The interpretation of broad diffuse maxima using ...€¦ · Introduction to superspace...
Transcript of The interpretation of broad diffuse maxima using ...€¦ · Introduction to superspace...
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