X ray diffraction line profile analysis of cerium oxide nanoparticle by using double Voigt function method

(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 E-Mail: [email protected]

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 (12.4964nm) and the emotion was equal to (0.006819). 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, size-strain plot (SSP) method and Halder-Wagner method) and their results are as follows Sherrer crystallite size (7.6386 nm) and lattice strain (0.01039), SSP method crystallite size (59.8956 nm) and lattice strain (0.00157), and Halder-Wagner method crystallite size (9.2287nm) and lattice strain (0.00267). 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

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. Size-induced and strain-Size-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).

(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:

βfL= βhL–βgL (2 a)

And

β2

fG= β2hG -β2gG (2 b)

βiG and βiL are the Gaussian (G) and the Lorentzian (L) component of the profile i(x) of the integral breadth [7].

The integral breadth β = A / Io (3)

Where A is the area of the peak and Io 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 0.15406 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 βL and use the equation. (6)

to gives the strain by replacing β by βG, βL for the Gaussian and Lorentzian component of the integral breadth.

To stop the “hook” effect accomplish the conditions must be taken [8][9]:

βCS= (π/2)1/2βGS (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 = 0.63662 for Lorentz profile (9)

And for Gaussian reflections the ratio can be:

2w/β = 2*(ln (2))1/2_{/ π}1/2

= 0.93949 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:

Dv= (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 (0.15406 nm) and Dv 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]:

βh k l= βs + βD (13)

βh 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:

βh k lcos θ = (K λ /D) + 4 ε sin θ (15)

Where K is the shape factor (0.89) and βh k l is

(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 βh k lcos θ)2 = (K/D) (d2βh k lcos θ) + (ε/2)2 (16)

K was the shape factor in the study the value was 0.89 and the plot was between (d βh k lcos θ)2and (d2βh 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 (0.15406 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 CeO2 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.

20 30 40 50 60 70 80

0 100 200 300 400 500 600 700

intensi

ty (AU)

2(dgree)

28.242

7

32.819

8

47.302

8

56.195

3

69.305

3

(111)

(200)

(220)

(311)

(222)

(4)

27.0 27.5 28.0 28.5 29.0 29.5 150

200 250 300 350 400 450 500 550 600 650

intensi

ty (AU)

2(dgree)

28.242

7

28.323

3

28.323

3

Figure-2. Fitting peak (111) of CeO2 nanoparticle.

35 100

120 140 160 180 200 220 240 260

intensi

ty(A.U)

2(dgree)

32.819

8

40 45 50

200 400

Intinsi

ty(A.U)

2(Dgree)

47.302

8

55 60

200

Intensity

(A.U)

2(Dgree)

56.195

3

(5)

70 90

100 110 120 130 140

Intensity

(A.U)

2(Dgree)

69.305

3

27.5 28.0 28.5 29.0 29.5 30.0

200 250 300 350 400 450 500 550 600 650 700

Intensity

(A.U)

2(Dgree)

Figure-7. After fitting for 40 steps peak (111) of CeO2 nanoparticle.

31.0 31.5 32.0 32.5 33.0 33.5 34.0 34.5 120

140 160 180 200 220 240 260

Intensity

(AU)

2(Dgree)

45.5 46.0 46.5 47.0 47.5 48.0 48.5 49.0 49.5

150 200 250 300 350 400 450

Intensity

(A.U)

2(Dgree)

54.5 55.0 55.5 56.0 56.5 57.0 57.5 58.0 150

200 250 300 350

Intensity

(AU)

2(Dgree)

67.5 68.0 68.5 69.0 69.5 70.0 70.5 71.0 71.5

95 100 105 110 115 120 125 130

Intensity

(AU)

2(Dgree)

(6)

Table-1. Results of CeO2 nanoparticle for peak (111).

