# Presentation: 090497

## Diffuse Scattering

- Anticipatory (trick) question: If you have an x-ray or neutron detector looking at a small sample volume, which will scatter more x-rays or neutrons into the detector 1 atom 100 atoms or 1000 atoms?

**Detector**

**X-ray or neutron beam**

***Answer******: Depends!***

## Diffuse Scattering

**Gene E. Ice **
*Materials Science and Technology Division*
*Oak Ridge National Laboratory, USA*

- National School on Neutron and X-ray Scattering
- October 2008

## Presentation concentrates year graduate-level course into 1 hour

- Skip mathematical complexities
- Expose to range of applications
- Develop ***intuition*** for length scales
- Talk like x-ray/neutron scattering guru
***Reciprocal space***
***Debye Temperature***
***Laue monotonic***
***Krivoglaz defects of 1st/2nd kinds***!

**Great for cocktail parties or impressing future employers**

## Distribution of atoms redistributes scattering 

**Familiar light example**
**Practical applications- zero background plates for powder diffraction**
**Some problems- finding reflections**

## Diffuse scattering poised for a revolution!

- Synchrotron sources /new tools enable new applications
- Rapid measurements for combinatorial and dynamics
- High energy for simplified data analysis
- Small (dangerous) samples
- Advanced neutron instruments emerging
- Low Z elements
- Magnetic scattering
- Different contrast
- New theories provide direct link beween experiments and first-principles calculations

**Experiment           Theory**

***Major controversies have split leading scientists in once staid community!***

## Diffuse scattering due to local (short ranged) correlations/ fluctuations

- Thermal diffuse scattering (TDS)
- Point defect
- Site substitution
- Vacancy
- Interstitial
- Precipitate
- Dislocations
- Truncated surface- more

**All have in common reduced correlation length!**

## Bragg reflection occurs when scattering amplitudes add constructively

- Orientation of sample
- Wavelength
- Bragg Peaks sharp! ~1/N (arc seconds)

## Length scales are inverted!

- Big realsmall reciprocal
- Small realbig reciprocal
***We will see same effect in correlation length scales***
- Long real-space correlation lengths scattering close to Bragg peaks

***If you remember nothing else!***

## Krivoglaz classified defects by effect on Bragg Peak

**Defects of 1st kind**
**Bragg width unchanged**
**Bragg intensity decreased **
**Diffuse redistributed in reciprocal space**
**Displacements remain finite**
**Defects of 2nd kind**
**No longer distinct Bragg peaks**
**Displacements  continue to grow with crystal size**

**No defects**

**Defects of 1st kind**

**Defects of 2nd kind**

****

**Intensity**

## Thermal motion-Temperature Diffuse Scattering- (TDS) -defect of 1st kind

- Atoms coupled through atomic bonding
- Uncorrelated displacements at distant sites
- (finite)
- Phonons (wave description)
- Amplitude
- Period
- Propagation direction
- Polarization (transverse/compressional)

**Sophisticated theories from**
**James, Born Von Karmen, Krivoglaz**

## A little math helps for party conversation

- Decrease in Bragg intensity scales like e-2M, where
***Small***  ***Big*** reflections
- e-2M  shrinks (***bigger*** effect)  with   (q)

- Displacements, us depend on ***Debye Temperature ************D*** - ***Bigger*** *********D*** ***smaller*** displacements !

