How to solve a structure - Laurence Marks Research Group to solve a... · 2019-09-26 · How to...
Transcript of How to solve a structure - Laurence Marks Research Group to solve a... · 2019-09-26 · How to...
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How to solve a structure?
Guess, then refine– Will always give something, but if the guess is wrong GIGO
Use Patterson function– Difficult for complicated structures (in a moment)
Get an image– STM is hard to interpret– HREM, can be ambiguous
Use DFT– If the original guess is wrong, GIGO– Functionals can be inaccurate for TMO’s
Try something else?
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(x, y, z)
xy
(u v w)
uv
(u v w)
Patterson Function I (FT of Diffraction Pattern)
rdurruP
ehklIV
uvwPhkl
kwlvhui
3
)(2
)()()(
)(1)(
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xy uv
Patterson map will contain points corresponding to vectors between atoms in the real cell
Real Cell Patterson Cell
Patterson Function II
Solids normally contain well-separated atoms, and majority of scattering is near the core -- peaked
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Real Cell Patterson Cell
1) Patterson is symmetric about origin (centrosymmetry)
3) Contains N(N-1) peaks (not counting origin) gets complicated!2) Can see pattern of real cell in Patterson cell repeated N times
Patterson Function III
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Basics
We know the amplitudes We want to find the phases Problem is insolvable without additional
information – constraints Use an iterative approach
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F(h) = |F(h)|exp[2i(h)]FFTSurface (r)
Unmeasured
Measured diffraction intensities|F(h)| = [I(h)]1/2
Diffraction Phase Problem
An equal opportunity problem – true for x-ray and electron diffraction
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Phase: Apples & Oranges
FT Aa exp(-i a)
Ao exp(-i a) IFT
FT Ao exp(-i o)+
{ Oranle ?Appge ?
Phase of Apple + Amplitude of Orange = ?
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Phase of Apple = Apple
FT-1 {Ao exp(-i a) } Apple
Phase is more important than amplitude
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The importance of phase information
Correct Modulus Random Phases
Correct Phase Random Modulus
Suzy
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Role of error in phases (degrees)
0 10 20
30 40 50
We would like to find the phases exactly, but we don’t have to
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Phase and Modulus Errors0 10° 20° 30° 40°
10
R 0 26% 52% 78% 104%
Phase Error
Modulus Error
We only need approximately correct phases We can tolerate modulus errors
Modulus Correct
Phase Correct
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Demonstration: how resolution works in reciprocal space: If we can add beams at large distance from center of patterns with the correct phase, we can reconstruct the structure with very high definition:
Example: simple test structure of repeated molecule
Computed diffraction pattern of structure at left.
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Low Resolution …
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Higher resolution …
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Higher resolution …
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Higher resolution …
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Higher resolution …
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This is the goal of direct methods. Given measurement of amplitudes, obtain phases using educated guesswork. As illustrated, good phases give accurate representation of structure.
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Direct Methods vs. Indirect Methods
Indirect Methods:“Trial and Error”
Direct Methods:Using available informationto find solutions
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Infinite Number of Possible Arrangements of
Atoms
Finite
R2, structure, DFT and chemistry
Direct Methods
Implementation
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Caveat: Not Physics
This is probability, not an exact “answer”
All one can say is that the “correct” answer will be among those that are found
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What do D.M. give us
With the moon in the right quarter -- real space potential/charge density
In other cases:– Atom positions may be wrong (0.1-0.2 Å)– Peak Heights may be wrong– Too many (or too few) atoms visible
But... this is often (not always) enough to complete the structure
Chris Gimore
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Additional Information Available
Physical nature of experiment– Limited beam or object size
Physical nature of scattering– Atomic scattering
Statistics & Probability– Minimum Information/Bias = Maximum
Entropy
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Symmetry
Has to be determined a-priori– CBED– HREM (maybe)– Spot Pattern (can be tricky)
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First Step: Origin Definition
Not all phases are unknown– Translating the crystal has no physical
significance– Can therefore fix an origin for the crystal –
equivalent to fixing certain reflections– Relevant for crystallographic phase (not
absolute phase of wavefunction which is not important)
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Origin Definition c2mm
Mirror Planes
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Origin Definition c2mm
eam Defined
Origin Definition c2mm
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Origin Definition p2mm
& (10) eam Defined
Origin Definition c2mm
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Next Steps: Basic Ideas
There are certain relationships which range from exact to probably correct.