2θ intensity

intensity-Background Area=(y1+y2)/2*(x2-x1)

27.01076 154.41055 0 0.3623782215

27.11041 161.68357 7.27302 0.8818764697

27.18728 170.08215 15.6716 1.4007745476

27.25562 179.73323 25.32268 2.2245809734

27.32679 191.60244 37.19189 2.1415301937

27.37804 200.79057 46.38002 3.255383574

27.44068 211.96998 57.55943 4.3175742013

27.50901 223.22531 68.81476 5.2698728707

27.58019 233.66752 79.25697 6.5408648408

27.65706 245.33349 90.92294 7.6411101192

27.73394 262.26779 107.85724 5.2604852325

27.77949 277.52961 123.11906 4.8601966868

27.8165 293.93384 139.52329 5.0883495217

27.85067 312.71281 158.30226 6.3111406128

27.88768 337.15885 182.7483 6.0978433284

27.919 361.05199 206.64144 12.4534507675

27.9731 408.15546 253.74491 16.8847731291

28.03289 465.46822 311.05767 24.5420023178

28.10406 533.02556 378.61501 33.9168032968

28.18663 597.32402 442.91347 63.5386838862

28.3233 641.3088 486.89825 58.9224913902

28.44857 608.24019 453.82964 51.3804301049

28.571 539.9247 385.51415 41.6200955076

28.68773 481.99664 327.58609 18.5464583039

28.74752 447.21109 292.80054 11.0590565528

28.78738 416.50497 262.09442 7.77001437

28.8187 388.48563 234.07508 5.0862972146

28.84148 366.89361 212.48306 4.0450353505

28.86141 347.85176 193.44121 3.6695914197

28.88134 329.21735 174.8068 3.7500533482

28.90411 308.9892 154.57865 3.3090898871

28.92689 290.35779 135.94724 3.242974476

28.95251 271.62291 117.21236 3.082005208

28.98099 253.63114 99.22059 3.3139525659

29.018 234.27413 79.86358 2.8500888528

29.05786 217.55193 63.14138 2.8025675562

29.10911 200.63766 46.22711 2.2200809202

29.16605 186.1631 31.75255 1.7277793973

29.23438 173.22962 18.81907 1.05707117

29.31126 163.09073 8.68018 0.4708190782

29.41945 154.43393 0.02338

Σ Area = 442.9156274657

Io/2 Area under the

curve FWHM B=Area/Io FWHM / B

442.9156274657 0.8477593557 0.9096677334 0.9319439666

2θ-1 2θ-2 Intensity

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2θ intensity

intensity-Background Area=(y1+y2)/2*(x2-x1)