## TDS makes beautiful patterns reciprocal space

- Iso- intensity contours
- Butterfly
- Ovoid
- Star
- Transmission images reflect symmetry of reciprocal space and TDS patterns

**Experiment             Theory**

**sodium**

**Rock salt**

**Chiang et al. Phys. Rev. Lett. 83 3317 (1999)**

## X-rays scattering measurements infer phonon dispersion from quasi-elastic scattering 

- Phonon energies ***milli-eV***
- Synchrotron based high-E resolution X-ray beamlines can measure phonons *in some cases*
- Emerging area for high-brilliance x-ray sources

**Phonon spectrum gives natural vibration frequencies in different crystal directions!**

## Inelastic neutron/x-ray scattering directly measures phonon spectra in symmetry directions

**Inelastic neutron scattering confirms origins of negative Grüneisen coefficient in cubic ZrW****2****O****8**** (negative thermal expansion)-disordering phase transition.**
**Unusual thermal displacements often associated with phase transitions.**

***Phonon energies similar to meV neutron energies.***

## High-energy Synchrotron X-rays are revolutionizing TDS measurements

- Small samples
- Fast (time resolved/ combinatorial)
- Experiments in seconds rather than days
- Materials that cannot be studied with neutrons

**Pu experiment**

**Pu theory**

## Neutrons uniquely sensitive to low Z

- Deuterium cross section large
- Phonon energy comparable to neutron energy
- New instruments will accelerate data collection

**Welberry et al. ISIS**

**splitting**

**Experiment                      Theory**

## Often TDS mixed with additional diffuse scattering

- Experiment
- Strain contribution
- Combined theoretical TDS and strain diffuse scattering

**TDS must often be removed to reveal other diffuse scattering**

## Alloys can have another type 1 defect-site substitution

- Long range
- Ordering (unlike neighbors)
- Phase separation (like neighbors)
- Short ranged
- Ordering
- Clustering (like neighbors)

- Cu3Au
- L12

- CuAu
- L10

- Alternating planes of Au and Cu

- Each Au has 8 Cu near-neighbors

## Redistribution depends on kind of correlation

- Clustering intensity  fundamental sites

- Short-range ordering  superstructure sites

- Random causes ***Laue monotonic***

## Neutron/ X-rays Complimentary For Short-range Order Measurements

- Chemical order diffuse scattering **proportional to contrast (f****A****-f****B****)****2**
-  Neutron scattering cross sections
- Vary wildly with isotope
- Can have + and - sign
- Null matrix
- Low Z , high Z comparable
- X-ray scattering cross section
- Monotonic like Z2
- Alter by anomalous scattering

## Neutrons can select isotope to eliminate Bragg scattering

**Total scattering c****a****f****a****2****+**** ****c****b****f****b****2 ****   = 3**
**Bragg scattering (c****a****f****a****+**** ****c****b****f****b****)****2**** = 0**
**Laue (diffuse) scattering**
**c****a****c****b****(f****a- ****-f****b****)****2**** =3**

***Isotopic purity important as different isotopes have distinct scattering cross sections- ***_***only one experiment ever done***_***!***

## X-ray anomalous scattering can change x-ray contrast

- Chemical SRO scattering scales like (fa-fb)2
- Static displacements scale like (fa-fb)
- TDS scales like ~faverage2

## Atomic size (static displacements) affect phase stability/ properties

*Ionic materials (Goldschmidt)*
- Ratio of Components
- Ratio of radii
- Influence of polarization
*Metals and alloy phases (hume-Rothery)*
- Ratio of radii
- Valence electron concentration
- Electrochemical factor

- a

- c

- a

***Grand challenge -include deviations from lattice in modeling of alloys***

## Measurement and theory of atomic size are hard!

- Theory- violates repeat lattice approximation- every unit cell different!
- Experiment
- EXAFS marginal (0.02 nm) in dilute samples
- Long-ranged samples have balanced forces

**Important!**

## Systematic study of bond distances in Fe-Ni alloys raises interesting questions

- Why is the Fe-Fe bond distance stable?
- Why does Ni-Ni bond swell with Fe concentration?
- Are second near neighbor bond distances determined by first neighbor bonding?

