Magnetic Groups

Magnetic Groups

Daniel B. Litvin

Department of Physics

The Eberly College of Science

The Pennsylvania State University

Penn State Berks Campus

P.O. Box 7009

Reading, PA 19610-6009, U.S.A.

http://www.bk.psu.edu/faculty/litvin/

[email protected]

What are we going to do?

• What are magnetic groups?

• Some History

• Tables of Properties of Magnetic Groups

• Magnetic Groups/Black and White Groups

ATOM ARRANGEMENT

Scalar density function

ρ(r)

Symmetry Group of Atom Arrangement ρ(r)

Operations:

(G|t) = (Rotation or Rotation-inversion|translation)

(G|t) acts on ρ(r): (G|t)ρ(r) = ρ((G|t)-1r)

(G|t) is a symmetry of ρ(r) if:

(G|t)ρ(r) = ρ((G|t)-1r) = ρ(r)

i.e. ρ((G|t)r) = ρ(r)

Symmetry Group of Atom Arrangement ρ(r):

The set {(G|t)} of all symmetries of ρ(r).

{(G|t)} = space group F.

Translations

Body centered

I lattice

(1|100)

(1|010)

(1|001)

(1|½½½)

Rotations/Rotation-inversions

−

−

1

z z z

1

z zz

1,4 ,2 ,4

1,4 ,m ,4

F = I4/m

Magnetic Dipole (Spin) Arrangement

Axial Vector Function

S(r)

Symmetry Group of Spin Arrangement S(r)

Operations:


(1|0)’= time inversion

Action on S(r): (G|t)S(r) = δGGS((G|t)-1r)

δG = +1 if G = R (rotation)

δG = -1 if G = (rotation-inversion)

(1|0)’S(r) = -S(r)

(G|t)’S(r) = -δGS((G|t)-1r)

1 R

Symmetries of S(r):

(G|t): (G|t)S(r) = δGGS((G|t)-1r) = S(r) →

δGGS(r) = S((G|t)r)

(G|t)’:(G|t)’S(r) = -δGGS((G|t)-1r) = S(r) →

δGGS(r) = -S((G|t)r)

(1|0)’: (1|0)’S(r) = -S(r) = S(r) → S(r) = -S(r)

Symmetry Group of Spin Arrangement S(r): The set

M of all primed and unprimed operations (G|t) which

are symmetries of S(r).

A magnetic space group M is a group of unprimed

or primed operators (G|t) such that the set of

operators {(G|t)} constitutes a space group F.

M = F is a “trivial” magnetic space group

M = F(H) is a “non-trivial” magnetic space group,

i.e. half the operations of F, i.e. those in H, are not

primed, the other half are primed.

Translational symmetry

(1|100)

(1|010)

(1|001)

(1|½½½)’

IP lattice

Rotational Symmetry

IP4/m


• What are magnetic groups?

• Some History.

Related Groups

Notation

3 Dimensional Space Groups

Fedorov

Schoenflies

Barlow

After 3 dimensions comes 4!

History: Related Groups

Heesch (1930): 3d SG x {1,m4}

3d SG x {1,A}

A2

= 1

“A” commutes with all SG operations

3d SG x {(1)(2), (12)} (12)2

= (1)(2)

2 charges + –

2 atoms Na Cl

2 colors

Black and White

Arrangement

f(r)

a function which

takes one of two

values:

Symmetry Group of Black and White Arrangements f(r)

Operations:


(12) = black/white permutation

Symmetry Group of Black and White Arrangement

f(r): The set of all operations (G|t) and (G|t)(12)

which are symmetries of f(r).

Black and White

Arrangement

(4z|000)

(mz|000)

(E|100)

(E|010)

(E|001)

(E|½½½ )(12)

4PI /m

Shubnikov (1951):

2-color groupsblack and white groupsdichromatic groupsantisymmetry groups

Zamorzaev (1953,1957):

2-color space groupsShubnikov groups

Belov, Neronova, & Smirnova (1955, 1957):

2-color space groupsblack and white space groups1651 Shubnikov space groups

230 one-color groups1191 two-color (black and white) groups230 grey groups

1651

Opechowski & Guccione (1965)

Magnetic groups

1421

Landau and Lifschitz (1951, 1957):

{(1)(2), (12)} {1,1'}→

Numerology:

Magnetic Space

Groups

M = F 230

M = F(H) 1191

1421

Non-Magnetic F1' 230

1651

Black and White

Groups

one-color

two-color

grey

Magnetic space group notation:

(Zamorzaev, 1976)

Noncoordinated F(H) ShubnikovBNS

History: Notation

OG Notation (Opechowski & Guccione)

