Diffuse Scattering

Neutron School on Diffraction, Dec 2007

Diffuse ScatteringDiffuse Scattering

D. J. GoossensAINSE Research Fellow,

Research School of Chemistry & Department of Physics

ANU


What is diffuse scattering?What is diffuse scattering?Diffuse scattering is the scattered intensity that lies between the Bragg peaks.

It tells you about short-range order in the crystal. The Bragg peaks tell you about the unit cell -- the regular, long-range order. But that may not be the whole story.

Some example of diffuse scattering:

X-ray diffuse scattering from benzil, C14H10O2

Bragg peak

Diffuse intensity

Bragg peaks only occupy a few pixels at the centre of each bright region. The rest of the pattern is ‘diffuse scattering’ and conventional analysis ignores it all, and ignores all the information in it…


(h k 1)

Neutron diffuse scattering from PZN, PbZn1/3Nb2/3O3

Neutron diffuse scattering from paraterphenyl, C18D14

X-ray diffuse scattering from PCNB, C6Cl5NO2

…etc…h

k

Yttria stabilised cubic zirconia, hk0.5, X-rays

Examples of diffuse scattering.


What is diffuse scattering?What is diffuse scattering?Usually when you do a structural study you measure the Bragg reflections. In powder diffraction, you might get a pattern that looks something like this:

Powder diffraction pattern of deuterated benzil C14D10O2 at 100K. Inset shows boxed peak as a function of temperature.


In single crystal diffraction, you measure a bunch of integrated intensities of Bragg reflections.

Each reflection is due to a set of planes of atoms in the crystal.

The set of all possible reflections makes up a grid of points in reciprocal space.

The Reciprocal lattice


So say we have a perfect (simple cubic) crystal.

2-d cut through a simple cubic crystal, looking down (say) c

at the ab plane

a

b

We could measure the Bragg reflections that come off it,

and we would get a lattice of reflections in reciprocal space.

a*

b*

210 reflectio

n

A perfect crystal


This diffraction pattern is like a slice or cut through reciprocal space, and we can index the diffraction spots as usual with h, k and l

(2-d cut so we’ll take l = 0)

0 1 2 3 4 (h)

(k) 43210

All the intensity is localised on the reciprocal lattice points, an we can calculate the expected intensity for a given point in the usual way:

€

I ∝ F *F

F = fme2πi hx +ky +lz( )

m

∑

Structure factor


What happens when we introduce disorder (static or thermal)?

First: what can disorder look like?

a

b

Disorder in positions (‘Displacive disorder’)

Disorder in occupancies (‘Occupational disorder’)

And plainly both can occur at once.

Adding Disorder...


If our scatterers are a bit more complicated, we can have other forms of disorder:

If our scatterer is say a molecule, then we can have

orientational disorder:

And these can occur along with displacive and occupational disorder.

Or bits within the molecule can rotate or twist or

whatever…

Other types of disorder


No disorder. Random displacements Displacements, short-range correlated

Direct space

(crystal)

Reciprocal space

(diffraction)

Three examples


Random displacements Displacements short-range correlated

If we subtract out the scattering from the Bragg peaks and scale up, what is left?

Looks the same?


Random displacements Displacements short-range correlated, Bragg scattering subtracted…

If we subtract out the scattering from the Bragg peaks and scale up, what is left?

Looks the same...but it is not!


Displacements short-range correlated, Bragg scattering subtracted…

That’s why we’re interested in diffuse scattering.

Things that look the same to Bragg scattering look different to diffuse scattering.

The local ordering that diffuse scattering can study is what is truly reflective of the crystal chemistry and physics -- an individual atom does not care what ‘average’ it is supposed to obey, just how it interacts with its neighbours.

The average may be completely non-physical. So if we really want to understand how the structures (and properties) arise, sometimes we need to ‘get inside’ the average using diffuse scattering.

Implications...


Displacements short-range correlated, Bragg scattering subtracted…

The average may be completely non-physical. So if we really want to understand how the structures (and properties) arise, sometimes we need to ‘get inside’ the average using diffuse scattering.

Diffuse scattering lets us look at the population of local configurations that go into making up the average. We can tackle questions like:

Are atoms tending to push apart? Pull together? Are vacancies clustering or anticlustering? What sorts of defects do we have and how do they interact? How does the position/conformation/attitude of one molecule affect the next? What are the key interactions in propagating the correlations?

More implications


Positively correlated occupancies

Random occupancies Negatively correlated occupancies

Other Effects...


Positively correlated occupancies

(Bragg removed, diffuse on Bragg positions)

Random occupancies (Bragg removed, no structured diffuse)

Negatively correlated occupancies

(Bragg removed but positions indicated by

white dots)

Other Effects (2)


Like letting occupancy and displacement interact…

-ve occ. corr.

+ve occ. corr.

