A to Z of crystalline size and micro-strain calculation  by Williamson-Hall and Scherrer formula

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CoAl2O4 nanoparticles were prepared by a sol-gel process using citric acid as chelating agent with different calcination temperatures of 600 to 900 °C. The crystalline spinel cubic phase was confirmed by X-ray diffraction results. High-resolution scanning electron microscopy (HRSEM) revealed that nanoparticles of CoAl2O4 morphology showed spherical forms with a certain degree of agglomeration. The Williamson-Hall (W-H) method and size-strain method to evaluate the size of crystallites and strain in the CoAl2O4nanoparticles peak broadening were applied.

Physical parameters such as strain and stress values were calculated for all XRD reflection peaks corresponding to the cubic spinel phase of CoAl2O4 in the range of 20 to 70° from the modified plot shape by W-H plot assuming a uniform deformation model (UDM), uniform stress deformation model (USDM), uniform deformation energy density model (UDEDM) and by the size-strain plot method (SSP). The CoAl2O4 NPs crystal size calculated on the W-H plots and the SSP method are in good agreement with the HRSEM Scherrer method.

1. Introduction

Spinel oxide type AB2O4, where A and B represent two different sites tetrahedral and octahedral and the oxygen ions form a cubic closed structure, are a class of chemically and thermally stable materials that are suitable for a wide-ranging applications such as catalyst and magnetic materials [1–6]. The distribution of cations A and B, these two sites are influenced by the combination and nature of the two cations and is strongly dependent on the preparation and processing conditions [7, 8]. Spin cation distribution has been given a lot of attention because it allows understanding the correlations between structure and properties such as color, diffusivity, magnetic behavior and optical properties, which are heavily based on better occupation with these two metal sites [9, 10].

In the class of nanomaterials, cobalt aluminate nanocrystalline spinel (CoAl2O4) is known as blue Thenard, commonly used as a catalyst, color filter for automotive lamps and pigment layer on luminescent materials due to their optical properties, thermal chemical, peculiar stability and photochemistry [11, 12]. In recent years, much work has been done in preparation for nanoscale CoAl2O4 for optical properties. A variety of methods such as combustion [13], Pechni [14], sol-gel [15, 17], microemulsion [16] and plant extract- assisted combustion method [18, 19] have been successfully treated for the preparation of CoAl2O4nanoparticles.

It is known that the crystallite size and strain are of a pattern that the two main key factors lead to the expansion of XRD diffraction peaks [20, 21]. The crystallite size introduced for the first time by Scherrer in 1918 [20], which is mentioned to the size of a single crystal inside the particles or grains, and it is different from particle size or grain size where the single particle or grain can be comprised into several crystal aggregates. Lattice strain is affected by crystal imperfection such as dislocations and point defects [22]. Other two more types are lattice strains in a crystal, that is, uniforms and non-uniform strains. However, only the non-uniform strain will cause the broadening of the Peak [23].There are some methods for deformation sizing in materials represented by Scherrer [20, 24], Williamson-Hall [25, 26], Strain-size plot and Warren-Averbach methods [27, 28]. Scherrer and Williamson-Hall methods set both maximum half width (FWHM) values and integral breaths, while the Warren-Averbach method refers to the Fourier coefficient profile [29]. From the Warren-Averbach method is a bit of time method and mathematically cumbersome [27, 30].

In this study, CoAl2O4 nanoparticles were prepared by the sol-gel method. X-ray diffraction analysis (XRD) and high resolution scanning electron microscopy (HRSEM) to examine nanoparticles Structural and morphological behavior of CoAl2O4 have been used. X-ray peak profile (XPPA) analysis was determined to calibrate the size of the crystallites and strain of CoAl2O4 nanoparticles based on UDM, USDM and UDEDM models. This work discussed the importance of W-H models and SSP method in calibration of crystalline size and strain parameters for CoAl2O4 nanoparticles.