BNS Notation (Belov, Nerovova, & Smirnova)

We will look at only two notations:

OG symbols

I4/mI4/m1'I4'/mI4/m'I4'/m'IP4/mIP4'/mIP4'/m'

BNS symbols

I4/mI4/m1'I4'/mI4/m'I4'/m'Ic4/m

Space group: F

Magnetic space group: M

M = F OG & BNS symbols the same

I4/m

P4/nmm

P42/nnm

M = F(H) : H unprimed elementsF-H primed elements

MT : no primed translations

OG & BNS symbols the same

I4/m'

P4/n'mm

P42/nn'm'

M = F(H) : H unprimed elementsF-H primed elements

MR : half of translations are primed

OG & BNS symbols are not the same

IP4/m PI4/m

P2c4'/n'm'm Pc42ncm

PI42/nnm Ic41/amd

F(H) = I4/m(P4/m)

OG: I 4/m

BNS: P4/m

P

I

COMPARISON OF OG AND BNS

MAGNETIC GROUP TYPE SYMBOLS

Magnetic Frieze Groups 1-dimensional Magnetic Space Groups

Magnetic Rod Groups 2-dimensional Magnetic Space Groups

Magnetic Layer Groups 3-dimensional Magnetic Space Groups

(see The_Magnetic _Group_Tables_J11.pdf)

Bertaut 1975:

There are errors in OG list.

Opechowski and Litvin 1977:

No there aren't!

Problem: meaning of symbols

Space group F =

(E|T1) (E|T2) (E|T3) (E|T4)

(E|000), (R2|t2), (R3|t3), …., (Rn|tn)

coset representatives

(E|000)T + (R2|t2)T + (R3|t3 )T+….+ (Rn|tn)T

Coset representatives are not unique!!

(Ri|ti) can be replaced by (Ri|ti+T) where

T is any translation of T.

(E|000), (2x|000), (my|0½½), (mz|0½½),

(Ē|000), (mx|000), (2y|0½½), (2z|0½½)

1983

Cmma

Red Book

Volume A

(2z|½00)

(2z|0½0)

modified OG list of symbols to use with Volume A

equivalent to original OG list using Red book

Litvin 1998:

�� Bad Bad Idea!! ��

Litvin 2001

New presentation of original OG list of magnetic space group symbols

F

H

coset representatives

M = F(H)

SURVEY OF MAGNETIC GROUP TYPES




1999 2001


Hans Grimmer

Comments on tables of magnetic space groups

Acta Cryst. A65, 145-55 (2009)

Opechowski-Guccione-like symbols labelling magnetic

Space groups independent of tabulated (0,0,0)+ sets

Acta Cryst. A66, 284-291 (2010)

Survey of 3-Dimensional Magnetic Space Groups

Harold T. Stokes and Branton J. Campbell

Brigham Young University

http://stokes.byu.edu/magneticspacegroups.html

Explanation of Table

Table 1. BNS order. (magnetic_table_bns.txt)

Table 2. OG order. (magnetic_table_og.txt)

Data file (magnetic_data.txt)

Fortran code for reading data file (read_magnetic_data.f)

2010

July 20113-dimensional Magnetic Space Groups


• Some History


1966

R.P. Ozerov 1969

TABLES OF PROPERTIES OF MAGNETIC GROUPS




2005 2008


diagram: generators of translations

color (colour) coding

Symmetry operationsin Seitz notation

Primes

coordinates of atomic position

components of the symmetry allowed

magnetic dipole (spin)

at that position

SURVEY OF MAGNETIC GROUP TYPES

1-dimensional Magnetic Point Groups






TABLES OF PROPERTIES OF MAGNETIC GROUPS




COMPARISON OF OG AND BNS MAGNETIC GROUP TYPE SYMBOLS




MAXIMAL SUBGROUPS OF INDEX < 4




The_Magnetic _Group_Tables_J11.pdf


• Magnetic Groups/Black and White Groups

Are magnetic space groups and black and

white space groups the same?

The operators are denoted with the

same symbols.

The groups are denoted with the

same symbols.

The two types of groups are isomorphic.

Abstract Group: { 1, A ; A2 = 1 }

{ 1, 2z ; 2z2 = 1}

{ 1, my ; my2 = 1 }

"Magnetic groups …..describe the symmetry of

crystals having a certain property, associated

with each crystal site, that can take one of two

possible values."

Are magnetic space groups and black and

white space groups the same?

Operators act on

different types of functions: f(r) and S(r).

Operators act in different manner:

No, they are not.

http://www.bk.psu.edu/faculty/litvin

THANK YOU

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Magnetic Groups