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Theory of Calculation of Images of Thick Specimens

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Abstract

This chapter describes the theory of calculating transmission electron microscope image of thick specimens (more than a few atoms thick), including the effects of multiple (or plural) scattering. Two popular methods are presented: Bloch wave methods and multislice methods. These approximations are typically good for specimens up to a few thousand Angstroms thick.

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References

  1. G. P. Agrawal. Nonlinear Fiber Optics. Academic Press, San Diego, 2nd edition, 1995.

    MATH  Google Scholar 

  2. L. J. Allen, A. J. D’Alfonso, and S. D. Findlay. Modeling the inelastic scattering of fast electrons. Ultramicroscopy, 151:11–22, 2015.

    Article  Google Scholar 

  3. L. J. Allen, H. M. L. Faulkner, and H. Leeb. Inversion of dynamical electron diffraction data including adsorption. Acta Cryst., A56:119–126, 2000.

    Article  Google Scholar 

  4. L. J. Allen, S. D. Findlay, M. P. Oxley, and C. J. Rossouw. Lattice-resolution contrast from a focused coherent electron probe. Part I. Ultramicroscopy, 96:47–63, 2003.

    Article  Google Scholar 

  5. L. J. Allen, T. W. Josefsson, and H. Leeb. Obtaining the crystal potential by inversion from electron scattering intensities. Acta Cryst., A54:388–398, 1998.

    Article  Google Scholar 

  6. L. J. Allen, H. Leeb, and A. E. C. Spargo. Retrieval of the projected potential by inversion from the scattering matrix in electron-crystal scattering. Acta Cryst., A55:105–111, 1999.

    Article  Google Scholar 

  7. J. G. Allpress, E. A. Hewat, A. F. Moodie, and J. V. Sanders. n-beam lattice images. I. experimental and computed images of W4Nb26O77. Acta Cryst., A28:528–536, 1972.

    Google Scholar 

  8. J. G. Allpress and J. V. Sanders. The direct observation of the structure of real crystals by lattice imaging. J. Appl. Cryst., 6:165–190, 1973.

    Article  Google Scholar 

  9. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999.

    Google Scholar 

  10. G. R. Anstis and D. J. H. Cockayne. The calculation and interpretation of high-resolution electron microscope images of lattice defects. Acta Cryst., A35:511–524, 1979.

    Article  Google Scholar 

  11. N. W. Ashcroft and N. D. Mermin. Solid State Physics. Holt, Rinehart and Winston, New York, 1976.

    Google Scholar 

  12. J. Barthel. Dr. Probe: A software for high-resolution STEM image simulation. Ultramicroscopy, 193:1–11, 2018.

    Article  Google Scholar 

  13. B. W. Batterman and H. Cole. Dynamical diffraction of X-rays by perfect crystals. Reviews of Modern Physics, 36:681–717, 1964.

    Article  ADS  MathSciNet  Google Scholar 

  14. M. J. Beeching and A. E. C. Spargo. A method for crystal potential retrieval in HRTEM. Ultramicroscopy, 52:243–247, 1993.

    Article  Google Scholar 

  15. M. J. Beeching and A. E. C. Spargo. Inversion of nonperiodic wavefields to determine localized defect structure. J. Microscopy, 190:262–266, 1998.

    Article  Google Scholar 

  16. H. Bethe. Theorie der beugung von elektronen an kristallen. Annalen der Physik, 87:55–129, 1928.

    Article  ADS  Google Scholar 

  17. D. M. Bird. Theory of zone axis electron diffraction. J. of Elect. Micros. Tech., 13:77–97, 1989.

    Article  Google Scholar 

  18. M. Born and E. Wolf. Principles of Optics. Pergamon Press, Oxford, 6th edition, 1980.

    MATH  Google Scholar 

  19. L. A. Bursill and A. R. Wilson. Electron-optical imaging of the hollandite structure at 3 Å resolution. Acta Cryst., A33:672–676, 1977.

    Article  Google Scholar 

  20. C. Cai and J. Chen. An accurate multislice method for low-energy transmission electron microscopy. Micron, 43:374–379, 2012.

    Article  Google Scholar 

  21. Can Ying Cai, Song Jun Zeng, Hong Rong Liu, and Qi Bin Yang. Computational comparison of the conventional multislice method and the real space multislice method for simulating exit wavefunctions. Micron, 40:313–319, 2009.

