The interpretation of broad diffuse maxima usingsuperspace crystallography

Introducing disorder in superspace

Ella M. Schmidt and Reinhard B. NederInstitute for Crystallography and Structural Physics, FAU Erlangen-NürnbergMarch 20, 2019

Diffraction pattern of a modulated structure

-4 -2 0 2 4

h

Inte

nsi

ty q

Bragg reflections: h ∈ ZSatellite reflections: h±mq h,m ∈ Z

a

λ = 1q

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 1

Introduction to superspace crystallography

• Describe long-range ordered, modulated structure withfew parameters

• Modulation wave vector q = (α,β ,γ) Paul B. KlarFZU Prague

1-dimensional piano structure in two-dimensional superspace:

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 2

Introduction to superspace crystallography

Changing α ↔ Changing real space periodicity

Paul B. KlarFZU Prague

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 3

Introduction to superspace crystallography

as1

as2

qa1

A

Modulation function:u+(a) =Acos(2πqa)

• q determines periodicity ofmodulation function

• A determines the amplitude ofthe modulation function

• Modulation functions foroccupancy and/or displacement

• Real space≡ cut throughdirect superspace

• Reciprocal space≡ projectionof reciprocal superspace

-4 -2 0 2 4

h

Intensity q

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 4

[1] Van Smaalen, Sander. Incommensurate crystallography. Vol. 21. Oxford University Press, 2007.[2] Janssen, Ted, Gervais Chapuis, and Marc De Boissieu. Aperiodic Crystals: From Modulated Phases to Quasicrystals: Structure and Properties.

Oxford University Press, 2018.

Introduction to superspace crystallography

as1

as2

qa1

A

Modulation function:u+(a) =Acos(2πqa)

• q determines periodicity ofmodulation function

• A determines the amplitude ofthe modulation function

• Modulation functions foroccupancy and/or displacement

• Real space≡ cut throughdirect superspace

• Reciprocal space≡ projectionof reciprocal superspace

-4 -2 0 2 4

hIntensity q

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 4

[1] Van Smaalen, Sander. Incommensurate crystallography. Vol. 21. Oxford University Press, 2007.[2] Janssen, Ted, Gervais Chapuis, and Marc De Boissieu. Aperiodic Crystals: From Modulated Phases to Quasicrystals: Structure and Properties.

Oxford University Press, 2018.

(1+1)D ordered superspace

as1

as2

qa1

A

Modulation function:u+(a) =Acos(2πqa)

as1

as2

qa1

−A

Modulation function:u−(a) =−Acos(2πqa)

→ Sharp satellite reflections at h±q.→ u+(a) and u−(a) cannot be distinguished experimentally.

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 5

(1+1)D disordered superspace

as1

as2

qa1

A

• Phase domains in superspace• Positive correlation along as1

• Warren-Cowley short rangeorder parameter αs

1 > 0

• What happens to the sharpsatellite reflections?

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 6

(1+1)D disordered superspace

as1

as2

qa1

A

• Phase domains in superspace• Positive correlation along as1

• Warren-Cowley short rangeorder parameter αs

1 > 0• What happens to the sharp

satellite reflections?

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 6

(1+1)D disordered superspace - Occupational disorder

• Au/Ag 1:1 crystal

• q =√

77 ≈ 0.378

• Modulation functions:p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

• Create disordered superspace structure with• positive correlation

(αs1 = 0.85)

• mp+ = mp− = 0.5

-4 -2 0 2 4

h

Intensity q

q =√77

αs1 = 0.85

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 7

(1+1)D disordered superspace - Occupational disorder

• Au/Ag 1:1 crystal

• q =√

77 ≈ 0.378

• Modulation functions:p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

• Create disordered superspace structure with• positive correlation

(αs1 = 0.85)

• mp+ = mp− = 0.5

-4 -2 0 2 4

h

Intensity q

q =√77

αs1 = 0.85

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 7

(1+1)D disordered superspace - Peak position tuning

1D occupational disorder• p+(a) = 0.5+ 0.5cos(2πqa)• p−(a) = 0.5−0.5cos(2πqa)

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 8

(1+1)D disordered superspace - Peak width tuning

1D occupational disorder• p+(a) = 0.5+ 0.5cos(2πqa)• p−(a) = 0.5−0.5cos(2πqa)

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 9

(2+1)D superspace for occupational modulations

• 2D basic structure• q = (σ1,σ2)

