introduction to modulated structuresthe next truly incommensurate crystalline phases were alloys of...

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Introduction to Modulated Structures

Olivier Gourdon

Denver X-Ray Conference - Workshop


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Goal of this workshop

Slide 2

What is a modulation? What is a modulated structure?How can we recognize a modulated structure?What are the issues?How we deal with modulated structure ?In which material can we observe modulation?Why are they important ?


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Slide 3

What is a crystal ?

It is a macroscopic object where the atoms or molecules are perfectly ordered in a pattern which

repeats in the three directions of the space.

Diamond Structure

7 crystal systems

230 Space groups…

Graphite StructureTrue until …..

the nineteenth century is characterized by the development of the mathematical aspect of crystal structures


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Slide 4

Until ...

Some systems have additional diffraction spots –satellites reflections

These structures are known as aperiodic structures


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Slide 5

Until ...

Satellites are “regularly” spaced but most of the time they cannot be indexed with three reciprocal vectors

The problem

The consequences

One or more additional modulation vectors have to be added to index all diffraction spots. Diffraction pattern has not a 3d lattice character anymore the basic property of crystal 3d translation symmetry is

violated BUT in a specific regular way.

New definition

….a crystal is an object with a finite number of basic periodicities .


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Slide 6

History

A crystal structure with a displacively modulated structure was first considered by Dehlinger (1927)

The next truly incommensurate crystalline phases were alloys of transition elements and rare earth metals with so-called

magnetic helix /spiral structures (1959-1961)


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Slide 7

History


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Occupational modulation was described later by Korekawa & Jagodzinski(1967), Schweitz.Miner.Petrogr.Mitt., 47, 269-278.Composite crystals approach was introduced by Makovicky & Hide, Material Science Forum, 100&101, 1-100.

Structural analysis of modulated crystals is based on theoretical works of P.M. de Wolff, A.Janner and T.Janssen.

History

AperiodicCrystallography

Knowledge

SoftwareImprovement

HardwareImprovement

The number of studied modulated crystals grew with improving of experimental facilities – imaging plate, CCD.


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Moreover the real importance of modulations in crystals has been demonstrated by studies of organic conductors and superconductors (e.g. (BEDT-TTF)2I3 ) and high temperature Bi superconductor.

As superconductor

Thermoelectrics

Battery Materials, Fuel materials

Materials under extreme conditions

History

Interest• “Real” Materials with long range order structures possessinteresting physical properties. Their properties are usuallygoverned or linked to their “atypical” structure (modulated waves).


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Slide 10

History (superspace group)Development of the Mathematical tools & Super space group

Sander Van Smaalen for the Composite Structure(1990-1991)


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Slide 11

History (Quasicrystals)

Dan Shechtman discoveredquasiperiodic crystals in April 1982,as a visiting scholar at the NationalBureau of Standards in Maryland,USA. This new form of matter - alsoknown as quasicrystals orShechtmanite - possesses someunique and remarkablecrystallographic and physicalproperties, embodying a novel kindof crystalline order. His findingsdemonstrated a clear diffractionpattern with a fivefold symmetry. Thepattern was recorded from analuminum-manganese (Al-Mn) alloywhich he had rapidly cooled aftermelting.


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History

Report of the number of publications associated with the “aperiodic” order


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Slide 13

“The classical modulated structures”

N=1

Intergrowth compounds or

composite structuresN>1

QuasicrystalsN>1

position

The term “aperiodic crystal” covers modulated, composite crystals and quasicrystals.

Spinoccupation

Aperiodic Structures

Non-crystallographic rotation symmetry


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Slide 14

a

b

a*

b*

Direct Space

The reason of the satellite reflections- Periodic Deviation

Modulation on the atomic position of the

white atoms

Satellite reflections

The origin of the modulation may be:•An atomic displacement• A modification of occupation•A magnetic moment

Reciprocal Space

Main Reflections

The loss of information coming from the satellite reflections leads to the refinement of an average structure

Existence of a Modulation


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Slide 15


Modulated StructureWe have to use and to explain the intensities of the satellite

reflections in order to solve the real structure

Commensurability Incommensurability

The modulation w(q) is commensurate with the periodicity of the unit cell

The modulation w(q) is not commensurate with the periodicity of

the unit cell

w(q) w(q)

a’=3a a’=2.403..a

Treatment as supercellTreatment using superspace group (SSG) with an additional dimension

(2+1) vectors (a*, b* and q)How we do that ?


