Transmission Electron Microscopy 3. Elastic Scattering EMA 6518 Spring 2009 01/21/09 EMA 6518: Transmission Electron Microscopy C. Wang Outline •Particles and Waves •Mechanisms of Elastic Scattering •Scatter from Isolated Atoms •The Rutherford Cross Section •Modifications to the Rutherford Cross Section •Coherency of the Rutherford-Scattered Electrons •The Atomic Scattering Factor •Diffraction equation •Inelastic scattering •Beam damage EMA 6518: Transmission Electron Microscopy C. Wang 1 Particles and Waves • Electrons are particles---X-ray and electron spectrometry – They have a scattering cross section and differential scattering cross section – They can be scattered through particular angles – The electrons interact with the nucleus through Coulomb forces – We can relate this process to scattering of other particles, such as α particles, so lots of analysis can carry over form other systems EMA 6518: Transmission Electron Microscopy C. Wang Particles and Waves • Electrons have a wave nature and the electron beam is almost a plane wave---imaging, HRTEM, and diffraction patterns – Waves are diffracted by atoms or “scattering centers” – How strongly a wave is scattered by an atom is determined by the atomic scattering amplitude – We can relate the process to the scattering of X-rays, so lots of analysis already exists EMA 6518: Transmission Electron Microscopy C. Wang 2 Mechanisms of Elastic Scattering EMA 6518: Transmission Electron Microscopy C. Wang Scattered by an Isolated Atom • The electron may interact with the electron cloud, resulting in a small angular deviation • If an electron penetrates the electron cloud and approaches the nucleus, it will be strongly attracted and may be scattered through a larger angle that in rare cases in the TEM can approach 180º (backscattering) EMA 6518: Transmission Electron Microscopy C. Wang 3 Collective Scattering from Many Atoms A plane coherent electron wave generates secondary wavelets from a row of scattering centers (e.g., atoms in the specimen). The secondary wavelets interfere, resulting in a strong direct (zero-order) beam and several orders of coherent beams scattered (diffracted) at specific angles. EMA 6518: Transmission Electron Microscopy C. Wang Mechanisms of Elastic Scattering (1) Electron scattering from isolated single atoms: •Both two cases involve Coulomb forces • In fact, many electron-electron interactions are inelastic. We will ignore any inelastic effects now (2) Elastic scattering when the electron wave interacts with the specimen as a whole: • Diffraction is particularly important at low angles •The low angle elastic scattering distribution is modified by the crystal structure of the specimen, and intense diffracted beams emerge at certain specific angles EMA 6518: Transmission Electron Microscopy C. Wang 4 Scatter from Isolated Atoms • Elastic electron-electron interactions usually result in a relatively low scattering angle, while electron- nucleus interactions cause higher-angle scattering e σ = πr 2 = π ( )2 electron e Vθ 2 Ze 2 σ = πr n = π ( ) nucleus Vθ EMA 6518: Transmission Electron Microscopy C. Wang Rutherford Cross Section • High angle electron-nucleus interaction is analogous to the backscattering of α particles (composed of two protons and two neutrons) from a thin metal foil. 1911, undergraduate research result! • The positive matter in atoms was concentrated in an incredibly small volume and gave birth to the idea of the nuclear atom. EMA 6518: Transmission Electron Microscopy C. Wang 5 Rutherford Cross Section dσ (θ ) e4Z 2 = θ (Rutherford differential cross section) dΩ 16 (E )2 sin 4 0 2 −24 Z 2 2 θ (single isolated atom) σ nucleus = 62.1 ×10 ( ) cot E0 2 ρ −24 ρ Z 2 2 θ Qnucleus t = (N0 t)σ = 62.1 ×10 (N0 t)( ) cot A A E0 2 (scattering from atoms in a TEM specimen of thickness t) θ The beam energy (E 0), the angle of scattering ( ), and the atomic number (Z) all affect the probability that an electron will be scattered by the nucleus. EMA 6518: Transmission Electron Microscopy C. Wang Modifications to the Rutherford Cross Section • When the electron does not pass close to the nucleus, the scattering angle will be small (<5º) because 2 2 2 screening is important. sin (θ )2/ → sin (θ )2/ + (θ0 )2/ 3/1 .0 117 z θ When the scattering angle is greater than 0, θ0 = 2/1 we can neglect electron-electron interactions E0 and the nuclear interaction is dominant screening 2 parameter h ε 0 a0 = 2 Bohr radius of the scattering atom, 0.5Å πm0e dσ (θ) λ 4Z 2 = R Screened relativistic θ θ dΩ 64 π 4 (a )2 (sin 2 + ( 0 )2 )2 Rutherford cross section 0 2 2 EMA 6518: Transmission Electron Microscopy C. Wang 6 Screened Relativistic Rutherford Cross Section Limitation: voltages < 300-400V; Z<30 • Scattering is most likely to occur in the forward direction • Higher-energy electrons are less likely to be scattered EMA 6518: Transmission Electron Microscopy C. Wang Screened Relativistic Rutherford Cross Section If you keep your specimen below 100-nm thickness, you can then approach the ideal of single scattering EMA 6518: Transmission Electron Microscopy C. Wang 7 Coherency of the Rutherford-Scattered Electrons High-angle Rutherford-scattered electrons are incoherent: The phases of the electron waves are not in step • Z-contrast images: The high-angle forward scattering can be used to form exceptionally high-resolution images of a crystalline specimen in which the image contrast is due to the value of Z, not the orientation of the specimen EMA 6518: Transmission Electron Microscopy C. Wang Coherency of the Rutherford-Scattered Electrons The high-angle backscattered electrons can be used to form images of the beam entrance surface of the specimen in which the contrast is not only due to differences in Z but also to changes in surface topography of the specimen Backscattered electron images are rarely used in the TEM! For example: Cu, 10 3 incident electrons Only 0.3% are backscattered from foil specimen 30% are backscattered from bulk sample SEM EMA 6518: Transmission Electron Microscopy C. Wang 8 Atomic Scattering Factor Classical Rutherford differential • Atomic scattering factor f( θ) cross section ignores the wave nature of the EB 2 dσ (θ ) f (θ ) = dΩ •f( θ) is a measure of the amplitude of an electron wave scattered from an isolated atom 2 •f (θ ) is a proportional to the scattered intensity 2 E0 1( + 2 ) m c λ f (θ ) = 0 (Z − f ) 8 2a θ x π 0 sin 2 EMA 6518: Transmission Electron Microscopy C. Wang Atomic Scattering Factor •f( θ) depends on λ, θ, and Z: •f( θ) as θ •f( θ) as λ •f( θ) as Z EMA 6518: Transmission Electron Microscopy C. Wang 9 Remember! Both the differential cross section and the scattering factor are simply a measure of how the electron scattering intensity varies with θ EMA 6518: Transmission Electron Microscopy C. Wang Diffraction Equation Bragg equation EMA 6518: Transmission Electron Microscopy C. Wang 10 Inelastic Scattering • Historically, the conventional TEM only used two elastic signals, namely, the direct beam and the diffracted beams. • Inelastic scattering: – Processes that generate X-rays – Processes that generate other (secondary) electrons • Slow secondary electrons, Auger electrons, electron- hole pairs and cathodoluminescence – Processes that result form collective interactions with many atoms • Plasmons and phonons EMA 6518: Transmission Electron Microscopy C. Wang Beam Damage • Electron dose • Specimen heating • Beam damage in polymers • Beam damage in covalent and ionic crystals • Beam damage in metals • Sputtering EMA 6518: Transmission Electron Microscopy C. Wang 11.