31.69131 119.00487 0.01888 0.106341741

31.84751 120.32872 1.34273 0.3794994434

31.97838 123.4429 4.45691 0.8824039601

32.09659 129.45851 10.47252 1.0526282981

32.17258 136.21786 17.23187 1.1014687656

32.22746 141.89511 22.90912 1.2093976782

32.2739 148.16118 29.17519 1.3709795337

32.31611 154.77074 35.78475 1.48684025

32.35411 161.45599 42.47 1.5439557887

32.38788 167.95548 48.96949 1.5368922495

32.41743 174.03628 55.05029 1.983597069

32.45121 181.3778 62.39181 2.5347072362

32.4892 190.03494 71.04895 2.2019308088

32.51875 196.96789 77.9819 2.7688907774

32.55252 204.98933 86.00334 4.2482775618

32.59896 215.94034 96.95435 4.2954581558

32.64118 225.51142 106.52543 4.683819987

32.68339 234.38996 115.40397 5.5588538358

32.72983 242.98141 123.99542 8.6978855967

32.79738 252.51494 133.52895 14.4076300407

32.90292 258.48395 139.49796 13.8526988766

33.00424 252.93254 133.94655 8.7323752637

33.07178 243.62325 124.63726 5.5914126876

33.11822 235.15031 116.16432 4.7186902237

33.16044 226.35034 107.36435 4.3309002775

33.20265 216.82895 97.84296 4.6544698225

33.25331 204.89628 85.91029 2.434903452

33.28286 197.87458 78.88859 2.2292434564

33.31242 190.92578 71.93979 2.5678825735

33.35041 182.23349 63.2475 2.24608633

33.38841 173.95356 54.96757 1.534515543

33.41796 167.87734 48.89135 1.3603766543

33.44751 162.16751 43.18152 1.3562974183

33.48128 156.13005 37.14406 1.5530216346

33.52772 148.72486 29.73887 1.2335757354

33.57416 142.37269 23.3867 1.1134935224

33.62904 136.1785 17.19251 0.9637491974

33.69658 130.3321 11.34611 0.8033627736

33.78946 124.93882 5.95283 0.5300506067

33.92455 120.88053 1.89454 0.183959834

34.11875 118.98599 0

Σ Area = 124.0425246607

Io / 2 Area Under The

Curve FWHM B=Area/Io FWHM / B

2θ-1 2θ-2 intensity

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2θ intensity intensity- Background Area=(y1+y2)/2*(x2-x1)

45.96 105.238587 0.162689 0.374449301

46.1611973 108.635419 3.559521 1.334366486

46.3346432 116.902921 11.827023 2.0445493464

46.4525865 127.918913 22.843015 2.8928713559

46.5497162 141.800067 36.724169 2.3236984335

46.6052189 152.084536 47.008638 2.9464447496

46.6607216 164.240284 59.164386 3.1699482344

46.7092865 176.456339 71.380441 3.7986482069

46.7578514 190.13141 85.055512 4.4963474099

46.8064162 205.189373 100.113475 4.4567529199

46.8480432 219.090413 114.014515 5.0516689172

46.8896703 233.771969 128.696071 6.6843964477

46.9382351 251.657248 146.58135 6.4293075307

46.9798622 267.394641 162.318743 8.3318900012

47.028427 285.881805 180.805907 9.2227975152

47.0769919 304.083293 199.007395 10.088206307

47.1255568 321.521075 216.445177 12.5227920253

47.1810595 339.880553 234.804655 13.476061693

47.2365622 355.871464 250.795566 19.7850517386

47.3128784 372.782347 267.706449 36.0761729564

47.4446973 384.729091 279.653193 34.2505875723

47.5695784 373.953869 268.877971 19.896637623

47.6458946 357.624246 252.548348 15.2135577668

47.7083351 339.825306 234.749408 12.5195474484

47.7638378 321.459406 216.383508 10.0851219259

47.8124027 304.017941 198.942043 7.957196905

47.8540297 288.443265 183.367367 9.5913842892

47.9095324 267.327187 162.251289 6.4265213664

47.9511595 251.590839 146.514941 6.6812309455

47.9997243 233.708016 128.632118 4.2491143116

48.0344135 221.425752 116.349854 5.2548678504

48.0829784 205.132043 100.056145 3.8947780339

48.1246054 192.147244 87.071346 4.3959390818

48.1801081 176.409065 71.333167 3.1677832597

48.228673 164.1984 59.122502 2.6105987114

48.2772378 153.463309 48.387411 2.3940454972

48.3327405 142.9562 37.880302 2.8135124196

48.4229324 129.585066 24.509168 2.2993597042

48.5478135 117.391513 12.315615 1.2776395728

48.7004459 109.501677 4.425779 0.5526961499

48.9502081 105.075898 0

Σ Area = 311.0385420107

Io/2 Area Under The

311.0385420107 1.053099 3113

1.112229539

3 0.9468363086

46.91790713

17 47.971006443

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2θ intensity intensity-Background Area=(y1+y2)/2*(x2-x1)

