Different synthesis methods used to create nanostructured materials can result in a variety of structural flaws, including lattice strain. The diffraction peaks may get broader in crystals with nanometer-sized particles in the range of 10 nm to 100 nm. The crystallite size and lattice strain are determined by the widening of the diffraction peak. Scherrer's formula can be used to determine the crystallite size. In addition to these methods, the Williamson-Hall method was also used to determine the size of crystallites. The lattice strain is also calculated using this method. As a result, ZnO nanopowders were created in this work by precipitation and calcined at various temperatures. The Williamson-Hall (W-H) method was used to determine the crystallite size and lattice strain..

Thin film microstrain

Measurement of strain or deformation in the individual grains of thin films is referred to as thin film microstrain. To carry out microstrain measurements, different techniques like EBSD, -HRXRD, Raman microspectrometry, and X-Ray diffraction (XRD) can be employed. The values of microstrain measured in thin films containing multiple crystals can offer insights into various structural parameters, including crystallite size, dislocation density, and the sensitivities of the techniques employed for mapping microstrain in thin films. [1].

Williamson-Hall (W-H) Analysis

X-ray profile analysis is a simple and powerful tool to estimate the crystallite size and lattice strain. This method is attributed to G. K.Williamson and his student, W. H. Hall [2]. It relies on the principle that the approximate formulae for size broadening, BL, and strain broadening, ße, vary quite differently with respect to Bragg angle, 0.

Literature Review

ZnO nanopowders were created by Mote et al. using coprecipitation methods. At 450 °C, grains with an average size of roughly 50 nm were created. Using W-H analysis and several models, such as the uniform deformation model, the uniform deformation stress model, and the uniform deformation energy density mode, the physical parameters such as strain, stress, and energy density values were also computed. [3].

Fe-doped ZnO nanoparticles with a hexagonal wurtzite shape have been created by Prabhu et al. Williamson-Hall (W-H) analysis and a size-strain plot were used to examine the effects of crystallite sizes and lattice strain on the peak broadening of Fe doped ZnO nanoparticles. All of the samples' XRD peaks had their strain, stress, and energy density characteristics computed using the size-strain plot method (SSP), the uniform stress deformation model (USDM), the uniform deformation energy density model (UDEDM), and the uniform deformation model (UDM). studies from W-H analysis, SSP, and TEM studies were corroborated by the mean particle size of Fe doped ZnO-NPs[4].

Using nanocrystalline Ni and Cu powders, Brandstetter et al. utilized Williamson-Hall analysis to determine grain size and lattice strain[5].

Similar to this, Zak et al. used the sol-gel process to create ZnO nanopowders. The sample has a hexagonal wurtzite phase and is crystalline in form. XRD was used to investigate the ZnO nanoparticles' phase evolution. To investigate the individual effects of crystallite sizes and lattice strain on the peak broadening of the ZnO nanoparticles, the Williamson-Hall analysis and size-strain plot approach were utilized. ZnO-NPs that were calcined at 750 °C showed a nonuniform strain and an average particle size of roughly 20 nm in the TEM image. The TEM results and the W-H technique results were in good agreement[6].

In order to measure the lattice strain on ZnO particles produced by a straightforward, surfactant-assisted combustion synthesis, Prabhu et al. also used the W-H method[7].

ZnO flat thin film was created by Thool et al. using chemical bath deposition methods. The Scherrer method and the Williamson-Hall method were used to evaluate the crystallite size and lattice strain from X-ray line broadening[8].

After heating the precursor at 350 degrees Celsius, Sarma et al. generated ZnO nanopowders using an easy, cost-effective precipitation technique. The W-H method was used to analyze the line broadening of ZnO nanoparticles caused by the small crystallite size and strain[9].

Aim of this study

The objective is to calculate strain and crystallite size from XRD data using Willium- Hall analysis.

We will calculate crystallites size and strain from XRD data using Williamson-Hall (W-H) plot method. The broadening ()of the peaks in the XRD data is caused by the combined effects of crystallite size () and microstrain ().

Total broadening = Broadening due to crystallites size + Broadening due to strain

Where:

is the total broadening

is broadening due to crystallite size

is the broadening due to strain

From the Scherer equation, we know that,

Where

is FWHM (ie broadening of the peak) in radians is the shape factor, is the wavelength of X-ray source, D is the crystal size, is the peak position in radians similarly.

The XRD peak broadening due to micro strain is given by

Where . is broadening due to strain and is the peak position in radians.

Substituting equation (2-2) and (2-3) in equation (2-1),

we obtain:

As we know that

Therefore, equation (2-4) can be written as

Multiplying both sides by ,

Equation (2-5) represent a straight line, in which is the gradient (slope) of the line and is the y- intercept.

Consider the standard equation of a straight line,

Where is the slope of line and is the y- intercept

Comparing equation (2-5) with equation (2-6), we have:

i

ii

iii

iv

We will plot () on x-axis and () on y-axis.

The value of m which represent gradient (slope) of the line it will be the value of the strain ɛ. Finally, we will calculate crystallites size from the y-intercept

Figures 3-1 show XRD images of the Cu samples for various hot water treatment time.

1- From the calculation of strain, we can observe that increasing treatment time increases the strain of the thin films,

2- The calculated crystal size was in good agreement with the one measured by reference [10].

Recommendation

Determine other relevant physical parameters such as stress and energy density.