Magnetic symmetry of the plain domain walls in ferro- and ferrimagnets

ARTICLE IN PRESS Physica B 404 (2009) 4018–4022

Contents lists available at ScienceDirect

Physica B journal homepage: www.elsevier.com/locate/physb

Magnetic symmetry of the plain domain walls in ferro- and ferrimagnets B.M. Tanygin, O.V. Tychko  Kyiv Taras Shevchenko National University, Radiophysics Faculty, Glushkov av. 2, build. 5, Kyiv 01022, Ukraine

a r t i c l e in f o

a b s t r a c t

Article history: Received 4 November 2008 Received in revised form 15 June 2009 Accepted 14 July 2009

Magnetic symmetry of all possible plane domain walls in ferro- and ferrimagnets is considered. Magnetic symmetry classes of non 1801 (including 01) domain walls are obtained. The domain walls degeneracy is investigated. The symmetry classification is applied for research of all possible plane domain walls in crystals of the hexoctahedral crystallographic class. & 2009 Elsevier B.V. All rights reserved.

PACS: 61.50.Ah 75.60.Ch Keywords: Domain wall type Symmetry transformation Magnetic symmetry class Degeneracy

1. Introduction

2. Domain wall symmetry in the magnetically ordered media

The investigation of static and dynamic properties [1,2] of domain walls (DWs) in magnetically ordered media is of considerable interest for the physical understanding of medium behavior and it is also important for applications. For sequential examination of these properties it is necessary to take into account the magnetic symmetry [3,4] of the media. Determination of the DW magnetic symmetry allows to characterize qualitatively some elements of the DW structure and their change. The complete symmetry classification of plane 1801 DWs (1801-DWs) in magnetically ordered crystals [5] and similar classification of these DWs with Bloch lines in ferromagnets and ferrites [6] were carried out earlier. The plane DWs with width d [1,7] exceeding the characteristic size a of a unit magnetic cell were considered. Properties of these DWs in ferro- and ferrimagnets are described by the density of magnetic moment M [8]. Their symmetry can be characterized by the magnetic symmetry classes (MSCs) [9] of a crystal containing a DW [5]. The building of a totality of the MSCs of all possible [1] plane (i.e. DW with r0bd, where r0 is the curvature radius of the DW [5]) DWs in ferro- and ferrimagnets is the purpose of this work.

Let m be the unit time-odd axial vector [9] along the magnetization vector M: m ¼ M/M, where M is the saturation magnetization. Then m1 and m2 are unit time-odd axial vectors along magnetization vectors M1 and M2 in neighboring domains: m1 ¼ M1/M, m2 ¼ M2/M. The vectors m1 and m2 coincide with different easy magnetization axes (EMA) of the medium. The angle 2a between these vectors determines the DW type (2a-DW): 2a ¼ arccos(m1m2). A unit polar time-even vector nW indicates the DW plane normal. It is directed from domain with m1 to domain with m2 . In order to define the unified co-ordinate system we introduce the vectors a1 and a2 as well as the parameters bS ¼ j½nW  mS j and bD ¼ j½nW  Dmj. The unit vectors of the co-ordinate system Ox~ y~ z~ are chosen as ½ex~ ; ey~ ; ez~  ¼ ½a2 ; a1 ; nW . Here the unit vector a1 coincides with the direction of the vector Dm  nW ðnW DmÞ (at bD a0 and bS ¼ 0) or ½a2  nW  (at bD ¼ 0 or bS a0). The unit vector a2 coincides with the direction of vector mS  nW ðnW mS Þ (at bS a0) or ½nW  a1  (at bD a0 and bS ¼ 0) or else with an arbitrary direction in the DW plane (a2 ? nW at bS ¼ bD ¼ 0). The time-odd axial vectors Dm and mS are determined by equalities Dm ¼ m2  m1 and mS ¼ m1 þ m2 , respectively. The MSC Gk (here k is a MSC number) of a 2a-DW is the magnetic symmetry group including all symmetry transformations (here and hereinafter all translations are considered as unit operations) that do not change the spatial distribution of magnetic moments in the crystal with DW [5]. The abovementioned group is a subgroup of the magnetic (Shubnikov’s)

 Corresponding author. 64 Vladimirskaya str., Taras Shevchenko Kyiv National University, Radiophysics Faculty, 01033 Kyiv, Ukraine. Tel./fax: +38 044 526 03 49. E-mail addresses: [email protected] (B.M. Tanygin), [email protected], [email protected] (O.V. Tychko).

