View source for Specific Process Knowledge/Characterization/XRD/XRD SmartLab/Instrumental broading in GiXRD

This page describes values of Instrumental broadening in GiXRD measurements, and can be used as a reference for peak broadening in 2θ - scans. Most four common slit configuration cases are chosen.
The instrumental broadening has been measured by using standard Si powder sample. 

Scherrer formula:

<math>D=\frac{K\cdot\lambda}{\beta_{sample}\cdot cos\theta}</math>

where:
K is Scherrer constant (typically close to unity K=0.9-1)<br> 
λ is a wavelength of X-ray radiation (λ Cu-K<sub>α1</sub> = 1.540562 Å, and in case both Cu-K<sub>α1</sub> and Cu-K<sub>α2</sub> are present λ Cu-K<sub>α1,2</sub> = 1.541871 Å) <br>
θ is Bragg angle <br>
β<sub>sample</sub> is a peak broadening. The technical point should be noted that intergal breadth and/or FWHM have to be inserted in radians.<br>

<br clear="all" />

<math>\beta_{sample}=\beta_{measure}-\beta_{instrumental}</math>    in the case where the peaks obey Lorentzian (or Cauchy) distribution profile.<br>

<math>\beta_{sample}^2=\beta_{measure}^2-\beta_{instrumental}^2</math>   in the case of Gaussian distribution profile.<br>

Experimental results implied crystallite size to cause a Cauchy-type broadening while strain broadening was associated with a Gauss profile. Real measurement contains contributions of both Lorentzian (Cauchy) and Gaussian distributions.


=Full scans=

<gallery caption="2θ scans of Si reference sample." widths="1200px" heights="500px" perrow="1">
image:eves_GiXRD_Si_reference_instrumental_brodening_spectra_20220319.png|Measured Si powder reference sample with different incident slit (IS) and incident angle ω values.
</gallery>

<gallery caption="Broadening functions." widths="800px" heights="500px" perrow="2">
image:eves_GiXRD_Si_reference_instrumental_brodening_FWHM_20220319.png|Calculated FWHM as a function of 2θ.
image:eves_GiXRD_Si_reference_instrumental_brodening_int_br_20220319.png|Calculated Integral breadth as a function of 2θ.
</gallery>

= case1  =

<b>IS=0.5 mm and ω=2&deg;</b> 


{| border="2" cellspacing="0" cellpadding="9" 

!colspan="1" border="none" style="background:silver; color:black;" align="center"|Peak number
|style="background:silver; color:black"|<b>2θ (&deg;)</b>
|style="background:silver; color:black"|<b>FWHM (&deg;)</b>
|style="background:silver; color:black"|<b>FWHM (rad)</b>
|style="background:silver; color:black"|<b>Integral breadth (&deg;)</b>
|style="background:silver; color:black"|<b>Integral breadth (rad)</b>

|-
!style="background:WhiteSmoke; color:black;" align="center" |1 
|style="background:WhiteSmoke; color:black" align="center"|28.4077
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2923</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0051
|style="background:WhiteSmoke; color:black" align="center"|0.3232
|style="background:WhiteSmoke; color:black" align="center"|0.0056

|-
!style="background:WhiteSmoke; color:black;" align="center" |2 
|style="background:WhiteSmoke; color:black" align="center"|47.2895
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3109</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0054
|style="background:WhiteSmoke; color:black" align="center"|0.3395
|style="background:WhiteSmoke; color:black" align="center"|0.0059

|-
!style="background:WhiteSmoke; color:black;" align="center" |3 
|style="background:WhiteSmoke; color:black" align="center"|56.1274
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3004</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0052
|style="background:WhiteSmoke; color:black" align="center"|0.3368
|style="background:WhiteSmoke; color:black" align="center"|0.0059

|-
!style="background:WhiteSmoke; color:black;" align="center" |4
|style="background:WhiteSmoke; color:black" align="center"|69.1426
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3082</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0054 
|style="background:WhiteSmoke; color:black" align="center"|0.3325 
|style="background:WhiteSmoke; color:black" align="center"|0.0058

|-
!style="background:WhiteSmoke; color:black;" align="center" |5
|style="background:WhiteSmoke; color:black" align="center"|76.3659
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2585</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0045 
|style="background:WhiteSmoke; color:black" align="center"|0.2936 
|style="background:WhiteSmoke; color:black" align="center"|0.0051

|-
!style="background:WhiteSmoke; color:black;" align="center" |6
|style="background:WhiteSmoke; color:black" align="center"|88.0122
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3174</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0055 
|style="background:WhiteSmoke; color:black" align="center"|0.3609 
|style="background:WhiteSmoke; color:black" align="center"|0.0063

|-
!style="background:WhiteSmoke; color:black;" align="center" |7
|style="background:WhiteSmoke; color:black" align="center"|94.9581
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3126</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0055 
|style="background:WhiteSmoke; color:black" align="center"|0.3434 
|style="background:WhiteSmoke; color:black" align="center"|0.0060

|-
!style="background:WhiteSmoke; color:black;" align="center" |8
|style="background:WhiteSmoke; color:black" align="center"|106.7351
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3317</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0058 
|style="background:WhiteSmoke; color:black" align="center"|0.3637
|style="background:WhiteSmoke; color:black" align="center"|0.0063

|- 
|}

= case2  =

<b>IS=0.5 mm and ω=1&deg;</b> 


{| border="2" cellspacing="0" cellpadding="9" 