## Entire new row of elements accessible to 3method:

**Entire new row of elements accessible to 3********method:**
** ****includes 4d metals**

**2nd generation K edges**
**3rd generation K**
**2nd generation L edges**

## Diffuse x-ray scattering measurements of AgPd alloy finds surprising lack of local ordering

- Negative deviation from Vegard’s law indicates ordering.
- First principle theory predicts ordering.- *Good system for theory*.
*Semi-empirical theory predicts clustering.*

## Diffuse x-ray scans show no evidence for ordering

- No superstructure reflections along h00
- Structure in diffuse scattering independent of x-ray scattering contrast

- Measurements only possible at APS
- 	-High energy/anomalous (24.350 keV)
- 	-small beam for polycrystalline sample

- Large mean-square displacement
- scattering. (4.9% atomic
- size difference or thermal?)

- 300

- 400

- 200

- 320

- 300

- 320

## High-energy x-ray measurements revolutionize studies of phase stability

- Data in ***seconds*** instead of ***days***
- Minimum absorption and stability corrections
- New analysis provides direct link to first-principle

***Max Planck integrates  diffuse x-ray scattering elements!***

## Measurements show competing tendencies to order

- Both L12 and Z1 present
- Compare with first principles calculations

- Reichert et al. Phys. Rev. Lett. **95** 235703 (2005)

**Au****3****Ni -001 plane**

## Vacancies, interstitials, small dislocation loops, coherent precipitates are additional type 1 defects

**Vacancies, interstitials, small dislocation loops, coherent precipitates are additional type 1 defects**

## Numerical calculations determine quantitative cross sections

- Sign of diffuse scattering reverses for vacancy/interstitials
- For interstitial loops- enhanced scattering at q=bH/4R
- For coherent precipitates enhancement at q=-H

- q

- (A/R)2 q4

- Vacancy

- Interstitial

- Iq4

- bH/4R

**Bragg Peak**

**Bragg Peak**

##  Micro-diffuse scattering applied to High Energy, Self-Ion Implantation in Si

**1-10**
**MeV**
**Si****+**

*cleaved edge*

- silicon
- substrate

**X-ray Microbeam**

- Vacancy

- Interstitial

- cleave sample in cross-section
- translate to probe depth dependence

** ****Micro-diffuse scattering applied to High Energy, Self-Ion Implantation in Si**

- Spatial separation of recoils and
- vacancies due to momentum transfer

*Yoon, Larson, Tischler, Haynes*

- Beam
- Stop

- Order
- Sorting
- Aperture

- Fresnel
- Zone
- Plate

- Sample

## X-ray Diffuse ScatteringHuang theory  for Q «1/R, I KbR2/Q4

## Depth Dependence of Size Distributions for Ion-Implanted Si

**Depth Dependence of Size Distributions for Ion-Implanted Si**

## Intense microbeams/area detectors provide new direction in diffuse scattering

- Tiny crysals (20 m)
- Natural polycrystals
- No special sample prep
- Combinatorial
- Dangerous samples

- Tiny microbeams

- Single-crystal-quality defect characterization

- Hazardous ***polycrystal*** samples

## Small irradiated volumes simplify handling/preparation

- Activity ~volume (10-5)
- Much less waste (10-7)
- Polycrystalline samples easier obtain- *closer to real materials*

- Traditional diffuse sample~300 mm3

- Microsample ~10-3 mm3
- 100-1000 samples

**Notes:**
The possibility of using very small (polycrystalline samples) will greatly simplify handling and preparation of samples.

For example, a typical single crystal sample has a minimum size of about 300 mm square for traditional measurements. With microbeam techniques however, a sample 5 orders of magnitude smaller with approximately 100-1000 individual grain orientqations can be studied with the same single-crystal precision..

This will correspond to five orders of magnitude less activiy for handling and 7 orders of magnitude less waste.

In addition, polycrystalline samples with large grains (~20 microns) should be easy to obtain and are the real materials used in reactors and fusion instruments.