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Additional diffraction spots : qHqaaaQ mmlkh *3

*2

*1

modulation vector q can be expressed as a linear combination of

*3

*2

*1 ,, aaa *

33*22

*11 aaaq qqq

321 ,, qqq

321 ,, qqq

..... all rational commensurate structure

..... at least one irrational incommensurate structure

It is rather difficult to prove irrationality only from measured values.The higher the denominators the smaller difference between commensurate andincommensurate approach. But there are clearly distinguished cases 1/2,1/3 wherecommensurability plays an important role.

Superspace approach – Metric considerations


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Modulated and composite crystals can be described in a (3+d)dimensional superspace. The theory has developed byP.M.DeWolff, A.Janner and T.Janssen (Aminoff prize 1998).This theory allows to generalize concept of symmetry and alsoto modify all method used for structure determination andrefinement of aperiodic crystals.

The basic idea is that a real diffraction pattern can be realizedby a projection from the (3+d) dimensional superspace.

Superspace approach


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eq

*4A

*3

*3 Aa *3a

*3R


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1. The important assumption is that all satellites are clearly separated. This is true for the commensurate case or for the incommensurate case when the intensities diminish forlarge satellite index.

2. The additional vector e is perpendicular to the real space and plays only an ancillary role.

All diffraction spots form a lattice in the four-dimensional superspace there is a periodic generalized (electron) density in the four-dimensional superspace.

Reciprocal base :qeAaA *

4** 3,2,1iii

Direct base : 4,3,2,1,* jiijji AA

eAeqaaA 43,2,1iiii


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)(~)(~44332211 AAAARR nnnn

Then the generalized density fulfill the periodic condition :

and therefore it can be expressed as a 4-dimensional Fourier series :

H

H HRR 2exp)(~ F

3214321

3214321,,,,,,,,,,hhhhhhhxxxxxxx

hHrR

where:

From the definition of the direct and reciprocal base it follows:

H

H eqheeqrrR mmxF 4.2exp)(~


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H

H qrrqhR .2exp)(~4xmmF

But the real diffraction pattern is a projection of the 4d patterninto R3 which means that the term qr.4 x has to be constant.

Conclusion : A real 3d density can be found as a section through the generalized density.


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Slide 22

1. The important assumption is that all satellites are clearly separated. This is true for the commensurate case or for the incommensurate case when the intensities diminish for large satellite index.

2. The additional vector t is perpendicular to the real space and plays only an ancillary role.

Atomic rope

R31A

1a

Where are the atoms ?

A real 3d density can be found as a section through the generalized density.

A4= t

The definition of the modulation functions is such that they are periodic in t (0-1).

Superspace approach- Direct space

n

ncn

ns nxAnxAAxp 4,4,04 2cos2sin


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Slide 23

Super Space group Approach

Can we do everything we want ?


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Slide 24

Super Space group- Symmetry

The symmetry is described by a (3+d)-dimensional space group. A 4D superspace group must be 3+1 reducible = the internal and external dimensions cannot mix together.

General form of asymmetry operation:

Example of superspace group operations:x1, -x2, 1/2+x3, -x4

-x1, -x2, x3, 1/2+x4

RE 0RM RI

E

I


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Slide 25

Super Space group- SymmetryHow can the symmetry be determined? The first three rows are the components of the basic space group. The sign of RI depends on the action of the symmetry operation on the

q-vector:

2-fold: -x1, x2, -x3, -x42-fold: -x1, x2, -x3

mirror: x1, -x2, x3, x4mirror: x1, -x2, x3


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Slide 26

Super Space group- Symmetry

The translational part is determined from the extinction conditions in complete analogy to the 3D case:

in general:hR = h, h. = integer

c-glide:x1, -x2, 1/2+x3: h0l, l=2n

“superspace c-glide” with shift along x4:x1, -x2, 1/2+x3, 1/2+x4: h0lm, l+m=2n


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Symmetry in the superspace

Basic property of 3+d dimensional crystal - generalized translation symmetry :

)()(3

1

d

iiin Arr

tR |ˆ S)()ˆ( rr S rrrr SS ˆˆ

basic property unitary operator matrix representation

d

iiin

3

1

, AtER

Trivial symmetry operator - translation symmetry :


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The rotational part of a general symmetry element

IM

R 0R

1. The right upper part of the matrix is a column of three zeros. It is consequence of the fact that the additional ancillary vector e cannot be transformed into the real space.