!colspan="1" border="none" style="background:silver; color:black;" align="center"|Peak number
|style="background:silver; color:black"|<b>2θ (&deg;)</b>
|style="background:silver; color:black"|<b>FWHM (&deg;)</b>
|style="background:silver; color:black"|<b>FWHM (rad)</b>
|style="background:silver; color:black"|<b>Integral breadth (&deg;)</b>
|style="background:silver; color:black"|<b>Integral breadth (rad)</b>

|-
!style="background:WhiteSmoke; color:black;" align="center" |1 
|style="background:WhiteSmoke; color:black" align="center"|28.4185
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2961</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0052
|style="background:WhiteSmoke; color:black" align="center"|0.3246
|style="background:WhiteSmoke; color:black" align="center"|0.0057

|-
!style="background:WhiteSmoke; color:black;" align="center" |2 
|style="background:WhiteSmoke; color:black" align="center"|47.2935
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2961</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0052 
|style="background:WhiteSmoke; color:black" align="center"|0.3214
|style="background:WhiteSmoke; color:black" align="center"|0.0056

|-
!style="background:WhiteSmoke; color:black;" align="center" |3 
|style="background:WhiteSmoke; color:black" align="center"|56.1085
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2881</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0050
|style="background:WhiteSmoke; color:black" align="center"|0.3206
|style="background:WhiteSmoke; color:black" align="center"|0.0056

|-
!style="background:WhiteSmoke; color:black;" align="center" |4
|style="background:WhiteSmoke; color:black" align="center"|69.1297
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3009</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0053 
|style="background:WhiteSmoke; color:black" align="center"|0.3203 
|style="background:WhiteSmoke; color:black" align="center"|0.0056

|-
!style="background:WhiteSmoke; color:black;" align="center" |5
|style="background:WhiteSmoke; color:black" align="center"|76.3485
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3110</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0054
|style="background:WhiteSmoke; color:black" align="center"|0.3525  
|style="background:WhiteSmoke; color:black" align="center"|0.0062

|-
!style="background:WhiteSmoke; color:black;" align="center" |6
|style="background:WhiteSmoke; color:black" align="center"|88.0310
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2972</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0052 
|style="background:WhiteSmoke; color:black" align="center"|0.3234 
|style="background:WhiteSmoke; color:black" align="center"|0.0056

|-
!style="background:WhiteSmoke; color:black;" align="center" |7
|style="background:WhiteSmoke; color:black" align="center"|94.9806
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2872</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0050 
|style="background:WhiteSmoke; color:black" align="center"|0.3114 
|style="background:WhiteSmoke; color:black" align="center"|0.0054

|-
!style="background:WhiteSmoke; color:black;" align="center" |8
|style="background:WhiteSmoke; color:black" align="center"|106.7100
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2838</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0050 
|style="background:WhiteSmoke; color:black" align="center"|0.3024
|style="background:WhiteSmoke; color:black" align="center"|0.0053

|- 
|}

= case3  =


<b>IS=1 mm and ω=2&deg;</b> 


{| border="2" cellspacing="0" cellpadding="9" 

!colspan="1" border="none" style="background:silver; color:black;" align="center"|Peak number
|style="background:silver; color:black"|<b>2θ (&deg;)</b>
|style="background:silver; color:black"|<b>FWHM (&deg;)</b>
|style="background:silver; color:black"|<b>FWHM (rad)</b>
|style="background:silver; color:black"|<b>Integral breadth (&deg;)</b>
|style="background:silver; color:black"|<b>Integral breadth (rad)</b>

|-
!style="background:WhiteSmoke; color:black;" align="center" |1 
|style="background:WhiteSmoke; color:black" align="center"|28.3469
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3041</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0053
|style="background:WhiteSmoke; color:black" align="center"|0.3237
|style="background:WhiteSmoke; color:black" align="center"|0.0056

|-
!style="background:WhiteSmoke; color:black;" align="center" |2 
|style="background:WhiteSmoke; color:black" align="center"|47.2945
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3009</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0053
|style="background:WhiteSmoke; color:black" align="center"|0.3242
|style="background:WhiteSmoke; color:black" align="center"|0.0057

|-
!style="background:WhiteSmoke; color:black;" align="center" |3 
|style="background:WhiteSmoke; color:black" align="center"|56.1208
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3064</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0053
|style="background:WhiteSmoke; color:black" align="center"|0.3322
|style="background:WhiteSmoke; color:black" align="center"|0.0058

