Chapter 8
Structure Factor Calculations
8.1 The Structure Factor
We know that the intensity of the diffracted beams depends on atomic scattering
factor and the position of each atom in the unit cell. Since, X-rays are scattered by
electrons of atom only and not by the nucleus, therefore X-rays scattered from one
part of an atom interfere with those scattered from other parts at all angles of
scattering (at 2h ¼ 0; all atoms scatter in phase). Further, the diffracted beams are
obtained as a result of combination of the scattered waves from the electrons of all
the atoms in the unit cell. This process involves two distinct contributions:
Scattering from electrons of the same atom (called the atomic scattering factor or
atomic form factor f).
The summation of this scattering from all atoms in the unit cell (called geometrical structure factor F).
The atomic scattering factor is a measure of the efficiency of an atom in scattering X-rays. It is defined as the ratio:
f ¼
amplitude of the X - ray wave scattered by an atom
amplitude of the X - ray wave scattered by one electron
Therefore, if the atoms are assumed to be points only, then the atomic scattering
factor is first equal to the number of electrons present (i.e., the atomic number of the
neutral atom, Z). The atomic scattering factor of an atom falls off with increasing
scattering angle or, more precisely with increasing value of ðsin h=kÞ. On the other
hand, neutrons are scattered by atomic nuclei rather than electrons around the
nucleus. Since, the nucleus is very small as compared to the size of the atom (it can
be treated as a point atom), the scattering for a non-vibrating nucleus (because it is
very heavy) is almost independent of scattering angle.
In order to know the intensity of the X-ray beam by one unit cell in a particular
direction where there is a diffraction maximum, it is necessary to sum the waves
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_8
299
Structure Factor Calculations
8.1 The Structure Factor
We know that the intensity of the diffracted beams depends on atomic scattering
factor and the position of each atom in the unit cell. Since, X-rays are scattered by
electrons of atom only and not by the nucleus, therefore X-rays scattered from one
part of an atom interfere with those scattered from other parts at all angles of
scattering (at 2h ¼ 0; all atoms scatter in phase). Further, the diffracted beams are
obtained as a result of combination of the scattered waves from the electrons of all
the atoms in the unit cell. This process involves two distinct contributions:
Scattering from electrons of the same atom (called the atomic scattering factor or
atomic form factor f).
The summation of this scattering from all atoms in the unit cell (called geometrical structure factor F).
The atomic scattering factor is a measure of the efficiency of an atom in scattering X-rays. It is defined as the ratio:
f ¼
amplitude of the X - ray wave scattered by an atom
amplitude of the X - ray wave scattered by one electron
Therefore, if the atoms are assumed to be points only, then the atomic scattering
factor is first equal to the number of electrons present (i.e., the atomic number of the
neutral atom, Z). The atomic scattering factor of an atom falls off with increasing
scattering angle or, more precisely with increasing value of ðsin h=kÞ. On the other
hand, neutrons are scattered by atomic nuclei rather than electrons around the
nucleus. Since, the nucleus is very small as compared to the size of the atom (it can
be treated as a point atom), the scattering for a non-vibrating nucleus (because it is
very heavy) is almost independent of scattering angle.
In order to know the intensity of the X-ray beam by one unit cell in a particular
direction where there is a diffraction maximum, it is necessary to sum the waves
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_8
299