Solving crystals structures from HREM by crystallographic image processing Xiaodong Zou Structural Chemistry, Stockholm University

SAED intensities changes with crystal thickness -AlFeCr: P6 3 /m, a = , c = Å

Why structure determination by electron microscopy is possible

1 r1r1 2r2r2 rjrj j Projected Electrostatic Potential Crystal Structure r

The crystal potential: The structure factor of a crystal for electrons The projected potential (in 2D): F c (hk0) - the structure factor amplitudes  F (hk0 ) – the structure factor phases

Non-centrosymmetric Centrosymmetric r j = (x j y j z j ) r j = (x j y j z j ) (-x j –y j -z j ) Phases can be any angles Phases must be 0 or 180 degrees This is valid only when the origin is at the center of symmetry F(h) Imaginary Real f1f1 f1'f1' f2f2 f2'f2' f3f3 f3'f3' f4f4 f4'f4' (h)(h) f4f4 Imaginary F(h ) Real f1f1 f2f2 f3f3 (h)(h) Fourier synthesis of the crystal structure factor F(u)

Zr2Se – Pnnm a = , b = c = Å

Zr2Se

HREM images may be distorted by: Electron optics – defocus & astigmatism Crystal misalignment Dynamic effects The distortions may be compensated by crystallographic image processing

CRISP Crystallographic image processing by CRISP Fourier transform Lattice refinement Symmetry determination & Origin refinement p4g Symmetry imposed (p4g) Experiment image Amplitudes Phases Determine coordinates

Relation 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):  (xy)/  (u): Projected potential in real/reciprocal space T(xy)/T(u): Contrast transfer function (CTF) in real/reciprocal space * :Convolution operator N z : Number of periods in projection  : Interaction constant

Crystal Structure Projected Potential  C ontrast T ransfer F unction T(u) =f(  f, C s, C c,  ) HRTEM Image HRTEM

All amplitudes equalReverse one strong reflection Strongest half of the reflections are used Lattice averaged (p1)Symmetry imposed (p4g) Effects of phases and amplitudes on structure determination Correct model: all strong reflections are included and their phases are correct.

How 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) = I im (hk)/2  N z T(hk) Calculate projected potetial  (xy) by At Scherzer defocus, T(hk)  -1 over a large range of resolution  (hk) =  (u) = -I im (hk)/2  N z Calculate a Fourier transform of the image & extract amplitudes and phases I im (hk) All the phases of  (hk) are shifted by 180 o from those of I im (hk)

Zr2Se – Calculated & Experimental Structure Factors

Extract Amplitudes & Phases from HREM images

At Scherzer defocus, T(hk)  -1 over a large range of resolution: The amplitudes of the crystal structure factors are proportional to the amplitudes of the reflections in the Fourier transform I im (hk) of the image All the phases are shifted by 180  from those of the Fourier transform I im (hk) of the image. 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 I im (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.

Contrast Transfer Functions - for Tecnai 30 (LaB6) and Tecnai F30 (FEG) at Scherzer defocus LaB 6 FEG Information limit Scherzer resolution 1.4 Å 2.0 Å T(u)=D(u)sin  (u) D(u)=exp[-½  2  2 2 u 4 ]exp[-  2  2 u 2 (  +C s 2 2 u 2 ) 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, I im (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

Zr 2 Se - Pnnm, a = 12.64, b = & c = 3.60 Å

+ 525 Å Å Contrast Transfer Function - Philips CM30/ST Å

Zr2Se – Calculated & Experimental Structure Factors

Reconstructed potential maps -Zr 2 Se CTF compensated & symmetry imposed Lattice averaged Structure model

Compensating for Crystal Tilt by imposing Symmetry Original images After imposing symmetry 5o5o 0o0o KNb 7 O 18 P4/mbm a = 27.5 c = 3.94Å Z=8 [001]