• Internal dimension as3 perpendicularto external, physical spacespanned by a1 and a2

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 10

as3

as2

as1

a1

a2σ2

σ1

(2+1)D disordered superspace - Occupational disorder

• 2D basic structure• q = (σ1,σ2)

• Modulation functions:

p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 11

as3

as2

as1

a1

a2σ2

σ1

(2+1)D disordered superspace - Occupational disorder

• Au/Ag 1:1 crystal• Plane space group p1• One atom per unit-cell at (0,0)

• q =(

12 ,√

77

)• Modulation functions:

p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

• Create disordered super spacestructure with• positive correlations along as1

and as2:αs(1,0) = 0.7

αs(0,1) = 0.5

• mp+ = mp− = 0.5

-2 -1 0 1 2-2

-1

0

1

2

h

k

0

max

Intensity

+q

−q

as3

as2

as1

a1

a2σ2

σ1

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 12

(2+1)D disordered superspace - Occupational disorder

• Au/Ag 1:1 crystal• Plane space group p1• One atom per unit-cell at (0,0)

• q =(

12 ,√

77

)• Modulation functions:

p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

• Create disordered super spacestructure with• positive correlations along as1

and as2:αs(1,0) = 0.7

αs(0,1) = 0.5

• mp+ = mp− = 0.5

-2 -1 0 1 2-2

-1

0

1

2

h

k

0

max

Intensity

+q

−q

as3

as2

as1

a1

a2σ2

σ1

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 12

(2+1)D disordered superspace - Occupational disorder

• Au/Ag 1:1 crystal• Plane space group p1• One atom per unit-cell at (0,0)

• q =(

12 ,√

77

)• Modulation functions:

p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

• Create disordered super spacestructure with• positive correlations along as1

and as2:αs(1,0) = 0.7

αs(0,1) = 0.5

• mp+ = mp− = 0.5

-2 -1 0 1 2-2

-1

0

1

2

h

k

0

max

Intensity

+q

−q

as3

as2

as1

a1

a2σ2

σ1

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 12

(2+1)D disordered superspace - Occupational disorder

• Au/Ag 1:1 crystal• Plane space group p1• One atom per unit-cell at (0,0)

• q =(

12 ,√

77

)• Modulation functions:

p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

• Create disordered super spacestructure with• positive correlations along as1

and as2:αs(1,0) = 0.7

αs(0,1) = 0.5

• mp+ = mp− = 0.5

-2 -1 0 1 2-2

-1

0

1

2

h

k

0

max

Intensity

+q

−q

as3

as2

as1

a1

a2σ2

σ1

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 12

(2+1)D disordered superspace - Diffuse rods

• Au/Ag 1:1 crystal• Plane space group p1• One atom per unit-cell at (0,0)

• q =(

12 ,√

77

)• Modulation functions:

p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

• Create disordered super spacestructure with• positive correlation along as2:

αs(1,0) = 0.0

αs(0,1) = 0.95

• mp+ = mp− = 0.5

-2 -1 0 1 2-2

-1

0

1

2

h

k

0

max

Intensity

+q

−q

as3

as2

as1

a1

a2σ2

σ1

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 13

(2+1)D disordered superspace - Diffuse rods

• Au/Ag 1:1 crystal• Plane space group p1• One atom per unit-cell at (0,0)

• q =(

12 ,√

77

)• Modulation functions:

p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

• Create disordered super spacestructure with• positive correlation along as2:

αs(1,0) = 0.0

αs(0,1) = 0.95

• mp+ = mp− = 0.5

-2 -1 0 1 2-2

-1

0

1

2

h

k

0

max

Intensity

+q

−q

as3

as2

as1

a1

a2σ2

σ1

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 13

(2+1)D disordered superspace - Diffuse rods

• Au/Ag 1:1 crystal• Plane space group p1• One atom per unit-cell at (0,0)

• q =(

12 ,√

77

)• Modulation functions:

p+(a) = 0.5+ 0.5cos(2πqa)p−(a) = 0.5−0.5cos(2πqa)

• Create disordered super spacestructure with• positive correlation along as2:

αs(1,0) = 0.0

αs(0,1) = 0.95

• mp+ = mp− = 0.5

-2 -1 0 1 2-2

-1

0

1

2

h

k

0

max

Intensity

+q

−q

as3

as2

as1

a1

a2σ2

σ1

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 13

(2+1)D disordered superspace - Reflection condition

• Basic structure shows symmetry• Superspace group pg1( 1

2 ,β )Site A: (+0.1,0.0)Site B: (−0.1,0.5)