|-
!style="background:WhiteSmoke; color:black;" align="center" |4
|style="background:WhiteSmoke; color:black" align="center"|69.1388
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3363</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0059 
|style="background:WhiteSmoke; color:black" align="center"|0.3598 
|style="background:WhiteSmoke; color:black" align="center"|0.0063
|-
!style="background:WhiteSmoke; color:black;" align="center" |5
|style="background:WhiteSmoke; color:black" align="center"|76.3682
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2689</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0047
|style="background:WhiteSmoke; color:black" align="center"|0.3100 
|style="background:WhiteSmoke; color:black" align="center"|0.0054

|-
!style="background:WhiteSmoke; color:black;" align="center" |6
|style="background:WhiteSmoke; color:black" align="center"|88.0168
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3443</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0060 
|style="background:WhiteSmoke; color:black" align="center"|0.3811 
|style="background:WhiteSmoke; color:black" align="center"|0.0067

|-
!style="background:WhiteSmoke; color:black;" align="center" |7
|style="background:WhiteSmoke; color:black" align="center"|94.9673
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3018</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0053 
|style="background:WhiteSmoke; color:black" align="center"|0.3289 
|style="background:WhiteSmoke; color:black" align="center"|0.0057

|-
!style="background:WhiteSmoke; color:black;" align="center" |8
|style="background:WhiteSmoke; color:black" align="center"|106.7248
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3344</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0058 
|style="background:WhiteSmoke; color:black" align="center"|0.3655
|style="background:WhiteSmoke; color:black" align="center"|0.0064

|- 
|}

= case4  =


<b>IS=1 mm and ω=1&deg;</b> 


{| border="2" cellspacing="0" cellpadding="9" 

!colspan="1" border="none" style="background:silver; color:black;" align="center"|Peak number
|style="background:silver; color:black"|<b>2θ (&deg;)</b>
|style="background:silver; color:black"|<b>FWHM (&deg;)</b>
|style="background:silver; color:black"|<b>FWHM (rad)</b>
|style="background:silver; color:black"|<b>Integral breadth (&deg;)</b>
|style="background:silver; color:black"|<b>Integral breadth (rad)</b>

|-
!style="background:WhiteSmoke; color:black;" align="center" |1 
|style="background:WhiteSmoke; color:black" align="center"|28.4186
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2981</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0052
|style="background:WhiteSmoke; color:black" align="center"|0.3187
|style="background:WhiteSmoke; color:black" align="center"|0.0056

|-
!style="background:WhiteSmoke; color:black;" align="center" |2 
|style="background:WhiteSmoke; color:black" align="center"|47.2990
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2998</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0052
|style="background:WhiteSmoke; color:black" align="center"|0.3192
|style="background:WhiteSmoke; color:black" align="center"|0.0056

|-
!style="background:WhiteSmoke; color:black;" align="center" |3 
|style="background:WhiteSmoke; color:black" align="center"|56.0990
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2991</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0052
|style="background:WhiteSmoke; color:black" align="center"|0.3418
|style="background:WhiteSmoke; color:black" align="center"|0.0060

|-
!style="background:WhiteSmoke; color:black;" align="center" |4
|style="background:WhiteSmoke; color:black" align="center"|69.1210
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2992</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0052 
|style="background:WhiteSmoke; color:black" align="center"|0.3318
|style="background:WhiteSmoke; color:black" align="center"|0.0058
|-
!style="background:WhiteSmoke; color:black;" align="center" |5
|style="background:WhiteSmoke; color:black" align="center"|76.3549
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3070</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0054
|style="background:WhiteSmoke; color:black" align="center"|0.3386
|style="background:WhiteSmoke; color:black" align="center"|0.0059

|-
!style="background:WhiteSmoke; color:black;" align="center" |6
|style="background:WhiteSmoke; color:black" align="center"|88.0328
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.3053</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0053 
|style="background:WhiteSmoke; color:black" align="center"|0.3437
|style="background:WhiteSmoke; color:black" align="center"|0.0060

|-
!style="background:WhiteSmoke; color:black;" align="center" |7
|style="background:WhiteSmoke; color:black" align="center"|94.9712
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2838</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0050 
|style="background:WhiteSmoke; color:black" align="center"|0.3127 
|style="background:WhiteSmoke; color:black" align="center"|0.0055

|-
!style="background:WhiteSmoke; color:black;" align="center" |8
|style="background:WhiteSmoke; color:black" align="center"|106.7197
|style="background:WhiteSmoke; color:black" align="center"|<b><span style="color: red">0.2799</span></b>
|style="background:WhiteSmoke; color:black" align="center"|0.0049
|style="background:WhiteSmoke; color:black" align="center"|0.2980
|style="background:WhiteSmoke; color:black" align="center"|0.0052

|- 
|}