Simple case, Unitary Sayre Equation, 1 type
Divide by N, #atoms & f(k), atomic scattering factors
Constraint
l
l )ik.r(f(k)kF 2exp)(
2)()(
)(1)( );.2exp(1)(
rNuru
rrNrurikNkUl l
ll
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Real/Reciprocal Space
-2 1
U(r)
-
18
1
U(r)2
U( ) U( )U( )h k h kk
U(r) U(r)2
Reinforces strong (atom-like) features
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Cochran Distribution (2): I
Definition: Consider the product
If the atoms are randomly distributed,
(exponential terms average to zero if m l)
l
lmm
m rrihrikNhUhkNU )).(2exp().2exp()1()()(
1)).(2exp( l
lm rrih
)().2exp()1()()( kUrikNhUhkUNm
m
).2exp()1()( m
mrikNkU
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Cochran Distribution: II
Consider next Average is zero
Known
Known Average must be 2n
)()()(cos2
)()()(222
2
hhkkh)U(h)|N|U(k)U(k--)||U(k-h)U(h)|2N|U(k)|
kUhUhkNU
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Cochran Distribution: III
We have a distribution of values. The Central Limit theorem: all distributions tend towards Gaussian. Hence a probability:
P(U(k) - NU(k-h)U(h)) ~ Cexp(-|U(k) - NU(k-h)U(h)|2)~ Cexp(2|U(k)U(k-h)U(h)|cos[(k)- (k-h)- (h)])
Compare to exp(-x2/22)– 2 = 1/4|U(k)U(k-h)U(h)|
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Form of Distribution
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
-180 -120 -60 0 60 120 180
Degrees
Strong Beams
Weak Beams
Note: this is more statistics than the presence of atoms[(k)- (k-h)- (h)]
Probability
~ 1/2|U(k)U(k-h)U(h)|
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2 Triplet
For reflections h-k, k and h:(h) (k) + (h-k)
W. Cochran (1955). Acta. Cryst. 8 473-8.
1
h-k
k h
= known structure amplitude and phase= known structure amplitude and unknown phase
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Example: Si(111) 3x3 Au
3 ~ 360n degrees =0,120 or 240 =0 has only 1 atom 120 or 240 have 3
Only one strong reflection
(220)
(1,0)
Other information3 Au
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Inequalities
| aibi|2 < |ai|2 |bi|2 (Triangle Inequality)ai = 1/sqrt(N)cos(2kri) ; bi = 1/sqrt(N) aibi = U(k) |bi|2 = 1/N = 1 for N atoms |ai|2 = 1/N cos(2kri)2
= 1/2N (1+cos(2[2k]ri) ) = ½ + U(2k)
Hence U2(k) < ½ +U(2k)/2If U(k) is large – can set U(2k)
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Quartets
Phase relationships involving 4 terms for weak reflections– Positive and Negative– Very useful for x-ray diffraction– Rarely useful with TEM; dynamical effects can
make weak reflections stronger than they should be
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More subtle statistics
Better statistics (Information Theory) Entropy of a distribution is more
fundamental (as is Kullback-Liebler or relative entropy)
Most probable distribution maximizes entropy
S = - u(r) ln u(r) dr
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Last step - Refinement
Fit atom positions via:– Rn = |Icalc-Iexpt|n/ Iexpt
n (or Fcalc, Fexpt)– n = |Icalc-Iexpt|n/n
– n=1 for Robust Estimation Should use dynamical Icalc for electrons R1 < 0.01 for most x-ray structures, < 0.1
currently for TED. R1~0.5 for random variables
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Crystallographic Direct Methods Structure TriangleData
TrueStructure
Ideal World
TrialStructure
Direct Methods Map or Image
Structure Completion (non-trivial)
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Implementation
1. Chose phases to define origin2. Guess phases for some reflections3. Generate from these phases for others and
improved phases for initial set4. Test consistency of predicted amplitudes and
phases5. Iterate, so long as consistency is improving
Note: permuting phases has lower dimensions than permuting atom positions
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General Formalism as dual
1. Initial (r)2. Project onto “Real Space Constraint” 2(r)3. FFT4. Project amplitudes onto Observed5. FFT
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In Reciprocal Space: Tangent Formula
If U(r) = U(r)2 = U'( r) Important part is the phase U(u) = |U(u)|exp(i); we know |U(u)| but
not exp(i) = exp(i'); Tan() = Tan(') Replace old by new one
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Algorithm Overview (Gerschberg-Saxton)
Feasible Solution
Fourier Transform
InverseFourier Transform
