Research School of Chemistry & Diffuse Scattering D. J. Goossens AINSE Research Fellow, Research School of Chemistry & Department of Physics ANU

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: 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… Bragg peak Diffuse intensity X-ray diffuse scattering from benzil, C14H10O2

Examples of diffuse scattering. X-ray diffuse scattering from PCNB, C6Cl5NO2 (h k 1) Neutron diffuse scattering from PZN, PbZn1/3Nb2/3O3 Neutron diffuse scattering from paraterphenyl, C18D14 …etc… Yttria stabilised cubic zirconia, hk0.5, X-rays

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.

The Reciprocal lattice 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.

A perfect crystal b b* a a* 210 reflection So say we have a perfect (simple cubic) crystal. We could measure the Bragg reflections that come off it, and we would get a lattice of reflections in reciprocal space. a* b* b a 2-d cut through a simple cubic crystal, looking down (say) c at the ab plane 210 reflection

(2-d cut so we’ll take l = 0) Structure factor 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) 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: (k) 4 3 2 1 0 1 2 3 4 (h)

Adding Disorder... What happens when we introduce disorder (static or thermal)? First: what can disorder look like? b a Disorder in occupancies (‘Occupational disorder’) Disorder in positions (‘Displacive disorder’) And plainly both can occur at once.

Other types of disorder If our scatterers are a bit more complicated, we can have other forms of disorder: Or bits within the molecule can rotate or twist or whatever… If our scatterer is say a molecule, then we can have orientational disorder: And these can occur along with displacive and occupational disorder.

Displacements, short-range correlated Three examples Direct space (crystal) Reciprocal space (diffraction) No disorder. Random displacements Displacements, short-range correlated

Displacements short-range correlated Looks the same? If we subtract out the scattering from the Bragg peaks and scale up, what is left? Random displacements Displacements short-range correlated

Looks the same...but it is not! If we subtract out the scattering from the Bragg peaks and scale up, what is left? Random displacements Displacements short-range correlated, Bragg scattering subtracted…

Displacements short-range correlated, Bragg scattering subtracted… Implications... 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. Displacements short-range correlated, Bragg scattering subtracted…

Displacements short-range correlated, Bragg scattering subtracted… More implications 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? Displacements short-range correlated, Bragg scattering subtracted…

Other Effects... Random occupancies Positively correlated occupancies Negatively correlated occupancies

Other Effects (2) Random occupancies (Bragg removed, no structured diffuse) Positively correlated occupancies (Bragg removed, diffuse on Bragg positions) Negatively correlated occupancies (Bragg removed but positions indicated by white dots)

Other Effects (3) -ve occ. corr. +ve occ. corr. Like letting occupancy and displacement interact… -ve occ. corr. +ve occ. corr. Type 1 atoms pull together Type 2 push apart Like atoms push apart Unlike atoms pull together Unlike atoms push apart Like atoms pull together

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 data Diffuse 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; 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... Collecting neutron diffuse scattering… At a spallation source and; At a reactor (here!)

Collecting Diffuse Scattering at a Spallation Source (ISIS) 11 detectors 64  64 pixels per detector complete t.o.f. spectrum per pixel

Neutron Time of Flight Geometry angle subtended by 90detector bank volume of reciprocal space recorded simultaneously with one detector bank. A-A’ and B-B’ given by detector bank B-A and B’-A’ given by time-of-flight

Benzil Diffuse Scattering 3 crystal orientations 1 detector symmetry applied 3 crystal orientations 4 detectors symmetry applied 3 crystal orientations 1 detector 1 crystal orientation 6 detectors 1 crystal orientation 2 detectors 1 crystal orientation 3 detectors 1 crystal orientation 1 detector

PZN Diffuse Scattering 10 crystal settings 8 detectors (h k 0) apply m3m symmetry (h k 0.5) (h k 1) nb. full 3D volume

Cu1.8Se (Thanks to Andrew Studer and Sergey Danilkin, ANSTO) At a Reactor... Cu1.8Se (Thanks to Andrew Studer and Sergey Danilkin, ANSTO) Wombat

...still at a reactor Easiest to picture if we just thing of the equatorial pixels on the 2-d detector…  = sample angle Some trigonometry

Data Analysis No unit cells!

Considerations 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

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.

MC algorithm

In Summary 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!

More examples of diffuse scattering CaCSZ PCNB CePdSb DCDNB YCSZ 33’benzil Benzil Fe1-xO PZN CMA Oxide Molecular Intermetallic

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