Towards Precession for Dummies

1. Towards Precession for Dummies L. D. Marks, J. Ciston, C. S. Own & W. Sinkler (+ A couple of slides from Paul Midgley)

2. Scan Non-precessed Precessed De-scan Specimen (Ga,In)2SnO5Intensities 412Å crystal thickness Conventional Diffraction Pattern Precession Diffraction Pattern Precession… (Diffracted amplitudes)

3. PrecessionSystem US patent application: “A hollow-cone electron diffraction system”. Application serial number 60/531,641, Dec 2004.

4. Some Practical Issues Projector Spiral Distortions (60 mRad tilt) Bi-polar push-pull circuit (H9000)

5. e- e-   Block Diagram ‘Aberrations’

6. Better Diagram Postfield Misaligned Remember the optics Idealized Diagram Scan Objective Prefield Sample Objective Postfield Descan

7. Remember the optics Idealized Diagram Correct Diagram Both Misaligned Scan Objective Prefield Sample Objective Postfield Descan

8. One Consequence • Prefield/Postfield displacements of beam dPre = 1/(2p) ÑcPre(s-tPre); s = Scan dPost=1/(2p) ÑcPost(s-tPost) -sDq sD = DeScan Ñc(u)/(2p) = Dzq+Csq3 Total apparent displacement is the sum dNett = dPre+dPost = DzPre(q-qPre)+DzPost(q-qPost) + CsPre(q-qPre)3+CsPost(q-qPost)3 -sDq

9. Probe and Displacements (nm) Prefield misalignment 1 mRad; Postfield -1.0 mRad Caveat: this ignores 3-fold astigmatism in pre/post field which is probably not appropriate, and any projector distortions

10. Overview • Part I: What is the theory of precession • Early Models 1) Kinematical/Bragg’s Law 2) Blackmann (2-beam all angles) 3) Lorentz corrected Blackmann • More Recent Models 4) 2-beam integrated correctly • Good Models 5) Full Dynamical

11. Overview • Part II: What, if any generalizations can be made? • Role of Precession Angle • Systematic Row Limit • Importance of integration • Phase insensitivity • Important for which reflections are used • Fast Integration Options

12. Four basic elements are required to solve a recovery problem 1. A data formation model Imaging/Diffraction/Measurement 2. A priori information The presence of atoms or similar 3. A recovery criterion: A numerical test of Goodness-of-Fit 4. A solution method. Mathematical details Patrick Combettes, (1996). Adv. Imag. Elec. Phys. 95, 155

13. Scan Specimen De-scan k+s k s sz g 0 (Diffracted amplitudes) Ewald SphereConstruction z

14. Levels of theory • Precession integrates each beam over sz • Full dynamical theory • All reciprocal lattice vectors are coupled and not seperable • Partial dynamical theory (2-beam) • Consider each reciprocal lattice vector dynamically coupled to transmitted beam only • Kinematical theory • Consider only role of sz assuming weak scattering • Bragg’s Law • I = |F(g)|2

15. Some Accuracy Please! • Kinematical Theory is not I=|F(g)|2 • This is Bragg’s Law • This is the limit for weak scattering of thick, perfect crystals • Kinematical Theory includes the Ewald Sphere curvature • Dynamical Theory also includes multiple scattering • Please see Cowley, Diffraction Physics

16. Also, beware of other approximations • Column Approximation • Approximate sample of varying thickness as an average of intensities from different thicknesses • Correct in the limit of slowly varying sample thickness • Perhaps not be correct for PED from nanoparticles or precipitates • Inelastic/Absorption (needed for thick samples)

17. Early Models Iobs depends upon |F(g)|, g, f (precession angle) which we “correct” to the true result Options: 0) No correction at all, I=|F(g)|2 1) Geometry only (Lorentz, by analogy to x-ray diffraction) corresponds to angular integration 2) Geometry plus multiplicative term for |F(g)|

18. 100 Å; R=0.25 50 Å; R=0.19 3.2 Å; R=0.02 800 Å; R=0.88 400 Å*; R=0.78 200 Å; R=0.49 Bragg’s Law fails badly (Ga,In)2SnO5

19. Kinematical Lorentz Correction I(g) =ò |F(g) sin(ptsz)/(psz) |2 dsz sz taken appropriately over the Precession Circuit t is crystal thickness (column approximation) f is total precession angle I(g) = |F(g)|2L(g,t,f) K. Gjønnes, Ultramicroscopy, 1997.

20. Kinematical Lorentz correction:Geometry information is insufficient Fcorr Fkin Need structure factors to apply the correction!

21. 2-Beam (Blackman) form Limits: Ag small; Idyn(k) µ Ikin(k) Ag large; Idyn(k) µÖIkin(k) = |Fkin(k)| But… This assumes integration over all angles, which is not correct for precession (correct for powder diffraction)

22. Blackman Form

23. Blackman+Lorentz Comparison with full calculation, 24 mRad Angstroms Alas, little better than kinematical

24. Two-Beam Form I(g) = ò |F(g) sin(ptseffz)/(pseffz) |2 dsz sz taken appropriately over the Precession Circuit szeff = (sz2 + 1/xg2)1/2 Do the proper integration over sz (not rocket science)

25. Two-Beam Integration: Ewald Sphere Small (hkl) Medium (hkl) Large (hkl)

26. 100 Å;R=0.24 3.2 Å; R=0.02 50 Å; R=0.19 800 Å; R=0.67 400 Å*; R=0.62 200 Å; R=0.44 2-Beam Integration better

