Electron crystalography, Xiadong Zou 2011

Scope :

• EM and ED of inorganic molecules
• HRTEM and SAED, nanoED

Chap1 : Introduction

History

• fig1.1first ED solved crystal BaCl2.H2O (VainshteinPsinker 1949)
• theory of ED, dynamical effects : CowleyMoodie1953-1957
• direct method for phasing(HauptmannKarle1953) applied for ED(1976,1995)
• 3D reconstruction from FT of images : digitised image processing (RosierKlug1968)
• X-ray membrane protein ,Nobel Prize (MichelDeisenhoferHubber1988)
• cryoEM reducing radiation damage on proteins (Henderson1986)
• EM for protein book(Glaeser2007)
• fig1.2 Ti2Nb10O29, SHRLI Computer simu(Cs) O'Keefe(1973)
• fig 1.3 EM of K7Nb15W13O80, FT and electron density map reconstruction with simulation
• image processsing method-IMP (Klug1978)
• IMP+image deconvolution+phase-extension by max entropy+direct method (Hu1992)
• 5 HRTEM images => 3D emap reconstruction (Wenk1992)
• HRTEM(phase) + ED(VHR) => bacteriorhodopsin (Henderson1975)
• electron holography(Litche1986), convergent beam ED(Hoier1993) are other techniques

EC advantages over X-ray

• strong interaction => small crystals
• lens with magnets => imaging => phase
• electron density distribution(X-ray) vs electrostatic potential(elec)
• Combining EM and ED
• defects, no periodicity => EM
• Ed higher resolution, precession 3D ED => high quality data

Future of Crystallography

• Improved HR through-focus exit-wave reconstruction (Coene1992,1996)
• ED with high sampling frequency(large sampling with zero padding) allows iterative phase retrieval(Miao2002)
• Lens-aberration-corrected EM : spherical aberration(Haider1995)
• Ultrafast EM femtosec transition states(Nobel prize Zewail1999)

Challenges

• Answers :
  • What info about crystal in HRTEM and ED?
  • How extract info for structure analysis ?
  • How to obtain good data for ED ?
• Questions :
  • X-ray vs ED principles the same ?
  • multi-scattering and non-linear effects too problematic?
  • Phase problem in EC?

Chap2 : Crystallography basics

History and nobel prizes in crystallography

Date	Name	Discovery	Nobel
1830	Miller (1801-1880)	Identification of macroscopic crystal types
1850	Bravais (1811-1863)	Theory of 14 crystal lattices
1891	Fedorow,Schonflies	Theory 230 space groups
1912	Laue,Bragg	X-ray diffraction	1914
1912	Bragg	Bragg's law and $NaCl$ structure	1915
	Perutz,Kendrew	first protein	1962
	Watson	DNA	1962
	Hodkings	B12	1964
	Klug	Electron crystallo	1982
	Hautpman,Karle	direct methods	1985
	Ramakrishmann	ribosome	2009

Symmetry elements and effect in reciprocal space

• mirror symmetry as projected through :
  • perp to m plane => cannot be seen
  • parallel to m plane => mirror or 2-fold rotation
  • parallel to m plane => glide plane=screw axis perpendicular to plane
• 2 fold as projected :
  • along the axis => centrosymmetric
  • perpendicular : miroir symmetry
• inversion : phase=0 if origin same as center of inversion (not always the case $p4_2/nmc$)
• screw axis,glide plane : systematic absences h+k=2n which may exist in ED
• centring : systematic absences h+k=2n+1
• Summary : find positions of unique atoms and apply symmetries

Chap 3 : Crystal structure factors and symmetry

Electrostatic potential and structure factors

Structure factor and scattering factor are related by (kinematic) :

$F(hkl) = \sum_{j=1}^{N} f_j(hkl)\exp 2i\pi(\mathbf u\cdot\mathbf r )$

• $\mathbf u=h\mathbf a^{\star}+k\mathbf b^{\star}+l\mathbf c^{\star}$ reciprocal lattice vectors
• $\mathbf r_j=x_j\mathbf a+y_j\mathbf b + z_j\mathbf c$ is the position of atom $j$ in real space
• So that $\mathbf r_j\cdot\mathbf u = hx_j+ky_j+z_jl$
• $F(hkl)$ is the structure factor of the crystal
• $f_j(hkl)$ is the scattering factor of atom $j$

The potential and structure factor are relatedby the Born approximation :

$\Phi(\mathbf u) = \frac{\lambda}{\sigma\Omega}F(\mathbf u)$

• $\sigma=2\pi me\lambda/h^2 (V^{-1}A^{-1})$ interaction constant
• $\Omega$ the volume of the unit cell
• $\Phi$ the Fourier transform of the potential (V)
• $F$ the structure factor (A)

Crystallo structure factor phase and atom positions

HRTEM gives the projection in a plane perpendicular to the beam direction. For example for a beam going in $\mathbf c$ direction the projection of the potential in the $(xy)$ plane is : $$\begin{eqnarray} \varphi_p(x,y)&=&\int_0^t\varphi(x,y,z)dz \\ &=&\frac{\sigma\Omega}{\lambda}\sum_{h,k} F(hk0)\cos(2\pi(hx+ky)-\phi_F(hk0)) \end{eqnarray}$$

• So that maximum of the potential(position of atoms) are expected at $h+k=2n$ when structure factor phase is zero.
• Planes 5,2,0 and -5,2,0 have same amplitude but opposite phases which is an effect if symmetry following rules in the ITCr
• Phase relations between symmetry related reflections are determined by the space group (see space group explorer spg on terminal)
• The projection pattern is created by the summing the contributions of the projected structure factor phases

Amplitudes and phases of symmetry related reflections

• Symmetry related reflections $\mathbf u'=\mathbf u\cdot\mathbf R$ different than equivalent positions $\mathbf x'=\mathbf R\cdot\mathbf x$
• Translation part seen for relative phases : $F(\mathbf u')=e^{-2i\pi\mathbf u\cdot\mathbf t}F(\mathbf u)$
• Symmetry related reflections have same amplitudes but may differ in phase if translations are involved

References

Books

• Electron Crystallography of Biological Macromolecules, Glaeser2007

Questions

• image processsing method ?
• direct method ?
• patterson nethod ?
• max entropy likelihood
• iterative phase retrieval
• Field emssion gun
• contrast transfer function ?