X-RAY DIFFRACTION LINE PROFILEANALYSIS OF CERIUM OXIDE NANO PARTICLE BY USING DOUBLE VOIGT FUNCTION METHOD

Transcription

1 X-RAY DIFFRACTION LINE PROFILEANALYSIS OF CERIUM OXIDE NANO PARTICLE BY USING DOUBLE VOIGT FUNCTION METHOD Mustafa Mohammed Abdullah and Khalid Hellal Harbbi Department of Physics, College of Education (Ibn Al-Haitham), University of Baghdad, Iraq ABSTRACT In this research, the double Voigt method was used to analyze the X-ray lines and then to use the Williamson-Hall method for estimate the particle size and lattice strain of cerium oxide nanoparticle. The value of the crystallite size was equal to ( nm) and the emotion was equal to ( ). In addition, other methods have been used in addition to the double Voigt method for the calculation of crystallite size and lattice strain. These methods are (Sherrer method, sizestrain plot (SSP) method and Halder-Wagner method) and their results are as follows Sherrer crystallite size ( nm) and lattice strain ( ), SSP method crystallite size ( nm) and lattice strain ( ), and Halder-Wagner method crystallite size (9.2287nm) and lattice strain ( ). The double Voigt method combined with Williamson Hall method gave very accurate results in calculating both crystallite size and lattice strain by taking a full diffraction curve during calculations. Keywords: crystallite size, lattice strain, doubles Voigt method, Sherrer method, size-strain plot method, Halder-Wagner method. INTRODUCTION X-ray diffraction is a gadget for the investigation of the particular magnificent structure associated with the issue. This technique considered the precise origins in von Laue's finding in 1912, that will deposit diffract x-rays. The exclusive way with the diffraction exposing the unique arrangement associated with the specific crystal. At first, x-ray diffraction was utilized only for the dedication of crystal structure. Other uses had been developed; today the particular method is applied. Not unique to the structure dedication, but two such diverse problems as chemical evaluation and stress measurement to the study of stage equilibria and the dimension of particle size. In order to the determination of the particular inclination of one amazingly or the ensemble associated with directions in a polycrystalline aggregate [1]. The investigation of the peaks developed by x-ray diffraction their shapes is the well-developed and valuable technique for the study associated with the magnificent structure of crystalline materials. This method is identified as Diffraction Peak profile Analysis (DPPA). It is usually a statistical method since uses information via the single diffraction style, which comprehends details through many grains. From this particular pattern, a DPPA technique quantifies the magnificent structure associated with a sample [2]. A pure crystal would extend almost everywhere to infinity; however, definitely, no crystals are perfect because of their finite size. This change from perfect crystalline the reason for broadening of the particular diffraction peaks of components. You can find two main characteristics extracted from peak width analysis they are crystallite volume and lattice strain. Crystallite dimension calculation is linked to the size of the specification coherently diffracting domain. Since well as the crystallite size of the exacting grains is not genuinely usually the particular same as the specific particle bulk due to the presence of polycrystalline aggregates. Lattice strain is usually a measure of the selective distribution of lattice constants as a result of crystal imperfections, this rather as lattice dislocation [3]. One of diffraction peak profile analysis is Scherrer s formula it is the particular most of the known technique for extracting size facts from powder designs (namely, from Bragg peaks width). It is the particular straightforward method, but accurate only to the order of magnitude. Nevertheless, provided that Scherrer s work selection, profile analysis distributes made huge progress [4]. In a variety of ways, Bragg peak will be influenced by crystallite dimension and lattice stress. Usually raise the peak width plus strength shifting the 2θ top position accordingly. The particular crystallite bulk changes because 1/cos θ and strain vary as tan θ from the peak size. The scale and stress results on the peak increasing are usually known from the difference of 2θ. W-H analysis is emphatically an integral breadth method. Sizeinduced and strain-induced increasing are intricate simply by considering the peak size as a function associated with 2θ [5]. Double Voigt is a diffraction peak profile analysis method, used the Voigt function that derived equations by Langford to convolution the curves and showed that the Cauchy broadened and Gaussian broadened can be easily determined from the ratio of FWHM of the broadened profile to the integral breadth [6]. Theory Double Voigt method The double Voigt method use a Voigt function for fitting line profile, this function can find by convolution of the Gaussian (G) and Lorentzian (L) functions [6]. Additional to the line broadening there is a source broadening called instrument broadening in this broadening the correction can made by suppose that experimental profile h(x) is involution of the sample profile f(x) and instrumental assistance g(x). h(x) = f(x) * g(x) (1) 8173

