Electron Nanodiffraction

Part of the Springer Handbooks book series (SHB)


This chapter introduces the practice and theory of electron nanodiffraction. After a brief introduction, the chapter provides a comprehensive description of electron diffraction techniques and their use for nanodiffraction. This is followed by discussions on electron probe properties, electron energy filtering and electron diffraction data analysis. Throughout the chapter, we emphasize different electron nanoprobes that can be formed inside an electron microscope, from a focused beam to parallel illumination, and how these probes can be used to extract structural information from different materials. For this purpose, we outline the electron diffraction theories based on both kinematic approximation and dynamic diffraction, which serve as the basis for the interpretation of electron nanodiffraction patterns. The principles and applications of scanning electron nanodiffraction and coherent diffraction imaging are covered in detail with applications for orientation mapping, imaging strain, 3-D nanostructure determination, and study of defects.

electron diffraction

nanostructure analysis convergent beam electron diffraction (CBED) diffractive imaging four-dimensional scanning transmission electron microscopy (4-D STEM) strain mapping atomic resolution tomography 



The writing of this chapter was made possible with the support by US Department of Energy, Grant DEFG02-01ER45923 and NSF DMR 1410596. The work described here would not have been possible without the outstanding efforts of students and postdoc students, especially work by Weijie Huang, Yifei Meng, Yu-Tsun Shao, Kyouhyun Kim, Xiangwen Lu, Wenpei Gao, and Piyush Vivek Deshpande has contributed directly to the writing of this chapter.


  1. G. Möllenstedt: My early work on convergent-beam electron-diffraction, Phys. Status Solidi (a) 116, 13–22 (1989)Google Scholar
  2. C.H. MacGillavry: Examination of the dynamic theory of electron diffraction on lattice, Physica 7, 329–343 (1940)Google Scholar
  3. J.M. Cowley: Electron nanodiffraction, Microsc. Res. Tech. 46, 75–97 (1999)Google Scholar
  4. J.M. Cowley: Applications of electron nanodiffraction, Micron 35, 345 (2004)Google Scholar
  5. J.C.H. Spence, J.M. Zuo: Large dynamic-range, parallel detection system for electron-diffraction and imaging, Rev. Sci. Instrum. 59, 2102–2105 (1988)Google Scholar
  6. J.M. Zuo: Electron detection characteristics of a slow-scan CCD camera, imaging plates and film, and electron image restoration, Microsc. Res. Tech. 49, 245–268 (2000)Google Scholar
  7. L. Reimer (Ed.): Energy-Filtering Transmission Electron Microscopy (Springer, New York 1995)Google Scholar
  8. J.C.H. Spence, J.M. Zuo: Electron Microdiffraction (Plenum, New York 1992)Google Scholar
  9. H.E. Elsayed-Ali, P.M. Weber: Time-resolved surface electron diffraction. In: Time-Resolved Diffraction, ed. by J.R. Helliwell, P.M. Rentzepis (Oxford Univ. Press, New York 1997) pp. 284–322Google Scholar
  10. W.E. King, G.H. Campbell, A. Frank, B. Reed, J.F. Schmerge, B.J. Siwick, B.C. Stuart, P.M. Weber: Ultrafast electron microscopy in materials science, biology, and chemistry, J. Appl. Phys. 97, 111101 (2005)Google Scholar
  11. B.J. Siwick, J.R. Dwyer, R.E. Jordan, R.J.D. Miller: An atomic-level view of melting using femtosecond electron diffraction, Science 302, 1382–1385 (2003)Google Scholar
  12. J.M. Zuo, M. Gao, J. Tao, B.Q. Li, R. Twesten, I. Petrov: Coherent nano-area electron diffraction, Microsc. Res. Tech. 64, 347–355 (2004)Google Scholar
  13. G. Deptuch, A. Besson, P. Rehak, M. Szelezniak, J. Wall, M. Winter, Y. Zhu: Direct electron imaging in electron microscopy with monolithic active pixel sensors, Ultramicroscopy 107, 674–684 (2007)Google Scholar
  14. D. Contarato, P. Denes, D. Doering, J. Joseph, B. Krieger: Direct detection in transmission electron microscopy with a 5 μm pitch CMOS pixel sensor, Nucl. Instrum. Methods Phys. Res. A 635, 69–73 (2011)Google Scholar
