

0 1 1 )

0 , 2,2)

1 1 0)

2 , 2 , 0)

110)

2 , 2 , 0)

1,0, 0),

1 1 1)

2 , 2 , 1)

0 1 1 )

0 , 2,2 )

1 0 1 )

2 , 0, 2 )

1,0, 0), 0,1, 0), 1,0, 0),

^2' 2 11 1

(1, 2, 0), (0,0,1) (0,1, 0), (0,0,1)

(1, 1, 0), (0,0,1) (0,1, 2), (1,1,1) (1, 0,2), (2, 0,2)

22

(2, 0,2), (2,2, 0)

(1, 2, 0), (0,0,1) (0,1,0), (1, 0,2)

(0,1, 2), (1,1,1)

22

(0,1,0), (2, 0

10 2 )

2

(1, 0,2), (2, 0,2) (2, 0,2), (2, 2, 0)

(0,2, 2), (1,0,0) (1, 1, 0)( 2,2, 0) (0,1, 2), (1,1,1) (2, 0,2), (2, 2, 0)

(0,2, 2), (1,0,0) (0, 0,1), (2,2, 0)

(0,1, 2), (1,1,1)

(0,0,1), (2, 2, 0) (1, 2, 0), (2, 2, 0) (2, 0,2), (2, 2, 0)

(2, 0,2), (0,1,0) (0,1, 0), (0,0,1) (0,0,1), (1,0,0) (0,1, 0), (0,0,1)

2, 0, 2), (2, 0,2), (0,1,0) 0,1, 0), (0,0,1), (1,0,0)

Critical IR

Critical values of order parameters

Subgroup Schoenflies International

Basis

10-3, X+

10-4, X-

10-5, X+

10-6, X-

10-7, X+

(a, -a, a)

10-8, X-

a, 0 a, b a, b

a, 0 a, 0

a, 0 a, 0 a, a a, 0 a, b a, b

a, 0 a, 0

a, 0 a, 0 a, a a, 0 a, b a, b

a, 0 a, a a, b a, b

a, 0 a, 0, a (a, -a, a)

O1

D4h

Di Di

D6

D4h D12 D4h

T2 Th

D2h

D4h

D I0

D4h T

Td D2sh D2d D2

d44

D4h Th D2h D15

D4h

Ds

D4h T

Td D24h D2d D2

D9

D4h

O4

O4h

D12

D4h

D2

D2h

D12 D4h D10 D4h O2

P 432 Pban P 422 P222

P 4/mnc P42/nnm Pn3 Pnnn

P4/nmm P42/mcm P 43m Pccm P 42m P222

P42/mnm P4/mmm Pm3 Pmmm

P42/nmc P4/nbm P 43m Pban P 42m P222

P42/mmc Pn3m P42/nnm Pnnn

P42/nnm P42/mcm P4232

1, 0

0, 1

0, 1

1, 0

2, 0

0, 1

1, 0

1, 0

1, 0

0, 1

1, 0

0, 1

0, 1

1, 0

1, 0

0, 1

1, 0

1, 0

1, 0

0, 1

1, 0

0, 1

0, 1

1, 0

1, 0

1, 0

0, 1

1, 0

1, 0

0, 1

1, 0

0), 0), 0), 0),

2 ), 0),

0),

0),

2 ), 0),

0),

0),

0),

0),

2 ), 0),

0),

0),

2 ), 0),

0),

0),

0),

0),

2 ), 0),

0),

0),

2 ), 0),

0),

0, 1, 0),

0, 0, 1),

0, 0, 1),

0, 1, 0),

( 2, 0,2 ) 0, 0, 1), 0, 1, 0), 0, 1, 0),

( 2, 0,2 ) 0, 0, 1), 0, 1, 0), 0, 0, 1), 0, 0, 1), 0, 1, 0),

( 2, 0,2 ) 0, 0, 1), 0, 1, 0), 0, 1, 0),

( 2, 0,2 ) 0, 0, 1), 0, 1, 0), 0, 0, 1), 0, 0, 1), 0, 1, 0),

( 2, 0,2 ) 0, 1, 0), 0, 0, 1), 0, 1, 0),

( 2, 0,2 ) 0, 0, 1), 0, 1, 0),

0, 0, 1)

