modulated materials with electron diffraction
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Electron diffraction of commensurately and incommensurately modulated
materials
Joke Hadermann
www.slideshare.net/johader/
Modulation
•commensurate
•incommensurate
Modulation
One atom type A
ab
One atom type A
010
100
ab
[001]
One atom type A
Alkhi
AI fefF )000(2
010
100
ab
[001]
Alternation A and B atoms
ab
Alternation A and B atoms
ab
010
100
[001]
Alternation A and B atoms
ab
010
100
*bm
Gg 2Reflections at
[001]
010
100
[001]
Extra reflections
SupercellModulation
vector*
2b
mGg
*2
1'* bb *
2
1bq
qmclbkahg ***
010
100
[001]
Extra reflections
SupercellModulation
vector*
2b
mGg
*2
1'* bb *
2
1bq
qmclbkahg ***
010
100
[001]
Extra reflections
SupercellModulation
vector*
2b
mGg
*2
1'* bb *
2
1bq
qmclbkahg ***
ab
010
100
[001]
a’
Extra reflections
SupercellModulation
vector*
2b
mGg
*2
1'* bb *
2
1bq
qmclbkahg ***
010
100
b’
ab
010
100
[001]
Extra reflections
SupercellModulation
vector*
2b
mGg
*2
1'* bb *
2
1bq
qmclbkahg ***
q
[001]
100
010b’a’
ikBAII effF
)02
10(2)000(2 lkhi
Blkhi
AII efefF
[001]
100
010b’a’
ikBAII effF
BAII ffF BAII ffF
If k=2n If k=2n+1
)02
10(2)000(2 lkhi
Blkhi
AII efefF
[001]
100
010b’a’
*bn
mGg
*1
'* bn
b nbb '
Extra ref.:
If the periodicity of the modulation in direct space is
nb:
Can use supercell:
010
*2
bm
Gg Extra reflections
*2
1'* bb
010
100
bb 2'
[001]
b’a’
010
100
a’b’
*3
bm
Gg
*3
1'* bb bb 3'
Extra ref.:
010
[001]
010
100
a’b’
*4
bm
Gg
*4
1'* bb bb 4'
010
[001]
Extra ref.:
Modulation nót along main axis of basic structure
ab a
b
ab a
b
(110)
Modulation nót along main axis of basic structure
a
b
(110)
Modulation nót along main axis of basic structure
a
b
(110)
010
100 110
],,[mGg 03131
[001]
Modulation nót along main axis of basic structure
010
100 110
1/3 1/3 0
2/3 2/3 0
[001]
010
100 110
030
300
1 1 0
2 2 0
330
[001]
010
100 110
120-
100
010
[001]
010
100 110
120-
100
010
[001]
200
300210-
110
b*b’*
[001]
a’*
a*
100
011
012
P
*
*
*
*'
*'
*'
c
b
a
c
b
a
P
b*b’*
[001]
a’*
a*
100
011
012
P
*
*
*
*'
*'
*'
c
b
a
c
b
a
P
Pcbacba '''
baa 2'
bab 'cc '
ab
a’
b’
100
011
012
P
Pcbacba '''
baa 2'
bab 'cc '
ab
a’
b’
100
011
012
P
Pcbacba '''
,,=p/n Càn take supercelle.g. n x basic cell parameter
],,[mGg
,,=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
The q-vector approach
qclbkahG 0***
qmclbkahg ***
*** cbaq
Basic structure reflections
Allreflections
hkl0
hklm
010
*2
bm
Gg
100
ab
[001]
010
*2
bm
Gg
100
ab
qmclbkahg ***
*** cbaq
*2
1bq
[001]
010
100
*2
1bq
0001
0100
1000
1001
[001]
q
010
100
q
*458.0. bmGg
*458.0 bq
010
100
q
0001
0101-
0100
1000
*458.0. bmGg
*458.0 bq
0100
1000
0100
1000
0100
1000
0100
1000
010
100
]0,3
1,
3
1[mGg
[001]
*0*3
1*
3
1cbaq
0001
0100
1000
0002
q
Advantages of the q-vector method:
- subcell remains the same
- also applicable to incommensurate modulations
Incommensurately modulated materials
Loss of translation symmetry
LaCaCuGa(O,F)5: amount F varies sinusoidally
Example of a compositional modulation
Hadermann et al., Int.J.In.Mat.2, 2000, 493
Example of a displacive modulation
Bi-2201
Picture from Hadermann et al., JSSC 156, 2001, 445
Projections from 3+d reciprocal space & “simple” supercell in 3+d space
(Example in 1+1 reciprocal space)
q
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
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
Basis vectors of the reciprocal lattice
*a*a1
*b*a2
*c*a3
qe*a 44
*c*b*aq
Example: q= γc*(Displacive modulation along c)c
0 1
u
x 4
z
c
t
c
1
e4=a4
Example: q= γc*(Displacive modulation along c)c
0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γc
1
e4=a4
a3 = c - γe4
a3
Example: q= γc*(Displacive modulation along c)c
0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γc
1
e4=a4
a3 = c - γe4
a3
Example: q= γc*(Displacive modulation along c)c
0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γc
1
e4=a4
a3 = c - γe4
a3
Example: q= γc*(Displacive modulation along c)
0
c
1
c
0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γc
1
e4=a4
a3 = c - γe4
a3
Example: q= γc*(Displacive modulation along c)
0
c
cModulation function u
z = z0 + u(x4)
0 1
u
x 4
x 3x 3
= 0
z
c
a 3
t
γc
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
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
{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 + )
Example
Pnma(01/2)s00
Separate the basic reflections (m=0) from the satellites (m≠0)
Separate the basic reflections (m=0) from the satellites (m≠0)
-should form a regular 3D lattice
-highest symmetry with lower volume
Hint from changes vs. composition, temperature,…
Separate the basic reflections (m=0) from the satellites (m≠0)
Select the modulation vector
Possibly multiple solutions
ri qqq
** 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 0002
0101
2002-0003
2403-
2400
x
0103
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
0003
2403-
2400
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.
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.
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
Summary
Commensurate modulations:supercellq-vector
Incommensurate modulations(Commensurate approximation)q-vector
q-vector -> (3+1)D Superspace
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