modulated materials with electron diffraction

70
Electron diffraction of commensurately and incommensurately modulated materials Joke Hadermann www.slideshare.net/johader/

Upload: joke-hadermann

Post on 20-Jan-2015

526 views

Category:

Education


0 download

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

Page 1: Modulated materials with electron diffraction

Electron diffraction of commensurately and incommensurately modulated

materials

Joke Hadermann

www.slideshare.net/johader/

Page 2: Modulated materials with electron diffraction

Modulation

Page 3: Modulated materials with electron diffraction

•commensurate

•incommensurate

Modulation

Page 4: Modulated materials with electron diffraction

One atom type A

ab

Page 5: Modulated materials with electron diffraction

One atom type A

010

100

ab

[001]

Page 6: Modulated materials with electron diffraction

One atom type A

Alkhi

AI fefF )000(2

010

100

ab

[001]

Page 7: Modulated materials with electron diffraction

Alternation A and B atoms

ab

Page 8: Modulated materials with electron diffraction

Alternation A and B atoms

ab

010

100

[001]

Page 9: Modulated materials with electron diffraction

Alternation A and B atoms

ab

010

100

*bm

Gg 2Reflections at

[001]

Page 10: Modulated materials with electron diffraction

010

100

[001]

Extra reflections

SupercellModulation

vector*

2b

mGg

*2

1'* bb *

2

1bq

qmclbkahg ***

Page 11: Modulated materials with electron diffraction

010

100

[001]

Extra reflections

SupercellModulation

vector*

2b

mGg

*2

1'* bb *

2

1bq

qmclbkahg ***

Page 12: Modulated materials with electron diffraction

010

100

[001]

Extra reflections

SupercellModulation

vector*

2b

mGg

*2

1'* bb *

2

1bq

qmclbkahg ***

Page 13: Modulated materials with electron diffraction

ab

010

100

[001]

a’

Extra reflections

SupercellModulation

vector*

2b

mGg

*2

1'* bb *

2

1bq

qmclbkahg ***

010

100

b’

Page 14: Modulated materials with electron diffraction

ab

010

100

[001]

Extra reflections

SupercellModulation

vector*

2b

mGg

*2

1'* bb *

2

1bq

qmclbkahg ***

q

Page 15: Modulated materials with electron diffraction

[001]

100

010b’a’

Page 16: Modulated materials with electron diffraction

ikBAII effF

)02

10(2)000(2 lkhi

Blkhi

AII efefF

[001]

100

010b’a’

Page 17: Modulated materials with electron diffraction

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’

Page 18: Modulated materials with electron diffraction

*bn

mGg

*1

'* bn

b nbb '

Extra ref.:

If the periodicity of the modulation in direct space is

nb:

Can use supercell:

Page 19: Modulated materials with electron diffraction

010

*2

bm

Gg Extra reflections

*2

1'* bb

010

100

bb 2'

[001]

b’a’

Page 20: Modulated materials with electron diffraction

010

100

a’b’

*3

bm

Gg

*3

1'* bb bb 3'

Extra ref.:

010

[001]

Page 21: Modulated materials with electron diffraction

010

100

a’b’

*4

bm

Gg

*4

1'* bb bb 4'

010

[001]

Extra ref.:

Page 22: Modulated materials with electron diffraction

Modulation nót along main axis of basic structure

ab a

b

Page 23: Modulated materials with electron diffraction

ab a

b

(110)

Modulation nót along main axis of basic structure

Page 24: Modulated materials with electron diffraction

a

b

(110)

Modulation nót along main axis of basic structure

Page 25: Modulated materials with electron diffraction

a

b

(110)

010

100 110

],,[mGg 03131

[001]

Modulation nót along main axis of basic structure

Page 26: Modulated materials with electron diffraction

010

100 110

1/3 1/3 0

2/3 2/3 0

[001]

Page 27: Modulated materials with electron diffraction

010

100 110

030

300

1 1 0

2 2 0

330

[001]

Page 28: Modulated materials with electron diffraction

010

100 110

120-

100

010

[001]

Page 29: Modulated materials with electron diffraction

010

100 110

120-

100

010

[001]

200

300210-

110

Page 30: Modulated materials with electron diffraction

b*b’*

[001]

a’*

a*

100

011

012

P

*

*

*

*'

*'

*'

c

b

a

c

b

a

P

Page 31: Modulated materials with electron diffraction

b*b’*

[001]

a’*

a*

100

011

012

P

*

*

*

*'