Like atoms push apartUnlike atoms pull together

Type 1 atoms pull togetherType 2 push apart

Unlike atoms push apartLike atoms pull together

Other Effects (3)


So...So...We study diffuse scattering because it give additional information compared to the Bragg peaks.

Particularly, it tells you about the disorder and short-range-order in the material.

There are many materials where disorder is crucial in determining physical properties…

Eg: Relaxor ferroelectrics like PZN, PbZn1/3Nb2/3O3

Colossal magnetoresistance manganites

Host-guest systems and molecular framework materials

Glassy systems

Molecular crystals


Collecting the dataCollecting the dataDiffuse scattering can be measured using electrons, X-rays and neutrons.

Neutron X-ray Electron

Weak sources (big crystals, slow data collections ~days)

Scattering does not depend on atomic number;

Sensitive to magnetism;

Quantitative data;

Good range of sample environments;

Can see inelastic effects

Bright sources (small crystals, faster experiments ~hours); Wide range of sample environments;

Quantitative data;

Can’t see inelastic effects

Bright sources (very small crystals or even grains, fast experiments);

Non-quantitative data

Limited sample environments

Etc…


This is a neutron school so...This is a neutron school so...Collecting neutron diffuse scattering…

(1) At a spallation source and;

(2) At a reactor (here!)


11 detectors

64 64 pixels per detector

complete t.o.f. spectrum per pixel

Collecting Diffuse Scattering at a Spallation Source (ISIS)


angle subtended by 90detector bank

A-A’ and B-B’ given by detector bank

B-A and B’-A’ given by time-of-flight

volume of reciprocal space recorded simultaneously with

one detector bank.

Neutron Time of Flight Geometry


1 crystal orientation1 detector

1 crystal orientation2 detectors1 crystal orientation3 detectors1 crystal orientation6 detectors3 crystal orientations1 detector

3 crystal orientations1 detectorsymmetry applied

3 crystal orientations4 detectorssymmetry applied

Benzil Diffuse Scattering


(h k 1)

(h k 0)

10 crystal settings8 detectors

(h k 0.5)

apply m3m

symmetry

nb. full 3Dvolume

PZN Diffuse Scattering


WombatWombat

Cu1.8Se

(Thanks to Andrew Studer and Sergey Danilkin, ANSTO)

Cu1.8Se

(Thanks to Andrew Studer and Sergey Danilkin, ANSTO)

At a Reactor...


Easiest to picture if we just thing of the equatorial pixels on the 2-d detector…

Some trigonometry

= sample angle

...still at a reactor


No unit cells!No unit cells!

Data Analysis


Unit cells cannot be considered identical.

Need to model a region of the crystal large enough to contain a statistically valid population of local configurations, and to avoid finite-size effects

Usually upwards of 32 × 32 × 32 unit cells

Maybe 150+ atoms per cell

= 32 × 32 × 32 × 3 × 150 = too many coordinates to fit directly

Considerations


The Approach Work with the parameters which determine the coordinates – the interatomic

interactions. These will be the same from cell to cell.

Use ‘contact vectors’ between atoms:

Use torsional springs within molecules:

Use Ising terms to model occupancies:

We equilibrate a real-space model crystal subject to the imposed interactions and then calculate its diffuse diffraction pattern and compare with the observed, then adjust the interactions accordingly.

( ) ...}{ 2112

molecules allintra +Δ= ∑ φFE

Einter = all contact

vectors

∑ Fi di −d0i( )2

€

E = JnnSiS j +nn

∑ J2nnSiS j +2nn

∑ J3nnSiS j +3nn

∑ ....


MC algorithm

Randomly select a molecule and

calculate its energy

Randomly modify configuration and


Is the new energy less than the old?

Save the new configuration

yes no

accept or reject according to some

probability





Is the new energy less than the old?


yes no


probability





Is the new energy less than the old?Is the new energy less than the old?


yes no


probability


Diffuse scattering contains information about short-range order that is not present in the Bragg peaks.

This information relates to the local environments of the atoms and molecules, so can be important in relating structure to function.

Diffuse scattering is demanding to measure and analyse, but it can be done and it can reveal important insights.

It also produces some quite pretty pictures!

Diffuse scattering contains information about short-range order that is not present in the Bragg peaks.

This information relates to the local environments of the atoms and molecules, so can be important in relating structure to function.

Diffuse scattering is demanding to measure and analyse, but it can be done and it can reveal important insights.

It also produces some quite pretty pictures!

In Summary


QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

CaCSZ PCNBCePdSb

DCDNBYCSZ33’benzil

PCNB

Benzil Fe1-xO

PZN

CMA

Molecular

Molecular

Molecular

Molecular

Molecular

Molecular

Oxide

Oxide

Oxide

Oxide

Intermetallic

More examples of diffuse scattering


Go back to Disordered Materials

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Diffuse Scattering

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