  15. J.M. Cowley (Ed.): Electron Diffraction Techniques, Vol. I, II (Oxford Univ. Press, Oxford 1992)Google Scholar
  16. L.M. Peng, S.L. Dudarev, M.J. Whelan: High-Energy Electron Diffraction and Microscopy (Oxford Univ. Press, Oxford 2004)Google Scholar
  17. J.P. Morniroli: Large-Angle Convergent Beam Electron Diffraction (Society of French Microscopists, Paris 2002), English VersionGoogle Scholar
  18. J.M. Zuo, J.C.H. Spence: Advanced Transmission Electron Microscopy, Imaging and Diffraction in Nanoscience (Springer, New York 2017)Google Scholar
  19. S. Morishita, J. Yamasaki, K. Nakamura, T. Kato, N. Tanaka: Diffractive imaging of the dumbbell structure in silicon by spherical-aberration-corrected electron diffraction, Appl. Phys. Lett. 93, 183103 (2008)Google Scholar
  20. K.D. Van der Mast, C.J. Rakels, J.B. Le Poole: A high quality multipurpose objective lens. In: Proc. Eur. Congr. Electron Microsc. (1980) pp. 72–73Google Scholar
  21. K. Ran, X. Mi, Z.J. Shi, Q. Chen, Y.F. Shi, J.M. Zuo: Molecular packing of fullerenes inside single-walled carbon nanotubes, Carbon 50, 5450–5457 (2012)Google Scholar
  22. J.T. McKeown, J.C.H. Spence: The kinematic convergent-beam electron diffraction method for nanocrystal structure determination, J. Appl. Phys. 106, 074309 (2009)Google Scholar
  23. R. Vincent: Techniques of convergent beam electron-diffraction, J. Electron Microsc. Tech. 13, 40–50 (1989)Google Scholar
  24. M. Tanaka, R. Saito, K. Ueno, Y. Harada: Large-angle convergent-beam electron-diffraction, J. Electron Microsc. 29, 408–412 (1980)Google Scholar
  25. J.A. Eades: Zone-axis diffraction patterns by the Tanaka method, J. Electron Microsc. Tech. 1, 279–284 (1984)Google Scholar
  26. I.K. Jordan, C.J. Rossouw, R. Vincent: Effects of energy filtering in LACBED patterns, Ultramicroscopy 35, 237–243 (1991)Google Scholar
  27. K.K. Fung: Large-angle convergent-beam zone axis patterns, Ultramicroscopy 12, 243–246 (1984)Google Scholar
  28. M. Terauchi, M. Tanaka: Simultaneous observation of zone-axis pattern and ±G-dark-field pattern in convergent-beam electron-diffraction, J. Electron Microsc. 34, 347–356 (1985)Google Scholar
  29. M. Tanaka, M. Terauchi, T. Kaneyama: Convergent Beam Electron Diffraction II (JEOL, Tokyo 1988)Google Scholar
  30. J.P. Morniroli: CBED and LACBED analysis of stacking faults and antiphase boundaries, Mater. Chem. Phys. 81, 209–213 (2003)Google Scholar
  31. J.P. Morniroli, F. Gaillot: Trace analyses from LACBED patterns, Ultramicroscopy 83, 227–243 (2000)Google Scholar
  32. J.P. Morniroli, R.K.W. Marceau, S.P. Ringerz, L. Boulanger: LACBED characterization of dislocation loops, Philos. Mag. 86, 4883–4900 (2006)Google Scholar
  33. C.T. Koch: Aberration-compensated large-angle rocking-beam electron diffraction, Ultramicroscopy 111, 828–840 (2011)Google Scholar
  34. W. Krakow, L.A. Howland: A method for producing hollow cone illumination electronically in the conventional transmission microscope, Ultramicroscopy 2, 53–67 (1976)Google Scholar
  35. J.A. Eades: Zone-axis patterns formed by a new double-rocking technique, Ultramicroscopy 5, 71–74 (1980)Google Scholar
  36. R. Vincent, P.A. Midgley: Double conical beam-rocking system for measurement of integrated electron-diffraction intensities, Ultramicroscopy 53, 271–282 (1994)Google Scholar
  37. J.M. Zuo, J. Tao: Scanning electron nanodiffraction and diffraction imaging. In: Scanning Transmission Electron Microscopy, ed. by S. Pennycook, P. Nellist (Springer, New York 2011)Google Scholar
  38. K.H. Kim, H. Xing, J.M. Zuo, P. Zhang, H.F. Wang: TEM based high resolution and low-dose scanning electron nanodiffraction technique for nanostructure imaging and analysis, Micron 71, 39–45 (2015)Google Scholar
  39. K.H. Downing, R.M. Glaeser: Improvement in high-resolution image quality of radiation-sensitive specimens achieved with reduced spot size of the electron-beam, Ultramicroscopy 20, 269–278 (1986)Google Scholar
  40. C.S. Own, L.D. Marks, W. Sinkler: Electron precession: A guide for implementation, Rev. Sci. Instrum. 76, 033703 (2005)Google Scholar