2 By assuming the (f,g,h) is a Voigt function the information of the f(x) can find, from equation. (2.3) we find: β fl = β hl β gl (2 a) And β 2 fg = β 2 hg - β 2 gg (2 b) β ig and β il are the Gaussian (G) and the Lorentzian (L) component of the profile i(x) of the integral breadth [7]. The integral breadth β = A / I o (3) Where A is the area of the peak and I o is the height of the line profile, the two equations above the line broadening is assigned to the action of the domain size. β = K λ /<L>cosθ (4) This equation showed that the finite size of the crystal made the broadening of very small crystals, the β was the width must be in radian and the k is constant 0.89 and λ is the wave length which is nm, θ is the Bragg angle and <L> is the average column length it is similar to the length of the circle but it was parallel to the diffracted plane in this status therefor there is a relation of the volume weighted crystal diameter and the volume weighted column length which is <D> = 4/3 <L> (5) The strain can be estimated via equation: β = η tanθ (6) Where η is the strain and the 2θ dependence in this relation was deferent from that in the equation. (2.6) it was the reason that allowed to separate the action of the strain and the size on peak broadening. β cos θ = (K λ/<l> ) + η sin θ (7) This equation is similar to the form y=b +mx, the η can obtain from the slope between sin θ and β cos θ and <L> can obtain from the intercept. In the single line profile process the crystallite size will give the Lorentz component and the strain can appears the Gaussian component. The equation. (4) can use to produce the apparent domain size by replacing β by β L and use the equation. (6) to gives the strain by replacing β by β G, β L for the Gaussian and Lorentzian component of the integral breadth. To stop the hook effect accomplish the conditions must be taken [8][9]: β CS = (π/2) 1/2 β GS (8) 2w/β ratio for both h and g profiles where 2w is the FWHM and β is the breadth, for Lorentz reflections the ratio [10] [11]: 2w/β = 2/π = for Lorentz profile (9) And for Gaussian reflections the ratio can be: 2w/β = 2*(ln (2)) 1/2 / π 1/2 = for Gaussian profile (10) The crystallite size and the strain in this method is calculated using the plot of W-h, additional also the size and the strain is calculated but with separation which is an analytical method that use equations and arithmetic mean. Sherrer method The broadening of line diffraction profile can be happen from the finite crystallite size, strain and the defects of deflection from ideal crystallinity, the crystallite size can be measure by the equation that produce by Debye Sherrer in 1918: D v = (K λ /FWHM) cos θ (11) Where K is a shape factor (0.89) and θ is the Bragg angle and λ is the wavelength ( nm) and D v is the volume weighted quantity and FWHM full width at half maximum of the intensity of the peak The strain ε can obtain from line broadening which measure by the equation [12]: ε= β cot θ /4 = β/4 tan θ (12) Where β is the integral breadth and ε is the strain Williamson-Hall method The crystallite size and lattice these two factors additional to the instrumental broadening are responsible for the total line broadening and the strain can estimate by the equation. (2.14), the total broadening can obtain by take the summation of these two factors in the material, by assuming that the strain is in the uniform state in the material so that by using the W-H equation we can estimate the crystallite size and the strain [13]: β h k l = β s + β D (13) β h k l = (K λ /D cos θ) +4 ε tan θ (14) And the equation. (2.16) is multiply by cos θ therefor the equation can be: β h k l cos θ = (K λ /D) + 4 ε sin θ (15) Where K is the shape factor (0.89) and β h k l is integral breadth and λ is the wavelength ( nm) and D is the crystallite size in nm and ε is the lattice strain. The size can obtain by the intercept and the strain can obtain by the slope [14]. 8174

3 Size-strain plot method (SSP) The estimation of the size and the strain can be done by observing the average size-strain plot (SSP), the feature of this method that at high reflection angles lower value was given, and less precision can be obtain, the crystallite size was considering to be qualified by Lorentz function and the strain was considering to be qualified by Gaussian function: (d β h k l cos θ) 2 = (K/D) (d 2 β h k l cos θ) + (ε/2) 2 (16) K was the shape factor in the study the value was 0.89 and the plot was between (d β h k l cos θ) 2 and (d 2 β h k l cos θ) for all peaks, and d was the spacing distance that can calculate from the Bragg law, the crystallite size D could be estimated from the slope of the data that fitted, and the root mean square (RMS) strain could be obtained from the Y-intercept from the plot [15]. Halder-Wagner method In this method the average crystallite size was estimated and the Lorentz function and Gaussian function could be used for qualified the integral breadth, the size and the strain could be estimated by the equation: (β cos θ/ sin θ) 2 = (K λ/d) (β / tan θ * sin θ) + 16 ε 2 (17) Where β is the integral breadth and K is the Sherrer constant (0.89) and λ is the wavelength ( nm) and D is the crystallite size in nm and ε is the weighted average strain. By the plotting between (β cos θ/ sin θ) 2 against (β / tan θ * sin θ) the slope could give the crystallite size and the Y- intercept gives the strain [16]. RESULTS AND DISCUSSIONS Double Voigt method The Voigt function was used to analysis the x-ray diffraction line profile of the cerium oxide nanoparticle, at first the values of the intensity and 2θ of CeO 2 nanoparticle were calculated by using Get Data Graph Digitizer program after getting the values therefor used this values to plot the pattern of cerium oxide nanoparticle by using Origin Pro Lab program, after that each peak in the pattern was fitting to get the pure line of the peaks, after fitting the peaks, 40 steps on the fitting line of each peak was made additional to the high intensity step for each peak to get most pure and accurate line of each peak in the pattern, area under the curve was estimated after subtracting intensity to get rid of background values for each peak, after that FWHM was estimated and the equation.(3) was used to calculate the integral breadth for each peak (111) intensity (AU) () (220) (311) (222) (dgree) Figure-1. XRD pattern CeO 2 nanoparticle by origin pro lab program. 8175

