towards precession for dummies

Towards Precession for Dummies

L. D. Marks, J. Ciston, C. S. Own & W. Sinkler

(+ A couple of slides from Paul Midgley)

Scan

De-scan

Specimen

Conventional Diffraction Pattern

Precession…Precession

Diffraction Pattern

(Ga,In)2SnO5 Intensities

412Å crystal thickness

Non-precessedPrecessed

(Diffracted amplitudes)

PrecessionSystem

US patent application:“A hollow-cone electron diffraction system”.

Application serial number 60/531,641, Dec 2004.

Bi-polar push-pull circuit (H9000)

Projector Spiral Distortions (60 mRad tilt)

Some Practical Issues

Block Diagram

‘Aberrations’

e-e-

SampleSample

Objective Objective PostfieldPostfield

Objective Objective PrefieldPrefield

Better Diagram

Postfield Misaligned

Idealized Diagram

DescanDescan

ScanScan

Remember the optics

SampleSample

Objective Objective PostfieldPostfield

Objective Objective PrefieldPrefield

DescanDescan

ScanScan

Correct Diagram

Both Misaligned

Idealized Diagram

Remember the optics

One Consequence

Prefield/Postfield displacements of beamdPre = 1/(2) Pre(s-tPre); s = ScandPost=1/(2) Post(s-tPost) -sD sD = DeScan

(u)/(2) = z+Cs3

Total apparent displacement is the sumdNett = dPre+dPost

zPre(-Pre)+zPost(Post)

+ CsPre(-Pre)3+Cs

Post(Post)3

-sD

Probe and Displacements (nm)

Prefield misalignment 1 mRad; Postfield -1.0 mRad

-8

-6

-4

-2

0

2

4

6

8

-10 -8 -6 -4 -2 0 2 4 6

Displ on Sample

No Descan

Apparent Probe Shift Caveat: this ignores 3-fold astigmatism in pre/post field which is probably not appropriate, and any projector distortions

Overview

Part I: What is the theory of precession Early Models

1) Kinematical/Bragg’s Law2) Blackmann (2-beam all angles)3) Lorentz corrected Blackmann

More Recent Models4) 2-beam integrated correctly

Good Models5) Full Dynamical

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

Four basic elements are required to solve a recovery problem

1. A data formation modelImaging/Diffraction/Measurement

2. A priori informationThe 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

Ewald SphereConstruction

z 0 g

k

s

k+s

sz

Scan

De-scan

Specimen

(Diffracted

amplitudes)

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

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

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)

Early Models

Iobs depends upon |F(g)|, g, (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 integration2) Geometry plus multiplicative term for |F(g)|

GITO [010] Precession intensities, 3.2 A

Multislice intensity

0.0000 0.0001 0.0002 0.0003 0.0004

Kin

em

atic

al i

nte

nsi

ty

0.0000

0.0001

0.0002

0.0003

0.0004

GITO [010] Precession intensities, 50 A

Multislice intensity

0.000 0.005 0.010 0.015 0.020 0.025 0.030

Kin

em

atic

al i

nte

nsi

ty

0.00

0.02

0.04

0.06

GITO [010] Precession intensities, 100 A

Multislice intensity

0.000 0.005 0.010 0.015 0.020

Kin

em

atic

al i

nte

nsi

ty

0.00

0.02

0.04

0.06

0.08

0.10

GITO [010] Precession intensities, 800 A

Multislice intensity

0.000 0.002 0.004 0.006 0.008 0.010

Kin

em

atic

al i

nte

nsi

ty

0

2

4

6

8

10GITO [010] Precession intensities, 200 A

Multislice intensity

0.000 0.005 0.010 0.015 0.020 0.025 0.030

Kin

em

atic

al i

nte

nsi

ty

0.0

0.2

0.4

0.6

GITO [010] Precession intensities, 400 A

Multislice intensity

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018

Kin

em

atic

al i

nte

nsi

ty

0.0

0.5

1.0

1.5

2.0

2.5

3.2 Å; R=0.02 50 Å; R=0.19 100 Å; R=0.25

200 Å; R=0.49 400 Å*; R=0.78 800 Å; R=0.88

Bragg’s Law fails badly (Ga,In)2SnO5

Kinematical Lorentz Correction

I(g) = |F(g) sin(tsz)/(sz) |2 dsz

sz taken appropriately over the Precession Circuit

t is crystal thickness (column approximation)

is total precession angle

I(g) = |F(g)|2L(g,t,) 2

021,,

R

ggtgL

K. Gjønnes, Ultramicroscopy, 1997.

Kinematical Lorentz correction:Geometry information is insufficient

Need structure factors to apply the correction!