2. Materials and Experimental

CoAl2O4 nanoparticles were synthesized using Co(NO3)2 . 6H2O (LOBA chemie Ltd), Al (NO3)3× 9H2O (Merck), citric acid and deionized water. All of the chemicals above were of analytical grade and were used directly without further purification.

CoAl2O4 were prepared by a sol-gel method using citric acid as a chelating agent. First, a certain amount of cobalt nitrate (Co(NO3)2 . 6H2O) and aluminum nitrate (Al(NO3)3 . 9H2O) dissolved in deionized water. After that, appropriate amount of citric acid was added to the solution above with magnetic stirring. The molar ratio of metal ions to citric acid was 1:2. The mixed solution was continuously stirred for 1 hour and then heated to 80 °C until a highly viscous gel was formed. Then the gels were dried in a 110 °C oven and then calcined at desired temperatures (600-900 °C) for 5 h.

X-ray diffraction pattern powder (XRD) was performed on a Bruker D8 Advance X-ray diffractometer. Morphological analysis and energy dispersion X-ray analysis was performed using EDAX attached to the high resolution scanning electron microscope (HRSEM, FEI, Quanta 200F).

3. Results and Discussions

3.1. Powder XRD analysis

The crystalline structure of the prepared samples was determined by XRD models reached between 2θ of (20) to (70°). Fig. 1 shows XRD spectra of CoAl2O4 nanoparticles annealed at 600-900 °C for 5 h. Different diffraction peaks are observed in the model and the location of these peaks is very well matched to spinel cubic CoAl2O4 (JCPDS Card No 10-458), indicating the cobalt aluminate phase formation. The intensity of the diffraction peaks increases with the increase in the annealing temperature, which is associated with increased crystallinity. Similar reports were reported by Umapathy et al. [31] and Valan et al [32]. No peaks corresponding to any other material or element are observed, which means that the firing temperature used (900 °C) was sufficient for the preparation of high purity CoAl2O4. Strong, sharp and narrow diffraction peaks have shown that the synthesized product was well crystallized.

Figure 1.

XRD pattern of CoAl2O4 nanoparticles annealed at 600–900 °C

3.2. Crystallite size and strain determination

3.2.1. Scherrer method

In general, X-ray diffraction analysis by peak width is due to instrumental amplification, increase in crystallite size and lattice strain due to dislocation [33]. Unbundling these contributions must collect a diffraction pattern of a standard material such as silicon to determine instrumental enlargement [34]. The corrected instrumental brodening βhkl [33] corresponding to peak CoAl2O4 diffraction was estimated using the equation,

β2D = [(β2measured) – (β2instrumental)]1/2 (1)

It is well known that the Scherrer formula provides only the lower limit of crystallite size. The size of crystalline nanoparticles is estimated by the Scherrer formula,


Where, D is the volume weighted crystallite size (nm), k is the shape factor (k = 0.94), λ is the X-ray wavelength (1.54056 Å), θ is the diffraction angle of Bragg and β is the expanded diffraction peak measured at FWHM (in radians). The size of crystallites of CoAl2O4 nanoparticles are shown in Table 1. Crystallite size is increased with increasing temperature of CoAl2O4 NPs. It is possible on firing effect, because the highly energy-consuming sol-gel method requires rather a more reaction time to form the ending products. The higher temperature calcination can increase the growth of the crystalline dimension in the nucleation centers, thus limiting the larger dimension [35].Table 1.Download as CSV 

Geometric parameters of the CoAl2O4 nanoparticles

SampleScherrer methodWilliamson-Hall methodSize-Strain methodHRSEM
D (nm)D(nm)ε no unit x10-3D(nm)ε no unit x10-3σ (MPa)D(nm)ε no unit x10-3σ (MPa)U(KJm-3)D(nm)ε no unit x10-3σ (MPa)D (nm)

3.2.2 Williamson-Hall Method

Uniform Deformation Model (UDM)

