X-rays typically have a wavelength on the order of 1 Å, which is a comparable dimension to the inter-atomic distance in most materials. If the material has a crystalline structure, the periodic features of the crystal will act as a three dimensional diffraction grating to x-rays. The smallest repeating unit in a crystal structure is known as the unit cell. If x-rays are incident on a surface of a crystalline material with all crystal planes parallel to the surface, separated by a distance d, then at certain
incident angles the path difference between x-rays diffracted from each plane will be such that constructive interference between the diffracted x-rays will occur. The angle between the x-ray and the surface normal at which constructive interference
(a)
X-Ray Source Detector
Goniometer φ χ z y x θ ω (b)
Figure 3.14: (a) A photograph and (b) a schematic of the Bruker D8 Advance
diffractometer. The sample is located on the circular goniometer in the centre of the image. The rotation axes (φ, χ) and the translation axes (x,y,z) of the goniometer
are shown. The separate line elements of the detector are shown in red; they are spatially separated along the θ direction. The sample-source angle is ω and the
sample-detector angle is θ. Here, both source and detector are making the same
angle with respect to the sample surface so that ω=θ.
occurs (θ) is given by Bragg’s Law
nλ= 2dsinθ (3.17)
where n is the order of diffraction, and λ is the wavelength of the incident x-rays.
Thus the measurement of θ gives d. In practice, the x-ray source and detector are
aligned to both make an angleθ to the surface of the sample; the angle of both is
then increased in a coupled fashion while the diffraction intensity is measured. This is demonstrated in Figure 3.14(a) and is known as a symmetric scan.
Measurement of the parallel-to-surface crystal plane spacing will give a value for the lattice constant in this direction. If the other lattice constants of the crystal are to be found, then the Bragg angle for other crystal planes that are not parallel to the surface must be measured. For these crystal planes, the source and detector need to be positioned symmetrically about the normal of the chosen crystal plane, then have the angle of both source and detector with respect to the crystal plane increase in the same coupled fashion as the previous case. This is not possible for all crystal planes as the source and detector can not be rotated to every angle due to their finite size. X-ray diffraction can be used to distinguish between single crystal, polycrystalline and amorphous materials; an amorphous material will show no intense diffraction pattern, only a broad diffuse background. A polycrystalline film consists of a huge amount of randomly oriented crystal domains. For each crystal plane, a small percentage of crystallites will be in the correct orientation to satisfy the Bragg condition (Equation 3.17). Every plane will thus be visible in a symmetric scan.
If a symmetric scan is performed on a single crystal or an epitaxial film, only the diffraction peaks for the crystal plane parallel to the surface will be seen. For an epitaxial film, this will be set by the crystal structure of the underlying substrate. Other crystal planes can be detected by rotation of the sample about different axes. The large amount of perfectly aligned crystal planes in an epitaxial or single crystal sample will result in very narrow diffraction peaks. By analysing the exact position and width of the peaks, information on the strain and the lattice constants of the film can be obtained.
The primary uses for XRD include lattice constant determination, texture analysis, and phase identification. Each crystal structure will produce diffraction peaks at different angles, so by comparing a measured diffraction pattern with a reference database pattern the crystal structure of the measured material can be determined4.
The x-ray diffractometer used is a Bruker D8 Advance with a Cu anode operated at an accelerating voltage of 40 kV and emission current of 40 mA, a Göbel mirror to manipulate the x-ray emission into a parallel beam, a 2-bounce monochromator to select theKα(with a wavelength of 1.54 Å) from the Cu x-ray emission spectrum, and
a solid state detector with 192 one-dimensional linear detecting elements of 0.075 mm
width. The detector is oriented so that each element is spaced along the θ direction.
The detector measures 2.6° along this direction simultaneously which enables faster
measurements compared to a point detector. The sample stage is an Euler cradle which allows a full 360° rotation about the sample normal (φ), a 90° rotation about
an axis perpendicular to the sample normal (χ), and has three translational axes
of movement (x,y,z), enabling precise sample alignment. The diffractometer and a
schematic showing these axes of movement is shown in Figure 3.14.