Solving crystals structures from HREM by crystallographic image processing

Solving crystals structures from HREM by crystallographic image processing

Xiaodong Zou

Structural Chemistry, Stockholm University

SAED intensities changes with crystal thicknessSAED intensities changes with crystal thickness

-AlFeCr: P63/m, a = 40.687, c = 12.546 Å

Why structure determination by Why structure determination by electron microscopy is possibleelectron microscopy is possible

1

r1

2r2rj

j

Projected Electrostatic PotentialCrystal Structure

( ) ( )jj

jr r rN

1

F( ) f ( )exp ( )j jj

N

u u u r 2

1

i

r

The crystal potential:

The structure factor of a crystal for electrons

The projected potential (in 2D):

( ) ( )jj

jr r rN

1

F( ) f ( )exp ( )j jj

N

u u u r 2

1

i

p 0

t

c Fk=-h=-

xy xyz dzt

F (hk0) (hk0) 2 (hx ky)( ) ( ) cos

Fc(hk0) - the structure factor amplitudes

F(hk0) – the structure factor phases

  Non-centrosymmetric Centrosymmetric rj = (xj yj zj) rj = (xj yj zj) (-xj –yj -zj)

Phases can be any angles Phases must be 0 or 180 degreesThis is valid only when the origin is at the center of symmetry

F(h)

Imaginary

Real

f1

f1'

f2 f2

'

f3

f3'

f4

f4'

(h)

f4

Imaginary

F(h)

Real

f1

f2

f3

(h)

Fourier synthesis of the crystal structure factor F(u)Fourier synthesis of the crystal structure factor F(u)

F( ) f ( )exp ( )j jj

N

u u u r 2

1

i

Structure factor CalculatedNo. h k l Fc(hkl)

F(hkl)

1 0 6 0 6031 1802 3 5 0 5981 1803 5 1 0 5570 04 5 2 0 4751 05 1 7 0 4262 06 4 0 0 4189 1807 3 6 0 3782 1808 5 3 0 3705 1809 5 4 0 3315 0

10 2 6 0 3285 180… … …26 3 1 0 1074 18027 4 1 0 1039 18028 2 0 0 1003 029 2 2 0 1000 180… … …34 2 4 0 712 180

35 5 5 0 691 036 1 4 0 688 18037 1 3 0 609 18038 3 7 0 559 18039 0 2 0 521 18040 2 7 0 405 041 0 4 0 387 18042 2 1 0 348 043 1 1 0 161 180

F( ) f ( )exp ( )j jj

N

u u u r 2

1

i

Zr2Se –Zr2Se – PnnmPnnma = 12.640, b = 15.79a = 12.640, b = 15.7977 c = 3.60 c = 3.6022 Å Å

Structure factor CalculatedNo. h k l Fc(hkl)

F(hkl)

1 0 6 0 6031 1802 3 5 0 5981 1803 5 1 0 5570 04 5 2 0 4751 05 1 7 0 4262 06 4 0 0 4189 1807 3 6 0 3782 1808 5 3 0 3705 1809 5 4 0 3315 0

10 2 6 0 3285 180… … …26 3 1 0 1074 18027 4 1 0 1039 18028 2 0 0 1003 029 2 2 0 1000 180… … …34 2 4 0 712 180

35 5 5 0 691 036 1 4 0 688 18037 1 3 0 609 18038 3 7 0 559 18039 0 2 0 521 18040 2 7 0 405 041 0 4 0 387 18042 2 1 0 348 043 1 1 0 161 180

(0 6 0) (6031 180 ) (5 2 0) (4751 0 ) (5 -2 0) (4751 180 )

(3 5 0) + (3 -5 0) = (3 ±5 0) (0 6 0)+(3 5 0) (0 6 0)+(3 5 0)+(5 2 0)

f+(5 1 0)+(4 0 0)+(1 7 0) 14 strongest reflections 43 reflections up to 1 .95 Å

Zr2SeZr2Se

HREM images may be distorted by:

Electron optics – defocus & astigmatism

Crystal misalignment

Dynamic effects

The distortions may be compensated by crystallographic image processing

Crystallographic image processing by CRISPCRISP

Fourier transform

Lattice refinement

Symmetry determination & Origin refinement

p4g

Symmetry imposed (p4g)

Experiment image

Amplitudes

Phases

Determine coordinates

I ( ) ) * ( )im xy xy xy 1 2N T(z

Relation between HREM Images & Structure ProjectionsRelation between HREM Images & Structure Projections

These relations are based on kinematical scattering and are only valid for very thin crystals (weak phase object). For thicker crystals, dynamical calculation is needed.