Fkin

Fco

rr

2-Beam (Blackman) form

)()( ;20

0 ktFkAdxxJtI g

A

Blackman

g

Limits:

Ag small; Idyn(k) Ikin(k)

Ag large; Idyn(k) Ikin(k) = |Fkin(k)|

But…This assumes integration over all angles, which is not correct for precession (correct for powder diffraction)

Blackman Form

Blackman+Lorentz

Alas, little better than kinematical

Comparison with full calculation, 24 mRad

Angstroms

Two-Beam Form

I(g) = |F(g) sin(tseffz)/(seff

z) |2 dsz

sz taken appropriately over the Precession Circuit

szeff = (sz

2 + 1/g2)1/2

Do the proper integration over sz (not rocket science)

g

Bcg F

V

cos

Two-Beam Integration: Ewald Sphere

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 45 90 135 180 225 270 315 360

Scan Angle

sz/ g

-1

-0.5

0

0.5

1

1.5

0 45 90 135 180 225 270 315 360

Scan Angle

sz/ g

0

2

4

6

8

10

12

0 45 90 135 180 225 270 315 360

Scan Angle

sz/ g

Small (hkl)

Large (hkl)Medium (hkl)

a) 3.2 Å, R=0.023.

Multislice intensities0.0000 0.0001 0.0002 0.0003

2-b

ea

m in

ten

sitie

s

0.0000

0.0001

0.0002

0.0003

c) 100 Å, R=0.236

Multislice intensities0.00 0.01 0.02 0.03 0.04

2-b

ea

m in

ten

sitie

s

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

d) 200 Å, R=0.444

Multislice intensities

0.000 0.005 0.010 0.015 0.020 0.025 0.030

2-b

ea

m in

ten

sitie

s

0.00

0.02

0.04

0.06

0.08

0.10

0.12e) 400 Å, R=0.626

Multislice intensities0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014

2-b

ea

m in

ten

sitie

s

0.00

0.02

0.04

0.06

0.08

0.10

f) 800Å, R=0.672

Multislice intensities

0.000 0.002 0.004 0.006 0.008

2-b

ea

m in

ten

sitie

s

0.00

0.02

0.04

0.06

0.08

0.10

b) 50 Å, R=0.186

Multislice intensities

0.000 0.005 0.010 0.015 0.020 0.025

2-b

ea

m in

ten

sitie

s

0.00

0.01

0.02

0.03

0.043.2 Å; R=0.02 50 Å; R=0.19 100 Å;R=0.24

200 Å; R=0.44 400 Å*; R=0.62 800 Å; R=0.67

2-Beam Integration better

Thickness [A]

0 200 400 600 800 1000

R-f

acto

r

0.0

0.2

0.4

0.6

0.8

1.0

12 mrad24 mrad36 mrad48 mrad60 mrad72 mrad

Thickness [A]

0 200 400 600 800 1000

R-f

acto

r

0.0

0.2

0.4

0.6

0.8

1.0

12 mrad24 mrad36 mrad48 mrad60 mrad72 mrad

R-factors for 2-beam model R-factors for kinematical model

See Sinkler, Own and Marks, Ultramic. 107 (2007)

Some numbers

Fully Dynamical: Multislice

“Conventional” multislice (NUMIS code, on cvs) Integrate over different incident directions 100-

1000 tilts = 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

2

t

Multislice Simulation

Multislice simulations carried out using 1000 discrete tilts (8 shown) incoherently summed to produce the precession pattern1

How to treat scattering?

1) Doyle-Turner (atomistic)

2) Full charge density string potential -- later

1 C.S. Own, W. Sinkler, L.D. Marks Acta Cryst A62 434 (2006)

Multislice Simulation: works (of course)

R1 ~10% without refinement of anything

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.)

R-factor, (Ga,In)2SnO4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 250 500 750 1000 1250 1500

t (Å)

R-f

ac

tor

On Zone

Precession

exp

exp

1 F

FFR

sim

0mrad10mrad24mrad75mrad

Quantitative Benchmark:

Multislice Simulation

Experimental dataset

Error

g

thickness

(Own, Sinkler, & Marks, in preparation.)