0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.07.150

ARTICLE IN PRESS B.M. Tanygin, O.V. Tychko / Physica B 404 (2009) 4018–4022

4019

Table 1 Magnetic symmetry classes of the plane 2a-DWs with 2aa1801. MSC number k Mutual orientations of the vectors m1, m2 and nW Symmetry elements

2

Coordinate dependences of mðz~ Þ components

DW center International MSC symbol

my~ ðz~ Þ

mx~ ðz~ Þ

mz~ ðz~ Þ (–)

–

mm0 20

– z~ ¼ 0 – z~ ¼ 0 z~ ¼ 0 z~ ¼ 0

m

aD ¼ bD ¼ aS ¼ 0

1; 2 10 ; 2 2 ; 2n0

(–)

(A,S)

1; 2 2 1, 210 , 22 , 2n0 1, 2n0

(–)

(A,S)

(–)

(A) (A,S) (A)

(S) (A,S) (–)

(–) (–) (S)

(A) (A)

(S) (A)

(S) (S)

(–)

(A,S)

(A,S)

(S) (A,S) (S)

(A) (A,S) (A)

– z~ ¼ 0 z~ ¼ 0

m0

(A) (A,S) (–) (S)

(S)

(A)

z~ ¼ 0

m0

(–) (–)

(–) (–)

(A,S) (A,S)

– –

2 m0 m0 2

(–) (–)

(–) (–)

(A,S) (A,S)

– –

3 3m0

(–) (–)

(–) (–)

(A,S) (A,S)

– –

4 4m0 m0

(–) (–)

(–) (–)

(A,S) (A,S)

– – z~ ¼ 0

6 6m0 m0

6

aS ¼ aD ¼ aC ¼ 0

7 8 9

aS ¼ aD ¼ 0 aS ¼ aD ¼ 0 aC ¼ aD ¼ bS ¼ 0

10 11

aD ¼ 0 aC ¼ aD ¼ bS ¼ 0

12

aC ¼ 0

13 16 17

aS ¼ 0 Arbitrary aC ¼ aS ¼ bD ¼ 0

18

aC ¼ aS ¼ bD ¼ 0

19 22

bD ¼ bS ¼ 0 bD ¼ bS ¼ 0

24 26

bD ¼ bS ¼ 0 bD ¼ bS ¼ 0

30 32

bD ¼ bS ¼ 0 bD ¼ bS ¼ 0

37 39

bD ¼ bS ¼ 0 bD ¼ bS ¼ 0

43

aD ¼ bD ¼ aS ¼ 0

(–)

aD ¼ bD ¼ aS ¼ 0

ð1; 2 10 ; 2 2 ; 2 n0 Þ  ð1; 1Þ (–) (–) 1; 210 ; 2 2 ; 2 n0

(S)

44

(S)

(–)

45

aD ¼ bD ¼ aS ¼ 0

1, 1, 22 , 2 2

(–)

(S)

(–)

46

aD ¼ bD ¼ aS ¼ 0

1, 1, 2n0 , 2n0

(S)

(S)

(–)

47

aD ¼ bD ¼ 0

1, 1,

(–)

(S)

(S)

48

aD ¼ bD ¼ 0

1, 1

(S)

(S)

(S)

49

aD ¼ bD ¼ bS ¼ 0

1, 1, 2n , 2 n

(–)

(–)

(S)

50

aD ¼ bD ¼ bS ¼ 0

1, 2 10 , 220 , 2n

(–)

(–)

(S)

51

aD ¼ bD ¼ bS ¼ 0

ð1; 210 ; 220 ; 2n Þ  ð1; 1Þ

(–)

(–)

(S)

52

aD ¼ bD ¼ bS ¼ 0

(–)