4 intensity (AU) (dgree) Figure-2. Fitting peak (111) of CeO 2 nanoparticle intensity(a.u) Intensity(A.U) (dgree) Figure-3. Fitting peak () of CeO 2 nanoparticle (Dgree) Figure-5. Fitting peak (311) of CeO 2 nanoparticle. Intinsity(A.U) (Dgree) Figure-4. Fitting peak (220) of CeO 2 nanoparticle. 8176

5 Intensity(A.U) Intensity(A.U) (Dgree) Figure-6. Fitting peak (222) of CeO 2 nanoparticle (Dgree) Figure-9. After fitting for 40 steps peak (220) of CeO 2 nanoparticle Intensity(A.U) Intensity(AU) (Dgree) Figure-7. After fitting for 40 steps peak (111) of CeO 2 nanoparticle (Dgree) Figure-10. After fitting for 40 steps peak (311) of CeO 2 nanoparticle Intensity(AU) Intensity(AU) (Dgree) Figure-8. After fitting for 40 steps peak () of CeO 2 nanoparticle (Dgree) Figure-11. After fitting for 40 steps peak (222) of CeO 2 nanoparticle. 8177

6 Table-1. Results of CeO 2 nanoparticle for peak (111). 2θ intensity intensity- Background Area=(y1+y2)/2*(x2-x1) Σ Area = Io/2 Area under the curve FWHM B=Area/Io FWHM / B 2θ-1 2θ-2 Intensity

7 Table-2. Results of CeO 2 nanoparticle for peak (). 2θ intensity intensity- Background Area=(y1+y2)/2*(x2-x1) Σ Area = Io / 2 Area Under The Curve FWHM B=Area/Io FWHM / B 2θ-1 2θ-2 intensity

8 Table-3. Results of CeO 2 nanoparticle for peak (220). 2θ intensity intensity- Background Area=(y1+y2)/2*(x2-x1) Σ Area = Io/2 Area Under The Curve FWHM B=Area/Io FWHM / B 2θ-1 2θ-2 intensity

9 Table-4. Results of CeO 2 nanoparticle for peak (311). 2θ intensity intensity-background Area=(y1+y2)/2*(x2-x1) Σ Area = Io/2 Area Under The Curve FWHM B=Area/Io FWHM / B 2θ-1 2θ-2 intensity

10 Table-5. Results of CeO 2 nanoparticle for peak (222). 2θ intensity intensity-background Area=(y1+y2)/2*(x2-x1) Σ Area = Io/2 Area Under The Curve FWHM B=Area/Io FWHM / B 2θ-1 2θ-2 intensity

11 The silicon standard of x- ray diffraction pattern was used as instrumental broadening for calibration with the CeO 2 nanoparticle pattern, the reason for chosen this standard was the intensities values are more acceptable with CeO 2 intensities, also the integral breath of the standard was estimated additional to the CeO 2 integral breaths for Voigt function calibration. After calculate the FWHM and integral breadth for all peaks as B h(x) and the standard pattern B g(x) from the equation. (3) also testing these profiles for clarification either Gaussian profiles or Lorentz profiles by the equation. (9) and equation. (10) the results of introduce that the peaks (111), (), (220), (311) and (222) were Gaussian profiles, after that for calibration the Gaussian profile equation. (2b) used to determine the integral breadth of sample profile B f(x), there for the crystallite size <D> and the apparent strain η was estimating in this study according to the equation. (5) and equation. (7) with the plot according to Williamson hall equation. (15), the results included in the tables below: Table-6. Results of silicon standard XRD pattern for the highest peak. 2θ intensity intensity-background Area=(y1+y2)/2*(x2-x1)

