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Electron diffraction of commensurately and incommensurately modulated
materials
Joke Hadermann
www.slideshare.net/johader/
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Modulation
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•commensurate•incommensurat
eModulation
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One atom type A
ab
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One atom type A
010
100
ab
[001]
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One atom type A
Alkhi
AI fefF )000(2
010
100
ab
[001]
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Alternation A and B atoms
ab
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Alternation A and B atoms
ab 010
100
[001]
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Alternation A and B atoms
ab 010
100
*bmGg 2Reflections at
[001]
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010
100
[001]
Extra reflections
Supercell Modulation vector*
2bmGg
*21'* bb *
21 bq
qmclbkahg ***
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010
100
[001]
Extra reflections
Supercell Modulation vector*
2bmGg
*21'* bb *
21 bq
qmclbkahg ***
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010
100
[001]
Extra reflections
Supercell Modulation vector*
2bmGg
*21'* bb *
21 bq
qmclbkahg ***
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ab
010
100
[001]
a’
Extra reflections
Supercell Modulation vector*
2bmGg
*21'* bb *
21 bq
qmclbkahg ***
010
100
b’
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ab
010
100
[001]
Extra reflections
Supercell Modulation vector*
2bmGg
*21'* bb *
21 bq
qmclbkahg ***
q
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[001]
100
010b’a’
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ikBAII effF
)0210(2)000(2 lkhi
Blkhi
AII efefF
[001]
100
010b’a’
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ikBAII effF
BAII ffF BAII ffF
If k=2n If k=2n+1
)0210(2)000(2 lkhi
Blkhi
AII efefF
[001]
100
010b’a’
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*bnmGg
*1'* bn
b nbb '
Extra ref.:
If the periodicity of the modulation in direct space is
nb:
Can use supercell:
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010
*2
bmGg Extra reflections
*21'* bb
010
100
bb 2'
[001]
b’a’
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010
100
a’b’
*3
bmGg
*31'* bb bb 3'
Extra ref.:
010
[001]
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010
100
a’b’
*4
bmGg
*41'* bb bb 4'
010
[001]
Extra ref.:
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Modulation nót along main axis of basic structure
ab a
b
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ab a
b
(110)
Modulation nót along main axis of basic structure
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ab
(110)
Modulation nót along main axis of basic structure
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ab
(110)
010
100 110
],,[mGg 03131
[001]
Modulation nót along main axis of basic structure
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010
100 110
1/3 1/3 02/3 2/3 0
[001]
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010
100 110
030
300
1 1 02 2 0
330
[001]
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010
100 110
120-
100
010
[001]
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010
100 110
120-
100
010
[001]
200
300210-
110
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b*b’*[001]
a’*a*
100011012
P
***
*'*'*'
cba
cba
P
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b*b’*[001]
a’*a*
100011012
P
***
*'*'*'
cba
cba
P
Pcbacba '''
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baa 2'bab '
cc '
ab
a’
b’
100011012
P
Pcbacba '''
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baa 2'bab '
cc '
ab
a’
b’
100011012
P
Pcbacba '''
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,,=p/n Càn take supercelle.g. n x basic cell parameter
],,[mGg
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,,=p/n Càn take supercelle.g. n x basic cell parameter
0.458=229/500 !