  41. D. Jacob, P. Cordier, J.P. Morniroli, H.P. Schertl: Precession electron diffraction for the characterization of twinning in pseudo-symmetrical crystals: Case of coesite. In: Proc. EMC 2008 14th Eur. Microsc. Congr, ed. by M. Luysberg, K. Tillmann, T. Weirich (Springer, Berlin, Heidelberg 2008) pp. 193–194Google Scholar
  42. M. Blackman: On the intensities of electron diffraction rings, Proc. R. Soc. A 173, 68–82 (1939)Google Scholar
  43. M. Horstmann, G. Meyer: Messung der Elektronenbeugungsintensitäten polykristalliner Aluminiumschichten bei tiefer Temperatur und Vergleich mit der dynamischen Theorie, Z. Phys. 182, 380–397 (1965)Google Scholar
  44. K. Gjonnes: On the integration of electron diffraction intensities in the Vincent-Midgley precession technique, Ultramicroscopy 69, 1–11 (1997)Google Scholar
  45. J. Hwang, J.Y. Zhang, J. Son, S. Stemmer: Nanoscale quantification of octahedral tilts in perovskite films, Appl. Phys. Lett. 100, 191909 (2012)Google Scholar
  46. J.C.H. Spence, J.M. Cowley: Lattice imaging in STEM, Optik 50, 129–142 (1978)Google Scholar
  47. J.C.H. Spence, J. Lynch: STEM microanalysis by transmission electron-energy loss spectroscopy in crystals, Ultramicroscopy 9, 267–276 (1982)Google Scholar
  48. J. Zhu, J.M. Cowley: Micro-diffraction from stacking-faults and twin boundaries in fcc crystals, J. Appl. Crystallogr. 16, 171–175 (1983)Google Scholar
  49. J.M. Cowley, J.C.H. Spence: Convergent beam electron microdiffraction from small crystals, Ultramicroscopy 6, 359–366 (1981)Google Scholar
  50. J.M. LeBeau, S.D. Findlay, L.J. Allen, S. Stemmer: Position averaged convergent beam electron diffraction: Theory and applications, Ultramicroscopy 110, 118–125 (2010)Google Scholar
  51. C. Mory, C. Colliex, J.M. Cowley: Optimum defocus for STEM imaging and microanalysis, Ultramicroscopy 21, 171–177 (1987)Google Scholar
  52. S.D. Berger, I.G. Salisbury, R.H. Milne, D. Imeson, C.J. Humphreys: Electron energy-loss spectroscopy studies of nanometer-scale structures in alumina produced by intense electron-beam irradiation, Philos. Mag. B 55, 341–358 (1987)Google Scholar
  53. J.M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L.A. Nagahara: Atomic resolution imaging of a carbon nanotube from diffraction intensities, Science 300, 1419–1421 (2003)Google Scholar
  54. A. Beche, J.L. Rouviere, L. Clement, J.M. Hartmann: Improved precision in strain measurement using nanobeam electron diffraction, Appl. Phys. Lett. 95, 123114 (2009)Google Scholar
  55. G. Botton: Analytical Electron Microscopy. In: Science of Microscopy, Vol. I, ed. by P. Hawkes, J.C.H. Spence (Springer, New York 2007)Google Scholar
  56. H. Lichte, M. Lehmann: Electron holography—Basics and applications, Rep. Prog. Phys. 71, 016102 (2008)Google Scholar
  57. R.F. Egerton: Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd edn. (Springer, New York 2011)Google Scholar
  58. P. Duval, N. Hoan, J. Brian, L. Henry: Réalisation d'un dispositif de filtrage en énergie des images de microdiffraction électronique, Nouv. Rev. Opt. Appl. 1, 221–228 (1970)Google Scholar
  59. M.M.J. Treacy, J.M. Gibson: The effects of elastic relaxation on transmission electron-microscopy studies of thinned composition-modulated materials, J. Vac. Sci. Technol. B 4, 1458–1466 (1986)Google Scholar
  60. P. Hirsch, A. Howie, R.B. Nicolson, D.W. Pashley, M.J. Whelan: Electron Microscopy of Thin Crystals (Krieger, Malabar 1977)Google Scholar
  61. J.M. Zuo, A.L. Weickenmeier: On the beam selection and convergence in the Bloch-wave method, Ultramicroscopy 57, 375–383 (1995)Google Scholar
  62. I.A. Sheremetyev, A.V. Turbal, Y.M. Litvinov, M.A. Mikhailov: Computer deciphering of Laue patterns: Application to white synchrotron x-ray topography, Nucl. Instrum. Methods Phys. Res. A 308, 451–455 (1991)Google Scholar
  63. H.R. Wenk, F. Heidelbach, D. Chateigner, F. Zontone: Laue orientation imaging, J. Synchrotron Radiat. 4, 95–101 (1997)Google Scholar
