Aperiodic crystal workshop 2013: TEM
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Transcript of Aperiodic crystal workshop 2013: TEM
Í Î ÂÛ Å Í ÈÇØÈÅ ÑÌ ÅØÀÍ Í Û Å ÁËÎ ×Í Û Å ÕÀËÜÊÎ ÃÅÍ ÈÄÛ Í ÈÊÅËß
Московский Государственный Университетим. М.В. ЛомоносоваФакультет Наук о Материалах
Исаева А.А.
Научные руководителид.х.н., проф. Б.А. Поповкинк.х.н., асс. ФНМ А.И. Баранов
КАФЕДРА НЕОРГАНИЧЕСКОЙ ХИМИИЛаборатория Направленного Неорганического Синтеза
Ë Î Ì Î Í Î ÑÎ Â -2 0 0 3
Diffraction by incommensurately modulated phases
Joke Hadermann
At the end of this lecture you should be able to:
UNDERSTAND and INDEX
the reciprocal lattice of an
incommensurately modulated material.
One atom type A
a b
Basic cell, one plane
One atom type A
010
100
a b
[001]
Basic cell EDP
One atom type A
010
100
a b
[001]
Basic cell EDP
a=3 Å b=5 Å
One atom type A
010
100
a b
[001]
Basic cell EDP
1/3Å 1/5Å
a=3 Å b=5 Å
Alternation A and B atoms
a b
double cell model
Alternation A and B atoms
a b
010
100
[001]
double cell, EDP=
Alternation A and B atoms
a b
010
100
*bm
Gg2
Reflections at
[001]
double cell, g vectors=
Alternation A and B atoms
a b
010
100
*bm
Gg2
Reflections at
[001]
double cell, g vectors=
q
*2
1bq
qmclbkahg ***
Gq0Gg
qmclbkahg ***
*** cbaq
Basic structure
reflections
All reflections
hkl0
hklm
qmGg
010
100
*2
1bq
0001
0100
1000
1001
[001]
q
double indexed with q double indexed with q
*bm
Gg2
010
100
q
*458.0. bmGg
*458.0 bq
0.458: q indicated
010
100
q
0001
0101 -
0100
1000
*458.0. bmGg
*458.0 bq
0.458 indexed with q
Projections from 3+d reciprocal space &
“simple” supercell in 3+d space
(Example in 1+1
reciprocal space)
a1*
a2*
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
*a*a1
*b*a2
*c*a3
qe*a 44
Basis vectors of the reciprocal lattice
*c*b*aq
0100
1000
0100
1000
0100
1000
0100
1000
all four with q
q=0.5b* q=0.458b*
q=0.25b* q=0.33b*
010
100
]0,3
1,
3
1[mGg
[001]
*0*3
1*
3
1cbaq
0001
0100
1000
0002
q
110, with q
010
100
q
0001
0101 -
0100
1000
0.458 indexed with q
hk0m: no conditions
{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
Superspace groups
Incommensurately modulated materials
Reële ruimte Reciproke ruimte R*
a
200
000
a*
b*
a
Reële ruimte Reciproke ruimte R*
3a
a* q=
1/3a*
a 3a
a*
q=1/3a* (3+Δ)a
q=(1/3-δ)a*
Reële ruimte Reciproke ruimte R*
Exercise
Index the following experimental
diffraction patterns and derive the
possible superspace group(s) from the
reflection conditions.
The example is incommensurately
modulated LaSrCuO4-x.
First, index the commensurate parent
structure LaSrCuO4, this will help you with
indexing the next, incommensurate one.
1/6.50 Å 1/1.85 Å
1/6.50 Å
1/2.61 Å
1/2.61 Å
(Simulated ED patterns)
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1/1.85 Å
1/1.85 Å 1/2.61 Å
1/2.61 Å
1/1.85 Å
1/6.50 Å 1/1.85 Å
1/6.50 Å
1/2.61 Å
1/2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1/1.85 Å
001
002
200
The reflection in the red box is:
1/6.50 Å
2.61 Å
2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1.85 Å
001
002
200
The reflection in the red box is:
1/1.85 Å 1/2.61 Å
1/2.61 Å
1/1.85 Å
1/6.50 Å
002 1/1.85 Å
002
1/2.61 Å
1/2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1/1.85 Å
010
020
110
002 1/1.85 Å
002
1/2.61 Å
1/2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1/1.85 Å
010
020
110
002 020
002
2.61 Å
2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1.85 Å
010
020
110
002 020
002
2.61 Å
2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1.85 Å
002 020
002
2.61 Å
2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1/1.85 Å
020
1/1.85 Å
002 020
002
2.61 Å
2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1/1.85 Å
100
200
020
020
002 020
002
2.61 Å
2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
1.85 Å
100
200
020
020
002 020
002
2.61 Å
2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
200
110
220
½ ½ 0
020
002 020
002
2.61 Å
2.61 Å
Given data:
cell parameters of LaSrCuO4: a=b=3.7 Å, c=13 Å.