55.13721 114.31044 0.07067 0.1028394737

55.23146 116.35137 2.1116 0.2488311675

55.30371 119.01623 4.77646 0.5835202474

55.3854 123.74952 9.50975 1.073036286

55.46708 131.00417 16.7644 1.14322611

55.52363 137.90777 23.668 1.382093218

55.5739 145.55857 31.3188 1.4294008168

55.61474 152.92101 38.68124 1.4590072512

55.6493 159.99182 45.75205 2.0597680546

55.69014 169.35785 55.11808 2.4642741648

55.73098 179.80113 65.56136 2.4323096448

55.76554 189.43707 75.1973 3.0474005185

55.80324 200.70828 86.46851 3.796274032

55.84408 213.68086 99.44109 4.6952595006

55.88806 228.31662 114.07685 5.3474789424

55.93204 243.34068 129.10091 7.3822345961

55.98545 261.57528 147.33551 9.9005233271

56.04828 282.057 167.81723 13.462415148

56.12368 303.51578 189.27601 18.0892030751

56.21479 322.04867 207.8089 22.6194666761

56.32161 329.93708 215.69731 19.3615554245

56.41271 323.60405 209.36428 20.0600197985

56.51325 303.92104 189.68127 12.9677861925

56.5855 283.52836 169.28859 10.4632842367

56.65148 262.11651 147.87674 7.0028055449

56.70175 244.97077 130.731 5.4187093902

56.74573 229.92575 115.68598 5.3935785204

56.79599 213.18087 98.9411 2.9512620622

56.82741 203.15749 88.91772 3.6191071632

56.87139 189.90173 75.66196 2.4478046208

56.90595 180.23317 65.9934 2.304806088

56.94365 170.51725 56.27748 2.2512741882

56.98763 160.33947 46.0997 1.5925390455

57.02533 152.6249 38.38513 1.5160667411

57.06932 144.78243 30.54266 1.4194543366

57.12273 136.85024 22.61047 1.1408872026

57.18242 129.85638 15.61661 0.903721785

57.25467 123.63968 9.39991 0.5078771712

57.32379 119.53538 5.29561 0.3122495624

57.40861 116.3068 2.06703 0.100664361

57.50601 114.23977 0

Σ Area = 204.4540156863

Io/2 Area Under The

204.4540156863 0.8993001555 0.9478746661 0.9487542897 55.869751166

4 56.7690513219

(10)

2θ intensity intensity-background Area=(y1+y2)/2*(x2-x1)

67.8018922 90.1357243 0 0.0233960908

67.9759961 90.4044844 0.2687601 0.0789392671

68.1226099 90.9437969 0.8080726 0.1792136455

68.2600604 91.9353344 1.7996101 0.2701102251

68.3700207 93.2489823 3.113258 0.3138362526

68.452491 94.6333578 4.4976335 0.3592041266

68.5212162 96.0914362 5.9557119 0.399543292

68.5807781 97.5960818 7.4603575 0.5386915677

68.6449216 99.4718175 9.3360932 0.4620599058

68.6907384 100.969521 10.8337967 0.6494671761

68.7457186 102.927421 12.7916967 0.6933002219

68.7961171 104.85676 14.7210357 0.870393564

68.8510973 107.076762 16.9410377 0.8199709082

68.8969141 108.988147 18.8524227 1.1003395361

68.9518943 111.310058 21.1743337 1.227285904

69.0068745 113.606041 23.4703167 1.4688706857

69.0664363 115.987983 25.8522587 1.7334766335

69.1305799 118.333336 28.1976117 2.1502092325

69.2038868 120.601328 30.4656037 3.0314884782

69.3001021 122.684807 32.5490827 3.6270819723

69.4100625 123.557336 33.4216117 3.3324907119

69.5108595 122.836928 32.7012037 3.1911677237

69.6116565 120.753225 30.6175007 2.1627912256

69.6849634 118.524708 28.3889837 1.8652545165

69.7536886 116.028271 25.8925467 1.2544462683

69.8040871 114.024273 23.8885487 1.4466584516

69.8682306 111.354114 21.2183897 1.1027593522

69.9232108 109.032116 18.8963917 0.7436708926

69.9644459 107.309132 17.1734077 0.8828913057

70.0194261 105.079017 14.9432927 0.7635030934

70.0744063 102.966179 12.8304547 0.6011741831

70.1248048 101.162098 11.0263737 0.4705259249

70.1706216 99.6488001 9.5130758 0.5138483377

70.2301835 97.8769113 7.741187 0.4435297529

70.294327 96.2238342 6.0881099 0.4096338943

70.3722156 94.5660701 4.4303458 0.3511704457

70.4684309 93.0050584 2.8693341 0.2559442308

70.582973 91.7353886 1.5996643 0.1691793883

70.7341685 90.7739492 0.6382249 0.0642694818

70.8945273 90.2990704 0.1633461 0.0101599616

71.0182327 90.1366388 0.0009145

Σ Area = 40.0319478284

Io/2 Area Under

The Curve FWHM B=Area/Io FWHM / B

(11)