12 Σ Area = Io/2 Area Under The Curve FWHM B=Area/Io 2W / B 2θ-1 2θ-2 intensity Table-7. Calculation of B 2 f G. Peak B h G B 2 h G B g G (Standard) B 2 g G B 2 f G= B 2 h G- B 2 g G Table-8. Results that used to plot B cos θ against sin θ. Peak 2θ θ B Sin θ Cos θ B Cos θ

13 B Cos Sin Figure-12. Relation between B cos θ and sin θ. Table-9. Results of crystallite size and apparent strain by the plot. the intercept 1 2 The Slope (η) K λ <L>vol= Kλ / the <D>= 4/3 intercept <L>vol X= ; X= ; Y= Y= nm Also by the separation Double Voigt method the crystallite size and the apparent strain could be obtained by using the equation. (4) and equation. (5) and equation. (6), the results in the table below: Table-10. Results of crystallite size and apparent strain by separation double Voigt method. Peak 2θ θ B Cos θ B Cos θ tanθ K λ <L>vol= K λ / B Cos θ <D>= 4/3 <L>vol η = B / tanθ nm

14 Average: Average: Average: Scherrer method According to Scherrer formula the crystallite size of cerium oxide nanoparticle in (nm) and the lattice strain were estimated by using equation. (11) and equation. (12) and the results in tables below: Table-11. Calculating the B and Cot θ from peaks. Peak 2θ θ B Cos θ Cot θ Table-12. Estimating crystallite size and lattice strain for Scherrer formula. K λ nm <D>V = (K λ /B) Cos θ η = B Cot θ ε= η / Average= Average= Average= Size-strain plot (SSP) method The crystallite size and the lattice strain were calculated by using the plot, by using the equation. (16) which the slope of the plot gives the crystallite size in (nm) and the intercept with y- axis gives the root mean square of the strain, the results are in the tables: Table-13. Calculation used for the size-strain plot. Peak 2θ θ B Cos θ Sin θ d=λ /2 Sin θ (d. B. Cos θ) 2 (d 2. B.Cos θ) E E E E E E E E

15 (d. B. Cos ) (d 2. B. Cos ) Figure-13. Estimating the crystallite size and lattice strain by SSP method. Table-14. Estimating crystallite size and lattice strain from SSP. K λ the intercept 1 2 X = E-4, X = , nm 6.24E-007 Y = E-6 Y = E-5 The Slope ε = (The cute)1/2 * 2 D = K / The slope The Halder-Wagner method The Halder-Wagner method was used the integral breadth of the peaks whether it be Gaussian or Lorentz functions to calculate the crystallite size and the lattice strain by using the equation. (17) the crystallite size in (nm) was obtained from the slope and the lattice strain obtained from the intercept with Y-axis and the results in the tables: Table-15. Calculation of Halder-Wagner plot. Peak 2θ θ B Cos θ Sin θ Tan θ (B/ tan θ) 2 B/(tan θ. Sin θ) E E

16 (B / tan ) B / (tan. Sin ) Figure-14. Halder-Wagner relation between (B / tan θ) 2 and B/(tan θ. Sin θ). Table-16. Estimating the crystallite size and lattice strain by Halder-Wagner plot. K λ The intercept 1 2 X = , X = , nm 1.14E-004 Y = Y = The slope ε= (the intercept /16) 1/2 D = K λ / the slope CONCLUSIONS a) The accuracy of the results given by the Voigt method for the crystallite size and lattice strain, because it depends on the analysis of the line of diffraction fully where the line tails into the calculations. b) The calculation of crystallite size and lattice strain in the Sherrer method is very important because this method gives the values of crystallite size and lattice strain quickly. But the calculations in this method are inaccurate because they depend on FWHM and not the integral breadth of the peaks. c) The use of other methods in the calculations to demonstrate the validity of the results given by the method used in the study through the implementation of the method of Double Voigt and already found that there is a high accuracy in the calculation of crystallite size and lattice strain in relation to the ratio of other methods, which each method is specific in the calculation of crystallite size and lattice strain. REFERENCES [1] B. D. Cullity Elements of X-Ray Diffraction, Notre Dame, Indiana: Add1son-Wesley. p. 1. [2] SIMM Thomas H The Use of Diffraction Peak Profile Analysis in Studying the Plastic Deformation of Metals. School of Materials, University of Manchester, Manchester. [3] Thomas, P. Bindu Sabu Estimation of lattice strain in ZnO nanoparticles: X-ray peak. J Theor Appl Phys. 8: [4] A. Cervellino, C. Giannini, A. Guagliardi and M. Ladisa. 5. Nanoparticle size distribution estimation by a full-pattern powder diffraction analysis. Phys. Rev. B. 72(3): [5] Y. T. Prabhu, K. Venkateswara Rao, V. Sesha Sai Kumar, B. Siva Kumari X-ray Analysis of Fe doped ZnO Nanoparticles by Williamson-Hall and 8188

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