100 Years

of X-RayCrystallography

1913

2013

Cleaning

out the

closet

Equal-atom

structures

Direct

methods

1950‘s

Triclinic

structures

Four-circle

diffractometers

1960‘s

Incommensurate

and Higher

dimensional 1980‘s Quasicrystal

structures

crystallography

Disordered

structures

Quantitative analysis

of diffuse scattering

Heavy-atom

structures

Patterson method

1930‘s

A caveat concerning routine crystal structure analysis

from Bragg data

Routine crystal structure analysis

does not provide

´crystal structures´

It shows

the scattering density

of a single unit cell which represents

an average

over the time of the experiment and all unit

cells of the crystal.

Beyond Average Structure Determination –

Diffuse Scattering, Disorder and

Materials Properties.

Cal, April 2014

Those

who

did

the

work

D. Chernyshov (SNBL@ESRF, F)M. Chodkiewicz (UZH, CH)J. Hauser (UNIBE, CH)C. Hoffmann (ORNL, USA)M. Hostettler (B/ECV, CH)D. Komornicka (ILTSR, PL)V. Lynch (ORNL, USA)T. Michels-Clark (UT, USA)Th. Weber (ETHZ, CH)

Overview

•

Single crystal structure analysis, potential and limitations

•

Some examples of disordered materials, pictures of their diffuse scattering

•

Why study diffuse scattering?

•

A mini-introduction into diffuse scattering

•

A real-life case study: Monte Carlo crystal builder Parameter opt. by differential evolution Zürich Oak Ridge Disorder Simulation

Example

I of disordered

materials: Pigment Red 170

Constituent

of spray paint, engineering

problem:used

in the

car

industry,

R. Warshamanage, A. Linden, M. U.Schmidt, H.-B. Bürgi, Acta Cryst. B70 (2014) 283–295

Light-fastness

M. U. Schmidt, D. W. M. Hofmann,C. Buchsbaum, Angew. Chem. Int. Ed. 2006, 45, 1313 –1317

2c

2Na/2Ln Na/2Ln/2

Ln

Example

II of disordered

materials: light up-conversion

(NaLnF4

, doped)

- Single crystal X-ray structure:two

Ln-sites, both

C3

-symmetric- UV/VIS spectroscopy:

two

Ln-sites, one

C3

-, one

C1

-symmetric Na Ln

F4

H

K

Example

III of disordered

materials: host-guest

inclusion

compound,

SHG active

-

Superposition [R-PHTP+S-PHTP]/2

- 5-fold positionaldisorder

of NPP

Perhydrotriphenylene2

*1-(4-Nitrophenyl)piperazine5

Example

IV of disordered

materials: Prussian

blue

analog of Mn,

mixed-valence

and magnetic

properties

H K 0H K 0

Mn3

[Mn(CN)6

]2

(H2

O)6

-NaCl

lattice: 3 Mn2+ occupyedges, 2{Mn3+(CN-)6

}

and (H2

O)6

clusters

occupy

cornersand face centres

of cube.

(H2

O)62{Mn3+(CN)6

}3{Mn2+}

Why

study

disorder

diffuse scattering?

-

Many materials owe whatever (interesting) properties

theyhave to disordered arrangements

of atoms and molecules

-

Disorder diffuse scattering

tends to be weak

compared toBragg scattering. With synchrotron radiation, intense neutronbeams and pixel detectors it can now be measured reliably

-

No general protocol

for determining disordered structures

-

Interpretation of diffuse scattering is computationallyintensive. With today’s computing power this is no longer a major problem

Schematic

representation

of relation

between

disorder

and scattering

ρ

= Occupational

disorder

Δρ

+ Diffuse scattering

<ρ> ‚Bragg‘

scattering

ρ

= Positional

disorder

Δρ

+ Diffuse scattering

<ρ> ‚Bragg‘

scattering

Bragg diffraction and diffuse scattering

Occupational disorder

Positional disorder

A general

strategy

1)

