Precession Diffraction: The Philosphers Stone of Electron Crystallography?1FocusMany methods exist for obtaining diffraction informationSelected AreaNanodiffraction and variantsCBEDAll are complicated to interpretReciprocal space is right, but intensities depend upon thickness, tilt etcWhat PED can doWe would like a method where not just the positions of the spots, but also the intensities could be used.Not rigorously equivalent to simple kinematical diffraction, but has many similaritiesIf the structure factor is large Intensity is largeUseful for fingerprinting structuresOften does not need calculations to interpret3

History Electron Precession (1993)

e-e-Advantages:4

ScanDe-scanSpecimenConventional Diffraction PatternPrecessionPrecession Diffraction Pattern(Ga,In)2SnO5 Intensities412 crystal thickness

Non-precessed

Precessed(Diffracted amplitudes)5PrecessionSystem

US patent application:A hollow-cone electron diffraction system. Application serial number 60/531,641, Dec 2004.6

Can be easily retrofitable to any TEM 100- 300 KV

precession is possible for any beam size 300 - 50 nm

Precession is possible for a parallel or convergent beam

precession eliminates false spots to ED pattern that belong to dynamical contributions

precession angle can vary continuously (0-3) to observe true crystallographic symmetry variation

Software ELD for easy quantification of ED intensities and automatic symmetry ( point, space group ) research

SPINNING STAR : UNIVERSAL INTERFASE FOR PRECESSION ELECTRON DIFFRACTION FOR ANY TEM ( 120 -200 -300 KV ) Easily interfaced to electron diffractometer for automatic 3D structure determination

7Examples:Complicated StructuresHard to interpret SAEDSimple to interpret PEDEDSElemental ratios depend upon orientation in standard modeWeak to no dependence with PED

APPLICATION : FIND TRUE CRYSTAL SYMMETRY PRECESSION OFF IDEAL KINEMATICAL (111)

UVAROVITE (111)

PRECESSION ON

Courtesy M.Gemmi Univ of Milano9

APPLICATION : PERFECT CRYSTAL ORIENTATION PRECESSION OFFPRECESSION ONCrystals specially minerals -usually grow in platelet or fiber shape and results dificult to orient perfectly in a particular zone axis; in this example olivine crystals are perfectly oriented after precession is on.

OLIVINE

Courtesy X.Zou, S.Hovmoller Univ Stockholm10

Carbide

EDS, on zone (SrTiO3)RepeatMeasurementsPractical UseTwo commercial systems (one hardware, another software) are availableNot complicated, and could probably be written in scripting languageAlignment can be tricky it always isNot rocket science, to use

Bi-polar push-pull circuit (H9000)Projector Spiral Distortions (60 mRad tilt)Some Practical Issues14

Block DiagramAberrationse-e-15SampleObjective PostfieldObjective PrefieldBetter DiagramPostfield MisalignedIdealized DiagramDescanScanRemember the optics16SampleObjective PostfieldObjective PrefieldDescanScanCorrect DiagramBoth MisalignedIdealized DiagramRemember the optics17

Alignment can be trickyWhy?Although PED has been around since 1992, and very actively used for ~10 years (mainly in Europe), there is no simple explanation (many have tried and failed)Explanation is a bit rocket scienceWhy?What, if any generalizations can be made?Role of Precession AngleSystematic Row LimitImportance of integration Phase insensitivityImportant for which reflections are usedFast Integration Options

20Ewald SphereConstruction

z0gksk+ssz

ScanDe-scanSpecimen(Diffracted amplitudes)21Levels of theoryPrecession integrates each beam over szFull dynamical theoryAll reciprocal lattice vectors are coupled and not seperablePartial dynamical theory (2-beam) Consider each reciprocal lattice vector dynamically coupled to transmitted beam onlyKinematical theoryConsider only role of sz assuming weak scatteringBraggs LawI = |F(g)|222Early ModelsIobs depends upon |F(g)|, g, f (precession angle) which we correct to the true resultOptions:0) No correction at all, I=|F(g)|2 1) Geometry only (Lorentz, by analogy to x-ray diffraction) corresponds to angular integration2) Geometry plus multiplicative term for |F(g)|23

3.2 ; R=0.0250 ; R=0.19100 ; R=0.25200 ; R=0.49400 *; R=0.78800 ; R=0.88Braggs Law fails badly (Ga,In)2SnO524Kinematical Lorentz CorrectionI(g) = |F(g) sin(ptsz)/(psz) |2 dszsz taken appropriately over the Precession Circuitt is crystal thickness (column approximation) f is total precession angleI(g) = |F(g)|2L(g,t,f)

