towards precession for dummies
DESCRIPTION
Towards Precession for Dummies. L. D. Marks, J. Ciston, C. S. Own & W. Sinkler (+ A couple of slides from Paul Midgley). Scan. Non-precessed. Precessed. De-scan. Specimen. (Ga,In) 2 SnO 5 Intensities 412Å crystal thickness. Conventional Diffraction Pattern. - PowerPoint PPT PresentationTRANSCRIPT
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?