Do the

best experiment

possible, both

with

respect

to Bragg

and diffuse scattering

-

high intensity

primary

beam

(synchrotrons, SNS at ORNL)-

low(no)-noise

pixel

detector

(Pilatus)

2)

Find best average

structure

and scrutinize

it

for

features that

contradict

the

principles

of chemistry

and physics

3)

Look for

diffuse scattering

and attempt

a qualitative interpretation

with

simple (analytical) models

4)

Develop

a quantitative model

of disorder

and optimize

its parameters

by

numerical

methods, e.g. a genetic

algorithm

5 NaF

·

9 LuF3

,

hk0 layerlaboratory source

P.P. Das, A. Linden, H.B.Bürgi, unpublished

synchrotron

same crystal!

Space group R3bar

Nitro-group disordered equally over six positions

Distinct diffuse Scattering

Anomalous dielectric properties

L. H. Thomas, T. R. Welberry, D. J. Goossens, A. P. Heerdegen, M. J. Gutmann, S. J.Teat, P. L. Lee, C. C. Wilson, J. M. Cole, Acta Cryst. (2007). B63, 663–673

Importance

of average

structure Pentachloronitrobenzene

I

Importance

of average

structure Pentachloronitrobenzene

II (ADPs

at 5, 100, 180, 298 K)

J.M. Cole, H.B. Bürgi, G. McIntyre, PRB 83

(2010) 224202

at 5 K

T-independent Thermal motion

U33

(C) 0.0391 Ǻ2

0.0367(4) Ǻ2

0.0024 Ǻ2

U33

(Cl) 0.0781 Ǻ2

0.0712(4) Ǻ2

0.0069 Ǻ2

z-displacement of molecule 0.16 Ǻ

molecular tilt 4.2o

Displacement of Cl(from ADPs)

-0.06 to

0.48 Ǻ

vs. -0.08 to

0.59 Ǻ

(diff. scat.)

Case study: Upconversion

phosphors

NaLaF4 : Yb3+, Er3+

and

NaGdF4

: Yb3+, Er3+

best materials for NIR → VIS conversion or green to blue

Polarized absorption spectra- NaGdF4

:10% Er3+

(right):

two sites: A (C3h ), B (C1

)- LaCl3

:0.1% Er3+

(left):

one site (C3h ) LaCl3

:0.1% Er3+ NaGdF4

:10% Er3+

- two unit cells || c shown

-

P , R ~ 1%

- two Ln-sites -

but both with C3h symmetry!

- one fully ordered (Ln) -

one disordered (Na/2, Ln/2)

6

2cBest structure

from Bragg

reflections

2Na/2LnNa/2Ln/2 Ln

NaLnF4

, diffuse scattering I

- Regular array of Bragg peaks

- in addition:sharp, horizontallines at half-integer L

Ln

Ln

Ln

2Na/2 LnNa/2Ln/2

-

translational period along c doubled

-

Columns with Ln…Na…Ln…Na

-

strictly alternatingalong c

Na

Ln

2c

L2.51.50.5

-0.5

L

L43210

NaLnF4

, diffuse scattering II

H

K -

honeycomb pattern of diffuse scattering

Ln

Na..Ln

Ln..Na

Ln..Na

- Column of ordered Ln-ions(...Ln...Ln...Ln...)

