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This work supported by the Director, Office of Science, Office of Basic Energy Sciences,

NNI Interagency Workshop January 27-29, 2004 Instrumentation and Metrology for Nanotechnology Grand Challenge Workshop National Institute of Standards and Technology, Gaithersburg, MD. Track 1- Instrumentation and Metrology for Nanocharacterization. Breakout Session: Current State of the Art.

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This work supported by the Director, Office of Science, Office of Basic Energy Sciences,

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  1. NNI Interagency WorkshopJanuary 27-29, 2004 Instrumentation and Metrology for Nanotechnology Grand Challenge Workshop National Institute of Standards and Technology, Gaithersburg, MD Track 1- Instrumentation and Metrology for Nanocharacterization Breakout Session: Current State of the Art Sub-Ångstrom Electron Microscopy for Materials Science Michael A. O'Keefe Materials Sciences Division Lawrence Berkeley National Laboratory, Berkeley, CA 94720 and Lawrence F. Allard High-Temperature Materials Laboratory Oak Ridge National Laboratory, Oak Ridge, TN 37831 This work supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Science Division, DOE under contract DE-AC03-76SF00098, and Asst. Sec. for EERE, Office of FreedomCAR and Vehicle Tech. for the HTML User Program, ORNL, managed by UT-Battelle, LLC for DOE under contract DE-AC05-00OR22725.

  2. NNI Interagency WorkshopJanuary 27-29, 2004 Instrumentation and Metrology for Nanotechnology Grand Challenge Workshop National Institute of Standards and Technology, Gaithersburg, MD The Role of Measurement Theory the circle of nano Measurement Construction The high-resolution electron microscope can provide essential feedback in the nano- theory/construction/measurement loop.

  3. Rose (1994) Measurement with the electron microscope OÅM -- 0.78Å (2001) • In 1994, in a paper on aberration correction [1], Harald Rose showed resolution over time. He predicted 0.5Å resolution by 2015. • Better microscope resolution leads to less de-localization of higher spatial frequencies, so better precision in measurement of atomic coordinates. • The OÅM demonstrated sub-Angstrom microscopy to 0.78Å resolution in 2001 [2], using hardware correction of three-fold astigmatism and software correction of spherical aberration. TEAM -- 0.5Å (2006?) • The next-generation TEAM is designed for sub-0.5Å resolution [3], using hardware correction with lens current stability of 0.1ppm (rms) and a mono-chromator to reduce FWHH beam-energy spread below 0.35eV at 300keV or 0.18eV at 200keV. • Better resolution allows characterization in more viewing directions, leading to atomic-resolution 3-D images -- locate every atom in place! [1] “Correction of aberrations, a promising means for improving the spatial and energy resolution of energy-filtering electron microscopes” H. Rose, Ultramicroscopy56 (1994) 11-25. [2] “Sub-Ångstrom resolution of atomistic structures below 0.8Å”, M.A. O’Keefe, E.C. Nelson, Y.C. Wang and A. Thust, Phil. Mag. B81 (2001) 11, 1861-1878. [3] “HRTEM at Half-Ångstrom Resolution: from OÅM to TEAM”, M.A. O’Keefe, Microscopy & Microanalysis 9 (2003) 2: 936-937.

  4. 1990: resolution extension by focal series reconstruction. Images of oxygen atoms on JEOL-ARM 1000 O Model 1.6ÅScherzer- focus image 1.4Åreconstruction from 5 images 1.4Å simulation "Resolution of oxygen atoms in staurolite by three-dimensional transmission electron microscopy", Kenneth H. Downing, Hu Meisheng, Hans-Rudolf Wenk, Michael A. O'Keefe, Nature 348 (1990) 525.

  5. 1.7Åresolution +1 +1 +1 +1 +1 CM300FEG/UT  = 36Å 0 0 0 0 0 1.07Å info limit -1 -1 -1 -1 -1 OÅM  = 20Å 0.78Å  = 0.25 millirad  1.1Å  n = 2 1.03Å n = 36  0.89Å 0 0 Spatial Frequency (Å-1) Spatial Frequency (Å-1) 1.0 1.0 1.5 1.5 Resolution, information limit, and focal series - CTFs show transfer of spatial frequencies.