• Bragg-reflection condition:0k : k = 2n

• q =(

12 ,√

77

)• αs

(1,0) = αs(0,1) = 0.7

• Modulation functions:

-2 -1 0 1 20

1

2

3

4

h

k

0

0.1· max

Intensity

q

Site A: pA(aA) = 0.5±0.5cos(2π(+σ1a1A +σ2a2A))Site B: pB(aB) = 0.5±0.5cos(2π(−σ1a1B +σ2a2B + a2B))

→ Diffuse satellites obey reflection condition as well

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 14

(2+1)D disordered superspace - Reflection condition

• Basic structure shows symmetry• Superspace group pg1( 1

2 ,β )Site A: (+0.1,0.0)Site B: (−0.1,0.5)

• Bragg-reflection condition:0k : k = 2n

• q =(

12 ,√

77

)• αs

(1,0) = αs(0,1) = 0.7

• Modulation functions:

-2 -1 0 1 20

1

2

3

4

h

k

0

0.1· max

Intensity

q

Site A: pA(aA) = 0.5±0.5cos(2π(+σ1a1A +σ2a2A))Site B: pB(aB) = 0.5±0.5cos(2π(−σ1a1B +σ2a2B + a2B))

→ Diffuse satellites obey reflection condition as well

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 14

(2+1)D disordered superspace - Reflection condition

• Basic structure shows symmetry• Superspace group pg1( 1

2 ,β )Site A: (+0.1,0.0)Site B: (−0.1,0.5)

• Bragg-reflection condition:0k : k = 2n

• q =(

12 ,√

77

)• αs

(1,0) = αs(0,1) = 0.7

• Modulation functions:

-2 -1 0 1 20

1

2

3

4

h

k

0

0.1· max

Intensity

q

Site A: pA(aA) = 0.5±0.5cos(2π(+σ1a1A +σ2a2A))Site B: pB(aB) = 0.5±0.5cos(2π(−σ1a1B +σ2a2B + a2B))

→ Diffuse satellites obey reflection condition as well

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 14

(2+1)D disordered superspace - Reflection condition

• Basic structure shows symmetry• Superspace group pg1( 1

2 ,β )Site A: (+0.1,0.0)Site B: (−0.1,0.5)

• Bragg-reflection condition:0k : k = 2n

• q =(

12 ,√

77

)• αs

(1,0) = αs(0,1) = 0.7

• Modulation functions:

-2 -1 0 1 20

1

2

3

4

h

k

0

0.1· max

Intensity

q

Site A: pA(aA) = 0.5±0.5cos(2π(+σ1a1A +σ2a2A))Site B: pB(aB) = 0.5±0.5cos(2π(−σ1a1B +σ2a2B + a2B))

→ Diffuse satellites obey reflection condition as well

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 14

(2+1)D disordered superspace - Reflection condition

• Basic structure shows symmetry• Superspace group pg1( 1

2 ,β )Site A: (+0.1,0.0)Site B: (−0.1,0.5)

• Bragg-reflection condition:0k : k = 2n

• q =(

12 ,1+

√7

7

)• αs

(1,0) = αs(0,1) = 0.7

• Modulation functions:

-2 -1 0 1 20

1

2

3

4

h

k

0

0.1· max

Intensity

q

qold

Site A: pA(aA) = 0.5±0.5cos(2π(+σ1a1A +σ2a2A))Site B: pB(aB) = 0.5±0.5cos(2π(−σ1a1B +σ2a2B + a2B))

→ Diffuse satellites obey reflection condition as well→ "Virtual" satellite reflection condition 0k : k = 2n+ 1

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 15

(2+1)D disordered superspace - Reflection condition

• Basic structure shows symmetry• Superspace group pg1( 1

2 ,β )Site A: (+0.1,0.0)Site B: (−0.1,0.5)

• Bragg-reflection condition:0k : k = 2n

• q =(

12 ,1+

√7

7

)• αs

(1,0) = αs(0,1) = 0.7

• Modulation functions:

-2 -1 0 1 20

1

2

3

4

h

k

0

0.1· max

Intensity

q

qold

Site A: pA(aA) = 0.5±0.5cos(2π(+σ1a1A +σ2a2A))Site B: pB(aB) = 0.5±0.5cos(2π(−σ1a1B +σ2a2B + a2B))