-

surrounding columns are …Na…Ln…Na…Ln

or

…Ln…Na…Ln…Na-

Coulomb

frustration

Numerical

approach, automated

- Monte Carlo crystal

builder

- Model parameters: correlation

between

Na…La

‘up‘

and ‘down‘

columns, displacement

of F atoms, ADPs

- Simultaneous

construction

–

unit

cell

by

unit

cell

–of N random

crystals

(phenotypes) from

N different parameter

sets

(= genes), each

with

thousands

of Na…La-columns

-

Energy minimization

-

Calculation

of intensities, comparison

with

experiment

- Optimization

of parameters

by

differential evolution. Fitness selection

against

experimental intensities

(R)

Simulation (and later analysis)

Courtesy

of Michal Chodkiewicz

Principles

of Monte Carlo Simulations 2D and 3D disorder

1)

Define a starting model

2)

Define interactions or correlations (RR, RB, BB)

3) Manipulate the structure

4) Calculate the change of the lattice energy

5) Accept or reject the new configuration according to the change of the lattice energy

Courtesy

of Thomas Weber

Genetic algorithm for optimisation of model parameters

Th. Weber, H.-B. Bürgi, Acta

Crystallogr. A58 (2002) 526-540. H.-B. Bürgi, J. Hauser, Th. Weber, R.B. Neder, Crystal Growth & Design 5 (2005) 2073-2083

Differential Evolution in Parameter/Fitness space

(schematic)

Parents: target

AND pc

‘(pa

, pb

, pc

)

Children: 1

OR 2

Fitness(1) > targetFitness(2) < target

pi

pj

R

Genetic algorithm for optimisation of model parameters

Differential evolution

in

Parameter/Fitness space

(schematic)

J

d

Parallelization

of parameter

optimization

Global Optimization –

population based method

p1 p2

2

p3 p4 p5 p6 pn...

pn

[2] pn

[3] pn

[4]pn

[1]

Initialize crystal

Equilibrate

Calc. Intensities

many sets ofparameters

many disordered Crystals

for the

same parameters

Courtesy

of Michal Chodkiewicz

ZODSZürich – Oak Ridge Disorder Simulations

Courtesy

of Michal Chodkiewicz

H

K

MODELEXPERIMENT

Correlation

between

disordered

columns

Na/-

Ln/Ln

Ln/Na

p = 1.00Na/Ln

p = 0.61Ln/Na

p = 0.51

Ln/Na

p = 0.57Na/Ln

p = 0.61

Ln/Ln

-

F¯

will not want to bemidway between Ln3+

and Na+,

shifted

towards

Ln3+

!

-

disordered Ln3+

:

local C3h symmetry

- Ln3+

in ordered column:

C1 symmetry!

-

Explains spectroscopic observation, provides a basis formodeling the high efficiency of upconversion

NaLnF4

, diffuse scattering III

A. Aebischer, M. Hostettler, J. Hauser, K. Krämer, Th. Weber,H. U. Güdel, H.-B. Bürgi, Angew. Chemie Int. Ed. 45

(2006) 2802

Na

Ln

Ln

Ln

Ln

A summary

1)

Do the

best experiment

possible, both

on Bragg

AND diffuse scattering

-

high intensity

primary

beam

(Synchrotrons)-

low(no)-noise

detector

(Pilatus)

2)

Find best average

structure

and scrutinize

for

features

that contradict

the

principles

of chemistry

and physics

3)

Qualitative interpretation

of diffuse scatteringwith

simple (analytical) models

(NaLnF4

)

4)

Quantitative model

of disorder

and parameteroptimization

by

numerical

methods

(ZODS)

5)

Evaluate

local

structure

Na

LnLn

Ln

Ln

Broader

context

and a bit

of an

outlook

-

Ordered crystals - Disordered crystalsPowder diffraction -

diffuse scattering

single-crystal diffraction -

1D-PDF, Discus

-

ZODS Software development and supercomputing (collaboration with Oak Ridge National Lab)

-

YELL program

to analyse

3D-PDF (Weber and Simonov, ETHZ),

-

Disordered crystals, nanoparticles, composites, glasses

-

Incoherent diffuse scattering → probabilistic models Coherent diffuse scattering (FEL) → imaging

Quantitative models: A recent book on the topic

T. R. Welberry Diffuse X-ray Scattering and Models of Disorder

(International Union of Crystallography, Monographs on Crystallography)

Penrose quilt

by

Kitty and Mark Spackman, Perth WA