Hand calculations of magnetic modes

A. Daoud-Aladine (ISIS)

Physical problem

Site A

Ideal cubic Perovskite

Pnma

[010]O

[001]C

[101]O

[010]C

La/Tb

n

n

aa

m

m

Mn O

LaMnO3 TN=150K k=(0 0 0)

TbMnO3 TN=41K k=(0.28 0 0)

(b)

(a)

Pnma

[101]O

[010]C m

m

LaMnO3 TN=150K k=(0 0 0)

TbMnO3 TN=41K k=(0.28 0 0)

Physical problem

(b)

(a)

a

b

c

1z

2y

3x

Linear algebra problem

Sk1

Sk2

Sk3

Sk4

Pnma

a

b

c

1z

2y

3x

Linear algebra problem

Sk1

Sk2

Sk3

Sk4

Hermitian vector space

(complex)of dimension 12

Pnma

The group G=Pnma

Zero blocks elements

The group of k: Gk

G = Pnmaa (x)

b (y)

c (z)

k

mxy mxz

(a) Gk = Pnma

(b) Gk=

P21ma

2x

Irreps of Gk

(a)

(b)

Representation of gGk

c

b

a

Sk1

Sk2

Sk3

Sk4

(a and b) Trivial Ex1: Identity operation 1

Representation of gGk

(a and b) Trivial Ex1: Identity operation 1

Representation of gGk

(a only) Trivial Ex2: Inversion -1

c

b

a

Sk1

Sk2

Sk3

Sk4

Representation of gGk

(a only) Trivial Ex2: Inversion -1

Representation of gGk

c

b

a

Sk1

Sk2

Sk3

Sk4

Point

Sk . SkSpin

Mn1: 0,0,1/2 1/2, 1/2, 0 = Mn3

(a and b) Ex1: 21 axis along x

Representation of gGk

1z

2y

3x 3x

ap(21,3)

1

1z

1y1x

1’c

b

a Sk1’

Sk1

(a and b) Ex1: 21 axis along x

Representation of gGk(a and b) Ex1: 21 axis along x

Representation of gGk(a only) Ex2: Glide plane n axis x

1z

2y

3x

Representation of gGk(a only) Ex2: Glide plane n axis x

1z

2y

3x

Representation of gGk(a only) Ex2: Glide plane n axis x

Decomposition

Decomposition

= 12

Decomposition

= 0 etc…

Decomposition

Projection

Projection

Projection

Projection

Magnetic structure

(b) TbMnO3 TN=41K k=(0.28 0 0)

Magnetic structure

Magnetic structure(b)

Example (a)

Magnetic structure

Magnetic structure