  64. S. Zaefferer: New developments of computer-aided crystallographic analysis in transmission electron microscopy, J. Appl. Crystallogr. 33, 10–25 (2000)Google Scholar
  65. E.F. Rauch, L. Dupuy: Rapid spot diffraction patterns identification through template matching, Arch. Metall. Mater. 50, 87–99 (2005)Google Scholar
  66. E.F. Rauch, A. Duft: Orientation maps derived from TEM diffraction patterns collected with an external CCD camera, Mater. Sci. Forum 495–497, 197–202 (2005)Google Scholar
  67. E.F. Rauch, M. Veron: Coupled microstructural observations and local texture measurements with an automated crystallographic orientation mapping tool attached to a TEM, Materialwiss. Werkstofftech. 36, 552–556 (2005)Google Scholar
  68. G. Wu, S. Zaefferer: Advances in TEM orientation microscopy by combination of dark-field conical scanning and improved image matching, Ultramicroscopy 109, 1317–1325 (2009)Google Scholar
  69. E.F. Rauch, J. Portillo, S. Nicolopoulos, D. Bultreys, S. Rouvimov, P. Moeck: Automated nanocrystal orientation and phase mapping in the transmission electron microscope on the basis of precession electron diffraction, Z. Kristallogr. 225, 103–109 (2010)Google Scholar
  70. Y. Meng, J.-M. Zuo: Improvements in electron diffraction pattern automatic indexing algorithms, Eur. Phys. J. Appl. Phys. 80, 10701 (2017)Google Scholar
  71. J.P. Lewis: Fast template matching, Vis. Interface 95, 120–123 (1995)Google Scholar
  72. D. Dingley: Progressive steps in the development of electron backscatter diffraction and orientation imaging microscopy, J. Microsc. 213, 214–224 (2004)Google Scholar
  73. R. van Bremen, D. Ribas Gomes, L.T.H. de Jeer, V. Ocelík, J.T.M. De Hosson: On the optimum resolution of transmission-electron backscattered diffraction (t-EBSD), Ultramicroscopy 160, 256–264 (2016)Google Scholar
  74. K.J. Ganesh, A.D. Darbal, S. Rajasekhara, G.S. Rohrer, K. Barmak, P.J. Ferreira: Effect of downscaling nano-copper interconnects on the microstructure revealed by high resolution TEM-orientation-mapping, Nanotechnology 23, 135702 (2012)Google Scholar
  75. E.F. Rauch, M. Véron: Automated crystal orientation and phase mapping in TEM, Mater. Charact. 98, 1–9 (2014)Google Scholar
  76. A.D. Darbal, K.J. Ganesh, X. Liu, S.B. Lee, J. Ledonne, T. Sun, B. Yao, A.P. Warren, G.S. Rohrer, A.D. Rollett, P.J. Ferreira, K.R. Coffey, K. Barmak: Grain boundary character distribution of nanocrystalline Cu thin films using stereological analysis of transmission electron microscope orientation maps, Microsc. Microanal. 19, 111–119 (2013)Google Scholar
  77. Y. Hu, J.H. Huang, J.M. Zuo: In situ characterization of fracture toughness and dynamics of nanocrystalline titanium nitride films, J. Mater. Res. 31, 370–379 (2016)Google Scholar
  78. J.L. Rouviere, A. Beche, Y. Martin, T. Denneulin, D. Cooper: Improved strain precision with high spatial resolution using nanobeam precession electron diffraction, Appl. Phys. Lett. 103, 241913 (2013)Google Scholar
  79. H.N. Chapman, A. Barty, S. Marchesini, A. Noy, S.R. Hau-Riege, C. Cui, M.R. Howells, R. Rosen, H. He, J.C.H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, D. Shapiro: High-resolution ab initio three-dimensional x-ray diffraction microscopy, J. Opt. Soc. Am. A 23, 1179–1200 (2006)Google Scholar
  80. H. Poulsen: An introduction to three-dimensional x-ray diffraction microscopy, J. Appl. Crystallogr. 45, 1084–1097 (2012)Google Scholar
  81. B.C. Larson, W. Yang, G.E. Ice, J.D. Budai, J.Z. Tischler: Three-dimensional x-ray structural microscopy with submicrometre resolution, Nature 415, 887–890 (2002)Google Scholar
  82. W. Ludwig, S. Schmidt, E.M. Lauridsen, H.F. Poulsen: X-ray diffraction contrast tomography: A novel technique for three-dimensional grain mapping of polycrystals. I. Direct beam case, J. Appl. Crystallogr. 41, 302–309 (2008)Google Scholar
  83. A.D. Rollett, S.B. Lee, R. Campman, G.S. Rohrer: Three-dimensional characterization of microstructure by electron back-scatter diffraction, Annu. Rev. Mater. Res. 37, 627–658 (2007)Google Scholar