Impose Fourier space
constraints (S2)
Impose real spaceconstraints (S1)
Observed Intensities(assigned phases)(Global Search)
Recovery Criterion
YES
NO
Atoms
Intensities
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More: 1970’s Mathematics
C -- Some constraints (e.g. atomicity, probabilities of triplets)
F -- Some function (e.g. a FOM) Minimize, e.g. Lagrangian
I = F + C
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1990’s Mathematics
We have constraints (e.g. atomicity, amplitudes)– Treat as sets
We are looking for the solution as intersection of several constraint sets
Amplitudes
Atoms
Positivity
Acta Cryst A55, 601 (1999)
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The $64,000 question
A set is convex if any point between two members is also a member– If all the sets are convex, problem has one solution– If they are not, there may be more than one local
minimum Amplitude measurements
do not form a convex set But…there still may only
be one solution. Unsolved mathematical problem
CF(k)|=const
B
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Multiple non-convex constraints
Overall Convex Overall Non-Convex
Consider the two sets “N” and “U”
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Crystallographic methodology
Overall Non-Convex
Overall Unique
Addition of additional convex constraints tends to give a unique solution
Structure Completion: add additional constraints as the phases become known
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Im
Re0,0
Orthogonal Projections
Im
Re0,0
Estimate
New Value
Modulus Only Part of U(k) known
Project: closest point in set
|U(k)| |U(k)|
Known
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Successive Projections
Iterate between projections
Other variants possible (see Combettes,Advances in Imaging and Electron Physics95, 155-270, 1996)
Set of |Uobs(k)|exp(i(k))
Set of U(k) that satisfysome constraints
Set of all U(k)
Start
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Over-relaxed Projections
Iterate between projections
Overshoot (deliberately)
Converges faster Sometimes better
solutions
Set of |Uobs(k)|exp(i(k))
Set of U(k) that satisfiesatomistic constraints
Set of all U(k)
Start
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Classic Direct Methods
Consider as an iterationUn(k) un(r)
Constraint Constraint
U’(k) un2(k)
Note the similarities– Tangent Formula Orthogonal Projection– Real space operator, effectively an eigenfunction (fixed point)
method
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ProbabilityContours
{S1: | F {x}|=|Xe|}
Set with some probability
Multiply-Connected Feasible Set
Three shaded regions common to both sets, 3 unique solutions
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Typical results3D Calibration Test (In 4x1 Model)
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0 0.05 0.1 0.15 0.2
Algorithm Figure of Merit
True
Acc
urac
y of
Pha
ses
A
B
C
Correct
Incorrect
= phase errorU(k)|{1-cos()}
U(k)|
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Types of Constraints
Convex – highly convergent– Multiple convex constraints are unique
Non-convex – weakly convergent– Multiple non-convex constraints may not be
unique
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More Constraints
SymmetrySupport for gradient
Bond anglesStatistics (e.g. 2)Anti-bumpingIntensities & errors
InterferenceA(k)=| B(k)+Known(k)|2
Least bias (MaxEnt)Bond LengthsAtoms at given positionsPresence of AtomsPositivity (weak)
Convex Non-Convex
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Atomistic Constraints
(r) known (convex if position is known)
Bonding –another atom
Bumping (r)=0
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Example I: Difference Map
We know all the moduli, |F(k)| We know part of the structure,
Fa(k) = |Fa(k)|exp(ia(k)) Project onto known moduli
D(k) = exp(ia(k)){|F(k)obs|-|Fa(k)|}Conventional Fourier Difference
Map Other methods (SIM wts) equivalent
to further projections.
D
Fa
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Operators as projections
Some operator O, apply to some current estimate (x in real space, X in reciprocal space)
Define a set for the cases where<O(x)-x> < some number
New estimate obtained by the iterationxn+1 = O(xn)
N.B., there are some important formal mathematical issues…..