    Article  Google Scholar 

  22. E. Carlino, V. Grillo, and P. Palazzari. Accurate and fast multislice simulations of HAADF image contrast by parallel computing. In A. G. Cullis and P. A. Midgley, editors, Springer Proc. in Phys., volume 120, pages 177–180, 2008.

    Google Scholar 

  23. J. H. Chen and D. Van Dyck. Accurate multislice theory for elastic electron scattering in transmission electron microscopy. Ultramicroscopy, 70:29–44, 1997.

    Article  Google Scholar 

  24. J. H. Chen, D. Van Dyck, and M. Op de Beck. Multislice method for large beam tilt with applications to HOLZ effects in triclinic and monoclinic crystals. Acta Cryst., A53:576–589, 1997.

    Article  Google Scholar 

  25. J. H. Chen, D. Van Dyck, M. Op de Beck, and J. Van Landuyt. Computational comparisons between the conventional multislice method and the third-order multislice method for calculating high-energy electron diffraction and imaging. Ultramicroscopy, 69:219–240, 1997.

    Article  Google Scholar 

  26. W. Coene and D. Van Dyck. The real space method for dynamical electron diffraction calculation in high resolution electron microscopy, II. critical analysis of the dependency on the input parameters. Ultramicroscopy, 15:41–50, 1984.

    Article  Google Scholar 

  27. W. Coene and D. Van Dyck. The real space method for dynamical electron diffraction calculations in high resolution electron microscopy III. a computational algorithm for the electron propagation with practical applications. Ultramicroscopy, 15:287–300, 1984.

    Article  Google Scholar 

  28. J. M. Cordes, A. Pidwerbetsky, and R. V. E. Lovelace. Refractive and diffractive scattering in the interstellar medium. The Astrophysical J., 310:737–767, 1986.

    Article  ADS  Google Scholar 

  29. J. M. Cowley. Diffraction Physics. North-Holland, Amsterdam, 2nd edition, 1975.

    Google Scholar 

  30. J. M. Cowley and A. F. Moodie. The scattering of electrons by atoms and crystals. I. a new theoretical approach. Acta Cryst., 10:609–619, 1957.

    Article  MathSciNet  Google Scholar 

  31. J. M. Cowley and J. C. H. Spence. Innovative imaging and microdiffraction in STEM. Ultramicroscopy, 3:433–438, 1979.

    Article  Google Scholar 

  32. D. Drouin, A. R. Couture, D. Joly, X. Tastet, V. Aimez, and R. Gauvin. CASINO V 2.42 - a fast and easy-to-use modeling tool for scanning electron microscopy and microanalysis users. Scanning, 29:92–101, 2007.

    Article  Google Scholar 

  33. D. Drouin, P. Hovington, and R. Gauvin. CASINO: A new Monte Carlo code in C language for electron beam interaction–part II: Tabulated values of the Mott cross section. Scanning, 19:20–28, 1997.

    Article  Google Scholar 

  34. D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing. Prentice Halls, New Jersey, 1984.

    MATH  Google Scholar 

  35. B. J. Dulong, R. D. Haynes, and M. D. Robertson. A study in the computation time required for the inclusion of strain field effects in Bloch-wave simulations of TEM diffraction contrast images. Ultramicroscopy, 108:415–425, 2008.

    Article  Google Scholar 

  36. C. Dwyer. Multislice simulation of scanning transmission electron microscope images. In L. N. Brewer, S. McKernan, J. P. Shields, F. E. Schmidt Jr, J. H. Woodward, and N. J. Zaluzec, editors, Microscopy and Microanalysis 2009, volume 15, suppl. 2, pages 754–755, Cambridge, UK, 2009. Cambridge Univ. Press.

    Google Scholar 

  37. C. Dwyer. Simulation of scanning transmission electron microscope images on desktop computers. Ultramicroscopy, 110:195–198, 2010.

    Article  Google Scholar 

  38. A. S. Eggeman, A. London, and P. A. Midgley. Ultrafast electron diffraction pattern simulations using GPU technology. applications to lattice vibrations. Ultramicroscopy, 134:44–47, 2013.

    Article  Google Scholar 

  39. M. D. Feit and J. A. Fleck. Light propagation in graded-index optical fibers. Applied Optics, 17:3990–3998, 1978.

    Article  ADS  Google Scholar 

  40. J. Fertig and H. Rose. Resolution and contrast of crystalline objects in high-resolution scanning transmission electron microscopy. Optik, 59:407–429, 1981.