  84. H.H. Liu, S. Schmidt, H.F. Poulsen, A. Godfrey, Z.Q. Liu, J.A. Sharon, X. Huang: Three-dimensional orientation mapping in the transmission electron microscope, Science 332, 833–834 (2011)Google Scholar
  85. A.S. Eggeman, R. Krakow, P.A. Midgley: Scanning precession electron tomography for three-dimensional nanoscale orientation imaging and crystallographic analysis, Nat. Commun. 6, 7267 (2015)Google Scholar
  86. P.A. Midgley, R.E. Dunin-Borkowski: Electron tomography and holography in materials science, Nat. Mater. 8, 271–280 (2009)Google Scholar
  87. Y. Meng, J.-M. Zuo: Three-dimensional nanostructure determination from a large diffraction data set recorded using scanning electron nanodiffraction, IUCrJ 3, 300–308 (2016)Google Scholar
  88. J.M. Zuo, A.B. Shah, H. Kim, Y.F. Meng, W.P. Gao, J.L. Rouviere: Lattice and strain analysis of atomic resolution Z-contrast images based on template matching, Ultramicroscopy 136, 50–60 (2014)Google Scholar
  89. S. Zaefferer: A critical review of orientation microscopy in SEM and TEM, Cryst. Res. Technol. 46, 607–628 (2011)Google Scholar
  90. G.T. Herman: Fundamentals of Computerized Tomography: Image Reconstruction from Projections, Advances in Pattern Recognition (Springer, London 2009)Google Scholar
  91. J. Amanatides, A. Woo: A fast voxel traversal algorithm for ray tracing, Eurographics 87, 3–10 (1987)Google Scholar
  92. S. Kaczmarz: Angenäherte Auflösung von Systemen linearer Gleichungen, Bull. Intern. Acad. Pol. Sci. Lett., Cl. Sci. Math. Nat. A 35, 335–357 (1937)Google Scholar
  93. F.L. Markley: Attitude determination using vector observations and the singular value decomposition, J. Astronaut. Sci. 38, 245–258 (1988)Google Scholar
  94. S.V. Fortuna, Y.P. Sharkeev, A.J. Perry, J.N. Matossian, I.A. Shulepov: Microstructural features of wear-resistant titanium nitride coatings deposited by different methods, Thin Solid Films 377/378, 512–517 (2000)Google Scholar
  95. W.-L. Pan, G.-P. Yu, J.-H. Huang: Mechanical properties of ion-plated tin films on AISI D-2 steel, Surf. Coat. Technol. 110, 111–119 (1998)Google Scholar
  96. A.-N. Wang, G.P. Yu, J.-H. Huang: Fracture toughness measurement on tin hard coatings using internal energy induced cracking, Surf. Coat. Technol. 239, 20–27 (2014)Google Scholar
  97. C.H. Ma, J.-H. Huang, H. Chen: Nanohardness of nanocrystalline tin thin films, Surf. Coat. Technol. 200, 3868–3875 (2006)Google Scholar
  98. P.H. Mayrhofer, C. Mitterer, J. Musil: Structure–property relationships in single- and dual-phase nanocrystalline hard coatings, Surf. Coat. Technol. 174/175, 725–731 (2003)Google Scholar
  99. P.H. Mayrhofer, F. Kunc, J. Musil, C. Mitterer: A Comparative study on reactive and non-reactive unbalanced magnetron sputter deposition of tin coatings, Thin Solid Films 415, 151–159 (2002)Google Scholar
  100. L.M. Peng, S.L. Dudarev, M.J. Whelan: High Energy Electron Diffraction and Microscopy (Oxford Univ. Press, Oxford 2004)Google Scholar
  101. D.M. Bird, Q.A. King: Absorptive form-factors for high-energy electron-diffraction, Acta Crystallogr. A 46, 202–208 (1990)Google Scholar
  102. A. Weickenmeier, H. Kohl: Computation of absorptive form-factors for high-energy electron-diffraction, Acta Crystallogr. A 47, 590–597 (1991)Google Scholar
  103. L.M. Peng: Anisotropic thermal vibrations and dynamical electron diffraction by crystals, Acta Crystallogr. A 53, 663–672 (1997)Google Scholar
  104. L. Sturkey: The use of electron-diffraction intensities in structure determination, Acta Crystallogr. 10, 858 (1957)Google Scholar
  105. D. Jacob, J.M. Zuo, A. Lefebvre, Y. Cordier: Composition analysis of semiconductor quantum wells by energy filtered convergent-beam electron diffraction, Ultramicroscopy 108, 358–366 (2008)Google Scholar
  106. C.J. Rossouw, M. Alkhafaji, D. Cherns, J.W. Steeds, R. Touaitia: A treatment of dynamic diffraction for multiply layered structures, Ultramicroscopy 35, 229–236 (1991)Google Scholar
