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 k0: Ψ(k0, r) = Ae2πik0·r (1.1) This plane wave is scattered from a single atom located in Rj, 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 Rj × scattering strength of atomj × phase factor for plane wave traveling from Rj to r′ Using equation 1.1 we get the following expression for the wavelet scattered from atom j in position Rj: Ψscatt,j(k, r = r′) = Ψ(k0, r = Rj) × fj × e2πik·(r′−Rj)) = Ae2πik0·Rj × fj × e2πik·(r′−Rj) = Afje2πi(k0·Rj+k·r′−k·Rj) = Afje2πik·r′e2πi(k0−k)·Rj = Afje2πik·r′e−2πi∆k·Rj (1.2) where we now have defined the scattering vector ∆k = k − k0. 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 Ψscatt is a coherent sum of each wavelet originating with the atoms: Ψscatt(k, r = r′) =

N

• j=0

Ψscatt,j(k, r = r′) (1.3) 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 fj and the phase relation e−2πi∆k·Rj. Ψscatt(k, r = r′) =

N

• j=0

Afje2πik·r′e−2πi∆k·Rj = c

N

• j=0

fje−2πi∆k·Rj (1.4) We now see that the wave at the detector only depends on the scattering vector ∆k and position

• f the scattering atoms. For a given collection of atoms, we therefore define the structure factor

F(∆k) =

N

• j=0

fje−2πi∆k·Rj (1.5) 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)

1The 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 Rj, 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