    Google Scholar 

  41. R. P. Feynman. An operator calculus having applications in quantum electrodynamics. Phys. Rev., 84:108–128, 1951.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. P. M. Fields and J. M. Cowley. Computed electron microscope images of atomic defects in fcc metals. Acta Cryst., A34:103–112, 1978.

    Article  Google Scholar 

  43. J. A. Fleck(Jr.), J. R. Morris, and M. D. Feit. Time-dependent propagation of high energy laser beams through the atmosphere. Appl. Phys., 10:129–160, 1976.

    Google Scholar 

  44. R. G. French and R. V. E. Lovelace. Strong turbulence and atmospheric waves in stellar occultations. Icarus, 56:122–146, 1983.

    Article  ADS  Google Scholar 

  45. Matteo Frigo and Steven G. Johnson. The design and implementation of FFTW3. Proc. of the IEEE, 93:216–231, 2005. www.fftw.org.

    Article  Google Scholar 

  46. F. Fujimoto. Dynamical theory of electron diffraction in Laue-case I. general theory. J. Physical Soc. Japan, 14:1558–1568, 1959.

    Article  ADS  Google Scholar 

  47. A. Gómez-Rodríguez, L. M. Beltrán-del-Río, and R. Herrera-Becerra. SimulaTEM: Multislice simulations for general objects. Ultramicroscopy, 110:95–104, 2010.

    Article  Google Scholar 

  48. P. Goodman and A. F. Moodie. Numerical evaluation of N-beam wave functions in electron scattering by the multislice method. Acta Cryst., A30:280–290, 1974.

    Article  Google Scholar 

  49. M. De Graf. Intro. to Conventional Transmission Electron Microscopy. Cambridge Univ. Press, Cambridge, UK, 2003.

    Google Scholar 

  50. M. A. Gribelyuk. Structure retrieval in HREM. Acta Cryst., A47:715–723, 1991.

    Article  Google Scholar 

  51. V. Grillo, E. Carlino, and F. Glas. Influence of the static atomic displacement on atomic resolution Z-contrast imaging. Phys. Rev. B, 77:054103, 2008.

    Article  ADS  Google Scholar 

  52. V. Grillo and F. Rossi. STEM_CELL: a software tool for electron microscopy: Part ii analysis of crystalline materials. Ultramicroscopy, 125:112–129, 2013.

    Article  Google Scholar 

  53. V. Grillo and E. Rotunno. STEM_CELL: a software tool for electron microscopy: Part i simulations. Ultramicroscopy, 125:97–111, 2013.

    Article  Google Scholar 

  54. G. R. Grinton and J. M. Cowley. Phase and amplitude contrast in electron microscopy of biological materials. Optik, 34:221–233, 1971.

    Google Scholar 

  55. S. C. Hiller, E. T. Robertson, G. D. Reid, R. D. Haynes, and M. D. Robertson. On the role of the second-order derivative term in the calculation of convergent beam diffraction patterns. Ultramicroscopy, 179:73–80, 2017.

    Article  Google Scholar 

  56. P. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan. Electron Microscopy of Thin Crystals. Krieger, Huntington, New York, second edition, 1977.

    Google Scholar 

  57. F. Hosakawa, T. Shinkawa, Y. Arai, and T. Sannomiya. Benchmark test of accelerated multi-slice simulation by GPGPU. Ultramicroscopy, 158:56–64, 2015.

    Article  Google Scholar 

  58. P. Hovington, D. Drouin, and R. Gauvin. CASINO: A new Monte Carlo code in C language for electron beam interaction—part I: Description of the program. Scanning, 19:1–14, 1997.

    Article  Google Scholar 

  59. P. Hovington, D. Drouin, R. Gauvin, D. C. Joy, and N. Evans. CASINO: A new Monte Carlo code in C language for electron beam interaction—part III: Stopping power at low energies. Scanning, 19:29–35, 1997.

    Article  Google Scholar 

  60. A. Howie and Z. S. Basinski. Approx. of the dynamical theory of diffraction contrast. Phil. Mag, 17:1039–1063, 1968.

    Google Scholar 

  61. A. Howie and M. J. Whelan. Diffraction contrast of electron microscope images of crystal lattice defects, II. the development of a dynamical theory. Proc. Royal Society of London, A263:217–237, 1961.