  107. J.M. Cowley, A.F. Moodie: The scattering of electrons by atoms and crystals. I. A new theoretical approach, Acta Crystallographica 10(10), 609–619 (1957)Google Scholar
  108. K. Ishizuka: Multislice formula for inclined illumination, Acta Crystallogr. A 38, 773–779 (1982)Google Scholar
  109. B.F. Buxton, J.A. Eades, J.W. Steeds, G.M. Rackham: Symmetry of electron-diffraction zone axis patterns, Philos. Trans. R. Soc. A 281, 171 (1976)Google Scholar
  110. M. Tanaka, R. Saito, H. Sekii: Point-group determination by convergent-beam electron-diffraction, Acta Crystallogr. A 39, 357–368 (1983)Google Scholar
  111. M. Tanaka, H. Sekii, T. Nagasawa: Space-group determination by dynamic extinction in convergent-beam electron-diffraction, Acta Crystallogr. A 39, 825–837 (1983)Google Scholar
  112. G.B. Hu, L.M. Peng, Q.F. Yu, H.Q. Lu: Automated identification of symmetry in CBED patterns: A genetic approach, Ultramicroscopy 84, 47–56 (2000)Google Scholar
  113. R. Vincent, T.D. Walsh: Quantitative assessment of symmetry in CBED patterns, Ultramicroscopy 70, 83–94 (1997)Google Scholar
  114. J.F. Mansfield: Error bars in CBED symmetry?, Ultramicroscopy 18, 91–96 (1985)Google Scholar
  115. K.H. Kim, J.M. Zuo: Symmetry quantification and mapping using convergent beam electron diffraction, Ultramicroscopy 124, 71–76 (2013)Google Scholar
  116. J.M. Kiat, Y. Uesu, B. Dkhil, M. Matsuda, C. Malibert, G. Calvarin: Monoclinic structure of unpoled morphotropic high piezoelectric PMN-PT and PZN-PT compounds, Phys. Rev. B 65, 064106 (2002)Google Scholar
  117. Y.-T. Shao, J.-M. Zuo: Fundamental symmetry of barium titanate single crystal determined using energy-filtered scanning convergent beam electron diffraction, Microsc. Microanal. 22, 516–517 (2016)Google Scholar
  118. J.M. Zuo, J.C.H. Spence: Automated structure factor refinement from convergent-beam patterns, Ultramicroscopy 35, 185–196 (1991)Google Scholar
  119. J.M. Zuo: Accurate structure refinement and measurement of crystal charge distribution using convergent beam electron diffraction, Microsc. Res. Tech. 46, 220–233 (1999)Google Scholar
  120. Y. Ogata, K. Tsuda, M. Tanaka: Determination of the electrostatic potential and electron density of silicon using convergent-beam electron diffraction, Acta Crystallogr. A 64, 587–597 (2008)Google Scholar
  121. P.N.H. Nakashima, B.C. Muddle: Differential convergent beam electron diffraction: Experiment and theory, Phys. Rev. B 81, 115135 (2010)Google Scholar
  122. J.M. Zuo: Quantitative convergent beam electron diffraction, Mater. Trans. JIM 39, 938–946 (1998)Google Scholar
  123. J. Friis, B. Jiang, J.C.H. Spence, R. Holmestad: Quantitative convergent beam electron diffraction measurements of low-order structure factors in copper, Microsc. Microanal. 9, 379–389 (2003)Google Scholar
  124. P.N.H. Nakashima: Improved quantitative CBED structure-factor measurement by refinement of nonlinear geometric distortion corrections, J. Appl. Crystallogr. 38, 374–376 (2005)Google Scholar
  125. K. Tsuda, M. Tanaka: Refinement of crystal structural parameters using two-dimensional energy-filtered CBED patterns, Acta Crystallogr. A 55, 939–954 (1999)Google Scholar
  126. B. Jiang, J.M. Zuo, J. Friis, J.C.H. Spence: On the consistency of QCBED structure factor measurements for TiO2 (Rutile), Microsc. Microanal. 9, 457–467 (2003)Google Scholar
  127. M. Saunders, D.M. Bird, N.J. Zaluzec, W.G. Burgess, A.R. Preston, C.J. Humphreys: Measurement of low-order structure factors for silicon from zone-axis CBED patterns, Ultramicroscopy 60, 311–323 (1995)Google Scholar
  128. B.T.M. Willis, A.W. Pryor: Thermal Vibrations in Crystallography (Cambridge Univ. Press, Cambridge 1975)Google Scholar
  129. G. Ren, J.M. Zuo, L.M. Peng: Accurate measurements of crystal structure factors using a FEG electron microscope, Micron 28, 459–467 (1997)Google Scholar
  130. J.M. Zuo: Measurements of electron densities in solids: A real-space view of electronic structure and bonding in inorganic crystals, Rep. Prog. Phys. 67, 2053–2103 (2004)Google Scholar