2. From the condition that symmetry operator has to conserve scalar product it follows 1det1det IR

MIR qq3.

These conditions show that any superspace group is a four-dimensional space group but on the other hand not every four-dimensional space group is a superspace group. The superspace groups are 3+1 reducible.This allows to derive possible rotations and translation is same way as for 3d case.


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Examples

1. Inversion centre

1

100010001

01 I

01 I

There is no modulation for the second case and therefore the inversion centre has to have 1I


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2. Two-fold axis along z direction

1

100010001

0,010,0,01

I

Imonoclinic axial case

monoclinic planar case


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3. Mirror with normal parallel to z direction

1

100010001

0,0,010,01

I

Imonoclinic axial case

monoclinic planar case


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Translation part Symbol Reflection condition for

21,0,21,21

0,0,21,2121,0,21,0

0,0,21,021,0,0,21

0,0,0,2121,0,0,0

0,0,0,0

sm

msb

bsa

asm

m

1

1

1

1

hk0m

nmkhnkhnml

nlnmh

nhnm

222

22

22


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Slide 33

Super Space group

C2/m(0)0s

Herman-Mauguinsymbol

of the basic space group

Symbol of the

q-vectorDefinition of the intrinsic shifts in the fourth dimension

s=1/2; t=1/3q=1/4; h=1/6Generators: -x1, x2, -x3, (1/2)-x4

x1, -x2, x3, 1/2+x4Centering: 1/2 1/2 0 0


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Super-space group symbols

There are three different notation

00210or11::or11 sPmnasPmnaBBPmnas

The rational part of the modulation vector represents an additional centring. It is much more convenient to use the centred cell instead of the explicit use of the rational part of the modulation vector.

*c 210q 00q

*b

*b

*b


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Slide 35

Super Space group & Refinement Procedure

3D (3+n)D

Step 1

Metric, symmetry, Bravais classExtinction rules

Possible space group

Metric, symmetry, Bravais classExtinction rules

Possible superspace group756 for n=1, 3355 for n=2 et 11764 for n=3

Ex: R-3mEx: R-3m(00)0s

Direct Method or Patterson method to find the atomic positions Direct Method, Patterson method, Super flip,…

to find the atomic positions for the average structure from the main reflections

Step 2

Step 3

Refinement of the modulation using the satellites reflections

Physical space t space


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Positional modulation - longitudinal

Modulation vector : 0,,

Modulation wave : rq 2cosxx Uu

HUrQrqUu

22exp)2cos(

mJifF


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Positional modulation longitudinal1st harmonic 0.1Å

Fourier map

-1.0

0.0

1.0

2.0x1

0.0

1.0

2.0

3.0

x4


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Positional modulation longitudinal1st harmonic 0.1ÅDiffraction pattern


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Positional modulation longitudinal1st harmonic 0.5Å

Fourier map

-1.0

0.0

1.0

2.0x1

0.0

1.0

2.0

3.0

x4


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Positional modulation longitudinal1st harmonic 0.5ÅDiffraction pattern


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Positional modulation - transversal

Modulation vector : 0,,

Modulation wave : rq 2coszz Au

HUrQrqUu

22exp)2cos(

mJifF


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Positional modulation transversal – 1st harmonic 0.5ÅFourier map

-1.0 0.0 1.0 2.0x30.0

1.0

2.0

3.0

x4


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Positional modulation transversal – 1st harmonic 0.5ÅDiffraction pattern l=0


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Positional modulation transversal – 1st harmonic 0.5ÅDiffraction pattern l=1


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Composite structure


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Composite structure – without modulationFourier map

-1.0

0.0

1.0

2.0x1

0.0

1.0

2.0

3.0

x4


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Composite structure – without modulationDiffraction pattern


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Composite structure – with modulationFourier map

-1.0

0.0

1.0

2.0x1

0.0

1.0

2.0

3.0

x4


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Composite structure – with modulationDiffraction pattern


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Slide 50

The periodic modulation function can be expressed as a Fourier expansion:

n

ncn

ns nxAnxAAxp 4,4,04 2cos2sin

The necessary number of used terms depends on the complexity of the modulation function.