→ Diffuse satellites obey reflection condition as well→ "Virtual" satellite reflection condition 0k : k = 2n+ 1

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 15

(2+1)D disordered superspace - Reflection condition

• Basic structure shows symmetry• Superspace group pm1(0,β )s0

with internal translationSite A: (+0.1,0.0)Site B: (−0.1,0.0)

• Satellite-reflection conditiona:hk : h 6= 0

• q =(

0,√

77

)• α(1,0) = α(0,1) = 0.7• Modulation functions:

-2 -1 0 1 20

1

2

3

4

h

k

0

0.1· max

Intensity

q

Site A: pA(aA) = 0.5±0.5cos(2π(σ2a2A))Site B: pB(aB) = 0.5±0.5cos(2π(σ2a2B +

12))

→ Diffuse satellites obey superspace reflection condition as well

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 16

a For occupational modulation functions with only one cosine-term.

(2+1)D disordered superspace - Reflection condition

• Basic structure shows symmetry• Superspace group pm1(0,β )s0

with internal translationSite A: (+0.1,0.0)Site B: (−0.1,0.0)

• Satellite-reflection conditiona:hk : h 6= 0

• q =(

0,√

77

)• α(1,0) = α(0,1) = 0.7• Modulation functions:

-2 -1 0 1 20

1

2

3

4

h

k

0

0.1· max

Intensity

q

Site A: pA(aA) = 0.5±0.5cos(2π(σ2a2A))Site B: pB(aB) = 0.5±0.5cos(2π(σ2a2B +

12))

→ Diffuse satellites obey superspace reflection condition as well

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 16

a For occupational modulation functions with only one cosine-term.

(2+1)D disordered superspace - Reflection condition

• Basic structure shows symmetry• Superspace group pm1(0,β )s0

with internal translationSite A: (+0.1,0.0)Site B: (−0.1,0.0)

• Satellite-reflection conditiona:hk : h 6= 0

• q =(

0,√

77

)• α(1,0) = α(0,1) = 0.7• Modulation functions:

-2 -1 0 1 20

1

2

3

4

h

k

0

0.1· max

Intensity

q

Site A: pA(aA) = 0.5±0.5cos(2π(σ2a2A))Site B: pB(aB) = 0.5±0.5cos(2π(σ2a2B +

12))

→ Diffuse satellites obey superspace reflection condition as well

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 16

a For occupational modulation functions with only one cosine-term.

Possible applications

Diffuse rods showing ’extinctionconditions’

Sharp satellites on top of diffusescattering

Phase transitions: Diffuse scattering turns into sharp satellite reflections

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 17

Possible applications

Diffuse rods showing ’extinctionconditions’

Sharp satellites on top of diffusescattering

Phase transitions: Diffuse scattering turns into sharp satellite reflections

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 17

Possible applications

Diffuse rods showing ’extinctionconditions’

Sharp satellites on top of diffusescattering

Phase transitions: Diffuse scattering turns into sharp satellite reflections

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 17

Summary

• Disordered superspace models allow to characterize width and position ofdiffuse maxima independently

• Unified model for diffuse maxima at different positions in reciprocal space• Superspace symmetry allows direct access to extinction rules

Outlook• Application to different disordered systems• Expansion to (3+ d)D• Possible peak shape tuning by introducing size-effect like relaxations?• Possibility of curved diffuse features?

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 18

Summary

• Disordered superspace models allow to characterize width and position ofdiffuse maxima independently

• Unified model for diffuse maxima at different positions in reciprocal space• Superspace symmetry allows direct access to extinction rules

Outlook• Application to different disordered systems• Expansion to (3+ d)D

• Possible peak shape tuning by introducing size-effect like relaxations?• Possibility of curved diffuse features?

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 18

Summary

• Disordered superspace models allow to characterize width and position ofdiffuse maxima independently

• Unified model for diffuse maxima at different positions in reciprocal space• Superspace symmetry allows direct access to extinction rules

Outlook• Application to different disordered systems• Expansion to (3+ d)D• Possible peak shape tuning by introducing size-effect like relaxations?• Possibility of curved diffuse features?

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 18

Acknowledgments

FAU ErlangenReinhard NederMarvin BerlinghofJohannes DallmannStefan DiezMatthias Weißer

FZU Prague/EHU BilbaoPaul B. Klar

FundingFAU Office for Gender and DiversityBavarian Equal OpportunitiesSponsorship

Ella Schmidt | FAU Erlangen | Disorder in Superspace March 20, 2019 19