    Google Scholar 

  62. C. J. Humphreys. The scattering of fast electrons by crystals. Rep. Prog. Phys., 42:1825–1887, 1979.

    Article  ADS  Google Scholar 

  63. K. Ishizuka. Multislice formula for inclined illumination. Acta Cryst., A38:773–779, 1982.

    Article  Google Scholar 

  64. K. Ishizuka. A practical approach for STEM image simulation based on the FFT multislice method. Ultramicroscopy, 90:71–83, 2002.

    Article  Google Scholar 

  65. K. Ishizuka. FFT multislice method - the silver anniversary. Microsc. and Microanalysis, 10:34–40, 2004.

    Article  ADS  Google Scholar 

  66. K. Ishizuka, 2006. www.hremresearch.com.

  67. K. Ishizuka and N. Uyeda. A new theoretical and practical approach to the multislice method. Acta Cryst., A33:740–749, 1977.

    Article  Google Scholar 

  68. B. K. Jap and R. M. Glaeser. The scattering of high-energy electrons. I. Feynman path-integral formulation. Acta. Cryst., A34:94–102, 1978.

    Article  ADS  MathSciNet  Google Scholar 

  69. Nicholas H. Julian, Tian T. Li, Robert E. Rudd, and Jaime Martian. MS-STEM-FEM: A parallelized multi-slice fluctuation TEM simulation tool. Ultramicroscopy, 194:117–125, 2018.

    Article  Google Scholar 

  70. R. Kilaas. Interactive simulation of high-resolution electron micrographs. In G. W. Bailey, editor, Proceedings of the 45th Annual Meeting of the Microscopy Society of America, pages 66–69. San Francisco Press, 1987.

    Google Scholar 

  71. R. Kilaas, 2006. www.totalresolution.com/index.html.

  72. R. Kilaas and R. Gronsky. Real space image simulation in high resolution electron microscopy. Ultramicroscopy, 11:289–298, 1983.

    Article  Google Scholar 

  73. R. Kilaas, M. A. O’Keefe, and K. M. Krishman. On the inclusion of upper Laue layers in computational methods in high resolution transmission electron microscopy. Ultramicroscopy, 21:47–62, 1987.

    Article  Google Scholar 

  74. E. J. Kirkland. Advanced Computing in Electron Microscopy. Plenum, New York, 1998.

    Book  Google Scholar 

  75. E. J. Kirkland, 2013. www.sourceforge.com/computem.

  76. E. J. Kirkland. Computation in electron microscopy. Acta Cryst. A, 72:1–27, 2016.

    Article  Google Scholar 

  77. E. J. Kirkland, R. F. Loane, and J. Silcox. Simulation of annular dark field STEM images using a modified multislice method. Ultramicroscopy, 23:77–96, 1987.

    Article  Google Scholar 

  78. C. Kittel. Intro. to Solid State Physics. Wiley, New York, 7th edition, 1996.

    Google Scholar 

  79. C. Koch, 2015. elim.physik.uni-ulm.de.

    Google Scholar 

  80. D. Koslof and R. Kosloff. A Fourier method solution for the time dependent Schrodinger equation as a tool in molecular dynamics. J. Comp. Phys., 52:35–53, 1983.

    Article  ADS  MATH  Google Scholar 

  81. R. Kosloff. Time-dependent quantum-mechanical methods for molecular dynamics. J. Phys. Chem., 92:2087–2100, 1988.

    Article  Google Scholar 

  82. Florian F. Krause, Knut Müller, Dennis Zillmann, Jacob Jansen, and Marco Schowalter. Comparison of intensity and absolute contrast of simulated and experimental high-resolution transmission electron microscopy images for different multislice simulation methods. Ultramicroscopy, 134:94–101, 2013.

    Article  Google Scholar 

  83. A. L. Lewis, R. B. Hammond, and R. E. Villagrana. The importance of second-order partial derivatives in the theory of high-energy-electron diffraction from imperfect crystals. Acta. Cryst., A31:221–227, 1975.

    Article  Google Scholar 

  84. I. Lobato and D. van Dyck. MULTEM: a new multislice program to perform accurate and fast electron diffraction and imaging simulations using graphics processing unit with CUDA. Ultramicroscopy, 156:9–17, 2015.