The modulation can generally affect all structural parameters – occupancies, positions and atomic displacement parameters (ADP),…. The set of harmonic functions used in the expansion fulfils the orthogonality condition, which prevents correlation in the refinement process.

In many cases the modulation functions are not smooth and the number of harmonic waves necessary for the description would be large. In such cases special functions or set of functions are used to reduce the number of parameters in the refinement.

Modulated Functions


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Slide 51

Crenel function

0.0 0.4 0.8 1.2 1.6 2.0t0.0

0.4

0.8

1.2

occ 2,20

2,2104

0444

04

0444

xxxxp

xxxxp

V.Petříček, A.van der Lee & M. Evain, (1995). Acta Cryst., A51, 529.

Modulated Functions(Special Functions)


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Slide 52

Examples


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Slide 53

R-3m(00)0sx/2

�

� �

Synthesis and structure determination of two new composite compounds in the hexagonal perovskite-like sulfide family: Eu8/7TiS3 and Sr8/7TiS3.

O. GOURDON, L. CARIO, V. PETRICEK and M. EVAIN, (2001) Z Kristallogr. , 216(10), 541-550 A new structural type in the hexagonal perovskite family. Structure determination of the commensurately modulated misfit compound Sr9/8TiS3O. GOURDON, V. PETRICEK and M. EVAIN, (2000) Acta Cryst. B56, 409-418.

Direct space

Reciprocal space

“Random” spin chainOxygen Vacancy controversy

Perovskite AxBX3- Composite Structure


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Slide 54

0,4

0,8

1,2

1,6

2,0 x3

0,0

0,4

0,8

1,2

1,6

2,0

x4

0,0

1000010000100001

W1=

MX3

W2=

0100100000100001

A

A

X M

Define a sub-network Choice

M. Evain, F. Boucher, O. Gourdon, V. Petricek, M. Dusek & P. Bezdicka Chem. Mater. 1998, 10, 3068-3076.

Perovskite AxBX3


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Slide 55

A

B

O

-0.10 0.00 0.10 0.20x2

-0.2

0.2

0.6

1.0

x4

AX3 = 1/2, X1 = 0.154

A

B

O

A

AB

B

b

a

Discontinuity Phenomenon

Crenel function with a centro-Superspace Group

R3m(00)Ukei et al. Model

Perovskite AxBX3


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Slide 56

AB

O

-0.10 0.00 0.10 0.20x2

-0.2

0.2

0.6

1.0

x4

AX3 = 1/2, X1 = 0.154

A

B

O

A

A B

B

b

az

x4

x3

Oh

Oh

Oh

Oh

Oh

Oh

Oh

OhTP

TP

TPTP

TP

TP

Sr

Sr

Sr1.2872NiO3

R3m(00)0s_

Perovskite AxBX3


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Slide 57

Perovskite AxNiX3


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Slide 58

Perovskite AxCoX3

0.15x2

x3=0.000, x4=0.250x3=0.000, x4=0.250

-0.15-0.15 -0.05-0.05 0.050.05 0.150.15x1x1-0.15-0.15

-0.05-0.05

0.050.05

0.15x2

Sr14/11CoO 3

x3=0.000, x4=0.250x3=0.000, x4=0.250

-0.15 -0.05 0.05 0.15-0.15

-0.05

0.05

0.15x2x2

x1x1

Classical treatment Anharmonic treatment

Gourdon, O., Petricek, V., Dusek, M., Bezdicka, P., Durovic, S., Gyepesova, D & Evain M. (1999). Acta Cryst. B55, 841-848.

Co behavior in the TP site


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Slide 59

R3m(00)0s_

•Same Superspace group•Same atomic position•Valid for commensurate and incommensurate case

Common featuresFrom (3+1)D to 3D in a commensurate approach

Perovskite AxBX3-Where is the family?


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Slide 60

Family tree…Farey tree


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Slide 61

Family tree…Farey tree


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Slide 62

According to Hume-Rothery’s suppositions, which are verified in the Zn-Pd system, the crystal structures of various noble metal alloys are invariant with respect to specific valence electron concentrations (vec) called “electron compounds”.

