The degeneration of the magnetic domain walls in ferro- and ferrimagnets with arbitrary crystallographic class

8/7/2019 The degeneration of the magnetic domain walls in ferro- and ferrimagnets with arbitrary crystallographic class

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The degeneration of the magnetic domain walls in ferro- and

ferrimagnets with arbitrary crystallographic class

B. M. Tanygin

Kyiv Taras Shevchenko National University, Radiophysics Faculty, Glushkov av.2,

build.5, Kyiv, Ukraine, 01022

E-mail: [email protected]

Abstract. Degeneration of the magnetic plain domain walls in ferro- and ferrimagnets has

been considered. Results are obtained for arbitrary crystal system and any type of the

domain wall except 0 degree walls (magnetization inhomogeneities). These results can be

considered as an indirect systematization of the Bloch lines and points which are

principally possible in magnetic materials.

PACS: 61.50 Ah, 75.60 Ch

Please, cite original work as:B.M. Tanygin and O.V. Tychko,Magnetic symmetry of the plain domain walls in ferro- and ferrimagnets,Physica B: Condensed Matter, Volume 404, Issue 21 (2009) Pages 4018-4022.

e-mail: [email protected]: http://sites.google.com/site/btanygin/


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1. Introduction

The investigation of static and dynamic properties [1,2] of domain walls (DWs) in magnetically

ordered media is a considerable interest for the physical understanding of medium behaviorandit is also

important for applications. For sequential examination of these properties it is necessary to take into

account a magnetic symmetry [3,4] of the media.

The complete symmetry classification of plane (i.e. DW with 0r >> where 0r is a DW radius of

curvature [5]) 180 degree DWs (1800

DWs) in magnetically ordered crystals [5] and similar classification

of these DWs with Bloch lines in ferromagnets and ferrites [6] where carried out earlier. The plane DWs

with width [1,7] exceeding the characteristic size a of a unit magnetic cell where considered. Properties

of these DWs in ferro- and ferrimagnets are described by a density of magnetic moment M [8]. Their

symmetry can be characterized with use of magnetic symmetry classes [9-12] of a crystal with DW [5].

The symmetry classification has been extended to the case of the 0 DWs [13] (magnetization

inhomogeneities [14]).

A determination of a DW magnetic symmetry allows to find the degeneration [6, 13] for the DW.

The corresponding degeneracy is the total number of possible magnetization distributions (i.e. DWs

states) which are principally possible in the given crystal. The degeneracy defines the totality of all Bloch

lines and Bloch points which are possible in such DW [6]. The purpose of this work is a building of tables

of degeneracy for all possible [1] (with exception of the 0 or 360 DWs) plane DWs in all ferro- and

ferrimagnetic crystals.


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2. Domain wall degeneracy

Crystal magnetic ordering is accompanied by phase transition and change of crystal magnetic

symmetry [3]. In magnetically ordered crystal kq multiply degenerate 2 DWs with fixed 2 (DW

type) can be realized [6, 13], where ( ) ( )kPk GGq /ordord= . Functions ( )PGord and ( )kGord give an order

[11] of a point group of magnetic symmetry of thecrystalparamagnetic phase [9,10] and 2 DW in this

crystal respectively. These 2 DWs have the same energy but different structures (magnetization

distribution, plane orientation, DW width, etc.). The minimum value of kq is 2 in accordance with the

invariance of energy for time reversal operationR.

At kG expansion for PG the lost transformations (members of adjacent classes) lg [6,12]

interrelate the above-mentioned kq multiply degenerate 2 DWs (i.e.l

g operation onto one of such 2

DWs converts it into another).

Degeneracy kq of 2 DW can be written in the form kBk qqq = ( kq kq ), where

( ) ( )kBk GGq /ordord= is the quantity of equal energy 2 DWs with fixed boundary conditions,

( ) ( )BPB GGq /ordord= is the quantity of possible boundary conditions of 2 DW in defined crystal of

certain symmetry. Here ( )BGord is an order of the point group of the maximum magnetic symmetry of

2 DW in given crystal at the given boundary conditions. It also has been named as a symmetry of

boundary conditions [6].

Degeneracy kq of 180 and 2 DWs in crystals of higher, medium and lower symmetry

categories (in conformity with terminology of [11]) are presented in Appendix. Signs - in certain

columns or absence of some rows (with some k) in appendix tables mean an absence of 2 DWs of such

magnetic symmetry classes in crystals of certain or all (mentioned in corresponding tables)

crystallographic classes.