(S)

aD ¼ bD ¼ bS ¼ 0 aD ¼ bD ¼ bS ¼ 0

6n 3n ; 210

(–)

53 54

6 n ; 210

(–) (–)

(–) (–)

(S) (S)

55

aD ¼ bD ¼ bS ¼ 0

3 n ; 2 10

(–)

(–)

(S)

56

aD ¼ bD ¼ bS ¼ 0

(–)

(–)

(S)

57 58

aD ¼ bD ¼ bS ¼ 0 aD ¼ bD ¼ bS ¼ 0

4n ; 2 n 4n ; 210 4n ; 2 10 ; 2 n

(–) (–)

(–) (–)

(S) (S)

59

aD ¼ bD ¼ bS ¼ 0

4n

(–)

(–)

(S)

60

aD ¼ bD ¼ bS ¼ 0

4 n ; 210

(–)

(–)

(S)

61

aD ¼ bD ¼ bS ¼ 0

(–)

(–)

(S)

62 63

aD ¼ bD ¼ bS ¼ 0 aD ¼ bD ¼ bS ¼ 0

6n ; 2 n 6n ; 210 6n ; 2 10 ; 2 n

(–) (–)

(–) (–)

(S) (S)

64

aD ¼ bD ¼ bS ¼ 0

3n

(–)

(–)

(S)

1, 210 , 2 20 , 2 n 1, 210 1, 2 n 1, 2 10 1, 22 1 1, 2 10 , 22 , 2 n0 1, 2 n0 1, 2n 1, 2 10 , 2 20 , 2n 3n 3n ; 2 10 4n 4n ; 2 10 6n 6n ; 2 10

210 ,

2 10

symmetry group of the crystal paramagnetic phase [10]. These transformations do not change DW boundary conditions and can be classified by two types [5]. The first type transformations g ð1Þ do not change the directions of the vectors m1, m2 and nW: g ð1Þ nW ¼ nW , g ð1Þ m1 ¼ m1 , g ð1Þ m2 ¼ m2 . The second type transformations g ð2Þ change these directions: g ð2Þ nW ¼ nW , g ð2Þ m1 ¼ m2 , g ð2Þ m2 ¼ m1 . In conformity with the terminology of [6] the MSC GB of DW boundary conditions is the totality of all transformations of the magnetic symmetry group of the crystal paramagnetic phase that satisfy the mentioned six conditions. It is the MSC of the maximum possible symmetry of a 2a-DW in the given crystal for a particular mutual orientation of the vectors m1, m2 and nW. The other possible MSCs Gk of a 2a-DW with fixed directions of the vectors m1, m2 and nW result by enumeration of the subgroups of GB: Gk DGB  GP , where Gp is the MSC of the crystal paramagnetic phase. The mutual orientation of the vectors m1, m2 and nW is determined by the set of parameters aS ¼ ðnW mS Þ, aD ¼ ðnW DmÞ, aC ¼ ðnW mC Þ, bS and bD, where time-even axial vector mc is determined by equality mC ¼ ½m1  m2 .

220 20 20 mm0 20 20 m 2 1 m0 m0 2

mm0 m0

z~ ¼ 0 z~ ¼ 0 z~ ¼ 0

mm0 20

z~ ¼ 0 z~ ¼ 0 z~ ¼ 0

20 /m0

z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0 z~ ¼ 0

2/m 20 /m0 1 2/m 220 20 mm0 m0 6 320 6m0 20 3m0 4/m 420 20 4=mm0 m0 4 420 m0 6/m 620 20 6=mm0 m0 3

The possible MSCs Gk (1k42) of 1801-DWs were found earlier [5]. All possible MSCs of 2a-DWs with 2aa1801 are presented in Table 1. For a certain 2a-DW the different MSCs are different groups of magnetic point symmetry transformations. Their representations [11,15] are written in the co-ordinate system Ox~ y~ z~ . All represented MSCs are not interrelated by a rotation over an arbitrary angle around nW. Also the above-mentioned MSCs are not reduced with each other by unit vectors transformation a1 2a2 . The possible transformations g ð1Þ or g ð2Þ (column ‘‘Symmetry elements’’ of Table 1) of 2a-DWs with 2aa1801 are rotations around two-fold symmetry axes 2n , 2n0 or 21, 210 or else 22 , 220 that are collinear with the unit vectors nW or a1 or else a2, respectively, reflections in planes 2 n , 2 n0 or 2 10 or else 2 2 , 2 20 that are normal to the above mentioned vectors, respectively, rotations around three-, four-, six-fold symmetry axes 3n, 4n, 6n that are collinear with the vector nW, rotations around three-, four-, six-fold inversion axes 3 n , 4 n , 6 n that are collinear with the vector nW, inversion in the symmetry center 1 and identity (symmetry