Image intensity (real space):

Fourier transform of image intensity (reciprocal space):I ( ) ( ) ) ( )im u u u u 2 N T(z

(xy)/(u): Projected potential in real/reciprocal spaceT(xy)/T(u): Contrast transfer function (CTF) in real/reciprocal space * :Convolution operatorNz: Number of periods in projection: Interaction constant

Crystal Structure

ProjectedPotential

Contrast Transfer Function

T(u) =f(f, Cs, Cc, )

HRTEM Image

HRTEM

All amplitudes equal Reverse one strong reflectionStrongest half

of the reflections are used

Lattice averaged (p1) Symmetry imposed (p4g)

Effects of Effects of phases and phases and amplitudes on amplitudes on structure structure determinationdetermination

Correct model: all strong reflections are included and their phases are correct.

How to Solve Structure from HREM Images of Thin CrystalHow to Solve Structure from HREM Images of Thin Crystal

• Determine contrast transfer function T(u)T(u) =D(u)sin(u)

• Determine crystal structure factors (hk)

(hk) = (u) = Iim(hk)/2NzT(hk)

• Calculate projected potetial (xy) by

At Scherzer defocus, T(hk)-1 over a large range of resolution(hk) = (u) = -Iim(hk)/2Nz

• Calculate a Fourier transform of the image & extract amplitudes and phases Iim(hk)

All the phases of (hk) are shifted by 180o from those of Iim(hk)

( ) ( ) exp ( )xyhk

hk i hx ky2

Structure factor Calculated Experimental No. h k l Fc(hkl)

F(hkl) Fo(hkl) ctf(hkl)sym(hkl) Fo(h-kl)ctf(h-kl)

sym(h-kl)

1 0 6 0 6031 180 5350 -157 180 5350 157 1802 3 5 0 5981 180 10000 137 180 8685 -145 1803 5 1 0 5570 0 6532 0 0 6766 1 04 5 2 0 4751 0 4268 21 0 4842 176 1805 1 7 0 4262 0 1768 -13 0 1382 -32 06 4 0 0 4189 180 3234 -173 180 3234 173 1807 3 6 0 3782 180 2415 136 180 1852 29 08 5 3 0 3705 180 1918 -167 180 2750 157 1809 5 4 0 3315 0 2750 24 0 2855 179 180

10 2 6 0 3285 180 3381 160 180 3536 -148 180… … … … 26 3 1 0 1074 180 2775 -106 180 2176 -92 18027 4 1 0 1039 180 37 -59 0 * 1084 140 18028 2 0 0 1003 0 2881 137 180 * 2881 -137 18029 2 2 0 1000 180 1865 161 180 1558 99 180… … … … 34 2 4 0 712 180 2011 140 180 1901 112 180

35 5 5 0 691 0 817 -125 180 * 801 175 18036 1 4 0 688 180 823 72 0 * 48 93 18037 1 3 0 609 180 2409 -15 0 * 1109 -35 038 3 7 0 559 180 681 133 180 368 -134 18039 0 2 0 521 180 2339 -167 180 2339 167 18040 2 7 0 405 0 334 139 180 * 753 19 041 0 4 0 387 180 2603 24 0 * 2603 -24 042 2 1 0 348 0 1290 99 180 * 1258 -65 043 1 1 0 161 180 988 152 180 801 24 180

Zr2Se –Zr2Se –Calculated & Calculated & Experimental Experimental Structure FactorsStructure Factors

Extract Amplitudes & Phases from HREM imagesExtract Amplitudes & Phases from HREM images

At Scherzer defocus, T(hk) -1over a large range of resolution: 

 

The amplitudes of the crystal structure factors are proportional to the amplitudes of the reflections in the Fourier transform Iim(hk) of the image

All the phases are shifted by 180 from those of the Fourier transform Iim(hk) of the image.