The degeneracy of the DWs in the crystals of lower symmetry (the corresponding 180 DWs

magnetic symmetry classes are determined by the numbers 1 k23 as long as others magnetic

symmetry classes have the n-fold symmetry axes with n>2) have not divisor 3 as a rule in contrast with


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others crystals. All magnetic symmetry classes (1 k42) of the 180 DWs are possible in the crystals

of middle and higher symmetry. All 2 DWs magnetic symmetry classes are possible separately in the

crystals of lower, middle and higher symmetry.

The 2 DWs of magnetic symmetry class 16G (magnetic symmetry class 1) have the maximum

degeneracyk

q . 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. Here, crystallographic class means the class of paramagnetic

phase by the convention [4]. Typical DWs degeneracy is 4 for crystals of the lower symmetry category

respectively. All possible 1800

- and 2 DWs with 00

< 2


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References

[1] Hubert A. 1974 Theorie der Domanenwande in Geordneten Medielen (Theory of Domain Walls in

Ordered Media) (Berlin, Heidelberg, New York: Springer); Hubert A and Shafer R 1998 Magnetic

Domains. The Analysis of Magnetic Microstructures (Berlin: Springer)

[2] Bokov V. and Volkov V. Dynamic of the domain wall in the ferromagnetics 2008 Physics of the

Solid State. 50 198

[3] Shuvalov L. 1959 Sov. Phys. Crystallogr.4 399

[4] Shuvalov L. 1988Modern Crystallography IV : Physical Properties of Crystals. (Berlin: Springer)

[5] Baryakhtar V., Lvov V. and Yablonsky D. 1984JETP87 1863

[6] Baryakhtar V., Krotenko E. and Yablonsky D. 1986JETP91 921

[7] Lilley B. 1950 Phil.Mag. 41 792[8] Andreev A. and Marchenko V. 1976JETP70 1522

[9] Landau L., Lifshitz E. and Pitaevskii L. 1984 Course of Theoretical Physics, vol.8. Electrodynamics

of Continuous Media (London: Pergamon Press)

[10] Kopcik V. A. 1966 Xubnikovskie Gruppy: Spravoqnik po simmetrii i fiziqeskim svostvam.

kristalliqeskih struktur[Shubnikovs groups:Handbook on the symmetry and physical properties of

crystalline structures, in Russian], (Moscow: Izdatelstvo Moskovskogo Universiteta).

Shubnikov A.V., Belov N.V. Colored symmetry (London: Pergamon Press)

Tavger B. and Zaitzev V. 1956JETP3 430

[11] Vanshtein B. 1994Modern Crystallography 1: Symmetry of Crystals, Methods of Structural

Crystallography (Berlin: Springer)

[12] Wigner E. 1959 Group Theory and its Application to the Quantum Mechanics of Atomic Spectra

(New York: Academic Press)

[13] B.M. Tanygin, O.V. Tychko 2009 Phys. B: Condens. Matter404 (21) 4018-4022.

[14] Heyderman L., Niedoba H., Gupta H. and Puchalska I. 1991J. Magn. Magn. Mater96 125.

Vakhitov R., Yumaguzin A 2000J. Magn. Magn. Mater. 215-216 52


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Appendix. The degeneracy kq of the 2 and 180 DWs in crystals of different symmetry categories.

Table A1. The degeneracy kq of the 2 and 180 DWs in crystals of lower symmetry

k

Singony (system)

Triclinic Monoclinic Orthorhombic

Crystallographic classes

1 1 2 m 2/m 222 mm2 mmm

1 -

- - - - - - 2

2 - - - - - - 2 4

3 - - - - - - 2 4

4 - - - - 2 - - 4

5 - - - - 2 - - 4

6 - - - 2 4 - 4 8

7 - - - - - 2 - 4

8 - - 2 - 4 4 4 8

9 - - - - - - 2 4

10 - - 2 - 4 4 4 8

11 - - - 2 4 - 4 812 - - - 2 4 - 4 8

13 - - 2 - 4 4 4 8

14 - - - - 2 - - 4

15 - 2 - - 4 - 4 8

16 2 4 4 4 8 8 8 16

17 - - - - - - 2 4

18 - - - 2 4 - 4 8

19 - - 2 - 4 4 4 8

20 - - - - 2 - - 4

21 - - - - - 2 - 4

22 - - - - - - 2 4

23 - - - - - - - 2


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Table A2. The degeneracy kq of the 2 and 180 DWs in crystals of medium symmetry

k

Singony (system)