ARTICLE IN PRESS 4020

B.M. Tanygin, O.V. Tychko / Physica B 404 (2009) 4018–4022

element 1). Here an accent at symmetry elements means a simultaneous use of the time reversal operation R [9]. For MSCs with 24rkr39 and 52rkr64 only generative symmetry elements [11] are represented in Table 1. There is a correspondence between MSCs of 1801-DWs (i.e. at m1 ¼ m2 [1]), 01-DWs (i.e. at m1 ¼ m2 [13]) and 2a-DWs with non-collinear orientation of vectors m1 and m2 [1] (hereinafter the last DWs will be marked as 2a0 -DWs). The above mentioned determinations of criterions for transformations g ð1Þ and g ð2Þ can be represented in another identical form: g ð1Þ nW ¼ nW , g ð1Þ mS ¼ mS , g ð1Þ Dm ¼ Dm and g ð2Þ nW ¼ nW , g ð2Þ mS ¼ mS , g ð2Þ Dm ¼ Dm. These criterions restrict an ensemble of MSCs symmetry transformations for an arbitrary 2a0 -DW. We have Dm ¼ 0 and mS ¼ 0 for 01- and 1801-DWs, respectively. A pair from the above mentioned criterions does not restrict the MSCs symmetry transformations of 01- or 1801-DWs. Therefore the magnetic symmetry of 2a0 -DWs does not exceed the magnetic symmetry of 01- and 1801-DWs generically. The MSCs of 1801DWs are the MSCs of 2a0 -DWs if their transformations do not break the symmetry of the vector mS of the 2a0 -DW (i.e. these MSCs must be subgroup of the group 1=mm0 m0 , where the infinite-fold symmetry axis is collinear with the vector mS). There is an analogy between MSCs of 1801- and 01-DWs: their transformations g ð1Þ are the same since they belong to a subgroup of axial time-odd vector symmetry group (MSC 1=mm0 m0 ), where the infinite-fold symmetry axis is collinear with Dm or mS for 1801- or 01-DWs, respectively. Therefore if MSCs consist of the transformations g ð1Þ only then these MSCs are common for 1801and 01-DWs. They are marked with sign ‘‘–’’ in column ‘‘DW center’’ of Table 1. A conversion of MSC of 1801-DW into MSC of 01DW is simply a change of the criterion g ð2Þ Dm ¼ Dm by the criterion g ð2Þ mS ¼ mS . The transformations of corresponding MSCs of these 2a-DWs are different by the substitution g ð2Þ -g ð2Þ  R only. Therefore, if a pair of MSCs of 1801-DWs and a pair of MSCs of 01-DWs is connected by the above-mentioned substitution, then these MSCs are common for 1801- and 01-DWs.

As a result the lists of MSCs of 01-, 1801- and 2a0 -DWs are intersected in general. Total number of MSCs of a 2a-DW with arbitrary 2a value (including 2a ¼ 1801) in ferro- and ferrimagnets is equal to 64. General enumeration of MSCs of 1801-DWs contains 42 MSCs: 1  k  42 [5]. This enumeration holds also for MSCs of 2a-DW with 2aa1801 (MSC numbers are bold type in column ‘‘MSC number k’’ of Table 1). There are 10 MSCs of 2a0 -DWs: 7  k  13 and 16  k  18. The general list of MSCs of 01-DWs includes all 42 MSCs of Table 1: k ¼ 2, 6  k  13, 16  k  19, k ¼ 22, 24, 26, 30, 32, 37, 39 and 43  k  64.

