Department of Chemistry

X-ray Diffraction Laboratory


Introduction

X-ray radiation may result from the interaction of high energy particles (like electrons) or photons with matter (Bremsstrahlung). Synchrotron radiation results from the acceleration of charged particles in circular orbits by strong electric and magnetic fields.1

X-ray radiation is a form of electromagnetic radiation, having an energy range from 0.1 to 100 keV (about 100 to 0.1 Angstroms).

X-rays with photon energies above 5 ‑ 100 keV (shorter than a few angstroms) are called hard X-rays, which are used in X-ray diffraction due to the wavelength is comparable to the size of atoms.

The typical X-ray spectra from molybdenum and copper targets are shown below:2

Cu Mo spectra

Note 1: Pure & Appl. Chem., Vol. 63, No. 5, pp. 735-746, 1991
Note 2: International Union of Crystallography http://www.mx.iucr.org/
Monochromatic X-ray radiation can be generated by different technologies:
  • Sealed-tube
  • Rotating anode
  • Synchrotron
X-ray tube and rotating anode are commonly used in laboratory X-ray instruments, whereas synchrotron systems are stadium-sized facilities which can produce high-energy (GeV) X-ray light beams.1

In a laboratory X-ray generating instrument, generally consists of two electrodes enclosed in an evacuated tube. A filament as the negative electrode (cathode) is heated in vacuum to produce electrons which are accelerated by a high electric field in the range 20 ‑ 60 kV and collide on a positively charged metal target (anode).

The corresponding electric current is in the range of 5 ‑ 100 mA. Heat is dissipated in metal target as only about 1% of the energy can be converted into X-rays.

tube



Note 1: Examples of synchrotron radiation facilities: Large Hadron Collider (LHC), by the European Organization for Nuclear Research (CERN); National Synchrotron Light Source (NSLS-II) by Brookhaven National Laboratory; Beijing Electron‑Positron Collider II (BEPC II) Institute of High Energy Physics, Chinese Academy of Sciences
When X-ray photons collide with electrons in atoms, some photons from the incident beam will be deflected away from the original direction. If the X-ray photons did not lose any energy, the wavelength of these scattered x-rays would be the same, and the process is called Thomson Scattering (elastic scattering). If the X-ray photons transferred energy to the electrons, the scattered X-rays will have different wavelength than the incident X-rays, and the process is called Compton Scattering (inelastic scattering).

Diffraction experiments are based on X-ray photons from Thomson Scattering.

The scattered X-rays provide information about the electron distribution in materials. Consider the two X-ray paths A and B in the figure below, the length difference of the paths is indicated by the red bar, which can be deduced by trigonometry. This leads to Bragg's law, which describes the condition for the constructive interference of the scattered X-ray photons:

2 d sin θ = n λ

where θ is the incident angle, d is the spacing of the lattice, n is a positive integer and λ is the wavelength of the incident wave.

A diffraction pattern is obtained by measuring the intensity of scattered X-ray photons against scattering angle. When the scattering angle satisfy Bragg condition,1 strong intensities known as diffraction peaks are obtained. diffraction


Note 1: Bragg W.L., 1913, The Diffraction of Short Electromagnetic Waves by a Crystal, Proc. Cambridge Phil. Soc., 17, 43-57.
Solid material can be classified as crystalline, polycrystalline and amorphous. In crystalline material or crystal, the atoms, ions or molecules are in a highly ordered, periodic arrangement, forming a three-dimentional crystal lattice (e.g. NaCl, diamond). Polycrystalline materials are composed of many different size of crystals (e.g. minerals, ceramics), whereas amorphous materials are solid without long-range order (e.g. glass).

In a crystal structure, the smallest repeating unit is called unit cell. Unit cell is characterized by the lengths of the three cell edges (a, b, c) and the angles between them (α, β, γ) Based on the features of unit cells, there are 7 lattice systems. With the consideration of the lattice points position (Primitive P, Base-Centered A B or C, Body-Centered I, Face-Centered F and Rhombohedral R), resulting 14 Bravais lattices.

Lattice systems Unit cell Bravais lattices
triclinic α, β, γ ≠ 90o Primitive
monoclinic α, γ = 90o, β ≠ 90o Primitive, Base-Centered
orthorhombic α, β, γ = 90o abc Primitive, Base-Centered, Body-Centered, Face-Centered
tetragonal α, β, γ = 90o a = bc Primitive, Body-Centered
rhombohedral α = β = γ ≠ 90o a = b = c Primitive or Rhombohedral
hexagonal α = β = 90o, γ = 120o a = bc Primitive
cubic α = β = γ = 90o a = b = c Primitive, Body-Centered, Face-Centered


The volume of the unit cell can be calulated as below:

Lattice systems Unit cell volume
triclinic a b c ( 1 - cos2 α - cos2 β - cos2 γ + 2 cos α - cos β - cos γ )1/2
monoclinic a b c sin β
orthorhombic a b c
tetragonal a2 c
rhombohedral a3 ( 1 - 3 cos2 α + 2 cos3 α )1/2
hexagonal 1/2 ( √3 a2 c )
cubic a3


Space groups in three dimenional space was studied by Barlow, Fedorov and Schöenflies. The combination of point group symmetry operations of reflection, rotation, rotation-inversion (improper rotation), screw axis, and glide plane, results in 230 space groups in crystallography.

There are several notations to represent Space groups: number, Hermann‑Mauguin notation (international symbol) and Hall notation are those commonly used in crystallography. In Hermann‑Mauguin notation, space group symbols are a combination of letters representing the Bravais lattices, point group identifier and translation operations.

Full list of space groups can be found here. From Cambridge Structural Database (CSD), of the Cambridge Crystallographic Data Centre (CCDC) by 6 January 2016, the space group frequency has the following order:

Rank Space group number Space group % of CSD Crystal System
1 14 P21/c 34.6 monoclinic
2 2 P1 24.5 triclinic
3 15 C2/c 8.4 monoclinic
4 19 P212121 7.2 orthorhombic
5 4 P21 5.2 monoclinic
6 61 Pbca 3.3 orthorhombic
7 33 Pna21 1.4 orthorhombic
8 62 Pnma 1.1 orthorhombic
9 9 Cc 1.0 monoclinic
10 1 P1 0.9 triclinic


This table indicates that more than 87.6 % of the deposited compounds in CSD are belongs to these ten space group.