1, 0, 0)

1, 0, 0)

0, 0, 1)

(0,1, 0) 1, 0, 0) 0, 0, 1) 0, 0, 1)

(0,1,0) 1, 0, 0) 0, 0, 1) 1, 0, 0) 1, 0, 0) 0, 0, 1)

(0,1,0) 1, 0, 0) 0, 0, 1) 0, 0, 1)

(0,1,0) 1, 0, 0) 0, 0, 1) 1, 0, 0) 1, 0, 0) 0, 0, 1)

(0,1,0) 0, 0, 1) 1, 0, 0) 0, 0, 1)

(0,1,0) 1, 0, 0) 0, 0, 1)

0

a

(

)

b

c

0

a

(

)

b

c

Critical IR

Critical values of order parameters

Subgroup Schoenflies International

Basis

10-9, X+

10-10, X5

(a, 0, b) (a, b, a) (a, b, c)

(0, 0, 0, a, -a, 0) (0, 0, 0, 0, a, 0) (a, -a, a, -a, a, -a) (0, a, 0, 0, -a, 0) (0, 0, 0, 0, a, -a) (a, 0, a, 0, a, 0) (0, 0, 0, a, b, 0) (0, a, 0, 0, b, 0) (0,0,0, 0, a, b) (a, b, a, b, a, b) (a, b, c, - c, -b, -a) (a, 0, b, 0, c, 0) (a, 0, b, 0, c, d) (a, b, c, d, e, f )

(a, -a, 0, 0, 0, 0) (0, a, 0, 0, 0, 0) (a, -a, a, -a, a, -a) ( a, a, a, a, a, a) (0, a, 0, 0, -a, 0) (0, a, -a, 0, 0, 0) (0, a, 0, a, 0, a) (a, b, 0, 0, 0, 0) (0, a, 0, 0, 0,b) (0, a, 0, 0, b, 0) (0, a, b, 0, 0, 0) ( a, b, 0, 0, -b, -a) (0, a, b, - b, -a, 0) (0, a, -a, 0, b, - b) (a, b, a, b, a, b)

DSh

D5

D1

D4h

D2h

D5

n3d

D5

n4h

D12 n2h

rr6 T h

D10 n2h

D9

n2h

C2h C 2 C3i

C2h

n15 n2h

C2h C1

n13

n2h

D1h D7

C35v

D4h

D14

n4h T 4

C2h

n16 n2h

D13 n2h

D12 n2h

C2h

15 C2v

14 C2v

C34

Pccm P4222 P222

P 42/ncm

Cmca

R3m

P4/mbm

Pnnm

Pa3

Pccn

Pbam

P21/c

R3

C2/m Pbca P21/c P1

Pmmn

Cmcm

R32

R3m

P4/nmm

P42/mnm

P213

P21/m

Pnma

Pmmn

Pnnm

C2/m

Abm2

Amm2

R3

1, 0) 1, 0) 0, 0)

0,1) 0, 0) 1,0) 1,0) 0,1) 0, 0) 0,1) 1,0) 1, 0), 1,0) 1,1) 0, 0) 1,0) 0, 0)

10 1 )

2 , 0 , 2 )'

0, 0,1) 1,1, 0) 1,1, 0) 0, 1, 0) 1, 0, 0) 1, 0, 0)

1 0 1'

2 , 0, 2

0,1, 0) 0, 1, 0) 1, 0, 0) 0,1,1) 1, 0, 0) 0, 0,1) 1,1, 0)

0, 0, 1) 0, 0, 1) 0,1, 0)

1,0, 0) 0,1, 0) 0,1,1) 0, 0, 1)

110)

2 , 2 , 0)

0,1, 0) 1,0, 0) 0,0,1) (0,0,1) 0,1,1) 0,1,1) 0,1, 0) 0, 0, 1) 0,1, 0)

(0,1,0) 1,0, 0) 0,1,1) 0,1,1) 0, 0, 1) 0,1, 0) 0,1, 0) (0,1,0) 0, 0, 1) 0, 0, 1) 0,1, 0) 0,1,1) 0,1,1) 1,1, 0) 0,1,1)