( ) I ( ) exp I ( )im imhk hk hk 1

2

1

2

N Nz zi

T(u)=D(u)sin(u)

D(u)

Sin(u)

(u< first crossover)

At Scherzer defocus, the projected potential (xy) is obtained from the Fourier transform Iim(hk) of the image

by:

 

 

Black features in HREM positives (low intensity) correspond to atoms (high potential). The corresponding image is called the structure image.

This is not valid for images with resolutions better than Scherzer resolution.

( ) I ( )exp ( )

I ( )

im

im

xy hk hx ky

xyhk

1

22

2

N

N

z

z

i

Contrast Transfer FunctionsContrast Transfer Functions - for Tecnai 30 (LaB6) and Tecnai F30 (FEG) at Scherzer defocus

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80u(Å-1)

T(u)

LaB6 FEG

Information limit

Scherzer resolution

1.4 Å

2.0 Å

( )u uu

23 4

2

C s

T(u)=D(u)sin(u)

D(u)=exp[-½222u4]exp[-22u2(+Cs2 2u2)2]

Effects of astigmatism and defocus on the image and FT

ab

u

v

u

v

Determine Contrast Transfer Function (CTF) from Fourier Transform

CTF D(u)sin(u) - defocus & astigmatism, Iim(hkl) = K·D(hkl)sin(hkl)·F(hkl)

 u = -757 Å v = -642Å x^u=7.3

D(u)sin(u)

u

u = -997 Å v = -867Å x^u=76.5

D(u)sin(u)

uu

D(u)sin(u)

060

400

Zr2Se - Pnnm, a = 12.64, b = 14.58 & c = 3.60 Å

       

+ 525 Å

- 165 Å

Contrast Transfer Function - Philips CM30/ST

- 850 Å

-1.0

-0.5

0.0

0.5

1.0

0.0 0.1 0.2 0.3 0.4 0.5u (Å-1)

 

 

Structure factor Calculated Experimental No. h k l Fc(hkl)

F(hkl) Fo(hkl) ctf(hkl)sym(hkl) Fo(h-kl)ctf(h-kl)

sym(h-kl)

1 0 6 0 6031 180 5350 -157 180 5350 157 1802 3 5 0 5981 180 10000 137 180 8685 -145 1803 5 1 0 5570 0 6532 0 0 6766 1 04 5 2 0 4751 0 4268 21 0 4842 176 1805 1 7 0 4262 0 1768 -13 0 1382 -32 06 4 0 0 4189 180 3234 -173 180 3234 173 1807 3 6 0 3782 180 2415 136 180 1852 29 08 5 3 0 3705 180 1918 -167 180 2750 157 1809 5 4 0 3315 0 2750 24 0 2855 179 180

10 2 6 0 3285 180 3381 160 180 3536 -148 180… … … … 26 3 1 0 1074 180 2775 -106 180 2176 -92 18027 4 1 0 1039 180 37 -59 0 * 1084 140 18028 2 0 0 1003 0 2881 137 180 * 2881 -137 18029 2 2 0 1000 180 1865 161 180 1558 99 180… … … … 34 2 4 0 712 180 2011 140 180 1901 112 180

35 5 5 0 691 0 817 -125 180 * 801 175 18036 1 4 0 688 180 823 72 0 * 48 93 18037 1 3 0 609 180 2409 -15 0 * 1109 -35 038 3 7 0 559 180 681 133 180 368 -134 18039 0 2 0 521 180 2339 -167 180 2339 167 18040 2 7 0 405 0 334 139 180 * 753 19 041 0 4 0 387 180 2603 24 0 * 2603 -24 042 2 1 0 348 0 1290 99 180 * 1258 -65 043 1 1 0 161 180 988 152 180 801 24 180

Zr2Se –Zr2Se –Calculated & Calculated & Experimental Experimental Structure FactorsStructure Factors

Reconstructed potential maps -Zr2Se

CTF compensated & symmetry imposed

      

 

Lattice averaged Structure model 

   

Compensating for Crystal Tilt by imposing SymmetryCompensating for Crystal Tilt by imposing Symmetry

Original images After imposing symmetry

5o

0o

KNb7O18 P4/mbm a = 27.5 c = 3.94Å Z=8

[001]

Compensating for Defocus and Astigmatism

p1 pmg p1 pmg