Lecture notes for FYS5310/9320 Electron Microscopy, Electron - PDF document

Lecture notes for FYS5310/9320 Electron Microscopy, Electron Diffraction and Spectroscopy II Øystein Prytz January 26, 2018

Chapter 1 Elastic scattering 1.1 Derivation of the expression for the structure factor We consider an incident plane wave with wave-vector k 0 : Ψ( k 0 , r ) = Ae 2 πi k 0 · r (1.1) This plane wave is scattered from a single atom located in R j , resulting in a scattered, spherical wavelet Ψ( k , r ) with wave vector k . We now imagine that we place a detector in r ′ , and the question we ask is: what is the intensity of this scattered wavelet at the detector? See figure 1.1. In order to determine this, we make two assumptions: • The detector is located far enough from the scattering atom for the scattered wavelet to be considered a plane wave • We neglect the 1 /r dependence of the wavelet amplitude In this case, we can find an expression for the wavelet as follows: scattered wavelet in r ′ = incident wave in R j × scattering strength of atom j × phase factor for plane wave traveling from R j to r ′ Using equation 1.1 we get the following expression for the wavelet scattered from atom j in position R j : Ψ scatt,j ( k , r = r ′ ) = Ψ( k 0 , r = R j ) × f j × e 2 πi k · ( r ′ − R j )) = Ae 2 πi k 0 · R j × f j × e 2 πi k · ( r ′ − R j ) = A f j e 2 πi ( k 0 · R j + k · r ′ − k · R j ) (1.2) = A f j e 2 πi k · r ′ e 2 πi ( k 0 − k ) · R j = A f j e 2 πi k · r ′ e − 2 πi ∆ k · R j where we now have defined the scattering vector ∆ k = k − k 0 . 1

2 Chapter 1. Elastic scattering The more relevant case is the situation where the incoming wave is scattered by several atoms, each giving rise to a separate wavelet which travels to the detector, see figure 1.2. At the detector, the total scattered wave Ψ scatt is a coherent sum of each wavelet originating with the atoms: N � Ψ scatt ( k , r = r ′ ) = Ψ scatt,j ( k , r = r ′ ) (1.3) j =0 Here we should substitute in the expression for the different wavelets, which we found in equa- tion 1.2. If the detector is located far from the scattering atoms, we can approximate all the (in principle) different k -vectors of the various wavelets with one common wave vector 1 . The only atom (j) dependent factors left are then the scattering factor f j and the phase relation e − 2 πi ∆ k · R j . N � A f j e 2 πi k · r ′ e − 2 πi ∆ k · R j Ψ scatt ( k , r = r ′ ) = j =0 (1.4) N � f j e − 2 πi ∆ k · R j = c j =0 We now see that the wave at the detector only depends on the scattering vector ∆ k and position of the scattering atoms. For a given collection of atoms, we therefore define the structure factor N � f j e − 2 πi ∆ k · R j F (∆ k ) = (1.5) j =0 which gives the amplitude and phase of the total wave with scattering vector ∆ k . The observed intensity at the detector is then given by I = | F (∆ k ) | 2 (1.6) 1 The vectors will all be pointing in approximately the same direction, only differing due to a very small parallax. Furthermore, since we are dealing with elastic scattering, the length of the k -vectors will be the same for all wavelets.

1.1. Derivation of the expression for the structure factor 3 Figure 1.1: Sketch showing the geomtery of a plane wave scattered by an atom in R j , with the detector located in r ′ . Image taken from Fultz & Howe [1]. Figure 1.2: Diffraction from a material is the the coherent sum of wavelets scattered from many scattering centres (atoms). Image taken from Fultz & Howe [1].

Bibliography [1] Brent Fultz and James Howe. Transmission Electron Microscopy and Diffractometry of Mate- rials . Springer, 4 edition, 2013. 5