Specific Process Knowledge/Characterization/XRD/XRD SmartLab/Instrumental broading in GiXRD

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This page is written by Evgeniy Shkondin @DTU Nanolab if nothing else is stated.
All images and photos on this page belongs to DTU Nanolab.
The fabrication and characterization described below were conducted in 2023 by Evgeniy Shkondin, DTU Nanolab.

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\frac{K\cdot\lambda}{\beta_{sample}\cdot cos\theta}}

where: K is Scherrer constant (typically close to unity K=0.9-1)
λ is a wavelength of X-ray radiation (λ Cu-K_α1 = 1.540562 Å, and in case both Cu-K_α1 and Cu-K_α2 are present λ Cu-K_α1,2 = 1.541871 Å)
θ is Bragg angle
β_sample is a peak broadening. The technical point should be noted that intergal breadth and/or FWHM have to be inserted in radians.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta_{sample}=\beta_{measure}-\beta_{instrumental}} in the case where the peaks obey Lorentzian (or Cauchy) distribution profile.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta_{sample}^2=\beta_{measure}^2-\beta_{instrumental}^2} in the case of Gaussian distribution profile.

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

• 2θ scans of Si reference sample.
• 
  Measured Si powder reference sample with different incident slit (IS) and incident angle ω values.

• Broadening functions.
• 
  Calculated FWHM as a function of 2θ.
• 
  Calculated Integral breadth as a function of 2θ.

case1

IS=0.5 mm and ω=2°

Peak number	2θ (°)	FWHM (°)	FWHM (rad)	Integral breadth (°)	Integral breadth (rad)
1	28.4077	0.2923	0.0051	0.3232	0.0056
2	47.2895	0.3109	0.0054	0.3395	0.0059
3	56.1274	0.3004	0.0052	0.3368	0.0059
4	69.1426	0.3082	0.0054	0.3325	0.0058
5	76.3659	0.2585	0.0045	0.2936	0.0051
6	88.0122	0.3174	0.0055	0.3609	0.0063
7	94.9581	0.3126	0.0055	0.3434	0.0060
8	106.7351	0.3317	0.0058	0.3637	0.0063

case2

IS=0.5 mm and ω=1°

Peak number	2θ (°)	FWHM (°)	FWHM (rad)	Integral breadth (°)	Integral breadth (rad)
1	28.4185	0.2961	0.0052	0.3246	0.0057
2	47.2935	0.2961	0.0052	0.3214	0.0056
3	56.1085	0.2881	0.0050	0.3206	0.0056
4	69.1297	0.3009	0.0053	0.3203	0.0056
5	76.3485	0.3110	0.0054	0.3525	0.0062
6	88.0310	0.2972	0.0052	0.3234	0.0056
7	94.9806	0.2872	0.0050	0.3114	0.0054
8	106.7100	0.2838	0.0050	0.3024	0.0053

case3

IS=1 mm and ω=2°

Peak number	2θ (°)	FWHM (°)	FWHM (rad)	Integral breadth (°)	Integral breadth (rad)
1	28.3469	0.3041	0.0053	0.3237	0.0056
2	47.2945	0.3009	0.0053	0.3242	0.0057
3	56.1208	0.3064	0.0053	0.3322	0.0058
4	69.1388	0.3363	0.0059	0.3598	0.0063
5	76.3682	0.2689	0.0047	0.3100	0.0054
6	88.0168	0.3443	0.0060	0.3811	0.0067
7	94.9673	0.3018	0.0053	0.3289	0.0057
8	106.7248	0.3344	0.0058	0.3655	0.0064

case4

IS=1 mm and ω=1°

Peak number	2θ (°)	FWHM (°)	FWHM (rad)	Integral breadth (°)	Integral breadth (rad)
1	28.4186	0.2981	0.0052	0.3187	0.0056
2	47.2990	0.2998	0.0052	0.3192	0.0056
3	56.0990	0.2991	0.0052	0.3418	0.0060
4	69.1210	0.2992	0.0052	0.3318	0.0058
5	76.3549	0.3070	0.0054	0.3386	0.0059
6	88.0328	0.3053	0.0053	0.3437	0.0060
7	94.9712	0.2838	0.0050	0.3127	0.0055
8	106.7197	0.2799	0.0049	0.2980	0.0052