The Physics of Scattering Experiments

The physics of scattering can be understood according to the Huygens-Fresnel principle of interference. Incoming radiation interacts with scatterers in the specimen (electrons in case of X-rays and atomic nuclei in case of neutrons), causing these to emit secondary, nearly spherical wavefronts. The superposition of these secondary waves yields the scattered radiation in the detector placed far from the sample. The scattering intensity recorded by the detector is described by the general formula:

\[I(\vec{q})=\left|\iiint \Delta\rho(\vec{r}) e^{-i\vec{q}\vec{r}}\mathrm{d}^3\vec{r}\right|^2\textrm{,}\]

which is formally a Fourier transform of $\Delta\rho(\vec{r})$, the scattering length density function describing the sample. For the X-ray case, this is the same as the number density function of the electrons, apart from a constant factor. The natural variable of the Fourier transform, $\vec{q}$ describes the angle dependence of the scattering. The relation of its magnitude to the scattering angle ($2\theta$) is $q=4\pi\sin{\theta}/\lambda$ where $\lambda$ is the X-ray wavelength. From the physical point of view, $\vec{q}$ is the vector difference of the wave vectors of the scattered and incident radiation, and is thus proportional to the momentum transfer vector, which is proportional to $\hbar \vec{q}$, the momentum gained by the X-ray photon upon interacting with the sample. This is shown in the next figure. The typical units of $q$ are nm-1.



The scattered intensity describes the scattering power of the sample. The physical quantity is the differential scattering cross-section in this case, which represents the amount of particles scattered in a given solid angle, normalized by the incident flux at the sample:

$$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\left(\vec{q}\right) = \frac{\mathrm{d}N(\vec{q})}{\mathrm{d}\Omega} \frac{A}{N_\mathrm{in}}\textrm{,}$$

where $\frac{\mathrm{d}N(\vec{q})}{\mathrm{d}\Omega}$ is the number of particles scattered from the sample in infinitesimal solid angle at the direction given by $\vec{q}$, $A$ is the irradiated cross-section of the sample and $N_\mathrm{in}$ is the number of incoming rays. For practical reasons, the differential scattering cross-section is usually normalized by the sample volume, thus yielding the commonly used cm-1 units.

In contrast to the calibration of scattering angles to $q$, obtaining the intensity on this absolute scale is more difficult. Luckily, it is not needed in most cases, only when absolute quantities, such as molecular weight, concentration, etc. is to be determined.