Tetragonal Trigonal


4 4 422 4mm 4/m 4 2m 4/mmm 3 3 32 3m 3 m

1 - - - - - - 4 - - - - -2 - - - 4 - 4 8 - - - - -

3 - - - 4 - 4 8 - - - - -

4 - - - - 4 - 8 - - - - 6

5 - - - - 4 - 8 - - - - 6

6 - - - 8 8 8 16 - - - 6 12

7 - - 4 - - 4 8 - - - - -

8 4 4 8 8 8 8 16 - - 6 - 12

9 - - - 4 - 4 8 - - - - -

10 4 4 8 8 8 8 16 - - 6 - 12

11 - - - 8 8 8 16 - - - 6 12

12 - - - 8 8 8 16 - - - 6 12

13 4 4 8 8 8 8 16 - - 6 - 1214 - - - - 4 - 8 - - - - 6

15 - - - - 8 - 16 - 6 6 - 12

16 8 8 16 16 16 16 32 6 12 12 12 24

17 - - - 4 - 4 8 - - - - -

18 - - - 8 8 8 16 - - - 6 12

19 4 4 8 8 8 8 16 - - 6 - 12

20 - - - - 4 - 8 - - - - 6

21 - - 4 - - 4 8 - - - - -

22 - - - - - 4 8 - - - - -

23 - - - - - - 4 - - - - -

24 - - - - - - - 2 4 4 4 826 - - - - - - - - - - 2 4

27 - - - - - - - - - 2 - 4

29 - - - - - - - - - - - 2

30 2 - 4 4 4 - 8 - - - - -

31 - - - - 2 - 4 - - - - -

32 - - - 2 - - 4 - - - - -

33 - - 2 - - - 4 - - - - -

34 - - - - - - 2 - - - - -

35 - 2 - - 4 4 8 - - - - -

36 - - - - - 2 4 - - - - -

42 - - - - - - - - 2 - - 4


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Table A3 (continue). The degeneracy kq of the 2 and 180 DWs in crystals of medium symmetry

k

Singony (system)

Hexagonal


6 6 622 6mm 6/m 6 m2 6/mmm

1 - - - - - - 62 - - - 6 - - 12

3 - - - 6 - - 12

4 - - - - 6 - 12

5 - - - - 6 - 12

6 - 6 - 12 12 12 24

7 - - 6 - - - 12

8 6 - 12 12 12 12 24

9 - - - 6 - - 12

10 6 - 12 12 12 12 24

11 - 6 - 12 12 12 24

12 - 6 - 12 12 12 24

13 6 - 12 12 12 12 2414 - - - - 6 - 12

15 - - - - 12 - 24

16 12 12 24 24 24 24 48

17 - - - 6 - - 12

18 - 6 - 12 12 12 24

19 6 - 12 12 12 12 24

20 - - - - 6 - 12

21 - - 6 - - - 12

22 - - - 6 - - 12

23 - - - - - - 6

24 4 4 8 8 8 8 1625 - 2 - - 4 4 8

26 - - - 4 - 4 8

27 - - 4 - - 4 8

28 - - - - - 2 4

29 - - - - - - 4

37 2 - 4 4 4 4 8

38 - - - - 2 - 4

39 - - - 2 - - 4

40 - - 2 - - - 4

41 - - - - - - 2

42 - - - - 4 4 8


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Table A4. The degeneracy kq of the 2 and 180 DWs in crystals of higher symmetry

k

Singony (system)

Cubic

Crystallographic classes23 432 4 3m m3 m3m

1 - - - 6 12

2 - - - 12 24

3 - - - 12 24

4 - - - 12 24

5 - - - 12 24

6 - - 24 24 48

7 6 12 12 12 24

8 12 24 24 24 48

9 - - - 12 24

10 12 24 24 24 48

11 - - 24 24 4812 - - 24 24 48

13 12 24 24 24 48

14 - - - 12 24

15 - - - 24 48

16 24 48 48 48 96

17 - - - 12 24

18 - - 24 24 48

19 12 24 24 24 48

20 - - - 12 24

21 6 12 12 12 24

22 - - - 12 24

23 - - - 6 12

24 8 16 16 16 32

25 - - - - 16

26 - - 8 - 16

27 - 8 - - 16

28 - - - - 8

29 - - - 4 8

30 - 12 12 - 24

31 - - - - 12

32 - - 6 - 12

33 - 6 6 - 12

34 - - - - 635 - - 12 - 24

36 - - - - 12

37 - - - - 16

38 - - - - 8

39 - - - - 8

40 - - - - 8

41 - - - - 4

42 - - - 8 16

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The degeneration of the magnetic domain walls in ferro- and ferrimagnets with arbitrary crystallographic class