    Article  Google Scholar 

  85. D. F. Lynch and A. F. Moodie. Numerical evaluation of low energy electron diffraction intensities I. the perfect crystal with no upper layer lines and no absorption. Surface Science, 32:422–438, 1972.

    Article  ADS  Google Scholar 

  86. D. F. Lynch and M. A. O’Keefe. n-beam lattice images II. methods of calculation. Acta Cryst., A28:536–548, 1972.

    Google Scholar 

  87. D. S. MacLagan, L. A. Bursill, and A. E. C. Spargo. Experimental and calculated images of planar defects at high resolution. Phil. Mag., 35:757–780, 1977.

    Article  ADS  Google Scholar 

  88. L. Marks and R. Kilass, 2006. www.numis.northwestern.edu/edm/documentation/edm.htm.

  89. H. Matsuhata, D. Van Dyck, J. Van Lanuyt, and S. Amelincjx. A practical approach to the periodic continuation method for the simulation of high resolution TEM images of isolated crystal defects. Ultramicroscopy, 13:343–348, 1984.

    Article  Google Scholar 

  90. W. Q. Ming and J. H. Chen. Validities of three multislice algorithms for quantitative low-energy transmission electron microscopy. Ultramicroscopy, 134:135–143, 2013.

    Article  Google Scholar 

  91. K. Mitsuishi, K. Iakoubovskii, M. Takeguchi, M. Shimojo, A. Hashimoto, and K. Furuya. Bloch wave-based calculations of imaging properties of high-resolution scanning confocal electron microscopy. Ultramicroscopy, 108:981–988, 2008.

    Article  Google Scholar 

  92. Gordon E. Moore. Cramming more components onto integrated circuits. Proc. IEEE, 86:82–85, 1998. reprinted from Electronics, April 19, 1965, p. 114–117.

    Article  Google Scholar 

  93. P. D. Nellist and S. J. Pennycook. Incoherent imaging using dynamically scattered coherent electrons. Ultramicroscopy, 78:111–124, 1999.

    Article  Google Scholar 

  94. P. D. Nellist and S. J. Pennycook. The principles and interpretation of annular dark-field Z-contrast imaging. In P. W. Hawkes, editor, Adv. in Imaging and Electron Physics, vol. 113, pages 147–203. Academic Press, San Diego, 2000.

    Google Scholar 

  95. H. Niehrs and E. H. Wagner. Die amplituden der wellenfelder bei elektroneninterferenzen im Laue-fall. Z. Physik, 143:285–299, 1955.

    Article  ADS  Google Scholar 

  96. Jan Oliver Oelerich, Lennart Duschek, Jürgen Belz, Andreas Beyer, Sergei D. Baranovskii, and Kerstin Volz. STEMsalabim: A high-performance computing cluster friendly code for scanning transmission electron microscopy image simulations of thin specimens. Ultramicroscopy, 177:91–96, 2017.

    Google Scholar 

  97. M. A. O’Keefe. Advances in image simulation for high resolution TEM. In G. W. Bailey, M. H. Ellisman, R. A. Hennigar, and N. J. Zaluzec, editors, Proc. Micros. and Microanal. 1995, pages 38–39, New York, 1995. Jones and Begell.

    Google Scholar 

  98. M. A. O’Keefe and P. R. Buseck. Computation of high resolution TEM images of materials. Trans. American Crystallography Assoc., 15:27–46, 1979.

    Google Scholar 

  99. M. A. O’Keefe, P. R. Buseck, and S. Iijima. Computed crystal structure images for high resolution electron microscopy. Nature, 274:322–324, 1978.

    Article  ADS  Google Scholar 

  100. M. A. O’Keefe and R. Kilaas. Advances in high-resolution image simulation. In P. W. Hawkes, F. P. Ottensmeyer, W. O. Saxton, and A. Rosenfeld, editors, Image and Signal Processing in Electron Microscopy, Scanning Microscopy, Supplement 2, pages 225–244, Chicago, 1988. Scanning Microscopy Intern.

    Google Scholar 

  101. M. A. O’Keefe and J. V. Sanders. n-beam lattice images. VI. degradation of image resolution by a combination of incident-beam divergence and spherical aberration. Acta Cryst., A31:307–310, 1975.

    Article  ADS  Google Scholar 

  102. N. L. O’Leary and L. J. Allen. Quantitative structure retrieval at atomic resolution. Acta Cryst., A61:252–259, 2005.