Absolute Error = simulation(t) - kinematical

50mrad

Bragg’s Law

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

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

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

Bragg’s Law Simulation

Precession Off6.5 mrad13 mrad18 mrad24 mrad32 mrad

Experimental

Multislice

Measured and Simulated Precession patterns

Decay of Forbidden Reflections Decay with increasing

precession angle is exponential– The non-forbidden (002)

reflections decays linearly

D(001) D(003)

Experimental 0.109

R2=0.991

0.145

R2=0.999

Simulated (102nm)

0.112

R2=0.986

0.139

R2=0.963

Simulated

28-126 nm

0.112±0.012 0.164±0.015

Rate of decay is relatively invariant of the crystal thickness

Slightly different from Jean-Paul’s & Paul’s – different case

)exp( DAI

Projection of

Incident beam

Precession

Quasi-systematic row

Systematic Row

Forbidden, Allowed by Double Diffraction

On-ZonePrecessed

Rows of reflections arrowed should be absent (Fhkl = 0)

Reflections still present along strong systematic rows

Known breakdown of Blackman model

Si [110]

Si [112]

e-e-

Evidence: 2g allowed

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 0.35º5 settings at intervals of 0.35º21 settings at intervals of 1º

For each g and thickness compute avg., stdev:

}{

),(50

1),(

tItI gg2

1

22

}{

),(),(50

1),(

tItIt ggg

+ + + +0 mrad 12 24

36++

+

+

Phase and Averaging

Single setting calculations

Thickness [A]

0 500 1000 1500 2000

Ave

rage

of

rela

tive

stan

dar

d de

via

tion

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1 setting, 0 mrad 1 setting, 12 mrad 1 setting, 24 mrad 1 setting, 36 mrad

Multiple settings calculations at =36 mrad

Thickness [A]

0 500 1000 1500 2000

Ave

rage

of

rela

tive

sta

nda

rd d

evia

tion

0.0

0.2

0.4

0.6

0.8

1.0

1.2

single setting

3 settings, 0.35º interval

5 settings, 0.35º interval

21 settings, 1º interval (approaching precession)

Phase independence when averaged

Sinkler & Marks, 2009

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

Why is works - Intensity mapping

Iff Iobs(k1)>Iobs(k2) when Fkin(k1) > Fkin(k2)

– Structure should be invertible (symbolic logic, triplets, flipping…)

Speculation…Full Method

Generate initial structure estimate– Intial Bragg’s Law Analysis or part of data

Two-Beam Integration: Ewald Sphere

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 45 90 135 180 225 270 315 360

Scan Angle

sz/ g

-1

-0.5

0

0.5

1

1.5

0 45 90 135 180 225 270 315 360

Scan Angle

sz/ g

0

2

4

6

8

10

12

0 45 90 135 180 225 270 315 360

Scan Angle

sz/ g

Small (hkl)

Large (hkl)Medium (hkl)

Preprocess?

Bragg’s Law (reference)

“Good” Region

Precession pattern (experiment)= 24mrad

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

Approximation of precession circuit by series of g-vector tilts:

g-vector tilts obtained from

g-vector tilts obtained from

First approximation; single eigen-solution

2nd approximation; two eigen-solutions

=> increasingly accurate precession calculation

Bloch, exact precession circuit

0.000 0.002 0.004 0.006 0.008 0.010 0.012

Fro

m S

-mat

rix c

olum

ns,

sing

le E

igen

valu

e ca

lc.

0.000

0.002

0.004

0.006

0.008

0.010

0.012

R-Factor agreement between exact precession circuit(1024 settings) and approximation using increasing numbers

of Bloch calculations in 1st Brillouin Zone

Thickness [A]0 500 1000 1500 2000

R-F

acto

r

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

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

21

4

8

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)

Hypothesis: Thickness ranges

(not so easy)

Bragg’s Law

(pseudo-Bragg’s Law)

Summary

PED is– Approximately Invertable– Pseudo Bragg’s Law– Approximately 2-beam, but not great– Properly Explained by Dynamical Theory

PED is not– Pseudo-Kinematical (this is different!)– Fully Understood by Dummies, yet

Questions?