Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Theory of Bright Field Scanning Transmisison Electron Microscopy for Tomography



Zachary H. Levine


Radiation transport theory is applied to electron microscopy of samples composed of one or more materials. The theory, originally due to Goudsmit and Saunderson, assumes only elastic scattering and an amorphous medium dominated by atomic interactions. For samples composed of a single material, the theory yields reasonable parameter-free agreement with experimental data taken from the literature for the multiple scattering of 300 keV electrons through aluminum foils up to 25 um thick. For thin films, the theory gives a validity condition for Beer's Law. For thick films, a variant of Moliere's theory of multiple scattering leads to a new form for the bright-field signal for foils in the multiple scattering regime. The signal varies as 1/(t \ln exp(1-2 gamma t/tau)) where t is the path length of the beam, tau is the mean free path for elastic scattering, and gamma is Euler's constant. The Goudsmit-Saunderson solution interpolates numerically between these two limits. For samples with multiple materials, elemental sensitivity is developed through the angular dependence of the scattering. From the elastic scattering cross sections of the first 92 elements, a singular value decomposition of a vector space spanned by the elastic scattering cross sections minus a delta function shows there is a dominant common mode, with composition-dependent corrections of about 2 %. A mathematically correct reconstruction procedure beyond 2 % accuracy requires the acquisition of the bright-field signal as a function of the scattering angle. Tomographic reconstructions are carried out for three singular vectors of a sample problem with four elements Cr, Cu, Zr, and Te. The three reconstructions are presented jointly as a color image; all four elements are clearly identifiable throughout the image.
Journal of Applied Physics
97 No3


material identification, radiation transport theory, scanning transmission electron microscop, tomograpy


Levine, Z. (2005), Theory of Bright Field Scanning Transmisison Electron Microscopy for Tomography, Journal of Applied Physics (Accessed February 28, 2024)
Created February 1, 2005, Updated February 17, 2017