  131. J.M. Rodenburg: Ptychography and related diffractive imaging methods, Adv. Imaging Electron Phys. 150, 87–184 (2008)Google Scholar
  132. W. Hoppe: Trace structure-analysis, ptychography, phase tomography, Ultramicroscopy 10, 187–198 (1982)Google Scholar
  133. J.C.H. Spence, U. Weierstall, M. Howells: Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction, Philos. Trans. R. Soc. A 360, 875–895 (2002)Google Scholar
  134. H. Lichte: Electron holography approaching atomic resolution, Ultramicroscopy 20, 293–304 (1986)Google Scholar
  135. A. Orchowski, W.D. Rau, H. Lichte: Electron holography surmounts resolution limit of electron-microscopy, Phys. Rev. Lett. 74, 399–402 (1995)Google Scholar
  136. S.G. Podorov, K.M. Pavlov, D.M. Paganin: A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging, Opt. Express 15, 9954–9962 (2007)Google Scholar
  137. I.L. Karle, J. Karle: The crystal and molecular structure of the alkaloid jamine from Ormosia jamaicensis, Acta Crystallogr. 17, 1356 (1964)Google Scholar
  138. J. Karle, I.L. Karle: The symbolic addition procedure for phase determination for centrosymmetric and noncentrosymmetric crystals, Acta Crystallogr. 21, 849 (1966)Google Scholar
  139. D. Gabor: A new microscopic principle, Nature 161, 777–778 (1948)Google Scholar
  140. J.C.H. Spence: Stem and shadow-imaging of biomolecules at 6 eV beam energy, Micron 28, 101–116 (1997)Google Scholar
  141. J.C.H. Spence, T. Vecchione, U. Weierstall: A coherent photofield electron source for fast diffractive and point-projection imaging, Philos. Mag. 90, 4691–4702 (2010)Google Scholar
  142. H. Nyquist: Certain topics in telegraph transmission theory, Trans. Am. Inst. Electr. Eng. 47, 617 (1928)Google Scholar
  143. C.E. Shannon: Communication in the presence of noise, Inst. Radio Eng. 37, 10 (1949)Google Scholar
  144. D. Sayre, H.N. Chapman, J. Miao: On the extendibility of x-ray crystallography to noncrystals, Acta Crystallogr. A 54, 232–239 (1998)Google Scholar
  145. J.C.H. Spence, U. Weierstall, M. Howells: Coherence and sampling requirements for diffraction imaging, Ultramicroscopy 101, 149–152 (2004)Google Scholar
  146. W.J. Huang, B. Jiang, R.S. Sun, J.M. Zuo: Towards sub-Å atomic resolution electron diffraction imaging of metallic nanoclusters: A simulation study of experimental parameters and reconstruction algorithms, Ultramicroscopy 107, 1159–1170 (2007)Google Scholar
  147. V. Elser: Phase retrieval by iterated projections, J. Opt. Soc. Am. A 20, 40 (2003)Google Scholar
  148. J.R. Fienup: Phase retrieval algorithms—A comparison, Appl. Opt. 21, 2758–2769 (1982)Google Scholar
  149. J.R. Fienup: Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint, J. Opt. Soc. Am. 6, 118 (1987)Google Scholar
  150. R.W. Gerchberg, W.O. Saxton: Practical algorithm for determination of phase from image and diffraction plane pictures, Optik 35, 237 (1972)Google Scholar
  151. G. Oszlanyi, A. Suto: Ab initio structure solution by charge flipping, Acta Crystallogr. A 60, 134–141 (2004)Google Scholar
  152. J.S. Wu, J.C.H. Spence: Reconstruction of complex single-particle images using charge-flipping algorithm, Acta Crystallogr. A 61, 194–200 (2005)Google Scholar
  153. J.M. Zuo, J. Zhang, W.J. Huang, K. Ran, B. Jiang: Combining real and reciprocal space information for aberration free coherent electron diffractive imaging, Ultramicroscopy 111, 817–823 (2011)Google Scholar
  154. R.P. Millane, W.J. Stroud: Reconstructing symmetric images from their undersampled Fourier intensities, J. Opt. Soc. Am. A 14, 568–579 (1997)Google Scholar
  155. C.C. Chen, J. Miao, C.W. Wang, T.K. Lee: Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method, Phys. Rev. B 76, 064113 (2007)Google Scholar
  156. R. Dronyak, K.S. Liang, Y.P. Stetsko, T.K. Lee, C.K. Feng, J.S. Tsai, F.R. Chen: Electron diffractive imaging of nano-objects using a guided method with a dynamic support, Appl. Phys. Lett. 95, 111908 (2009)Google Scholar