In many cases, X-ray diffraction patterns are influenced not only by the size of crystallites, but possibly also by lattice strain and lattice defects. Williamson-Hall analysis is a simplified integral breath method, clearly differentiates the armature size and strain induced deformation peak considering the broadening of the peak width as a function of 2 theta. Individual contribution to the line broadening of a Bragg reflection line can be expressed as:

βhkl = βs + βD (3)

Where βhkl represents the full width at half maximum (FWHM) of a radiant peak, and βs βD are the width due to the size strain, respectively. In the W-H relation it is assumed that the strain is uniform throughout the crystallographic direction, is given by βhkl

βhkl = (kλ/Dcosθ) + 4ε tanθ (4)

Rearranging Eq. (4) gives

βhklcosθ = (kλ/D) + 4εsinθ (5)

Here D and ε correspond to the value of the crystallite size and the value of the microstrain respectively. By potting 4sinθ, the average size of the crystallites and the strain can be estimated by the Y-intercept extrapolation and the slope of the line; See fig. 2.

Uniform Stress Deformation Model (USDM)

According to Hooke’s law, within the elastic limit, there exists a linear proportionality relation between the stress (σ) and strain (ε)

σ = E ε (6)

where E is the elasticity modulus or Young’s modulus. This equation is an approximation that is valid for the significantly small strain. Hence, by assuming that the lattice deformation stress is uniform in the second term of equation signifying UDM and is replaced by ε = (σ/E) and the modified Eq. (5) is given by

βhklcosθ = (Kλ/D) + (4σsinθ/Ehkl) (7)

Here Ehkl is Young’s modulus in the direction normal to the set of (hkl) planes. The slope of the straight line between 4sinθ/Ehkl and βhklcosθ gives the uniform stress and the crystallite size D easily determined from the intercept (Fig. 3). Young’s modulus Ehkl for samples with a cubic crystal phase is related to their elastic compliances Sij (for CoAl2O4 S11 = 6.07, S12 = -3.83 and S44 = 7.22 TPa-1) as [35]:

Figure 2.

UDM plot for CoAl2O4-NPs calcined at 600–900 °C.

Figure 3.

USDM plot for CoAl2O4 – NPs calcined at 600–900 °C.

1/Ehkl = S11 – 2(S11 – S12 – 0.5S44) (m12 m22 + m22m32 + m32m12) (8)


m1 = h (h2 + k2 + l2)-0.5,

m2 = k (h2 + k2 + l2)-0.5 and

m3 = l (h2 + k2 + l2)-0.5.

The Young’s modulus Ehkl value for cubic CoAl2O4 NPs was calculated as 2.44 TPa. Fig. 3 shows the plot of βhklcosθ values as a function of 4sin(θ)/Ehkl, the uniform deformation stress can be calibrated from the slope.

Uniform Deformation Energy Density Model (UDEDM)

The following model that can be used to find the energy density of a crystal called Uniform Deformation Energy Density Model (UDEDM). Previously, it is assumed that crystals are homogeneous and isotropic. Although in many cases, the assumption of homogeneity and isotropy is not justified. In addition, the proportional constants for strain stress relationship are not widely independent when studying deformation energy density (u). For an elastic system that follows Hooke’s law, the energy density (unit energy) can be calculated from the relation u=(ε2Ehkl)/2. Thus, the equation (9) can be rewritten according to the energy and strain relationship, that is,

βhklcosθ =(Kλ/D) + [4sinθ(2u/Ehkl)1/2] (9)

Plot of βhklcosθ versus 4sinθ(2u/Ehkl)1/2 is shown in Fig. 4. The anisotropic energy density (u) is estimated from the slope, and the crystallite size (D) of the Y-intercept.

Figure 4.

UDEDM plot for CoAl2O4-NPs calcined at 600–900 °C.