3. Domain wall structure The 2a-DWs with dba in ferro- and ferrimagnets are described by the macroscopic density of magnetic moment M(z~ ) [5]. The transformations g ð1Þ and g ð2Þ (g ð1Þ 2 Gk ; g ð2Þ 2 Gk ) impose restrictions on the kind of coordinate dependence of mðz~ Þ components (mðz~ Þ ¼ mx~ ðz~ Þ þ my~ ðz~ Þ þ mz~ ðz~ Þ) in the DW volume and allow to find this dependence [5]. For the determination of the kind of coordinate dependence of mðz~ Þ component of 01- and 2a0 -DWs for each MSC (column ‘‘Coordinate dependences of mðz~ Þ components’’ in Table 1) the next rules are used: (a) if an axial time-odd vector ~ y~ or z~ Þ is not an invariant of the along unit vectors er ðr  x; transformation g ð1Þ then there is no component mr ðz~ Þ (figure (-) in column ‘‘Coordinate dependences of mðz~ Þ components’’ of Table 1); (b) if the axial time-odd vector along er is inverted by the transformation g ð2Þ then the component mr ðz~ Þ is an odd (A) function of coordinate z~ ; (c) if the axial time-odd vector along er is an invariant of the transformation g ð2Þ then mr ðz~ Þ is an even (S) function of coordinate z~ ; (d) if the axial time-odd vector along er is an invariant of the transformation g ð1Þ then transformation g ð1Þ does not restrict the kind of function mr ðz~ Þ (A,S). If the MCS of a 2a-DW includes transformations that transpose adjacent magnetic domains then this DW has a center of symmetry [5]. These MSCs enclose the symmetry transformations

Table 2 Number k (degeneracy qB) of MSC of boundary conditions of arbitrary oriented plane 2a-DW (2a4901) in the cubic m3m crystals at selected domain magnetization directions. DW plane

(1 0 0) (0 1 0) (0 0 1) (111) ð1 1 1Þ

2a-boundary conditions 1801-DW

1801-DW

1801-DW

1201-DW

1091-DW

[1 0 0], ½1 0 0

[11 0], ½1 1 0

[111], ½1 1 1

[11 0], ½0 1 1

[111], ½1 1 1

34 (6) 1 (12) 1 (12) 14 (24) 14 (24)

14 14 1 14 4

14 (24) 14 (24) 14 (24) 29 (8) 14 (24)

16 13 16 16 13

9 13 13 12 12

(24) (24) (12) (24) (24)

(96) (48) (96) (96) (48)

(24) (48) (48) (48) (48)

ð1 1 1Þ

14 (24)

4 (24)

14 (24)

16 (96)

10 (48)

ð1 1 1Þ (11 0) (1 0 1) (0 11)

14 (24)

14 (24)

14 (24)

13 (48)

10 (48)

ð1 1 0Þ

14 14 1 14

(24) (24) (12) (24)

23 (12) 15 (48) 15 (48) 1 (12)

14 14 14 5

16 11 16 16

16 16 17 16

ð1 0 1Þ

14 (24)

15 (48)

5 (24)

13 (48)

ð0 1 1Þ (h h l) (h k h) (h k k)

1 (12)

15 (48)

5 (24)

16 (96)

(48) (48) (24) (48)

14 15 15 4

(24) (48) (48) (24)

14 14 14 15

(24) (24) (24) (24)

(24) (24) (24) (48)

16 16 16 16

(96) (48) (96) (96)

(96) (96) (96) (96)

(96) (96) (24) (96)

16 (96) 7 (24)

ðh h lÞ

15 15 14 15

16 16 12 16

(96) (96) (48) (96)

ðh k hÞ

15 (48)

15 (48)

15 (48)

13 (48)

16 (96)

ðh k kÞ

14 (24)

15 (48)

15 (48)

16 (96)

10 (48)

(h k 0), ðh k 0Þ

14 (24)

14 (24)

15 (48)

16 (96)

16 (96)

(h 0 l), ðh 0 lÞ

14 (24)

15 (48)

15 (48)

16 (96)

16 (96)

(0 k l), ð0 k lÞ (h k l), ðh k lÞ, ðh k lÞ, ðh k lÞ

4 (24)