The silicon standard of x- ray diffraction pattern was used as instrumental broadening for calibration with the CeO2 nanoparticle pattern, the reason for chosen this standard was the intensities values are more acceptable with CeO2 intensities, also the integral breath of the standard was estimated additional to the CeO2 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), (200), (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)

28.2907574 929.244335 0 0.5095278101

28.3055275 998.238832 68.994497 0.8537563022

28.3129126 1091.46033 162.215995 0.6530475831

28.3162695 1156.10611 226.861775 1.1175217334

28.3202977 1257.23174 327.987405 0.993173376

28.3229832 1340.91314 411.668805 1.237018253

28.3256687 1438.83284 509.588505 1.5200495969

28.3283542 1551.69798 622.453645 1.3489959826

28.3303683 1646.34283 717.098495 1.5481825068

28.3323824 1749.49007 820.245735 2.4045416338

28.3350679 1899.75759 970.513255 2.8260796409

28.3377534 2063.4264 1134.182065 3.2808784673

28.3404388 2238.55558 1309.311245 3.7630941229

28.3431243 2422.46083 1493.216495 4.2641831121

28.3458098 2611.73673 1682.492395 6.0472886463

28.3491667 2849.65262 1920.408285 6.8339530791

28.3525235 3080.54308 2151.298745 9.1776390311

28.3565518 3334.52729 2405.282955 10.1175395472

28.36058 3547.3165 2618.072165 14.5817957626

28.365951 3740.99744 2811.753105 15.2781790407

28.37132 3808.74809 2879.503755 15.2898564857

28.3766929 3741.21315 2811.968815 14.5834542737

28.3820639 3547.71837 2618.474035 10.1193719552

28.3860921 3335.03521 2405.790875 9.1795995418

28.3901203 3081.12167 2151.877335 6.836150361

28.3934772 2850.26185 1921.017515 6.0491636853

28.396834 2612.35199 1683.107655 4.2658205017

28.3995195 2423.065 1493.820665 3.7646854965

28.402205 2239.13657 1309.892235 3.2825165262

28.4048905 2063.97435 1134.730015 2.8274967927

28.407576 1900.26505 971.020715 2.4058432017

28.4102615 1749.95194 820.707605 1.5490763443

28.4122756 1646.76854 717.524205 1.3498162752

28.4142897 1552.08682 622.842485 1.521028126

28.4169752 1439.17275 509.928415 1.2378675692

28.4196607 1341.20575 411.961415 0.9938624985

28.4223461 1257.47991 328.235575 0.9571534341

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28.4297312 1091.60677 162.362435 0.7379626021

28.4357736 1011.14332 81.898985 0.5895722665

28.449201 935.161646 5.917311

Σ Area = 176.7112193845

Io/2 Area Under The

Curve FWHM B=Area/Io 2W / B

176.7112193845 0.057987114 0.0613686365 0.9448981961 28.342340591 28.400327705 2368.99

62125

Table-7. Calculation of B2f G.

Peak B h G B2 h G B g G (Standard) B2g G B2f G= B2h G- B2g G

111 0.9096677334 0.8274953852 0.0613686365 0.0037661096 0.8237292756 200 0.8892067286 0.7906886062 0.0613686365 0.0037661096 0.7869224967 220 1.1122295393 1.2370545481 0.0613686365 0.0037661096 1.2332884385 311 0.9478746661 0.8984663826 0.0613686365 0.0037661096 0.894700273 222 1.1977862764 1.4346919639 0.0613686365 0.0037661096 1.4309258543

Table-8. Results that used to plot B cos θ against sin θ.