  6. 1.0 Resolution (Å) 1.0 Resolution (Å) What does aberration-correction (CS-correction) do? Compare OÅM (CS = 0.6mm) with CS-corrected (0.02mm) OÅM with CS of 0.6mm and Delta of 20Å Info Limit (0.78Å) CS corrected OÅM with CS at 0.02mm and Delta of 20Å With CS corrected, phase reversals are gone. Better mid-range transfer Info Limit (0.78Å)

  7. NCEM Materials Sciences Division 1992-2002: the LBNL One Ångstrom Microscope Project Sub-Ångstrom Resolution by Image Reconstruction Principal Investigator: Michael A. O’Keefe 1992 -- 2002 OÅM team: J.-O. Malm 1992 -- 1993 E.C. Nelson 1995 -- 2002 C.J.D. Hetherington 1995 -- 1997 Y.C. Wang 1997 -- 1998 C. Kisielowski 1998 -- 2000 Aim: to produce sub-Ångstrom resolution for NCEM users. *Supported by DOE/SC/BES/DMS

  8. 1998: first sub-Ångstrom result from OÅM OÅM image shows 0.89Å spacings in test specimen of diamond 0.89Å Model of diamond structure in [110] orientation. Pairs of C atoms are separated by 0.89Å to form the ‘dumbbells’. OÅM image taken close to alpha-null defocus shows pairs of C atoms separated by 0.89Å in the diamond structure. Y.C. Wang, A. Fitzgerald, E.C. Nelson, C. Song, M.A. O’Keefe et al, Microscopy and Microanalysis 5 (1999) 2: 822-823.

  9. 004 (a) |A2| = 2.46m simulated OÅM image averaged 004 OÅM image averaged 1998: aberration correction -- three-fold astigmatism Before correction, diamond image shows effect of 3-fold astigmatism After correction, diamond image shows 0.89Å atom pairs in “dumbbells” (b) |A2| < 0.05m Zemlin tableaux -- O’Keefe, Wang & Pan, 1998 Images -- Wang & O’Keefe, 1998

  10. Experimental 0.78Å Transfer at 3kV Electron Gun Extraction Voltage Silicon structure model in [112] orientation. Pairs of Si atoms are separated by 0.78Å in ‘dumbbells’. Si622 (0.82Å) 0.78Å Si444 (0.78Å) Image taken near alpha-null defocus shows pairs of Si atoms separated by 0.78Å. Si531 (0.92Å) Diffractogram confirms transfer of spacings to 0.78Å. M.A. O’Keefe, E.C. Nelson, Y.C. Wang and A. Thust,Philosophical Magazine B81 (2001) 11: 1861-1878.

  11. 0.78Å “Last-Century” Cutting-Edge Resolution [112] Si images from STEM and TEM [112] Best possible STEM - HB603U - Best possible TEM - OÅM - 0.78Å [112] Si has become the “de facto” test specimen “Quantitative interpretation and information limits in annular dark-field STEM images”, P.D. Nellist & S.J. Pennycook, Microscopy and Microanalysis6, 2: (2000) 104-105. “Sub-Ångstrom resolution of atomistic structures below 0.8Å”, M.A. O’Keefe, E.C. Nelson, Y.C. Wang and A. Thust, Philosophical Magazine B81 (2001) 11, 1861-1878.

  12. 1.7 CdTe 1.62Å 1.6 AlSb 1.6 1.53Å 1.5 Ge 1.41Å 1.4 Si 1.4 1.36Å 1.3 -InN 1.24Å 1.2 -SiC 1.2 1.1 1.11Å 1.0 CdTe 1.0 diamond AlSb 0.94Å 0.9 0.89Å Ge 0.89Å Si 0.82Å 0.8 0.78Å -InN 0.8 0.72Å 0.7 -SiC 0.64Å 0.6 0.6 diamond 0.5 0.51Å 0.4 3.0 0.4 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 Testing Microscope Resolution (the A-OK test series) Atom-atom spacings for diamond-cubic test specimens from 1.62Å to 0.51Å [110] series Dumbbell Spacing (Å) [112] series Lattice Parameter (Å) [112] silicon [110] diamond 0.89Å 0.78Å OÅM images reconstructed from focal series of 20 component images