K. Gjnnes, Ultramicroscopy, 1997.25Kinematical Lorentz correction:Geometry information is insufficient

Need structure factors to apply the correction!FkinFcorr262-Beam (Blackman) form

Limits:Ag small; Idyn(k) Ikin(k)Ag large; Idyn(k) Ikin(k) = |Fkin(k)|ButThis assumes integration over all angles, which is not correct for precession (correct for powder diffraction)

27Blackman Form

28Blackman+LorentzAlas, little better than kinematicalComparison with full calculation, 24 mRadAngstroms29Two-Beam FormI(g) = | F(g) sin(ptseffz)/(pseffz) |2 dszsz taken appropriately over the Precession Circuitszeff = (sz2 + 1/xg2)1/2

Do the proper integration over sz (not rocket science)

30Two-Beam Integration: Ewald Sphere

Small (hkl)Large (hkl)Medium (hkl)31

3.2 ; R=0.0250 ; R=0.19100 ;R=0.24200 ; R=0.44400 *; R=0.62800 ; R=0.672-Beam Integration better32

R-factors for 2-beam modelR-factors for kinematical modelSee Sinkler, Own and Marks, Ultramic. 107 (2007)Some numbers33Fully Dynamical: Multislice

Conventional multislice (NUMIS code, on cvs)Integrate over different incident directions 100-1000 tiltsf = cone semi-angle0 50 mrad typicalt = thickness~20 50 nm typicalExplore: 4 150 nmg = reflection vector|g| = 0.25 1 -1 are structure-defining2ft34Multislice Simulation

Multislice simulations carried out using 1000 discrete tilts (8 shown) incoherently summed to produce the precession pattern1How to treat scattering?Doyle-Turner (atomistic)Full charge density string potential -- later1 C.S. Own, W. Sinkler, L.D. Marks Acta Cryst A62 434 (2006)35Multislice Simulation: works (of course)

R1 ~10% without refinement of anything36Global error metric: R1

Broad clear global minimum atom positions fixedR-factor = 11.8% (experiment matches simulated known structure)Compared to >30% from previous precession studiesAccurate thickness determination:Average t ~ 41nm (very thick crystal for studying this material)

(Own, Sinkler, & Marks, in preparation.)37Quantitative Benchmark:Multislice Simulation

0mrad10mrad24mrad75mradExperimental datasetErrorgthickness(Own, Sinkler, & Marks, in preparation.)Absolute Error = simulation(t) - kinematical50mradBraggs Law38Partial ConclusionsSeparable corrections fail; doing nothing is normally betterTwo-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 found39OverviewWhat, if any generalizations can be made?Role of Precession AngleSystematic Row LimitImportance of integration Phase insensitivityImportant for which reflections are used

40Role of Angle: AndalusiteNatural MineralAl2SiO5Orthorhombic (Pnnm)a=7.7942b=7.8985c=5.55932 atoms/unit cellSample PrepCrushm Disperse on holey carbon filmRandom Orientation

2 C.W.Burnham, Z. Krystall. 115, 269-290, 196141Measured and Simulated Precession patterns

Braggs Law SimulationPrecession Off6.5 mrad13 mrad18 mrad24 mrad32 mradExperimentalMultislice

42Decay of Forbidden ReflectionsDecay with increasing precession angle is exponentialThe non-forbidden (002) reflections decays linearly

D(001)D(003)Experimental0.109R2=0.9910.145R2=0.999Simulated (102nm)0.112R2=0.9860.139R2=0.963Simulated28-126 nm0.1120.0120.1640.015Rate of decay is relatively invariant of the crystal thicknessSlightly different from Jean-Pauls & Pauls different case

43

Projection of Incident beamPrecession Quasi-systematic rowSystematic Row

Forbidden, Allowed by Double DiffractionOn-ZonePrecessed44

Rows of reflections arrowed should be absent (Fhkl = 0)Reflections still present along strong systematic rowsKnown breakdown of Blackman modelSi [110]Si [112]e-e-Evidence: 2g allowed45For 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 0.355 settings at intervals of 0.3521 settings at intervals of 1

For each g and thickness compute avg., stdev:

++++0 mrad122436++++Phase and Averaging46

single setting3 settings, 0.35 interval5 settings, 0.35 interval21 settings, 1 interval (approaching precession)Phase independence when averagedSinkler & Marks, 200947Why does it work?If the experimental Patterson Map is similar to the Braggs Law Patterson Map, the structure is solvable Doug DorsetIf the deviations from Braggs Law are statistical and small enough the structure is solvable LDM48Why is works - Intensity mappingIff Iobs(k1)>Iobs(k2) when Fkin(k1) > Fkin(k2)Structure should be invertible (symbolic logic, triplets, flipping)49SummaryPED isApproximately InvertablePseudo Braggs LawApproximately 2-beam, but not greatProperly Explained by Dynamical TheoryPED is notPseudo-Kinematical (this is different!)Fully Understood by Dummies, yet50Questions ?51