X-ray Scattering Techniques

X-ray Interactions and Physics

2.3.1 X-rays and matters

X-rays are defined as electro-magnetic waves with much shorter wavelength (~0.1 nm) than that of visible light (~500 nm), which makes it possible to probe structures much smaller than what can be observed in a microscope. X-rays with high photon energies (above 5–10 keV, below 0.2–0.1 nm wavelength) are called hard X-rays, while those with lower energy are called soft X-rays. The hard X-rays are usually employed for materials imaging while the soft X-rays are easily absorbed by air or water with very short attenuation length [31].

The interaction of X-rays with matters are quite different compared with other radiological wave like ultraviolet, visible, infrared and microwaves. There are mainly two interactions between X-rays and matters: absorption and scattering, including Compton and Ryleigh scattering [31].

2.3.2 X-rays and structures

When X-rays are shined on the atoms, spherical waves comes out from the scattered position. Depending on the obersrevation angle 2θ , the orientation and the distance r, the interenfeces can be constructive meaning in phase, destrctive meaning out of phase. [32]

Figure 2-9: Illustration of X-ray interactions with structure and the detector brightness observation. [32]

The constructive and destructive interferences give different scattering patterns on the detector: bright spot and dark spot respectively. The scattering patterns can be expressed as a function of scattering angle θ and wavelength λ: [32]

Where the q is the momentum transfer or length of the scattering vector with the dimension is on over length.

2.3.3 Form factor and structure factor

The form factor is the measurement of the scattering amplitude of X-rays by a particle consists of many atoms. It is calculated by pair-distance distribution function p(r): [32]

Where r is the distance between particles. The observed scattering pattern corresponds to the form factor only when the particles are far away from each other or identical in shape. When the sample is dilute, the scattering pattern is the form factor multiplied by the number of particles which are illuminated by the X-ray beam. If the particles have various sizes, the form factor is calculated by summing up to achieve the scattering pattern for the sample [32]. The structure factor is an additional interference pattern multiplying the form factor of the single particles. Especially in crystallography it is called lattice factor which includes all the information regarding the positions of particles with respect to each other. At small angle scattering, the concentration effects become visible [32]. When the particles are highly ordered, there is a pronounced peak which is called a Bragg peak. The Bragg’s law gives the distance between ordered particles in the following formula. [33]