  13. Resolution of light atoms -- imaging lithium • LiCoO2 is the most commonly used positive electrode materials for lithium rechargeable batteries • Energy storage  lithium insertion into and extraction from LixCoO2 • Ultra high resolution is needed to resolve light elements in a heavy matrix • Conventional HRTEMs with resolutions to 1.6Å can routinely image the heavier metal atoms in structures such as oxides. • The OÅM (One-Ångstrom Microscope) at the NCEM has achieved resolutions to 0.8Å and, in addition to heavy atoms, has previously imaged columns of lighter atoms, including O, N, and C. • In this work, we have used the OÅM to image all the component atoms, including columns of Li atoms in a matrix of CoO2.

  14. Schematic of Layered LiCoO2 Structure Single unit cell projected in the [110] orientation Li atoms Co atoms O atoms CoO6 octahedra

  15. Reconstructed Exit-Surface Wave of LiCoO2 Comparison of simulated and experimental ESWs shows that Li atom columns are visible at 0.9Å resolution in the OÅM. Experimental Simulation O Co O Li O is strong Co is “fuzzy” Li is weak The reconstructed exit-surface wave shows that the specimen is tilted away from exact [110] zone axis orientation and also reveals buckling and possible electron beam damage.

  16. a 6 atom column a b 11 atom column b 0.286 radian a b 6 7 8 9 10 11 10 9 8 7 6 # atoms in columns Simulated Pd cube-octahedron analysis -- Line trace shows peaks in ESW phase -- Model ESW phase ESW phase (peak height) is proportional to the number of atoms in the column producing the peak. Line trace shows the one-atom difference between adjacent columns.

  17. Analysis of experimental image of 70Å Au nanoparticle Phase shows white atom columns Single image at -2600A underfocus FSR of particle

  18. Twinning in ESW phase becomes clearer after application of a high-pass filter Particle image High-pass image

  19. Edge Center Analysis of 70Å gold nanoparticle by peak profile 9 7 7 5 Zero? Line trace of ESW phase shows initial increase from outer edge, followed by groups of peaks with very similar heights. “Quantization” of ESW phase peak steps suggests that height differences may be due to different integral numbers of atoms. The technique of profile tracing of phase to measure peak heights suffers from the lack of a well-defined zero level, especially for supported nanoparticles.

  20. Atomic structure O-K Mn L 14 II/III 12 1 Ti Ti 10 2 8 3 6 Sr Sr 4 4 5 6 0.2 nm 2 0 550 600 650 Energy Loss (eV) Z-Contrast Microscopy and electronic structure 1 2 4 5 Detector 3 6 Spectrometer Courtesy of S. Pennycook

  21. No spherical aberration FWHM ~ 0.8 Å Current density is concentrated into central maximum Electron Microscopy in 2003 -- aberration-corrected STEM STEM Probe Size is Limited by Spherical Aberration VG Microscope’s HB501UX, 100 kV Aberration limited Significant current is lost in probe “tails” FWHM ~2 Å Aberration correction can achieve the smaller brighter probe Courtesy of S. Pennycook

  22. Spectroscopic identification of a single atom within a bulk material. 8% collection efficiency La M4/5 Intensity 820 850 880 Energy (eV) Single Atom Spectroscopy 5 Å La in CaTiO3 grown by MBE Courtesy of S. Pennycook

  23. First Column Single Au Single Au Au to Au spacing 2.88 Å Carbon film background Linetrace of STEM Intensities Courtesy of S. Pennycook

  24. Electron Microscopy in 2003 Advanced TEM Diebold et al. (2003). Measurement of gate-oxide width • Diebold et al. (2003), compared measurements of gate-oxide width using TEM and STEM. • TEM shows silicon [110] dumbbells (left) up to nitrided gate oxide, then oxide, then poly silicon. • STEM (HAADF) with 10 millirad aperture agrees with TEM • STEM with 13 millirad aperture shows oxide as wider • STEM with larger aperture shows even “wider” oxide “Thin Dielectric Film Thickness Determination by Advanced Transmission Electron Microscopy”, A.C. Diebold et al., Microscopy & Microanalysis 9 (2003) 493–508.