27. Some numbers R-factors for kinematical model R-factors for 2-beam model See Sinkler, Own and Marks, Ultramic. 107 (2007)

28. 2f t Fully Dynamical: Multislice • “Conventional”multislice (NUMIS code, on cvs) • Integrate over different incident directions 100-1000 tilts • f = cone semi-angle • 0 – 50 mrad typical • t = thickness • ~20 – 50 nm typical • Explore: 4 – 150 nm • g = reflection vector • |g| = 0.25 – 1 Å-1 are structure-defining

29. Multislice Simulation Multislice simulations carried out using 1000 discrete tilts (8 shown) incoherently summed to produce the precession pattern1 • How to treat scattering? • Doyle-Turner (atomistic) • Full charge density string potential -- later 1 C.S. Own, W. Sinkler, L.D. Marks Acta Cryst A62 434 (2006)

30. Multislice Simulation: works (of course) R1 ~10% without refinement of anything

31. Global error metric: R1 • Broad clear global minimum – atom positions fixed • R-factor = 11.8% (experiment matches simulated known structure) • Compared to >30% from previous precession studies • Accurate thickness determination: • Average t ~ 41nm (very thick crystal for studying this material) (Own, Sinkler, & Marks, in preparation.)

32. Quantitative Benchmark:Multislice Simulation Bragg’s Law Absolute Error = simulation(t) - kinematical Error thickness g 10mrad 24mrad 0mrad 75mrad 50mrad Experimental dataset (Own, Sinkler, & Marks, in preparation.)

33. Partial Conclusions • Separable corrections fail; doing nothing is normally better • Two-beam correction is not bad (not wonderful) • Only correct model is full dynamical one (alas) • N.B., Other models, e.g. channelling, so far fail badly – the “right” approximation has not been found

34. Overview • Part II: What, if any generalizations can be made? • Role of Precession Angle • Systematic Row Limit • Importance of integration • Phase insensitivity • Important for which reflections are used

35. Role of Angle: Andalusite • Natural Mineral • Al2SiO5 • Orthorhombic (Pnnm) • a=7.7942 • b=7.8985 • c=5.559 • 32 atoms/unit cell • Sample Prep • Crush • Disperse on holey carbon film • Random Orientation 2 C.W.Burnham, Z. Krystall. 115, 269-290, 1961

36. Experimental Multislice Measured and Simulated Precession patterns Bragg’s Law Simulation Precession Off 6.5 mrad 13 mrad 18 mrad 24 mrad 32 mrad

37. Decay of Forbidden Reflections • Decay with increasing precession angle is exponential • The non-forbidden (002) reflections decays linearly Rate of decay is relatively invariant of the crystal thickness Slightly different from Jean-Paul’s & Paul’s – different case

38. Precession Projection of Incident beam Quasi-systematic row Precessed On-Zone Systematic Row Forbidden, Allowed by Double Diffraction

39. e- e-   Evidence: 2g allowed Si [110] Si [112] Rows of reflections arrowed should be absent (Fhkl = 0) Reflections still present along strong systematic rows Known breakdown of Blackman model

40. + + + + + + 36 + 24 12 0 mrad + Phase and Averaging For each of the incident condition generate 50 random phase sets {f} and calculate patterns: Single settings at 0, 12, 24 and 36 mrad (along arbitrary tilt direction). At fixed semi-angle (36 mrad) average over range of a as follows: 3 settings at intervals of 0.35º 5 settings at intervals of 0.35º 21 settings at intervals of 1º For each g and thickness compute avg., stdev:

41. Phase independence when averaged single setting 3 settings, 0.35º interval 5 settings, 0.35º interval 21 settings, 1º interval (approaching precession) Sinkler & Marks, 2009

42. Why does it work? • If the experimental Patterson Map is similar to the Bragg’s Law Patterson Map, the structure is solvable – Doug Dorset • If the deviations from Bragg’s Law are statistical and “small enough” the structure is solvable – LDM

43. Why is works - Intensity mapping • Iff Iobs(k1)>Iobs(k2) when Fkin(k1) > Fkin(k2) • Structure should be invertible (symbolic logic, triplets, flipping…)

44. Speculation…Full Method • Generate initial structure estimate • Intial Bragg’s Law Analysis or part of data

45. Two-Beam Integration: Ewald Sphere Small (hkl) Medium (hkl) Large (hkl)

46. Preprocess? “Good” Region Bragg’s Law (reference) Precession pattern (experiment) f= 24mrad

47. Speculation…Full Method • Generate initial structure estimate • Intial Bragg’s Law Analysis or part of data • Use 2-beam approximation to invert/correct • Partial refinement, using 1/2 kpoint Bloch-Wave method

48. Approximation of precession circuit by series of g-vector tilts: First approximation; single eigen-solution g-vector tilts obtained from 2nd approximation; two eigen-solutions g-vector tilts obtained from => increasingly accurate precession calculation

49. Precession calculations for(Ga,In)2SnO5, 36 mrad semi-angle, increasing numbers of eigensolutions. 2 1 4 8

50. Speculation…Full Method • Generate initial structure estimate • Intial Bragg’s Law Analysis or part of data • Use 2-beam approximation to invert/correct • Partial refinement, using 1/2 kpoint Bloch-Wave method • Final full refinement using 25-100 beam BW (+Bethe terms, e.g. Stadleman or NUMIS + dmnf, large residual code)