3.2.3. Size-strain plot method

The Williamson-Hall plot showed that the line brosening was essentially isotropic. This indicates that the diffraction domains were isotropic and there was also a MicroStrain contribution. However, in the case of isotropic line broadening, it is possible to obtain a better evaluation of the size-strain considering an average ‘‘size-strain plot” (SSP), which has relatively less weight gain, is given to high-angle reflections where Accuracy is generally lower. In this approach, the “crystalline dimension” profile is assumed to be described by a Lorentz function and the “strain profile” of a Gaussian function. As a result, we have:

(dhkl βhkl cosθ)2 = (d2hkl βhkl cosθ) + (ε/2)2 (10)

where K is a constant dependent on the shape of the particles; For spherical particles is given as 03/04. In Fig. 5, similar to W-H methods, the term (dhklβhkl cosθ)2 is plotted with respect to d2hkl βhkl cosθ for all CoAl2O4NPs orientation peaks. In this case, the particle size is determined by the slope of the linearly fitted data and the intercepted root yields strain. By varying the size of crystallites (D) with the result of the calcination temperature obtained by the Scherrer formula, W-H (UDM, USDM and UDEDM) models and SSP methods shown in Figs 6–8. The values of average crystallite size of different calcined temperature of CoAl2O4-NPs obtained from different models are more or less similar, implying that the inclusion of strain in various shapes has a slight variation in the average size of crystallites. By inspecting all the graphs, it is clear that the result of the SSP method is more accurate than UDM, USDM and UDEDM since data is more accurately set in this method, with all high intensity points touching linear adaptation. The geometric parameters of CoAl2O4 nanoparticles shown in Table 1.

Figure 5.

Size-strain plot of CoAl2O4-NPs calcined at 600–900 °C

Figure 6.

Variation of crystallite size (D) with calcination temperature obtained from Scherrer method.

Figure 7.

Variation of crystallite size (D) with calcination temperature obtained from W-H method

Figure 8.

Variation of crystallite size (D) with calcination temperature obtained from SSP method

3.3. Morphological Studies

The morphology of cobalt aluminate synthesized nanoparticles was studied by high-resolution scanning electron microscopy (HRSEM). From Fig. 9, it can be clearly demonstrated that the obtained nanoparticles are of spherical shape with some groups present in aggregation of spheres. It can be seen from SEM micrographs using ImageJ processing program confirmed the particle size ranges from 18.76 to 24.95 nm. Alida Mazzoli and Orlando Favoni evaluated the particle sizes and size distribution with the use of Scanning Electron Microscopy (SEM) and ImageJ processing program [36].

Figure 9.

HRSEM images of CoAl2O4 nanoparticles annealed at 600 °C (a), 700 °C (b), 800 °C (c) and 900 °C (d).

4. Conclusion

CoAl2O4 nanoparticles have been prepared by the sol gel method and are characterized by powder XRD and HRSEM analysis. From the XRD powder analysis, the peak line broadening of CoAl2O4 nanoparticles due to finite crystallite size and strain were analyzed by Scherrer’s formula, Williamson-Hall method based on UDM, UDSM and UDEDM models and Size strain plot method. W-H plot has been worked out and established to determine the crystallite size and strain-induced broadening due to lattice deformation. The W-H analysis based on the UDM, UDSM and UDEDM models are very helpful in calculating the estimation of crystallite size and strain. W-H has been developed and established to determine the size of crystallites and the elongated induced deformation due to deformation of the network. UDM-based U-DMS, UDSM and UDEDM analyzes are useful for calculating dimension estimation and deformation of crystallites. The size of crystallites and strain evaluated by the XRD powder measurements are in good agreement with HRSEM results. The elastic properties of the Young Sij (Ehkl) module were estimated by the values ​​of the plane of the lattice (h, k, l). The methods discussed above were very useful in determining the average size of deformation crystals, stress and energy density value, including the size-strain method; it is highly preferable to set crystal perfection.


The Authors are thankful to Dept. of Nuclear Physics, University of Madras, Chennai for XRD facility.


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Authors: H. Irfan, K Mohamed Racik, S. Anand