  25. Electron Microscopy in 2003 3-D STEM Work by P.A. Midgley and M. Weyland Cambridge U.

  26. P.A. Midgley and M. Weyland, Cambridge U. Fig. 2. Non-uniform sampling of Fourier space over-emphasizes lower frequencies, giving a blurred reconstruction. The greater density of low-frequency data is compensated by using weightedback-projection reconstruction. 2-D test object for simulation Fig. 3a. Result of adding successively more projections to the reconstruction, using direct (left) and weighted (right) back-projection over a tilt range of 90.

  27. Object Reconstruction Direct Weighted P.A. Midgley and M. Weyland, Cambridge U. Fig. 3b. Effect of tilt range. Limited tilt produces a missing wedge in Fourier space. Missing data limit the reconstruction resolution in the vertical direction, causing streaking. Figure shows tilt ranges from 10 to 60. Tilt axis is into the plane of the figure. Recent advances in tomographic specimen holders allow tilts to70 around two axes within the 2.2mm polepiece gap of modern ultra-high-resolution electron microscopes. With a tilt series in x and one in y, the “missing wedge” becomes a 20 “missing pyramid”.

  28. P.A. Midgley and M. Weyland, Cambridge U. 3-D image of nanoparticles. Reconstructed using weighted back projection from 55 STEM HAADF images of Pd6Ru6–MCM 41 catalysts. Tilts from +60 to -48 in 2 steps at 300kV. Metal particles have been colored red for clarity. An individual nanoparticle in the reconstructed data set can be isolated to show that it is anchored to the wall of a 3nm-diameter mesopore. The particle is about 1nm in diameter.

  29. NNI Interagency WorkshopJanuary 27-29, 2004 Instrumentation and Metrology for Nanotechnology Grand Challenge Workshop National Institute of Standards and Technology, Gaithersburg, MD Conclusion Theory the circle of nano Measurement Construction The electron microscope will continue to evolve and provide essential feedback in the nano- theory/construction/measurement loop.

  30. OÅM information limit is at sub-Ångstrom level (b) (a) • (a) Standard CM300FEG/UT* • spread of focus = 35Å • information limit = 1.05Å • (a) One Ångstrom Microscope • spread of focus = 20Å • information limit = 0.78Å Compare standard CM300FEG/UT with OÅM-spec CM300 1.05Å 0.8Å 1.05Å 0.8Å *Hans Bakker, Arno Bleeker, and Peter Mul, Ultramicroscopy64 (1996) 17-34.

  31. a a b b 1.01Å 1.01Å 0.80Å 0.80Å 0.68Å 0.68Å 0.1 0.1 1 1 2 2 3 3 0 0 0.03 0.03 0.05 0.05 0.1 0.1 A2 (m) A2 (m) A2 (m) A2 (m) d d A2 A2 1998: correction of OÅM three-fold astigmatism Before correction: mean = 2.46m After correction: means = 0.03m 1Å limit 1Å limit Correction method uses 2-fold stigmators to provide an approximation to a 3-fold field (D. Typke & K. Dierksen, Optik 99, 4: (1995) 155-166)

  32. O Li Co LiCoO2 Exit-Surface Wave with less Smoothing • Sub-Angstrom image of LiCoO2 battery material shows all atom species. • Superimposed model identifies the strong white peaks with the positions of oxygen atom columns, the strong fuzzy peaks with cobalt sites, and the weak white peaks at lithium positions.

  33. Thickness 39.4Å 45.1Å Resolution 50.7Å Li Li Li 0.8Å 0.9Å 1.0Å Exit-Surface Wave Simulation of LiCoO2 - the [110] Zone ESW simulations suggest that Li atom columns should be clearly visible for resolutions of 0.8 to 1.0 Ångstrom at specimen thickness of 40 to 50 Ångstrom.

