Page 464 - Numerical Methods for Chemical Engineering

Scattering theory 453

2

1

2 2
1

1

1 2
2
1
1

1 2

Figure 9.7 Power spectrum of f (x, y) obtained from a 2-D FFT.

λ
k i
e −
θ
incident wave
| k | = 2πλ
i
k s
scattered
wave | k | = | k | i
s

Figure 9.8 Generation of scattered light/X-ray radiation from an electron.

a light, X-ray, or neutron beam. In light and X-ray scattering, an incident beam of
electromagnetic radiation causes the electrons of the sample to oscillate and emit sec-
ondary “scattered” electromagnetic waves, each with the same frequency as the incident
beam. The intensity of this scattered radiation is measured as a function of orientation from
the incident beam. From the observed interference pattern, the relative spatial positions of
the electrons are extracted through Fourier analysis. In neutron scattering, the mechanism
for scattering is different, but the mathematical treatment is the same.
Consider a system in which an incident electromagnetic wave interacts with a single
electron (Figure 9.8). The wavelength is λ and its speed of propagation is c; thus, the
frequency in radians per second is ω = 2πc/λ. The incident electromagnetic wave imparts
a force to the electron and causes it to oscillate at the same frequency ω as the incident
radiation. As an accelerating charge emits its own electromagnetic wave, there arise waves
of scattered radiation from each electron, each at the same frequency ω as the incident
radiation, but propagating in all directions from the electron (although not uniformly – the
2
intensity varies as 1 + cos θ).