1,0,0 1,0,0 0, 0, 1

0,1,0 0, 0, 1 1,1,1 1,0,0 (1,2, 0) 0, 0, 1 0,1,0 1,0,0 (1,0, 0) 1,1,1 1,0,0 0, 0, 1 1,0,0 0, 0, 1

(1, 0,1 ) 0,1,0 1,1,1 1,1,1 1,0,0 0, 0, 1 0, 0, 1

(2, 0,1 ) 1,0,0 1,0,0 0, 0, 1 1,0,0 0,1,1 1,1,0 1,1,1

0

Critical IR

Critical values of order parameters

Subgroup

Schoenflies International

Basis

(0, a, 0, 0, b, c) C2h P 2i/m (0,1,0), (0,0,1), (1,0,0)

(0, a, b, c, 0,0) C2h P 2 i/c (0,1,0), (1,0,0), (0,1,1)

(a, b, c, —c, —b, -a) C3 C2 (0,1,1), (0,1,1), (1,0,0)

(a, b, c, c, - b, -a) C3 Cm (0,1,1), (0,1,1), (1,0,0)

(0, a, 0, b, 0, c) D P2 2 2 (1,0,0), (0,1,0), (0,0,1)

(0, a, b, 0, 0, c) C2v Pmn21 (0,0,1), (0,1,0), (1,0,0)

(a, b, 0, 0, c, d) C1 P1 (1,0,0), (0,1,0), (0,0,1)

(0, a, b, c, 0, d) C22 P21 (0,0,1), (1,0,0), (0,1,0)

(0, a, b, 0, c, d) Cl Pm (0,1,0), (0,0,1), (1,0,0)

(0, a, b, c, d, 0) C2 Pc (0,1,0), (1,0,0), (0,1,1)

(ab, c d, e, f) Cl P1 (1,0,0), (0,1,0), (0,0,1)

(a) 11-8, r,r, (a, a, 0), Imm2 (b) 11-10, r-, (a, a, 0), Imm2 (c) 11-10, r-, (a, 0, 0), 14mm

(d) 10-10, X-, (a,a,a,a,a,a), (e) 11-10, r-, (a,a,a), R3m (f) 11-8, r-, (a,a,a), R32 R3m

Fig. 2. Polyhedral representation of ordered phases in elpasolite A2BB'X6 structure

Conclusion

Group theoretical methods derived dissymmetric phases which appear in result of X anion ordering of elpasolite structure A2BB'X6. The maximum achievable disordering of X anion in 192/ position of cubic face-centered cell with space group Fm3m were considered in the parent phase. Among all critical IR only IR of center, point K11 — r, and boundary, point K10 — X, of Brillouin zone of space group Fm3m were investigated. 108 dissymmetric phases were modeled during this work. Ordering of X anion depend on number of components of OP which describes PT. Increasing of dimension of OP leads to decrease of influence of each component of OP on ordering process.

The research is conducted within the framework of the state task of Ministry of Education and Science of the Russian Federation to Siberian Federal University on R&D performance in 2015 (Task 3.2534.2015/K).

References

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Фазы анионного упорядочения в структуре эльпасолита (упорядоченного перовскита)

Роман Г. Севрюков Иван Н. Сафонов Максим С. Молокеев Сергей В. Мисюль

Теоретико-групповыми методами получено 108 диссимметричных фаз, возникающих в результате упорядочения анионов X в структуре эльпасолита Á2BB'X6 (упорядоченного перовскита). Из критических параметров порядка рассмотрены только такие, которые преобразуются по неприводимым представлениям точек Г и X зоны Бриллюэна пространственной группы Oh — Fmim. Для диссимметричных фаз, переходы в которые могут сопровождаться возникновением сегнетоэлектричества, приведены полиэдрические изображения структур. Все результаты собраны в удобных для использования таблицах.

Ключевые слова: эльпасолит, структурные данные, полный конденсат параметров порядка, неприводимые представления, критические искажения, фазовый переход, диссимметричные фазы.