Chapter 3 Experimental Methods
3.2. Structural characterization 1. X-ray diffraction
Powder X-ray diffraction (XRD) is the most widely used non-destructive technique for general crystalline material characterizations. Two different powder X-ray diffractometers viz. Seifert 3003 T/T and high-power (18 kW) Rigaku TTRAX III were used in the thesis work depending upon the availability. Cu-Kα X-ray radiation with a wavelength of 1.54056 Å was used in all the cases. Figure 3.03 displays the typical photographic view of the Rigaku high-power (18 kW) TTRAX III X-ray diffractometer and Bragg-Brentano geometry of powder X-ray
diffractometer. The theta-theta (θ–θ) goniometer was used in the reflection (Bragg-Brentano) geometry [CULL2001].
Figure 3.03: (left) Photographic view of Rigaku TTRAX III 18 kW X-ray diffractometer and (right) Bragg-Brentano diffraction geometry of a powder X-ray diffractometer.
Figure 3.04: Diffraction of X-ray by a crystal.
XRD technique allows identification of various crystalline phases present in the material and provides other structural information such as the size of the crystallites, strain present inside the crystallites, lattice constant, etc. It is well-known that an ideal crystal has a periodic arrangement of atoms as shown in Figure 3.04. Diffraction of X-ray occurs through constructive interference of X-ray scattered from atoms of a set of parallel planes in crystal lattice at a particular angular positions of the incident wave known as Bragg angles [CULL2001]. This condition for obtaining constructive interference is known as Bragg’s law given by the relation [CULL2001],
λ θ n dhklsin =
2 (3.01)
where, dhkl is inter-planer spacing, θ is the angle of incidence of the X-ray beam with the atomic plane, λ is wavelength of the X-ray and n is order of diffraction (we consider only the first order
diffraction, n =1, because the second order peaks are mostly difficult to detect experimentally).
A series of these angles can be used to determine the Miller indices (hkl) and the crystal structure can be identified from the systematic behavior of these indices. Figure 3.04 shows the diffraction of X-ray from crystal lattice planes illustrating Bragg’s law. The structural parameters such as the average size of the crystallites, d-spacing, lattice constant and strain present inside the crystallites, etc. can be determined by a careful analysis of the XRD patterns using various models [CULL2001]. All measurements were carried out at an accelerated voltage of 50 kV and current of 200 mA. XRD data were collected at a slow scan rate of 0.005
°/s. The exact peak position and full width half maxima (FWHM) of the XRD peak is obtained using Gaussian fitting to the experimental data. The crystallite size and the strain present in the crystallites can be estimated from the width of the diffraction peaks. This is analogous to the diffraction of light from a grating where the line width is proportional to the number of diffracting grooves in the grating. The broadening Δθsize due to crystallite size can be quantified by Scherrer’s formula [CULL2001],
θ θsize = Dcoskλ
Δ (3.02)
where Δθsize size is broadening due to crystallite size and D is the average size of the crystals.
Presence of strain also broadens the peak which can be quantified as [CULL2001], θ
η θ =4 tan
Δ strain (3.03)
where Δθstrain is broadening due to strain and η is the lattice strain. Williamson and Hall plot (WHP) method is used to estimate the effects of crystallite size and strain due to the broadening [WILL1953], which is the combination of both equations (3.02) and (3.03) for a set of Bragg peaks. This method is a linear representative of the total broadening effect expressed as
θ θ η
θ = λcos +4 tan Δ
DWHP
k (3.04)
or
θ λ η
θ
θcos = +4 sin Δ
DWHP
k (3.05)
In case of elastically anisotropic materials, certain Bragg peaks are more affected by strain than other peaks. In such cases, the use of WHP method for the analysis of size and strain is questionable, as the data extracted from XRD do not fall into straight line. In order to consider the different strain contribution to different peaks, Modified Williamson-Hall Plot (MWHP)
method [UNGA1991, UNGA1992] can be used to analyze the diffraction peaks. According to this model, the individual contribution to the broadening of XRD peaks can be expressed as
( )
K ChklB b
K D 2 2
2 2
2 9
.
0
+
=
Δ π ρ (3.06)
where ΔK [= (2cosθΔθ)/λ], Δθ is full width at half maximum (FWHM) of the Bragg reflections (in radian) after correcting instrumental broadening, K = 2sinθ/λ, b is modulus of Burgers vector of dislocations, B is a constant (taken as 10 for a wide range of dislocation distributions [SHEN2005]), Chkl =Ch00
(
1−qH2)
is dislocation contrast factor introduced to take care elastically anisotropic materials, where the residual strains affect some Bragg reflections more than the others, H2 =(
h2k2+k2l2+l2h2) (
/h2+k2+l2)
for a cubical system and q is a constant.In the MWHP formulation, Ungar et al [UNGA1991, UNGA1992] considered that dislocations/defect are the main contributors to the strain. The plots between (ΔK)2 and K2Chkl
can be fitted using a straight line and the values of D and ρ could be calculated from intercept and slope of the fitted straight line, respectively. Note that the dislocation density is one of the major structural parameters [HULL2001] influencing the final nanocrystalline microstructure and correlated to the induced strain and reduced crystal size as
3 2 ρDb
η = (3.07)
Figure 3.05: (left) Photographic view and (right) schematic view of field-emission scanning electron microscope.
3.3. Morphological and microstructural characterization