  34. EpicierCo50-Pt50_20-zoomed-3M6.jpg five-fold Pt-Co particle with an 'artistic' schema... Not as well-oriented, not as well-resolved, but significantly smaller (2.5 - 3 nm...; the scale is missing) http://cecm.insa-lyon.fr/people/people.php?name=epicier

  35. Simulation study of Pd cube-octahedron The nanoparticle model used in the test simulations has 561 atoms of palladium arranged as a cube-octahedron. Model ESW phase is proportional to the specimen potential projected through thickness H in the direction of the incident electron beam. Exit-surface wave In the image, large phase changes have produced white peaks in atom columns near the particle center. Scherzer image Delocalization has produced strong Fresnel fringes, masquerading as “white atoms”, near the particle edges. "Deceptive "Lattice Spacings" in High-Resolution Micrographs of Metal Nanoparticles", J.-O. Malm & M.A. O'Keefe, Ultramicroscopy68 (1997)13-23.

  36. Image formation and exit-wave reconstruction Ewald sphere g specimen shape function Image formation at the microscope information limit Incident electron beam 1. The incident electron beam passes through the specimen to produce the specimen exit-surface wave. 1. The incident electron beam passes through the specimen to produce the specimen exit-surface wave. Exit-surface wave resolution is limited only by the electron scattering described by the interaction of the Ewald sphere with the specimen shape function. For electron wavelength  and specimen thickness t, the scattering resolution is given by dscatt = (t/2) At 300keV,  = 0.02Å, and values of thickness of 100Å and 65Å give dscatt = 1Å and 0.8Å. 1. The incident electron beam passes through the specimen to produce the specimen exit-surface wave. Exit-surface wave resolution is limited only by the electron scattering described by the interaction of the Ewald sphere with the specimen shape function. Specimen Exit-surface wave Central maximum in shape function falls to zero at sin (gt)/(gt)  0 then gt  and t  1/ g 2/(u2) or u2 2/(t) and d2 t/2 Objective lens Diffraction amplitudes 2. The (complex) exit-surface wave is transferred to the image plane by the objective lens, forming a (complex) image amplitude. 2. The (complex) exit-surface wave is transferred to the image plane by the objective lens, forming a (complex) image amplitude. During transfer, the objective lens imposes phase changes on the components of the exit-surface wave due to the lens defocus. Lens transfer blocks the exit-surface components that describe specimen spacings finer than the information limit of the microscope. 2. The (complex) exit-surface wave is transferred to the image plane by the objective lens, forming a (complex) image amplitude. During transfer, the objective lens imposes phase changes on the components of the exit-surface wave due to the lens defocus. 2. The (complex) exit-surface wave is transferred to the image plane by the objective lens, forming a (complex) image amplitude. During transfer, the objective lens imposes phase changes on the components of the exit-surface wave due to the lens defocus. Lens transfer blocks the exit-surface components that describe specimen spacings finer than the information limit of the microscope. For electron wavelength  and microscope spread of focus of D, the information limit is given by d= (pl D/2) At 300keV,  = 0.02Å, and values of D of 35Å and 20Å give d = 1Å and 0.8Å. Image Image formation to microscope information limit

  37. Image formation and exit-wave reconstruction Inverse 2-D Fourier transform Project paraboloid to zero-focus plane Locus of linear image contributions Estimate of exit-surface wave to information limit Incident electron beam Specimen Fourier components of exit-surface wave Exit-surface wave Objective lens  (Å-1) Diffraction amplitudes Stack of image diffractograms u (Å-1) 3-D Fourier transform  (Å) Focal series of images x (Å) Image Reconstruction of exit-wave to microscope information limit Image formation to microscope information limit ”Direct Structural Retrieval from high-resolution electron micrographs", D. Van Dyck and M. Op de Beeck, in Computer Simulation of Electron Microscope Diffraction and Images, A TMS Publication, William Krakow and Michael A. O'Keefe (eds) (1989) 265-271.

