Electron diffraction of commensurately and incommensurately modulated

materials

Joke Hadermann

www.slideshare.net/johader/

Modulation

•commensurate•incommensurat

eModulation

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

*bmGg 2Reflections at

[001]

010

100

[001]

Extra reflections

Supercell Modulation vector*

2bmGg

*21'* bb *

21 bq

qmclbkahg ***

010

100

[001]

Extra reflections

Supercell Modulation vector*

2bmGg

*21'* bb *

21 bq

qmclbkahg ***

010

100

[001]

Extra reflections

Supercell Modulation vector*

2bmGg

*21'* bb *

21 bq

qmclbkahg ***

ab

010

100

[001]

a’

Extra reflections

Supercell Modulation vector*

2bmGg

*21'* bb *

21 bq

qmclbkahg ***

010

100

b’

ab

010

100

[001]

Extra reflections

Supercell Modulation vector*

2bmGg

*21'* bb *

21 bq

qmclbkahg ***

q

[001]

100

010b’a’

ikBAII effF

)0210(2)000(2 lkhi

Blkhi

AII efefF

[001]

100

010b’a’

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’

*bnmGg

*1'* bn

b nbb '

Extra ref.:

If the periodicity of the modulation in direct space is

nb:

Can use supercell:

010

*2

bmGg Extra reflections

*21'* bb

010

100

bb 2'

[001]

b’a’

010

100

a’b’

*3

bmGg

*31'* bb bb 3'

Extra ref.:

010

[001]

010

100

a’b’

*4

bmGg

*41'* 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

ab

(110)

Modulation nót along main axis of basic structure

ab

(110)

010

100 110

],,[mGg 03131

[001]

Modulation nót along main axis of basic structure

010

100 110

1/3 1/3 02/3 2/3 0

[001]

010

100 110

030

300

1 1 02 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*

100011012

P

***

*'*'*'

cba

cba

P

b*b’*[001]

a’*a*

100011012

P

***

*'*'*'

cba

cba

P

Pcbacba '''

baa 2'bab '

cc '

ab

a’

b’

100011012

P

Pcbacba '''

baa 2'bab '

cc '

ab

a’

b’

100011012

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

bmGg

100

ab

[001]

010

*2

bmGg

100

ab

qmclbkahg ***

*** cbaq

*21 bq

[001]

010

100

*21 bq

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

]0,31,

31[mGg

[001]

*0*31*

31 cbaq

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

γ c

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

γ c

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

γ c

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

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)

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

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 + )

ExamplePnma(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 00020101

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

00032403-

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