15 (48)

15 (48)

16 (96)

13 (48)

15 (48)

15 (48)

15 (48)

16 (96)

16 (96)

ARTICLE IN PRESS B.M. Tanygin, O.V. Tychko / Physica B 404 (2009) 4018–4022

g ð2Þ . They are marked by coordinate z~ ¼ 0 in column ‘‘DW center’’ of Table 1. As in the case of 1801-DWs [5], the 01- DWs can be pulsating (i.e. DW with collinear directions of vectors M and jMjaconst in its volume [5]) DWs. The MSCs with k ¼ 2, 6, 19–45, 49–64 describe symmetry of pulsating DWs only. In contrast with 1801and 01-DWs there are no pulsating DWs among the 2a0 -DWs, since 2a0 -DWs require the presence of two ‘‘nonzero’’ mðz~ Þ components. The 2a0 -DWs are rotary (i.e. DW with |M| ¼ const in its volume) or semi-rotary [5] DWs only. Among rotary or semirotary DWs there are DWs with only Bloch (i.e. DWs with MnW ¼ const) [1,14] (k ¼ 7, 8 or 46) and only Neel (i.e. DWs with m rotation in the plane containing nW) [1,15] (k ¼ 9, 12, 17 or 47) laws of m rotation in their volume. Crystal magnetic ordering is accompanied by phase transition and change of crystal magnetic symmetry [3]. In a magnetically ordered crystal qk-multiply degenerate 2a-DWs with fixed 2a can be obtained [6], where qk ¼ ord(Gp)/ord(Gk). Functions ord(Gp) and ord(Gk) give the order [11] of the magnetic point group of the crystal paramagnetic phase [9,10] and of a 2a-DW in this crystal, respectively. These 2a-DWs have the same energy but different structures (magnetization distribution, plane orientation, etc.). The minimum value of qk is 2 in accordance with the invariance of energy for time reversal operation R. At representation of the Gp as the totality of Gk (with fixed value k and different symmetry elements orientations) the lost transformations (members of adjacent classes) g1 [6,12] interrelate the above mentioned qk-multiply degenerate 2a-DWs (i.e. g1 operation converts an one of such 2a-DWs into another). The degeneracy qk of a 2a-DW can be written in the form qk ¼ qB qk0 ðqk0  qk Þ, where qk0 ¼ ordðGB Þ=ordðGk Þ is the number of equal-energy 2a-DWs with fixed boundary conditions, qB ¼ ord(GP)/ord(GB) is the number of possible boundary conditions. Here ord(GB) is the order of the point group of the maximum magnetic symmetry of the 2a-DW in the given crystal. The 2a-DWs of MSC G16 (MSC 1) have the maximum degeneracy qk. For 1801- and 2a0 -DWs it is equal to 16 (crystallographic class mmm), 48 (crystallographic class 6/mmm) and 96 (crystallographic class m3m) in crystals of lower, medium and higher symmetry singonies (in conformity with terminology of [11]), respectively. The 01-DWs are formed in spatially inhomogeneous media [13]. Conditions of occurrence and existence of such DWs demand to take into account medium peculiarities.

4. Magnetic symmetry classes of domain walls in hexoctahedral crystals As an example let’s consider MSCs of all possible DWs in magnetically ordered crystals of hexoctahedral class (crystallographic point symmetry group m3m in the paramagnetic phase [3]). This class is assumed to exhibit the largest variety of possible DWs. Furthermore it encompasses widely investigated and used magnetic media (all cubic symmetry metals, specifically iron and nickel [6], magnetic oxides, specifically ferrites with structures of spinel [4] and garnet [16], perovskite, magnetite and others). The magnetic anisotropy (MA) energy ek is the invariant of the initial paramagnetic phase of crystal. For the m3m crystal this energy is given by ek(a1, a2, a3) ¼ K1s+K2P+K3s2+K4sp+?, where K1, K2, K3 and K4 are first, second, third and fourth MA constants, s ¼ a21 a22 þ a22 a23 þ a21 a23 , p ¼ a21 a22 a23 , a1, a2 and a3 are the direction cosines of m [16]. The absolute minimum of this energy corresponds to EMAs. Signs of MA constants and relation between their values determine EMAs directions. In the framework of the (K1, K2, K3) approximation the EMAs directions can coincide with