  38. 1 atom column 11 atom column 1 3 5 7 9 11 9 7 5 3 1 a b Simulated Pd cube-octahedron analysis -- Line trace shows peaks in ESW phase -- a b a b Model ESW phase ESW phase (peak height) is proportional to the number of atoms in the column producing the peak. Line trace shows the two-atom difference between adjacent columns. 0.57 radian # atoms in columns

  39. Experimental Au nanoparticle analysis Color coding shows phase of normalized ESW ((x,y) - (1+0i)). (a) Wide view (b) Particle (c) Support Complex pixel map of large area (a) shows a pink peak near 3pi/4 phase due to the nanoparticle. Amorphous support (c) contributes random phase to particle pixel map (b). Complex pixel maps of (x,y) - 1

  40. +1 +1 +1 +1 +1 0 0 0 0 0 -1 -1 -1 -1 -1 0 Spatial Frequency (Å-1) 1.0 1.5 OÅM CTF shows transfer of 0.89Å spacings from diamond CM300FEG/UT  = 36Å 1.07Å OÅM  = 20Å 0.78Å  = 0.25 millirad  1.1Å  n = 2 1.03Å n = 36  0.89Å

  41. 1.7Åresolution +1 +1 +1 +1 +1 CM300FEG/UT  = 36Å 0 0 0 0 0 1.07Å info limit -1 -1 -1 -1 -1 OÅM  = 20Å 0.78Å  = 0.25 millirad  1.1Å  n = 2 1.03Å n = 36  0.89Å 0 0 Spatial Frequency (Å-1) Spatial Frequency (Å-1) 1.0 1.0 1.5 1.5 CTFs show transfer of spatial frequencies, resolution, information limit

  42. Reconstructed Exit-Surface Wave of LiCoO2 Comparison of simulated and experimental ESWs shows that Li atom columns are visible at 0.9Å resolution in the OÅM. Experimental Simulation O Co O Li O is strong Co is “fuzzy” Li is weak The reconstructed exit-surface wave shows that the specimen is tilted away from exact [110] zone axis orientation and also reveals buckling and possible electron beam damage.

  43. Transmission Electron Achromatic Microscope >>> the TEAM project <<< • TEAM STEM/TEM building blocks: • Probe CS corrector is required for sub-Å probe to allow sub-Å Z-contrast and accurate spectroscopic imaging. • Monochromator for high energy resolution (better than 0.05eV) to provide high energy resolution for chemical characterization and improved information limit for high-resolution sub-Å microscopy. • Biprism for holographic studies of phase at high resolution. • Energy filter (in-column or post-column) with better than 0.05eV resolution (hi-res GIF already there). Post-column filter also provides extra magnification for holography and sub-Å imaging (>5Mx). • Objective-lens CS corrector to extend microscope resolution to the information limit. • High-stability lens and HT power supplies. Lens to 0.1ppm (FEI UT already at 0.3ppm). HT to 0.25ppm (FEI now at 0.25ppm at 200keV). • Large CCD camera for sufficient field of view at high magnification and holographic reconstruction (Gatan UltraScan has 4k by 4k now). • Low drift stage with sub-Å piezo-electric control (JEOL has 0.05Å). • Automated (computerized) procedures for alignment, aberration correction, image acquisition, and focal-series reconstruction.

  44. Push TEAM information limit to 0.5Å level Information limit is set by temporal coherence damping function: ED(u) = exp{-½p2l2D2u4} where D = CC{(s 2(Ebeam)/E2 + 4s 2(I)/I2} • CC is the chromatic aberration coefficient for the objective lens • s(Ebeam)/E is the rms energy spread in the electron beam as a fraction of total beam energy over the time of image acquisition. • s(I)/I is the fractional rms ripple in lens current. ED(u) imposes an information limit for the microscope of d = 1/|u|D = (plD/2)at a level of exp(-2) or 13.5% d = 0.8Å requires D = 20Åand d = 0.5Å requires D = 8Å TEAM OÅM

  45. 0.4 0.3 0.2 dE (eV) 0.1 0 0.2 0.3 0.1 0 Objective Lens Current Stability (I)/I (rms ppm) Improved lens current stability allows greater energy spread Greater energy spread allows more incident beam current Allowed incident beam energy spread (FWHH) for 0.5Å resolution 300keV 200keV 120keV 80keV If an objective lens current stability of 0.1 (rms) ppm can be achieved, an information limit of 0.5Å can be achieved with an incident beam energy spread (FWHH) of up to 0.35eV at 300keV and up to 0.18eV at 200keV, allowing reasonable incident beam currents for HREM imaging.