Approximations: 5/9=0.444, 4/11=0.455, 6/13=0.462,…Different cells, space groups, inadequate for refinements,…
],,[mGg
*b.mGg 4580
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The q-vector approach
qclbkahG 0***
qmclbkahg ***
*** cbaq
Basic structure reflections
Allreflections
hkl0
hklm
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010
*2
bmGg
100
ab
[001]
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010
*2
bmGg
100
ab
qmclbkahg ***
*** cbaq
*21 bq
[001]
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010
100
*21 bq
0001
0100
1000
1001
[001]
q
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010
100
q
*458.0. bmGg
*458.0 bq
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010
100
q
0001
0101-
0100
1000
*458.0. bmGg
*458.0 bq
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0100
1000
0100
1000
0100
1000
0100
1000
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010
100
]0,31,
31[mGg
[001]
*0*31*
31 cbaq
0001
0100
1000
0002
q
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Advantages of the q-vector method:
- subcell remains the same
- also applicable to incommensurate modulations
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Incommensurately modulated materials
Loss of translation symmetry
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LaCaCuGa(O,F)5: amount F varies sinusoidally
Example of a compositional modulation
Hadermann et al., Int.J.In.Mat.2, 2000, 493
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Example of a displacive modulation
Bi-2201
Picture from Hadermann et al., JSSC 156, 2001, 445
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Projections from 3+d reciprocal space & “simple” supercell in 3+d space
(Example in 1+1 reciprocal space)
q
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Projections from 3+d reciprocal space & “simple” supercell in 3+d space
(Example in 1+1 reciprocal space)
a1*
a2*
q
e2
a2*=e2+q
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Projections from 3+d reciprocal space & “simple” supercell in 3+d space
(Example in 1+1 reciprocal space)
a1*
a2*
q
e2
a2*=e2+q
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Basis vectors of the reciprocal lattice
*a*a1
*b*a2
*c*a3
qe*a 44
*c*b*aq
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Example: q= γc*(Displacive modulation along c)c
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0 1
u
x 4
z
c
t
c
1
e4=a4
Example: q= γc*(Displacive modulation along c)c
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0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γ c
1
e4=a4
a3 = c - γe4
a3
Example: q= γc*(Displacive modulation along c)c
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0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γ c
1
e4=a4
a3 = c - γe4
a3
Example: q= γc*(Displacive modulation along c)c
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0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γ c
1
e4=a4
a3 = c - γe4
a3
Example: q= γc*(Displacive modulation along c)
0
c
1
c
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0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γ c
1
e4=a4
a3 = c - γe4
a3
Example: q= γc*(Displacive modulation along c)
0
c
cModulation function u
z = z0 + u(x4)
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0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γ c
1
e4=a4
a3 = c - γe4
a3
Example: q= γc*(Displacive modulation along c)
0
c
cModulation function u
z = z0 + u(x4)
In 3+1D: again unit cell, translation symmetry
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Basis vectors
*a*a1
*b*a2
*c*a3
qe*a 44
Basis vectors in reciprocal space
Basis vectors in direct space
41 eaa
42 eba
43 eca
*c*b*aq 44 ea
jiji *aa 44332211 axaxaxaxx
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{R|v} is an element of the space group of the basic structure is a phase shift and is ±1
Space group of the basic structure
components of q
symmetry-operators for the phase
Superspace groups: position and phase
(r,t) ( Rr + v, t + )
ExamplePnma(01/2)s00
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Separate the basic reflections (m=0) from the satellites (m≠0)
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Separate the basic reflections (m=0) from the satellites (m≠0)
-should form a regular 3D lattice
-highest symmetry with lower volume
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Hint from changes vs. composition, temperature,…
Separate the basic reflections (m=0) from the satellites (m≠0)
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Select the modulation vector
Possibly multiple solutions
ri qqq
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** baq hklm: h+k=2n, k+l=2n, h+l=2n
Fmmm(10)
*aq HKLm: H+K+m=2n, K+L+m=2n,
L+H=2nXmmm(00)
0200
20002200
0200
20002200
q q0001
0002 00020101
2002-0003
2403-
2400
x
0103
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Conditions for the basic cell and modulation vector
)0(')0(: mGmGR
)m('g)m(g:R 00
(qr,qi) in correspondence with chosen crystal system & centering basic cell
** baq
0200
20002200
q0001
0002
00032403-
2400
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Possible irrational components in the different crystal systems
Crystal system
qi Crystal system qi
Triclinic () Tetragonal Trigonal Hexagonal
(00) Monoclinic (-setting)
()
(0)
Orthorhombic (00)
(00)
(00)
Cubic none
Example of derivation: see lecture notes.
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Compatibility of rational components with centering types
Crystal system q Crystal system q Triclinic no rational
component Orthorhombic-P Orthorhombic-C Orthorhombic-A Orthorhombic-F
(1/2)
(1/2)
(10)
(1/2)
(10)
Monoclinic-P
Monoclinic-B (-setting)
()
(1/20)
(0, 1/2, )
Tetragonal-P Trigonal-P
(1/21/2)
(1/31/3)
Example of derivation: see lecture notes.
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Bulk Powder Diffraction• Difficulties in
determining periodicity
• Difficulties in determining symmetry
• Difficulty in detecting weak satellites due to modulations in light atoms
• Relative intensities reliable for refinements
Electron Diffraction• Clear determination
periodicity
• Clear determination symmetry
• Picks up also weak satellites due to modulations in the light atoms
• Relative intensities not as reliable for refinements
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Summary
Commensurate modulations:supercellq-vector
Incommensurate modulations(Commensurate approximation)q-vector
q-vector -> (3+1)D Superspace