4021

Table 3 Number k (degeneracy qB) of MSC of boundary conditions of arbitrary oriented plane 2a-DW (2ar901) in the cubic m3m crystals at selected domain magnetization directions. DW plane

2a-DW boundary conditions 901-DW

901-DW 711-DW 601-DW [1 0 0], ½0 1 0 [11 0], ½1 1 0 [111], ½1 1 1 [11 0], [0 11]

(1 0 0) (0 1 0) (0 0 1) (111) ð1 1 1Þ

12 12 7 13 10

(48) (48) (24) (48) (48)

9 17 7 16 16

(24) (24) (24) (96) (96)

17 10 10 12 12

(24) (48) (48) (48) (48)

16 10 16 10 16

(96) (48) (96) (48) (96)

ð1 1 1Þ

10 (48)

16 (96)

13 (48)

10 (48)

ð1 1 1Þ (11 0) (1 0 1) (0 11)

13 (48)

16 (96)

13 (48)

16 (96)

17 16 16 9

12 10 13 12

16 16 9 16

16 10 16 16

ð1 1 0Þ

(24) (96) (96) (24)

(48) (48) (48) (48)

(96) (96) (24) (96)

(96) (48) (96) (96)

ð1 0 1Þ

16 (96)

10 (48)

16 (96)

18 (48)

ð0 1 1Þ (h h l) (h k h) (h k k)

16 (96)

13 (48)

7 (24)

16 (96)

13 16 16 10

16 16 16 16

ðh h lÞ

(48) (96) (96) (48)

(96) (96) (96) (96)

16 16 12 16

(96) (96) (48) (96)

16 10 16 16

(96) (48) (96) (96)

ðh k hÞ

16 (96)

16 (96)

16 (96)

16 (96)

ðh k kÞ

16 (96)

16 (96)

13 (48)

16 (96)

(h k 0), ðh k 0Þ

12 (48)

12 (48)

16 (96)

16 (96)

(h0l), ðh 0 lÞ

16 (96)

10 (48)

16 (96)

16 (96)

16 (96) (0 k l), ð0 k lÞ (h k l), ðh k lÞ, ðh k lÞ, ðh k lÞ 16 (96)

13 (48)

10 (48)

16 (96)

16 (96)

16 (96)

16 (96)

both high-symmetric and low-symmetric crystallographic directions [17]. In the framework of the two-constant (K1 ,K2 ) approximation the EMA directions can coincide only with highsymmetric /111S or /11 0S or else /1 0 0S like crystallographic directions at K1rK2/3 or 0ZK1ZK2/2 or else K1Z0, respectively [1,18]. At that 711-, 1091- and 1801-DWs or 601-, 901-, 1201and 1801-DWs or else 901- and 1801-DWs are realized in a m3m crystal, respectively [1]. The MSCs and degeneracy qB of a 2a-DW boundary conditions with 2a4901 and 2ar901 are presented in tables 2 and 3, respectively. The earlier obtained MSCs of merely 1801-DWs (bold type numbers in Table 2) include elements [5]: k ¼ 1ð1; 2 1 ; 22 ; 2 n Þ ð1; 1 0 Þ; k ¼ 4ð1; 1 0 ; 210 ; 2 1 Þ; k ¼ 5ð1; 1 0 ; 2n0 ; 2 n Þ; k ¼ 14ð1; 1 0 ; 22 ; 2 20 Þ; k ¼ 15ð1; 1 0 Þ; k ¼ 23ð1; 21 ; 22 ; 2n Þ  ð1; 1 0 Þ; k ¼ 29 ð3 n0 ; 2 10 Þ; k ¼ 34ð4n ; 2 10 ; 2 n0 Þ. Only generative symmetry elements are presented for k ¼ 29 and 34. Other MSCs of Tables 2 and 3 are presented in Table 1. In these tables the DW plane orientation is assigned by different Miller indexes h,k,l41. A simultaneous change on negative and/or cyclic permutation of all indexes does not change MSCs. There are no common MSCs of maximum symmetrical 1801and 2a0 -DWs in the m3m crystal. It is connected with the presence of the 1 0 transformation (2a0 -DW vector mS is changed by this transformation) in the MSCs of such 1801-DW.

