Physical resolution limits of single particle 3D imaging with X-rays and electrons

Transcription

1 Physical resolution limits of single particle 3D imaging with X-rays and electrons Coherence 2005 Dirk Van Dyck and Sandra Van Aert June 15, 2005

2 Ultimate goal: imaging of atoms! Understanding properties of nanostructures biological structures (in)organic materials components! Matching experiment with theoretical, ab initio calculations! Modelling! Predicting and designing nanostructures

3 What is needed to match experiment with ab initio theory? 1. Precise atomic positions (±0.001 nm) 2. Complementary information - Prior information (e.g. substructures) - Local spectroscopic information chemical bonding electronic structure local configuration

4 How to characterize atomic structures?! Interaction with particles photons (X-rays) electrons neutrons protons! Requirements bright coherent sources easy to detect subångstrom wavelength X-rays (XDM, XD) and electrons (EM, ED, EDI)

5 Quantitative experiment source object detectors instrumental parameters! Detection of individual particles! Model based fitting

6 Model based fitting model space simulation experiments refinement

7 Model based fitting Observations number of particles hitting detector i (e.g. pixel in CCD) n i ni = i Model E p θ stochastic variables N [ n ] = Np ( θ ) i k i i k probability to hit pixel i model parameters structural parameters (atomic positions, ) instrumental parameters (fixed and tunable)

8 Model based fitting Requirements The model contains all ingredients needed to perform a simulation (structure, interaction, instrument, detection) The model is assumed to be correct The experiment is the ensemble of all experiments (focal images, tomographic series, DP, ) Only fitting with original experimental data (noise model)

9 Model based fitting Maximum likelihood estimator Lowest possible error bar (CRLB) Unbiased Likelihood function L = N! ln L = i i n p i n i i p ( θ ) n! i i k ( θ ) + C Maximum likelihood k ln L θ k = 0 k estimates ˆ θ k

10 Error bar on the estimated parameters σ 2 i Then σ 2 i 2 ln L E θk θl : variance of estimated parameter 1 ii θ i lower bound on variances 2 σ i can be used for experimental design (Cramér-Rao lower bound)

11 Resolution - precision resolution ρ σ = CR ρ N ρ = 1 Å N= σ CR = 0.01 Å dose σ CR Å The dose can be distributed over many images (dose fractionation theorem)

12 Model based fitting model space simulation experiments Iteration till best fit refinement

13 Model based fitting Problems: - convergence - local optima - uniqueness

14 How to avoid local optima? 1) Resolving the structure Obtain a good starting structure using general principles Direct imaging Image reconstruction Direct methods (phase constraints) Constrained optimisation: hybrid I-O 2) Refining the structure Convergence to global optimum (maximum likelihood)

15 Focus variation reconstruction ELECTRON MICROSCOPY FOR MATERIALS SCIENCE

16 Phase of total exit wave Σ 5 Al: Cu ELECTRON MICROSCOPY FOR MATERIALS SCIENCE Courtesy C. Kisielowski (NCEM,Berkeley)

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19 Uniqueness problem If number of parameters exceeds information capacity of imaging channel RECIPROCAL SPACE 1/a 1/ρ Requirement: P 2 πa < ρ 2 or P 3 < a ρ 2 2 parameters data Information content = 3 parameters per unit ρ 2

20 3D Tomography parameters P 4 < π ρ 1 3 a data P/a 3 < 4/ρ 3 Information content = 4 parameters per ρ 3 2 angstrom resolution sufficient in 3D

21 HREM of amorphous structures resolution = 1A 2D: 1.5 atom per A 2 not resolvable 3D: 1 atom per A 3 resolvable

22 Required dose Rose criterion Dose signal-to-noise ratio SNR = 10 imaging particles D = = volume 4( SNR) 3 ρ 2 N 3 ρ N 3 SNR = = N = Dρ N D = 3 D = 400 particles / ρ

23 Limitations of resolution D > 400 ρ 3 Sufficient brightness, time, stability of specimen (inorganic objects) Resolution limited by the instrument ultimate resolution = atom D < 400 ρ 3 Insufficient brightness, time Instability of specimen (radiation damage) (life-science objects) object resolution