Peak 2θ θ B Sin θ Cos θ B Cos θ

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0.2

0.3

0.4

0.5

0.6

0.80

0.85

0.90

0.95

1.00

1.05 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}

intercept

<D>= 4/3 <L>vol

0.838641686 X=0.350745946; Y=0.897555621

X=0.499762162;

Y=0.95581089 0.0068195988 0.89

0.15406

nm _9.3723028 _12.49640373

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

111 28.3233 14.16165 0.9075953259 0.9696093252 1.102932598 0.2523268311

200 32.90292 16.45146 0.8870865215 0.9590600037 1.0662810692 0.2952922092 220 47.4446973 23.72234865 1.1105352036 0.9155057409 1.2742461809 0.4394346088 311 56.32161 28.160805 0.9458859725 0.8816265092 1.0451614043 0.5353148593 222 69.4100625 34.70503125 1.1962131308 0.8220940484 1.2325087409 0.6925627504

K λ

<L>vol= K λ / B Cos θ <D>= 4/3 <L>vol η = B / tanθ 0.89 0.15406 nm

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7.367681338 9.8235751174 0.0524313885

6.1652287071 8.2203049428 0.0441078044

7.5165606979 10.0220809305 0.0308394663

6.3740068319 8.4986757758 0.0301457993

Average: 6.9092647315 Average: 9.2123529753 Average: 0.0440604544

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

111 28.3233 14.16165 0.014796192 0.9696093252 3.9631140118 200 32.90292 16.45146 0.0146778661 0.9590600037 3.3864760692 220 47.4446973 23.72234865 0.0183800503 0.9155057409 2.2756514391 311 56.32161 28.160805 0.0156957487 0.8816265092 1.8680594844 222 69.4100625 34.70503125 0.0197139612 0.8220940484 1.4439124821

Table-12. Estimating crystallite size and lattice strain for Scherrer formula.

K λ

<D>V = (K λ /B) Cos θ η = B Cot θ ͤε= η / 4

0.89 0.15406 nm

8.9851788259 0.0586389959 0.014659749

8.9590664354 0.0497062424 0.0124265606

6.8295843913 0.041826588 0.010456647

7.7016274082 0.0293205922 0.007330148

5.7177808756 0.0284652346 0.0071163086

Average= 7.6386475873 Average= 0.0415915306

Average= 0.0103978827

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 (d2. B .Cos θ)

111 28.3233 14.16165 0.0158405267 0.9696093252 0.2446584485 0.3148470878 2.34E-005 0.0015225297

200 32.90292 16.45146 0.0154825806 0.9590600037 0.2832029473 0.2719957569 1.63E-005 0.0010985337

220 47.4446973 23.72234865 0.0193824958 0.9155057409 0.4023049072 0.1914716888 1.15E-005 6.51E-004

311 56.32161 28.160805 0.0165088246 0.8816265092 0.4719477707 0.1632172134 5.64E-006 3.88E-004

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0.0000 0.0005 0.0010 0.0015 0.000005

0.000010 0.000015 0.000020 0.000025

(d

. B. Cos



)

2

(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

0.89 0.15406 nm 6.24E-007 X = 5.0044405E-4,

Y = 8.10065574E-6

X = 0.00124994449, Y = 1.9237623E-5

The Slope ε = (The cute)1/2 * 2 D = K / The slope

0.0148591871 0.0015796866 59.8956049728

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)

111 28.3233 14.16165 0.0158405267 0.9696093252 0.2446584485 0.2523268311 0.0039410539 0.2565936871

200 32.90292 16.45146 0.0154825806 0.9590600037 0.2832029473 0.2952922092 0.0027490505 0.185137157

220 47.4446973 23.72234865 0.0193824958 0.9155057409 0.4023049072 0.4394346088 0.0019454984 0.1096377489

311 56.32161 28.160805 0.0165088246 0.8816265092 0.4719477707 0.5353148593 9.51E-004 0.0653450831

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0.0 0.1 0.2 0.3 0.001

0.002 0.003 0.004

(B / t

an



)

2

B / (tan



. Sin



)

Figure-14. Halder-Wagner relation between (B / tan θ) 2and B/(tan θ. Sin θ).

Table-16. Estimating the crystallite size and lattice strain by Halder-Wagner plot.

K λ The intercept 1 2

0.89 0.15406 nm 1.14E-004 X = 0.100643872,

Y = 0.00160337705

X = 0.250111012, Y = 0.00382404098

The slope ε= (the intercept /16)1/2 D = K λ / the slope

0.0148572049 0.0026704209 9.2287479779

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.

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