  157. H.D. Jiang, C.Y. Song, C.C. Chen, R. Xu, K.S. Raines, B.P. Fahimian, C.H. Lu, T.K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, J.W. Miao: Quantitative 3-D imaging of whole, unstained cells by using x-ray diffraction microscopy, Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010)Google Scholar
  158. J. Gulden, O.M. Yefanov, A.P. Mancuso, V.V. Abramova, J. Hilhorst, D. Byelov, I. Snigireva, A. Snigirev, A.V. Petukhov, I.A. Vartanyants: Coherent x-ray imaging of defects in colloidal crystals, Phys. Rev. B 81, 224105 (2010)Google Scholar
  159. P.E. Batson, N. Dellby, O.L. Krivanek: Sub-Angstrom resolution using aberration corrected electron optics, Nature 418, 617–620 (2002)Google Scholar
  160. P.D. Nellist, M.F. Chisholm, N. Dellby, O.L. Krivanek, M.F. Murfitt, Z.S. Szilagyi, A.R. Lupini, A. Borisevich, W.H. Sides, S.J. Pennycook: Direct sub-angstrom imaging of a crystal lattice, Science 305, 1741–1741 (2004)Google Scholar
  161. R. Erni, M.D. Rossell, C. Kisielowski, U. Dahmen: Atomic-resolution imaging with a sub-50-pm electron probe, Phys. Rev. Lett. 102, 096101 (2009)Google Scholar
  162. H. Sawada, Y. Tanishiro, N. Ohashi, T. Tomita, F. Hosokawa, T. Kaneyama, Y. Kondo, K. Takayanagi: STEM imaging of 47-pm-separated atomic columns by a spherical aberration-corrected electron microscope with a 300-kV cold field emission gun, J. Electron Microsc. 58, 357–361 (2009)Google Scholar
  163. S. Van Aert, K.J. Batenburg, M.D. Rossell, R. Erni, G. Van Tendeloo: Three-dimensional atomic imaging of crystalline nanoparticles, Nature 470, 374–377 (2011)Google Scholar
  164. A.Y. Borisevich, A.R. Lupini, S.J. Pennycook: Depth sectioning with the aberration-corrected scanning transmission electron microscope, Proc. Natl. Acad. Sci. U.S.A. 103, 3044–3048 (2006)Google Scholar
  165. A.Y. Borisevich, A.R. Lupini, S. Travaglini, S.J. Pennycook: Depth sectioning of aligned crystals with the aberration-corrected scanning transmission electron microscope, J. Electron Microsc. 55, 7–12 (2006)Google Scholar
  166. H.L. Xin, D.A. Muller: Aberration-corrected ADF-STEM depth sectioning and prospects for reliable 3-D imaging in S/TEM, J. Electron Microsc. 58, 157–165 (2009)Google Scholar
  167. R. Ishikawa, A.R. Lupini, S.D. Findlay, S.J. Pennycook: Quantitative annular dark field electron microscopy using single electron signals, Microsc. Microanal. 20, 99–110 (2014)Google Scholar
  168. E.C. Cosgriff, P.D. Nellist: A Bloch wave analysis of optical sectioning in aberration-corrected STEM, Ultramicroscopy 107, 626–634 (2007)Google Scholar
  169. M.C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B.C. Regan, J. Miao: Electron tomography at 2.4-ångström resolution, Nature 483, 444–491 (2012)Google Scholar
  170. C.-C. Chen, C. Zhu, E.R. White, C.-Y. Chiu, M.C. Scott, B.C. Regan, L.D. Marks, Y. Huang, J. Miao: Three-dimensional imaging of dislocations in a nanoparticle at atomic resolution, Nature 496, 74 (2013)Google Scholar
  171. X.W. Lu, W.P. Gao, J.M. Zuo, J.B. Yuan: Atomic resolution tomography reconstruction of tilt series based on a GPU accelerated hybrid input-output algorithm using polar fourier transform, Ultramicroscopy 149, 64–73 (2015)Google Scholar
  172. J. Miao, T. Ohsuna, O. Terasaki, K.O. Hodgson, M.A. O'Keefe: Atomic resolution three-dimensional electron diffraction microscopy, Phys. Rev. Lett. 89, 155502 (2002)Google Scholar
  173. J. Keiner, S. Kunis, D. Potts: Using NFFT 3---A software library for various nonequispaced fast Fourier transforms, ACM Trans. Math. Softw. 36, 19 (2009)Google Scholar
  174. M. Fenn, S. Kunis, D. Potts: On the computation of the polar FFT, Appl. Comput. Harmon. Anal. 22, 257–263 (2007)Google Scholar
  175. S. Kunis, S. Kunis: The nonequispaced FFT on graphics processing units, Proc. Appl. Math. Mech. 12, 7–10 (2012)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dept. of Materials Science & Engineering and Materials Research LaboratoryUniversity of Illinois at Urbana-ChampaignUrbana, ILUSA

Personalised recommendations