24 Knock-on damage (inorganic objects) Inelastic cross section<elastic cross section resolution = atom Ionisation damage (life science objects) Instrument resolution < object resolution

25 Atomic resolution EM Point-spread function: pr () = p() r p() r p () r p() r p () r A T EM V D ρ A ρ T ρ EM ρ V ρd atom potential thermal motion electron microscope environment ρ = ρ + ρ + ρ + ρ + ρ R A T EM V D detector

26 ρ R ρ A ultimate resolution = intrinsic width of the atom image same information as DP (2x) Note knock-on damage: cross section smaller than ρ A

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28 z (Å) Example: amorphous tungsten Number of atoms used: 1859 C s = 0.5 mm E = 300 kev ε = Sch 180 images with 1º increment Dose/image = Å -2 y (Å) x (Å)

29 C s = 0.5 mm Cs = 0.05 mm x (Å) Number of atoms used: y (Å) x (Å) C s = 0.5 mm E = 300 kev ε = Sch 180 images with 1º increment Dose/image = Å -2

30 First atomic-resolution diffractive image reconstruction. Double-walled Nanotube SAD TEM Image reconstructed from electron-diffraction pattern by HiO ELECTRON MICROSCOPY FOR MATERIALS SCIENCE J.M.Zuo et al Science 300, 1420 (2003).

31 Object resolution for life-science objects

32 Resolution-dose for life-science objects Region of successful XDM experiments ELECTRON MICROSCOPY FOR MATERIALS SCIENCE Howells et al. 2005

33 How to improve object resolution?! Radiation damage " Electrons better than X-rays " Tune energy " Cryo protection " Averaging over identical objects " Inertial imaging (FELS)! Fitting known substructures " Small proteins " Alpha helices " Beta sheets! Specimen preparation (FIB, DIP-PEN)

34 Henderson, Quarterly Reviews of Biophysics 28, p. 171 (1995) ELECTRON MICROSCOPY FOR MATERIALS SCIENCE

35 Averaging identical objects ρ A ρ M 1. Crystalline: natural average, smaller structures 2. 2D, 1D periodic structures 3. Oriented objects (laser) 4. Random objects (minimal size)

36 Henderson, Quarterly Reviews of Biophysics 37, p. 3 (2004) ELECTRON MICROSCOPY FOR MATERIALS SCIENCE

37 Henderson, Quarterly Reviews of Biophysics 37, p. 3 (2004) ELECTRON MICROSCOPY FOR MATERIALS SCIENCE

38 Henderson, Quarterly Reviews of Biophysics 37, p. 3 (2004) ELECTRON MICROSCOPY FOR MATERIALS SCIENCE

39 Courtesy J. Spence What is the best that has been done by cryo -TEM methods? (not 2D xtals) Protein synthesis ( Life itself ) in the Ribosome: The ribosome structure determined to 1nm resolution by TEM (tomographic cryomicroscopy). J.Frank et al. A - T C - G T - A G - C T - A A - T word (codon) mrna trna polypep Adaptor plug DNA 4 nucleic acids 2 (4) base pairs mrna reads one side only 3 pairs per word (per amino) 4 3 = 64 possibilities per amino 20 amino acids. n words per gene (protein) Ribosome width 25nm (Cell,100, p.537 (2000)) Experimental e-coli ribsome reconstuction from TEM images of non-crystallised mols in ice. mrna bring 3-bit codons from DNA. trna adaptors (E,P,A) have plugs at one end to mrna codon, at the other to an amino acid, which is added to the polypetide chain as the ribo runs along the mrna. Chain will fold to become a new protein. (Simplified).

40 ! Source brightness " X-rays New generation synchrotrons Free Electron Laser Source (FELS) Bosons (NO fundamental limit) " Electrons Field emission sources Correctors Fermions: - limit of phase space (still 10 5 off) - Coulomb interaction

