Self-Assembly of Synthetic and Biological Polymeric Systems of ...

scattering process that incoming X-ray beams may undergo upon interaction with matter is the
energy transfer to an electron but, in this case, without a release of the electron from the
atom (phenomenon known as Compton scattering). Finally, X-rays may be elastically scattered
by electrons, process known as Thomson scattering. In this process, the electron oscillates at
the frequency of the incoming beam so the wavelength of the scattered radiation is conserved.
There are many theories and equations to correlate the diffraction pattern and the material
structure. The Bragg law is a simple way to describe the diffraction of X-rays by a crystal. In
Figure 2.23a, the incident X-rays reach the crystal planes with an incident angle θ and are
diffracted at an angle θ. The diffraction peak is observed when the Bragg condition is satisfied
(3)(51):
where λ is the wavelength of the incident X-rays, d is the distance between each adjacent
crystal plane (d-spacing), θ is the Bragg angle at which a diffraction peak is observed, and n (1,
2, ...) is an integer number, called the reflection order; this means that the Bragg condition can
be satisfied by various X–ray wavelengths. The Bragg law gives a simple relationship between
the diffraction pattern and the crystal structure, and many X-ray diffraction applications can be
easily explained by this law.
Figure 2.23. a) Incident and reflected X-rays form an angle θ which is symmetric to the crystal
normal plane; b) The diffraction peak is observed at a Bragg angle θ (52).
In X-ray diffraction using a single wavelength, the Bragg equation is typically expressed with n
= 1 for the first order of diffraction because higher order reflections can be considered as
2.52
65