modulated materials with electron diffraction
DESCRIPTION
This lecture was given at the International School of Crystallography in Erice 2011, on the topic of Electron Crystallography. It explains the very basics of how to index commensurately and incommensurately modulated materials. It was meant for a 40 minute lecture.TRANSCRIPT
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