    Article  Google Scholar 

  103. Colin Ophus. A fast image simulation algorithm for scanning transmission electron microscopy. Advanced Structural and Chemical Imaging, 3:13, 2017.

    Article  Google Scholar 

  104. S. J. Pennycook and D. E. Jesson. High-resolution incoherent imaging of crystals. Phys. Rev. Let., 64:938–941, 1990.

    Article  ADS  Google Scholar 

  105. S. J. Pennycook and D. E. Jesson. Atomic resolution Z-contrast imaging of interfaces. Acta Metall. Mater., 40:S149–S159, 1992.

    Article  Google Scholar 

  106. A. Pidwerbetsky and R. V. E. Lovelace. Chaotic wave propagation in a random medium. Physics Letters A, 140:411–415, 1989.

    Article  ADS  Google Scholar 

  107. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes. Cambridge University Press, Cambridge, 3rd edition, 2007.

    MATH  Google Scholar 

  108. M. J. Quinn. Parallel Programming in C with MPI and openMP. McGraw Hill, New York, 2004.

    Google Scholar 

  109. M. Radek, J.-G. Tenberge, S. Hilke, G. Wilde, and M. Peterlechner. STEMcl- a multi-GPU multislice algorithm for simulation of large structure and imaging parameter series. Ultramicroscopy, 188:24–30, 2018.

    Article  Google Scholar 

  110. L. Reimer. Transmission Electron Microscopy, volume 36 of Spring Series in Optical Sciences. Springer-Verlag, New York, third edition, 1993.

    Google Scholar 

  111. M. D. Robertson, J. C. Bennett, M. M. J. Burns, and D. Currie. The simulation of annular dark field images of InAs/InP quantum dots. In P. Kotula, M. Marko, J.-H. Scott, R. Gauvin, D. Beniac, G. Lucas, S. McKernan, and J. Shields, editors, Microscopy and Microanalysis 2006, volume 12, suppl. 2, pages 714–715, Cambridge, UK, 2006. Cambridge Univ. Press.

    Google Scholar 

  112. H. Rullgård, L.-G. Öfverstedt, S. Masich, B. Daneholt, and O. Öktem. Simulation of transmission electron microscope images of biological specimens. J. Microscopy, 243:234–256, 2011.

    Article  Google Scholar 

  113. Jason Sanders and Edward Kandrot. CUDA by Example, An Intr. to General-Purpose GPU Programming. Addison-Wesley, Boston, 2011.

    Google Scholar 

  114. Noah Schnitzer, Suk Hyun Sung, and Robert Hovden. Intro. to the ronchigram and its calculation with ronchigram.com. Microscopy Today, May:12–15, 2019.

    Google Scholar 

  115. P. G. Self, M. A. O’Keefe, P. R. Buseck, and A. E. C. Spargo. Practical computation of amplitudes and phases in electron diffraction. Ultramicroscopy, 11:35–52, 1983.

    Article  Google Scholar 

  116. A. E. C. Spargo, M. J. Beeching, and L. J. Allen. Inversion of electron scattering intensity for crystal structure analysis. Ultramicroscopy, 55:329–333, 1994.

    Article  Google Scholar 

  117. J. C. H. Spence. Direct inversion of dynamical electron diffraction patterns to structure factors. Acta Cryst., A54:7–18, 1998.

    Article  Google Scholar 

  118. J. C. H. Spence. High-Resolution Electron Microscopy. Oxford University Press, New York, fourth edition, 2013.

    Google Scholar 

  119. J. C. H. Spence, B. Calef, and J. M. Zuo. Dynamic inversion by the method of generalized projections. Acta Cryst., A55:112–118, 1999.

    Article  Google Scholar 

  120. J. C. H. Spence and J. M. Zuo. Electron Microdiffraction. Plenum Press, New York, 1992.

    Book  Google Scholar 

  121. P. A. Stadelmann. EMS - a software package for electron diffraction analysis and HREM image simulation in materials science. Ultramicroscopy, 21:131–146, 1987.

    Article  Google Scholar 

  122. P. A. Stadelmann. JEMS - EMS java version, 2004. www.cimewww.epfl.ch/people/stadelmann/jemsWebSite/jems.html.

  123. L. Sturkey. The calculation of electron diffraction intensities. Proc. Phys. Soc., 80:321–354, 1962.

    Article  ADS  MATH  Google Scholar 

  124. M. Tournaire. Recent developments of the matrical and semi-reciprocal formulation in the field of dynamical theory. J. of the Physical Society of Japan, Suppl. B II, 17:98–100, 1962.