5. Conclusions The full magnetic symmetry classification of all possible domain walls in ferro- and ferrimagnet crystals includes 64 magnetic symmetry classes: 42 classes of 01-DWs, 10 classes of 2a-DWs with 01o2ao1801 and 42 classes of 1801-DWs. Lists of magnetic symmetry classes of all above mentioned types of DWs are intersected in general case.

ARTICLE IN PRESS 4022

B.M. Tanygin, O.V. Tychko / Physica B 404 (2009) 4018–4022

01-DWs can be pulsating, rotary or semi-rotary DWs. The 2aDWs with 01o2ao1801 are rotary or semi-rotary DWs only. Among rotary or semi-rotary DWs there are DWs with Bloch or Neel laws of magnetization rotation in their volume. Pulsating, rotary or semi-rotary DWs can have a center of symmetry in their volume. All possible 1801- and 2a-DWs with 01o2ao1801 have even degeneracy (its value is between 2 and 96 in general case). Magnetic symmetry classes of maximum symmetrical 1801DWs do not meet with such classes of 2a-DWs with 01o2ao1801 in a m3m crystal.

[5] [6] [7] [8] [9] [10]

[11]

References

[12] [13]

[1] A. Hubert, Theorie der Domanenwande in Geordneten Medielen (Theory of Domain Walls in Ordered Media), Springer, Berlin, Heidelberg New York, 1974; A. Hubert, R. Shafer, Magnetic Domains. The Analysis of Magnetic Microstructures, Springer, Berlin, 1998. [2] V. Bokov, V. Volkov, Phys. Solid State 50 (2008) 198. [3] L. Shuvalov, Sov. Phys. Crystallogr. 4 (1959) 399. [4] L. Shuvalov, Modern Crystallography IV: Physical Properties of Crystals, Springer, Berlin, 1988.

[14] [15] [16] [17] [18]

V. Baryakhtar, V. Lvov, D. Yablonsky, JETP 87 (1984) 1863. V. Baryakhtar, E. Krotenko, D. Yablonsky, JETP 91 (1986) 921. B. Lilley, Phil. Mag. 41 (1950) 792. A. Andreev, V. Marchenko, JETP 70 (1976) 1522. L. Landau, E. Lifshitz, L. Pitaevskii, Course of Theoretical Physics, Electrodynamics of Continuous Media, vol. 8, Pergamon Press, London, 1984. V.A. Kopcik, Xubnikovskie Gruppy: Spravoqnik po simmetrii i fiziqeskim svostvam. kristalliqeskih struktur [Shubnikov’s Groups: Handbook on the Symmetry and Physical Properties of Crystalline Structures, in Russian], Izdatel’stvo Moskovskogo Universiteta, Moscow, 1966; A.V. Shubnikov, N.V. Belov, Colored Symmetry, Pergamon Press, London, 1964; B. Tavger, V. Zaitzev, JETP 3 (1956) 430. B. Vanshtein, Modern Crystallography 1: Symmetry of Crystals, Methods of Structural Crystallography, Springer, Berlin, 1994. E. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959. L. Heyderman, H. Niedoba, H. Gupta, I. Puchalska, J. Magn. Magn. Mater 96 (1991) 125; R. Vakhitov, A. Yumaguzin, J. Magn. Magn. Mater. 215–216 (2000) 52. L. Landau, E. Lifshitz, Sov. Phys. 8 (1935) 153. L. Neel, Compt. Rend. 2419 (1955) 533. A. Paoletti, Physics of Magnetic Garnets, Elsevier, Amsterdam, 1978. U. Atzmony, M. Dariel, Phys. Rev. B 13 (1976) 4006. K.P. Belov, A.K. Zvezdin, R.Z. Levitin, A.S. Markosyan, B.V. Mill’, A.A. Mukhin, A.P. Perov, JETP 41 (1975) 590.