*'

*'

c

b

a

c

b

a

P

Pcbacba '''

Page 32: Modulated materials with electron diffraction

baa 2'

bab 'cc '

ab

a’

b’

100

011

012

P

Pcbacba '''

Page 33: Modulated materials with electron diffraction

baa 2'

bab 'cc '

ab

a’

b’

100

011

012

P

Pcbacba '''

Page 34: Modulated materials with electron diffraction

,,=p/n Càn take supercelle.g. n x basic cell parameter

],,[mGg

Page 35: Modulated materials with electron diffraction

,,=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

Page 36: Modulated materials with electron diffraction

The q-vector approach

qclbkahG 0***

qmclbkahg ***

*** cbaq

Basic structure reflections

Allreflections

hkl0

hklm

Page 37: Modulated materials with electron diffraction

010

*2

bm

Gg

100

ab

[001]

Page 38: Modulated materials with electron diffraction

010

*2

bm

Gg

100

ab

qmclbkahg ***

*** cbaq

*2

1bq

[001]

Page 39: Modulated materials with electron diffraction

010

100

*2

1bq

0001

0100

1000

1001

[001]

q

Page 40: Modulated materials with electron diffraction

010

100

q

*458.0. bmGg

*458.0 bq

Page 41: Modulated materials with electron diffraction

010

100

q

0001

0101-

0100

1000

*458.0. bmGg

*458.0 bq

Page 42: Modulated materials with electron diffraction

0100

1000

0100

1000

0100

1000

0100

1000

Page 43: Modulated materials with electron diffraction

010

100

]0,3

1,

3

1[mGg

[001]

*0*3

1*

3

1cbaq

0001

0100

1000

0002

q

Page 44: Modulated materials with electron diffraction

Advantages of the q-vector method:

- subcell remains the same

- also applicable to incommensurate modulations

Page 45: Modulated materials with electron diffraction

Incommensurately modulated materials

Loss of translation symmetry

Page 46: Modulated materials with electron diffraction

LaCaCuGa(O,F)5: amount F varies sinusoidally

Example of a compositional modulation

Hadermann et al., Int.J.In.Mat.2, 2000, 493

Page 47: Modulated materials with electron diffraction

Example of a displacive modulation

Bi-2201

Picture from Hadermann et al., JSSC 156, 2001, 445

Page 48: Modulated materials with electron diffraction

Projections from 3+d reciprocal space & “simple” supercell in 3+d space

(Example in 1+1 reciprocal space)

q

Page 49: Modulated materials with electron diffraction

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

Page 50: Modulated materials with electron diffraction

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

Page 51: Modulated materials with electron diffraction

Basis vectors of the reciprocal lattice

*a*a1

*b*a2

*c*a3

qe*a 44

*c*b*aq

Page 52: Modulated materials with electron diffraction

Example: q= γc*(Displacive modulation along c)c

Page 53: Modulated materials with electron diffraction

0 1

u

x 4

z

c

t

c

1

e4=a4

Example: q= γc*(Displacive modulation along c)c

Page 54: Modulated materials with electron diffraction

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

Page 55: Modulated materials with electron diffraction

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

Page 56: Modulated materials with electron diffraction

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

Page 57: Modulated materials with electron diffraction

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)

Page 58: Modulated materials with electron diffraction

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

Page 59: Modulated materials with electron diffraction

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

Page 60: Modulated materials with electron diffraction

{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

Page 61: Modulated materials with electron diffraction

Separate the basic reflections (m=0) from the satellites (m≠0)

Page 62: Modulated materials with electron diffraction

Separate the basic reflections (m=0) from the satellites (m≠0)

-should form a regular 3D lattice

-highest symmetry with lower volume

Page 63: Modulated materials with electron diffraction

Hint from changes vs. composition, temperature,…

Separate the basic reflections (m=0) from the satellites (m≠0)

Page 64: Modulated materials with electron diffraction

Select the modulation vector

Possibly multiple solutions

ri qqq

Page 65: Modulated materials with electron diffraction

** 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

Page 66: Modulated materials with electron diffraction

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

Page 67: Modulated materials with electron diffraction

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.

Page 68: Modulated materials with electron diffraction

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.

Page 69: Modulated materials with electron diffraction

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

Page 70: Modulated materials with electron diffraction

Summary

Commensurate modulations:supercellq-vector

Incommensurate modulations(Commensurate approximation)q-vector

q-vector -> (3+1)D Superspace