-Brass Structures: Zn1-xPdx0.14 x 0.24

1.52 vec 1.72 ( = 1.618)

(Pd)

L

Weight Percent Zinc

Atomic Percent Zinc

20 40 60 80 100

20 40 60 80 100

200

600

1000

1400

Tem

pera

ture

ºC

(Zn)

PdZ

n 2

W. Hume-Rothery, J. Inst. Metals (1926), 35, 309.

In the Zn-Pd system we focused our research on the -brass phase because we considered that similarities with quasicrystalline phases might be explained by structural rules as well as electronic rules.

Motivation

-Brass Structures- Composite or not?


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Slide 63

-Brass Structures

“106.86 Å”Image

From IPDS Main reflection

Satellitereflection

Image From a Brucker

CCD


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Slide 64

-Brass Structures

HT

LT

’” ”’””””’

Intergrowth Compounds in the Zn-Rich Zn-Pd System: Toward 1D Quasicrystal Approximants.O. GOURDON and G. J. MILLER (2006) Chem. Mat ., 18(7) 1848-1856.

Zn1xPdx (x=0.14 to 0.24): A Missing Link Between Intergrowth Compounds And Quasicrystal ApproximantsO. GOURDON , Z. IZAOLA, L. ELCORO, V. PETRICEK and G. J. MILLER (2006) Phil. Mag. 86(3-5) 419-426.


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Slide 65

-Brass Structures

Orthorhombic average unit cell as = 12.929 (3) Åbs= 9.112 (4) Å,

cs = 2.5631 (7) Å

q = c*with a unique 1D modulated wave

w(q)

Family of intergrowth compounds with 2 composites such as defined:

1000010000100001

0100100000100001

First subsystem Second subsystem

Zn1 at (0, 0, 0) Zn2 at (1/4, 1/8, ¾)

Pd4/Zn4 at (1/4, 0, ¼)

Zn3/Pd3 at (~0.6, 1/6, ¾)

The problem is reduced to refine 4 atomic sites

Xmmm(00)0s0


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Slide 66

-Brass Structures

What are the modulations?


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Slide 67

-Brass Structures


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Slide 68

Why modulations are important ?

• For decades we have limited our knowledge of the atomic structure of materials to the average structure. However the “real materials” possess atomic arrangements which are usually more complicated than that and deviate from this standard description.

Interest• Materials with disordered structures and long range order

structures possess interesting physical properties and usually the properties are governed or linked to their atypical structure (local structure/defect or modulated waves).


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Slide 69

Rietveld refinement of the powder neutron diffraction pattern using an orthorhombic Fddd(α00) superspace approach

Solvothermal synthesis of LiMn2O4-

• High conductivity ?• Possible O vacancies ?

Questions

Answer• Clear Deviation from the

Fd-3m cubic symmetry• Addition of a modulation

Vacancy ordering in Battery Materials

Improved electrode kinetics in lithium manganospinel nanoparticles synthesized by hydrothermal methods: identifying and eliminating oxygen vacancies , Hao, Xiaoguang, Olivier Gourdon, Liddle, Brendan and Bartlett, Bart (2012) Chemical Science .


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Slide 70

Solvothermal synthesis of LiMn2O4/annealed

• Annealing the samplemodified the structure to an“average” Fd-3m cubicstructure.

• Addition of extra oxygens• The peak shape/Lorentzian

part indicated a possibledefect ordering into thematerial (confirmed bymicroscopymeasurements).




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Slide 71

Fddd(α00) with superspace approach α=1/3 a*

Organization of the O vacanciesConduction associated with the diffusion thru the vacancies




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Slide 72

Example K1/3Ba2/3AgTe2How an incorrect structure analysis leads to a non-understanding of the physical properties.

• Monoclinic Structure P21/m•Square Te-net•Semi-Conductor (gap ~eV)

Grand interest for thermoelectricitySemiconductor with small gap

-14

-12

-10

-8

-6

-4

-2

0

2

Y Γ X M Γ

px

py

DOS

Metal

Square [Te] net


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Slide 73

Example K1/3Ba2/3AgTe2

“Classical modulated” structures

P21()

q = 0.3248(6) a*- 0.071(8) c*

A unique distortion in K1/3Ba2/3AgTe2. X-ray diffraction determination and electronic band structure analysis of its incommensurately modulated structure

O. GOURDON, JASON HANKO, F.BOUCHER, V. PETRICEK, M.-H. WHANGBO, M. G. KANATZIDIS and M. EVAIN, (2000) Inorganic Chemistry 7, 1398-1409.