## Nondispersive tunable microbeam will measure diffuse scattering without moving sample

- Tunable beam moves <0.5 m
- Area detector collects data in parallel
- Cooling needed to reduce background
- Must eliminate harmonics

## Dislocations examples of defects of the 2nd kind where microbeams already provide new information

- Long-range effect on lattice sites
- Paired dislocations causes broadening of Bragg peak
- Unpaired causes macroscopic rotations of crystal planes with streaking of reflections

**Dislocation**

## Influence of number and orientation of dislocations can be quantified

-  ******=[1-1-2],   n=[1-11],   b=[110]**

-  ******=[-11-2],    n=[1-1-1],    b=[110]**

-  ******=[1-2-1],   n=[-1-11],   b=[101]**

-  ******=[-1-21],   n=[-111],    b=[101]**

**Notes:**
As shown in this figure, the excess dislocation orientation is pronounced and allows for rapid quantification of the local dislocation in real materials.

## X-ray diffuse scattering at Femtosecond Resolution

**Ultra-brilliant LCLS opens new experimental possibilities**
**Transient behaviors at femtosecond time scales demonstrated.**

**Lindenberg et al. PRL 100 135502 (2008)**

## Corelli SNS beamline specialized for diffuse scattering with elastic Discrimination

- Complex disorder and short-range correlations

## New directions in diffuse scattering

- Microdiffuse x-ray scattering
- Combinatorial
- Easy sample preparation
- Diffuse neutron data from every sample
- Interpretation more closely tied to theory
- Modeling of scattering x-ray/neutron intensity

**Experiment**

**Model**

## Intense synchrotron/neutron sources realize the promise envisioned by pioneers of diffuse x-ray scattering

- M. Born and T. Von Karman 1912-1946- *TDS*
- Andre Guiner (30’s-40’s)-*qualitative size*
- I. M. Lifshitz *J. Exp. Theoret. Phys. (USSR)* **8** 959  (1937)
- K. Huang *Proc. Roy. Soc*. **190A** 102 (1947)-*long ranged strain fields*
- J. M Cowley (1950) J. Appl. Phys.-*local atomic size*
- Warren, Averbach and Roberts *J. App. Phys* **22** 1493 (1954) -*SRO*
- Krivoglaz *JETP* **34** 139 1958 *chemical and spatial fluctuations*

- X-ray Diffraction- B.E. Warren Dover Publications New York 1990.
- _http://www.uni-wuerzburg.de/mineralogie/crystal/teaching/dif_a.html_
- Krivoglaz vol. I and Vol II.

- Other references:

## Diffuse scattering done by real people-small community

**Diffuse scattering done by real people-small community**

- Warren school
- Krivoglaz school

- Cullie Sparks
***ORNL***

**Bernie Borie,***ORNL*

**Simon Moss,***U. Of Houston*

- Rosa Barabash
***IMP, ORNL***

**Jerry Cohen, ***Northwestern*

**S. Cowley**, *Arizona St*.

- A. Gunier

**Peisl,** *U. München*

**Gitgarts,***Minsk*

**Rya Boshupka**, *IMP*

**B. Schoenfeld,** ETH Zurich

**W. Schweika, **KFA Jülich

## Diffuse scattering song

- I’ll teach you diffraction- what I says mostly true
- Between the Bragg Peaks lies a world where you see
- Fluctuations and defects- they stand out plane-ly
- Chorus
- For its dark as a dungeon between the Bragg peaks
- But here in the darkness- each defect speaks
- It gathers- from throughout- reciprocal space
- And re-distributes all over the place.
- Between the Bragg peaks - one thing that we see
- Is TDS on our CCD
- Intensity totals are conserved- you can’t win
- It steals from the Bragg peaks that stay very thin
- Substitutional alloys can cause quite a stir
- The shorter the length scale the greater the blur
- With care you can find out the bond length between
- Each atom pair type-the measurements clean
- Dislocations and other- type 2 defects
- Destroy the Bragg peaks -they turn them to wrecks
- But near the Bragg peaks- you still can see
- Intense diffraction continuously
- Many -are- the defects you find
- Between the Bragg peaks where others are blind
- So go tell your friends and impress your boss
- You’ve new understanding -with one hours loss