Download - Hand calculations of magnetic modes
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Hand calculations of magnetic modes
A. Daoud-Aladine (ISIS)
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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)
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(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
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(b)
(a)
a
b
c
1z
2y
3x
Linear algebra problem
Sk1
Sk2
Sk3
Sk4
Pnma
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a
b
c
1z
2y
3x
Linear algebra problem
Sk1
Sk2
Sk3
Sk4
Hermitian vector space
(complex)of dimension 12
Pnma
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The group G=Pnma
Zero blocks elements
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The group of k: Gk
G = Pnmaa (x)
b (y)
c (z)
k
mxy mxz
(a) Gk = Pnma
(b) Gk=
P21ma
2x
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Irreps of Gk
(a)
(b)
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Representation of gGk
c
b
a
Sk1
Sk2
Sk3
Sk4
(a and b) Trivial Ex1: Identity operation 1
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Representation of gGk
(a and b) Trivial Ex1: Identity operation 1
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Representation of gGk
(a only) Trivial Ex2: Inversion -1
c
b
a
Sk1
Sk2
Sk3
Sk4
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Representation of gGk
(a only) Trivial Ex2: Inversion -1
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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
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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
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Representation of gGk(a and b) Ex1: 21 axis along x
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Representation of gGk(a only) Ex2: Glide plane n axis x
1z
2y
3x
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Representation of gGk(a only) Ex2: Glide plane n axis x
1z
2y
3x
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Representation of gGk(a only) Ex2: Glide plane n axis x
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Decomposition
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Decomposition
= 12
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Decomposition
= 0 etc…
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Decomposition
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Projection
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Projection
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Projection
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Projection
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Magnetic structure
(b) TbMnO3 TN=41K k=(0.28 0 0)
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Magnetic structure
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Magnetic structure(b)
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Example (a)
Magnetic structure
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Magnetic structure