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Example II: Sayre Equation
Use O(x) x2 ; = scaling term Couple with known moduli as second set Iteration
– xn+1 = O(xn)= xn2
– |Xn+1| = |Xobserved| This is the Sayre equation (and tangent
formula)
Iterate
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Example III: Structure Completion
Explanation (pseudo-mathematical) of why structure completion strategies can solve, uniquely, problems when the initial maps are not so good
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Overall Convex
Structure Completion
Add a third set “O” Addition of additional
constraints tends to give a unique solution
Structure Completion: add additional constraints as the atoms become known
Consider the two non-convex sets “N” and “U”
Solution
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IV Convex Set for unmeasured |U(h,k,l)|
Phase of U(h,k,l) can be estimated from other reflections
Set of U(h,k,l) with a given phase is convex Hence |U(h,k,l)| is well
specified and can be (approximately) recovered
Remember, phase is more important than amplitude
C
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Support Constraint
Displacements decay as (+z)exp(-qz) into bulk1
Real space constraint– (z)=(z)w(z) w(z)=1, -L<z<L
=0, otherwise Convex constraint Has well documented properties
PRB 60, 2771 (1999)
(z)=0
(z)0
(z)=0
Biharmonic expansion of strain field, SS 294, 324 (1993)
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Unmeasured ReflectionsRecovery of Unmeasured Reflections
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0 5 10 15 20 25 30 35 40 45
l value
|U(0
,4,l)
|
TrueCalculated
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Restoration and Extension
0.3nm Image
+DP
0.05nm Image
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Example V: Diffractive Imaging
True diffraction pattern for small particle model(Non-Convex Constraint)
++ == ??Convex SupportConstraint
|(x,y)|=1
|(x,y)|=?
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Example V: Diffractive imaging
Constraint: part of real-space x is zero Convex constraint Iteration
– x = 0, part of map– |X| = |Xobserved|
Iterate
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Example V: Diffractive Imaging
True diffraction pattern for small particle model(Non-Convex Constraint)
++ == ??Convex SupportConstraint
|(x,y)|=1
|(x,y)|=?
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Example V: Diffractive imaging
Constraint: part of real-space x is zero Convex constraint Iteration
– x = 0, part of map– |X| = |Xobserved|
Iterate
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The flow chart of hybrid input and output algorithm for iterative phase retrieval (after Millane and Stroud, 1997).
Calculate real-space constrained image Cn
Apply Equation
Calculate amplitudes and phases; replace with experimental amplitudes
Inverse Fourier Transform
Forward Fourier Transform
fn
Cngn
gn+1Gn
G’n
rfrCif
rfrCifrfrCrg
rfrg
nn
nn
nnn
nn
1
The Algorithm
Charge flipping(Wu and Spence)
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0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
DW-4 Seed = 211
R-fa
ctor
Iteration
HIO-W
HIO HIO HIO HIO
ER-W
ERER
ER ER
HIO
ER-Error ReductionHIO-Hybrid Input OutputW-Whole Pattern
Convergence and the Missing Central Beam
R
100%Exp
Exp
F FR
F
• Missing central beam from IP saturation• Use low mag. TEM image • Reconstruction start with the whole pattern• Finish with as recorded diffraction pattern
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Phase Recovery for a Small Particle
Reconstructed exit wave after 3000 iterations
True real space exit wave for small particle model
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Electron Nanoprobe formation
e-
nm
0
200
400
600
800
e-
0 10 20 30 40nm
Lower ObjectiveSpecimen
Condenser Lens III
Back Focal Plane
10 m aperture -> 50 nm beamM = 1/200
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Coherence length > 15 nmConvergence angle <0.2 mrad
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BL29XUL, SPring-8
Coherent Diffraction Pattern
Coherent X
-rays
Sample
Unstained Human Chromosome
Fourier Transform
Phase Retrieval
Coherent X-ray Diffraction
Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, submitted (2008).
Sample ImageReconstruction
without the aid of Lenses
22 )(KFPrdd
e
rrK rK deF i )()(
Structure Factor
Differential Cross Section
(r) : Electron Density Distribution
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From 2D to 3D
-60º -30º 30º 60ºexposure time at each incident angle: 2700 s
Coherent diffraction measurement at 38 incident anglesfrom -70º to 70º at 2.5º intervals at the minimum
• normalize the diffraction data by using the total number of electrons in the 2D reconstruction
• use interpolation to obtain diffraction intensity in each voxel
• image reconstruction using 3D Fourier transformation
J. Miao, T. Ishikawa, B. Johnson, E.H. Anderson,B. Lai & K.O. Hodgson, PRL 89, 088303 (2002)
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Reconstructed Si structure
Nano structures can be reconstructed with atomic resolutionby electron diffractive imaging using SAND
SAND pattern
200 spot
・ Intensity ratio of 200 and the direct spots → thickness : 4 ~ 8 nm
direct spotamplitude phase
Multi-slice simulatedwave fields
reconstructed wave field
0.136 nm
amplitude・ Dumbbell structure with the separation of 0.136 nm is resolved clearly
→ We succeeded in reconstructing dumbbell structure in silicon
・ Lattice fringes can be seen, but dumbbell structure is not reconstructed
phase
4nm 6nm 8nmamp
phase