Theory of bright-field scanning transmission electron microscopy for tomography

Abstract

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 25μm thick. For thin films, the theory gives a validity condition for Beer’s law. For thick films, a variant of Molière’s theory [V. G. Molière, Z. Naturforschg. 3a, 78 (1948)] of multiple scattering leads to a form for the bright-field signal for foils in the multiple-scattering regime. The signal varies as [tln(e1−2γt∕τ)]−1 where t is the path length of the beam, τ is the mean free path for elastic scattering, and γ 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 that 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.