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Slide 74


� � ��

�

�

� � �

V pattern W pattern


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Slide 75


-14

-12

-10

-8

-6

-4

-2

0

2

Y Γ X M Γ

px

py

DOS

� � � � � � � � � �

� � � � � � � � � �

� � � � � �

� �

CDW and existence of nesting vectors in the Fermi surfaces


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Slide 76


� � � � � � � � � �

� � � � � � � � � �

� � � � � �

� �

2:1:2:1 2:2:1:1Polymerization

V pattern W pattern


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Slide 77

Example K1/3Ba2/3AgTe2V pattern W pattern


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Slide 78

Aperiodicity is everywhere….

Thermoelectric materialsSuperconductor MaterialsMagnetic MaterialsDrugs (ephedrine)


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Slide 79


profilin:actin complex


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Slide 80


Levyclaudite


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Slide 81



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In 1999 two papers reported incommensurate composite character of pure metals under high pressure :

Nelmes, Allan, McMahon & Belmonte, Phys.Rew.Lett. (1999), 83, 4081-4084. Barium IV.


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Schwarz, Grzechnik, Syassen, Loa & Hanfland, Phys.Rew.Lett. (1999), 83, 4085-4088. Rubidium IV.

Theory of aperiodic crystals and computer program – Jana2000 – has been later applied to solve and refine several analogical structures.


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Slide 84

GeochemistryGeophysics



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Slide 85

• Extra-small peaks are not always a secondary phase.

• Missing a modulation information can lead to a misunderstanding of the physical properties of your materials

The modulation can be present even in very simple compounds as oxides – PbO, U4O9, Nb2Zrx-2O2x+1.


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Slide 86


a

b

Direct Space

Example of an hypothetical 2D crystal

Modulation on the atomic position of the

white atoms

Satellite reflections

The origin of the modulation A magnetic moment

Reciprocal Space

Main Reflections

concept of the superspace can be generalized for magnetic structures

m

mcmsm mm rkMrkMrkM 2cos2sin ,,,Magnetic space and superspace groups, representation analysis: competing or friendly concepts? V. Petrícek, J. Fuksa and M. Dusek, Acta Cryst. (2010). A66, 649.

Modulated Approach to treat magnetic ordering


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Slide 87

structuresconcept of the superspace can be generalized for magnetic structures

m

mcmsm mm rkMrkMrkM 2cos2sin ,,,

Magnetic space and superspace groups, representation analysis: competing or friendly concepts? V. Petrícek, J. Fuksa and M. Dusek, Acta Cryst. (2010). A66, 649.

magnetic modulated structures using harmonic ordering.

Modulated Approach to treat magnetic ordering


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Slide 88

“Reinvestigation of the Antiferromagnetic Structures in Mn5Si3: Inverse MagnetocaloricEffect Material” co-authored by M. Gottschlich, O. Gourdon, M. Ohl, J. Persson, C. de la Cruz, V. Petricek and Th. Brueckel (2011) Chemistry of Materials.

Refinement using the magnetic super space groupCcmm1’(00)00sswith q=(0,b,0)=b*.

1/3 of the magnetic atoms are magnetically ordered

Mn5Si3 family: Magnetic Structure


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Slide 89


Refinement using the magnetic super space groupC21/m1’(00)0sswith q=(0,b,0)=b*.

2/3 of the magnetic atoms are magnetically ordered

Relation from group to subgroupa= 6.8943(3) Å, b=11.9131(5) Å, c= 4.80742(5) Å and b=90.268(3)



UNCLASSIFIED

Slide 90

Process of Inverse MCE


C21/m1’(00)0sswith q=(0,b,0)=b*.

Entropy Change between the two

magnetic configurations

Ccmm1’(00)00sswith q=(0,b,0)=b*.



UNCLASSIFIED

Slide 91

Acknowledgements

Jana2006 team

Vasek Petricek, Michal Dusek and Lukas Palatinus

introduction to modulated structuresthe next truly incommensurate crystalline phases were alloys of...

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