  46. Series of Simulated “Weak-Phase-Object” Images of LiCoO2 Co O 1.8Å 1.6Å 1.4Å 1.2Å Li Li O 1.0Å 0.8Å 0.6Å 0.4Å • Positions of Co atom columns should be seen clearly at resolutions as poor as 1.8Å. • Resolution of 1.4Å is required to make the O atom columns visible. • Li atoms can be seen at 1.0Å and become more visible at a resolution of 0.84Å. • In this approximation, atom columns appear black and proportional to atom mass (scattering cross section). • Resolutions of 0.6Å and 0.4Å are not attainable experimentally (yet).

  47. Simulated Pd cube-octahedron analysis -- Argand pixel map shows ESW amplitude and phase for every pixel -- 6 7 5 8 4 9 3 10 2 1 11 1+0i Im 2i i -3 -2 Re -1 2 -i Phase angle advances in “ticks” of 0.282 radian -2i Phase of ESW, (x,y) Complex pixel map of ESW,(x,y) Phase of ESW, (x,y) is color coded from light blue (zero phase) to red (pi radian). ESW, displayed in the form of an Argand plot with the same color coding, shows 11 atom columns. Argand plots can be used to show the trajectory of a complex function as specimen thickness is increased (O’Keefe, Ph.D. thesis, 1975). In this case the Argand plot of the ESW (Tomaszewicz, 2003), taken over every pixel in the (marked) frame containing the particle, shows 11 quantized traces (numbered) corresponding to the 11 different column heights making up the particle. The light blue area near the 1+0i point on the Argand plot contains the background (“vacuum”) pixels (since exp {i0} = 1+0i). The 11 traces contain all the pixels making up the ESW phase peaks at the positions of the atom columns. The highest phase change of each trace corresponds to the central pixel of each phase peak. Because only one atom column contains 11 atoms, trace number 11 is much weaker. ESW analysis program by T. Tomaszewicz (2003, to be published).

  48. Simulated Pd cube-octahedron analysis -- Argand pixel map shows ESW amplitude and phase for every pixel -- Im 6 7 5 8 2i 4 9 3 i 10 2 1 11 0+0i -1 -2 Re 2 1 -i -2i Phase of ((x,y) - (1+0i)) Complex pixel map of (x,y) - (1+0i) Phase of (ESW minus 1) is color coded from dark blue (zero phase) to red (pi radian). ESW, displayed in the form of an Argand plot with the same color coding, shows 11 atom columns. The constant complex “vacuum” amplitude can be subtracted out of the ESW and the resulting function ((x,y) -1) plotted in the Argand plane. These plots, and the resultant shifted trajectory have been used by Sinkler and Marks (1999) in an implementation of a direct-methods structure refinement. With the constant complex “vacuum” amplitude subtracted out, the background pixels become black, and the Argand plot of the “normalized” ESW ((x,y) -1) now shows the 11 column traces emanating from the origin (Tomaszewicz, 2003). "Dynamical Direct Methods for Everyone", W. Sinkler & L.D. Marks, Ultramicroscopy75 (1999)251-268. ESW analysis program by T. Tomaszewicz (2003, to be published).

  49. Experimental Au nanoparticle analysis -- Argand pixel map shows ESW amplitude and phase for each pixel -- The Argand plot of the ESW is taken over every pixel in the (marked) frame containing the particle (Tomaszewicz, 2003). Im i -1 Re -i Experimental ESW, as an Argand plot with the same color coding, shows elongation but no individual atom columns. Phase of experimental ESW, (x,y) is color coded from light blue (zero phase) to dark blue (pi/4 radian). ESW analysis program by T. Tomaszewicz (2003, to be published).

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