41 TABLE 1. Comparison of synchrotron soft X-ray and field-emission electron sources. All values are for 500 ev X-rays, or the 300 kev electron beams which are typically used to study ELNES at around 500 e V. ELNES uses parallel detection, XANES serial. ALS Undulator U5 e - Cold FEG at 300 kv Brightness 6.9 X particles /sec /cm 2 /sr /ev (1.1 X Ph/s /mm 2 /mr 2 / 0.1%BW)* 1.3 X particles /sec /cm 2 /sr /ev ( 6 X 10 9 A/cm 2 /sr.) Electron brightness values from Speidel et al Optik 49, 173. Nanotip at RT is 55 times brighter (Qian, Scheinfein, Spence J.Appl Phys.73, p ELECTRON MICROSCOPY FOR MATERIALS SCIENCE Degeneracy δ X 10-5 Coherent flux j c 2.0 X 10 7 Energy spread E (un-monochromated). ph/s/0.1%bw 4.6 ev 0.28 ev Source size. 307x23 µm 2nm Resolution of focussing element. Flux into focussed probe 4.0 x nm 0.1 nm ph/s/0.1%bw 1nA into 1nm diameter. Higher with aberration corrector. * This is the long time average value. Instantaneous values are about D=100 times greater. Spence and Howells, Ultramic.93, p.213 (2003)

42 Scanning Electron Microscopy & HREM & Spectroscopy A STEM / HRTEM : Tecnai G 2 Current technology: HAADF-image Local energy spectrum On core Off core Counts (arb. unit) nm Energy Loss (ev) Dislocation core in GaN [0001] Focused e-beam Scanning coils Sample HAADF Detector Image Filter ELECTRON MICROSCOPY FOR MATERIALS SCIENCE Upgrade to NCEM in 2002 First instrument of this kind in the US Probe size 0.13 nm (currently at NCEM: ~1 nm) Energy resolution: mev (currently: ~1eV) Information Limit : < kv Phase Contrast & Z-Contrast & Spectroscopy on identical areas N. Browning, C. Kisielowski, LDRD,

43 Courtesy H. Chapman We are entering a new era in x-ray science Linac Coherent Light Source Tesla Test Facility SPPS femtosecond photons Angstrom wavelength LLNL Thompson source This short-pulse high-fluence x-ray regime is completely unexplored. What ever we do with these sources will be new science

44 Combine damage and classification results to determine the required pulse parameters Resolution (A) Preliminary Results fs 20fs photons in (0.1 µm) 2 15fs 4fs 1fs Radius of particle (A) Maximum Pulse Length versus Radius, Resolution, and Sample Size; Limited by Classification and Damage (R max = 0.25) Resolution from damage modeling worsens with increasing fluence and increasing pulse length For a given radius and fluence, we can determine the maximum pulse length Atomic resolution can be achieved for optimized XFEL parameters (short pulses, high fluences) ELECTRON MICROSCOPY FOR MATERIALS SCIENCE Courtesy H. Chapman

45 Resolution - precision resolution ρ σ = CR ρ N ρ = 1 Å N= σ CR = 0.01 Å dose σ CR Å EXAMPLE: coherent HREM better than HAADF STEM

46 Precision radiation damage Precision: Cramér-Rao Lower Bound σ = CR ρ δσ el ρ = resolution δ = incident dose/å σ el = elastic cross section 2 Probability for displacive damage of an atom: p = δσ in σ in = inelastic cross section Figure of merit σ 2 p CR = 2 ρ σ σ in el

47 σ el () 2 2πeV r = p d r Eλ V p ( r) = projected potential E = incident electron energy λ = electron wavelength α σ = π in 2E 2 Ze α = 4πε T = 4m M 0 E 2 T 1 ε Z = atom number m = electron mass M = atom mass ε = threshold energy for displacive radiation damage

48 Simple model σ p c CR = 1000Zε E With Z = atomic number E = incident energy (ev) ε = treshold energy for displacement damage

49 σ 2 p CR 4 10 Zε = 10eV Zε = 20eV Zε = 50eV E =50 kv E =20 kv E =10 kv Optimal energy: function of Zε 1 E

50 Experimental design At low accelerating voltages: Do we need a Cs-corrector? Do we need a Cc-corrector? Do we need a monochromator?

51 Precision of a Si atom position as a function of Cs Accelerating voltage = 50 kev Lower bound on standard deviation of the position 0.4 without monochromator, without C c -corrector with monochromator with C c -corrector C s (mm)

52 Conclusion Both electron and X-ray methods can ultimately resolve individual atoms. Quantitative refinement can yield precisions that are needed for theoretical understanding.