200
110
220
½ ½ 0
020
002 200
002
110 200
110
020
002 200
002
110 200
110
020
Determine the reflection condition for h0l.
002 200
002
110 200
110
h0l: h+l=2n
h = 2n
l = 2n
002 200
002
110 200
110
h0l: h+l=2n
Determine the reflection condition for h0l.
002 200
002
110 200 110
hhl: h+l=2n
h = 2n
l = 2n
h0l: h+l=2n
Determine the reflection condition for hhl.
002 200
002
110 200
110
hhl:
l = 2n
h0l: h+l=2n
Determine the reflection condition for hhl.
002 200
002
110 200
110
hhl: l = 2n h0l: h+l=2n hk0: h+k=2n
h = 2n
k = 2n
Determine the reflection condition for hk0.
002 200
002
110 200
110
hhl: l = 2n h0l: h+l=2n hk0: h+k=2n
Determine the reflection condition for hk0.
Solved reflection conditions
002 200
002
110
h0l:h+l=2n hhl:l=2n hk0:h+k=2n
Also (from rest of the zones) hkl: h+k+l=2n.
110 -
200
110
Determine space group
• International Tables: I--- • Most symmetrical I4/mmm
Incommensurate vs. basic cell
Hadermann et al., Journal of Materials Chemistry, 17, 22, 2007, 2344-2350
LaSrCuO3.52
Identify and index the subcell reflections.
Identify and index the subcell reflections.
Propose a modulation vector.
q=αa*
α<0.5
q=αa*
α>0.5
200
002
[010]
q=αa*
α<0.5
Index the satellite indicated in green.
- 001
0001
100
0001
Index the satellite indicated in green.
Index the next satellite indicated in green.
0002
2002 -
2001 -
-
2002 -
Which of the indicated vectors corresponds to the
modulation vector chosen on the previous slides?
Which of the indicated vectors corresponds to the
modulation vector chosen on the previous slides?
Is the proposed
vector still valid?
yes
no
Is the proposed
vector still valid?
no
You need
q1
q2
q1 ànd q2
q1=αa*+βb*
q2= -αa*+βb*
α=β<0.25
You need
q1 ànd q2
q1=αa*+βb*
q2= -αa*+βb*
α=β<0.25
20011
20011 -
20011 -
Index the reflection indicated in red.
20011 -
Index the reflection indicated in red.
Index the others patterns consistently with this new choice.
002 [010] 200
002
[010]
200
[110] [001]
020 200
110
002 -
Index the reflection indicated in yellow.
20011
20011
20011 -
-
00010
00001
- 20011
Index the reflection indicated in yellow.
20011 -
00010
00001
- 20011
Index the reflection indicated in yellow.
00010
00001
- 20011
10110
10111
10111 -
Index the reflection indicated in yellow.
00010
00001
- 20011
10111 -
Index the reflection indicated in yellow.
00010
00001
- 20011
00002
00002
00001 -
-
Index the reflection indicated in yellow.
00010
00001
- 20011
00002 -
00010
00001
- 20011
Derive the reflection conditions for hklmn.
hklmn:
h+k+l+m+n=2i
h+k+l=2i
m+n=2i
hhl:l=2n
hkl:h+k+l=2n
00010
00001
- 20011
hklmn:
h+k+l=2i
hhl:l=2n
hkl:h+k+l=2n
Derive the reflection conditions for hklmn.
00010
00001
- 20011
Derive the reflection conditions for hhlm0.
hhlm0: l=2i
h=2i
l,m=2i
hhl:l=2n
hkl:h+k+l=2n
00010
00001
- 20011
Derive the reflection conditions for hklm.
hhl:l=2n
hkl:h+k+l=2n
hhlm0:
l,m=2i
but
hhlm0:m=2i
is sufficient
00010
00001
- 20011
Derive the reflection conditions for hklm.
hklmn:h+k+l=2i
hhlm0:m=2i
(-hhl0n: n=2i)
hhl:l=2n
hkl:h+k+l=2n
00010
00001
- 20011
Determine the superspace group.
I4/mmm(α α0, -αα0)00mg
http://stokes.byu.edu/iso/ssg.php
Stokes et al., Acta Cryst. A67, 45-55 (2011)
You know cell parameters,
modulation vector and
superspace group.
At the end of this lecture you should be able to:
UNDERSTAND and INDEX
the reciprocal lattice of an
incommensurately modulated material.
Recap? http://www.slideshare.net/johader