    Google Scholar 

  125. W. van den Broek, X. Jiang, and C. T. Koch. FDES, a GPU-based multislice algorithm with increased efficiency of the computation of the potential. Ultramicroscopy, 158:89–97, 2015.

    Article  Google Scholar 

  126. D. van Dyck. The path integral formalism as a new description for the diffraction of high-energy electrons in crystals. Phys. Stat. Sol., B72:321–336, 1975.

    Article  ADS  Google Scholar 

  127. D. van Dyck. On the optimisation of methods for the computation of many-beam structure images. In J. M. Sturgess, V. I. Kalnins, F. P. Ottensmeyer, and G. T. Simon, editors, Electron Microscopy 1978, Vol. 1, Ninth Intern. Congress on Electron Microscopy (Toronto), pages 196–197, Ontario, 1978. The Imperial Press.

    Google Scholar 

  128. D. van Dyck. Improved methods for the high speed calculation of electron microscopic structure images. Phys. Stat. Sol., A52:283–292, 1979.

    Article  ADS  Google Scholar 

  129. D. van Dyck. Fast computational procedures for the simulation of structures in complex or disordered crystal: A new approach. J. of Microscopy, 119:141–152, 1980.

    Article  Google Scholar 

  130. D. van Dyck. High-speed computation techniques for the simulation of high resolution electron micrographs. J. of Microscopy, 132:31–42, 1983.

    Article  Google Scholar 

  131. D. van Dyck. Image calculations in high-resolution electron microscopy: Problems, progress, and prospects. In P. W. Hawkes, editor, Advances in Electronics and Electron Physics, Vol. 65, pages 295–355. Academic Press, Orlando, 1985.

    Google Scholar 

  132. D. van Dyck and W. Coene. The real space method for dynamical electron diffraction calculation in high resolution electron microscopy, I. principles of the method. Ultramicroscopy, 15:29–40, 1984.

    Article  Google Scholar 

  133. C. Wacker and R. R. Schröder. Multislice algorithms revisited: Solving the Schrödinger equation numerically for imaging electrons. Ultramicroscopy, 151:211–223, 2015.

    Article  Google Scholar 

  134. K. Watanabe. n-beam dynamical calculations. In P. W. Hawkes, editor, Advances in Electronics and Electron Physics, Vol. 86, pages 173–224. Academic Press, San Diego, 1993.

    Google Scholar 

  135. K. Watanabe, Y. Kikuchi, K. Hiratsuka, and H. Yamaguchi. A new approach for n-beam lattice image calculation. Phys. Status Solidi, A109:119–126, 1988.

    Article  ADS  Google Scholar 

  136. K. Watanabe, Y. Kikuchi, K. Hiratsuka, and H. Yamaguchi. A new approach for n-beam dynamical calculations. Acta. Cryst., A46:94–98, 1990.

    Article  Google Scholar 

  137. K. Watanabe, T. Yamazaki, I. Hashimoto, and M. Shiojiri. Atomic-resolution annular dark-field STEM image calculations. Phys. Rev. B, 64:115432, 2001.

    Article  ADS  Google Scholar 

  138. G. H. Weiss and A. A. Maradudin. The Baker-Hausdorff formula and a problem in crystal physics. J. Math. Phys., 3:771–777, 1962.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  139. R. M. Wilcox. Exponential operators and parameter differentiation in quantum physics. J. Math. Physics, 8:962–982, 1967.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  140. A. R. Wilson and A. E. C. Spargo. Calculation of the scattering from defects using periodic continuation methods. Phil. Mag., A46:435–449, 1982.

    Article  ADS  Google Scholar 

  141. J. M. Zuo. Web electron microscopy applications software (WebEMAPS), 2009. emaps.mrl.uiuc.edu/.

    Google Scholar 

  142. Jian Min Zuo and John C. H. Spence. Advanced Transmission Electron Microscopy, Imaging and Diffraction in Nanoscience. Springer, New York, 2017.

    Google Scholar 

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Kirkland, E.J. (2020). Theory of Calculation of Images of Thick Specimens. In: Advanced Computing in Electron Microscopy. Springer, Cham. https://doi.org/10.1007/978-3-030-33260-0_6

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