Specific Process Knowledge/Characterization/XRD/XRD SmartLab/Instrumental broading in T2T

This page describes the values of instrumental broadening in θ/2θ measurements and can be used as a reference for peak broadening in grain size calculation. The most common slit configuration cases are chosen, both with and without the Ge220 monochromator.

The instrumental broadening has been measured by using standard Si and LaB₆ powder.

Scherrer formula:

$D={\frac {K\cdot \lambda }{\beta _{sample}\cdot cos\theta }}$

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.

$\beta _{sample}=\beta _{measure}-\beta _{instrumental}$ in the case where the peaks obey Lorentzian (or Cauchy) distribution profile.

$\beta _{sample}^{2}=\beta _{measure}^{2}-\beta _{instrumental}^{2}$ in the case of Gaussian distribution profile.

Experimental results implied that crystallite size causes Cauchy-type broadening, while strain broadening is associated with a Gaussian profile. Real measurements contain contributions from both Lorentzian (Cauchy) and Gaussian distributions.

Both LaB₆ and Si powder provided the best fit with a Gaussian function. Therefore, all the results below are presented using a Gaussian fit.

It also seems necessary to use a high-quality single crystal to include both Lorentzian and Gaussian contributions and fit with a Pseudo-Voight approach.

• Reference samples used in instrumental broadening analysis.
• 
  LaB₆ and Si powders mounted on a Silicon-(100) wafer chip using double sticky Kapton tape.

Full scans

Here the full scans of LaB₆ and Si powder are presented. Each sample was measured with and without the Ge220 monochromator on the incident side, with the Incident Slit (IS) set to either 0.5 mm or 1 mm. On the receiving side, RS1 = 0.5 mm and RS2 = 0.525 mm were used for IS = 0.5 mm, and RS1 = 1 mm and RS2 = 1.125 mm were used for IS = 1 mm. The step width was 0.01° and the speed was 2°/min for all scans. A Soller slit of 5° was selected as the Receiving Parallel Slit (RPS) for all scans. In the case of the IPS adapter, the Soller slit of 5° was used.

• 2θ scans of Si reference sample.
• 
  Measured LaB₆ powder reference sample in a range of 20-120° of 2θ. Area marked as gray contains a main 004 Si peak from a silicon chip which is a holder for powder sample.
• 
  Measured LaB₆ powder reference sample in a range of 20-120° of 2θ. Area marked as gray contains a main 004 Si peak from a silicon chip which is a holder for powder sample.
• 
  Measured LaB₆ powder reference sample in a range of 20-120° of 2θ. Area marked as gray contains a main 004 Si peak from a silicon chip which is a holder for powder sample.
• 
  Measured LaB₆ powder reference sample in a range of 20-120° of 2θ. Area marked as gray contains a main 004 Si peak from a silicon chip which is a holder for powder sample.
• 
  Measured Si powder reference sample in a range of 20-120° of 2θ. Area marked as gray contains a main 004 Si peak from a silicon chip which is a holder for powder sample.
• 
  Measured Si powder reference sample in a range of 20-120° of 2θ. Area marked as gray contains a main 004 Si peak from a silicon chip which is a holder for powder sample.
• 
  Measured Si powder reference sample in a range of 20-120° of 2θ. Area marked as gray contains a main 004 Si peak from a silicon chip which is a holder for powder sample.
• 
  Measured Si powder reference sample in a range of 20-120° of 2θ. Area marked as gray contains a main 004 Si peak from a silicon chip which is a holder for powder sample.

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

• 2θ scans of Si reference sample.
• 
  Gaussian fitted FWHM of measured LaB₆ and Si powder reference samples using Ge220 monochromator with IS = 0.5 mm.
• 
  Gaussian fitted FWHM of measured LaB₆ and Si powder reference samples using Ge220 monochromator with IS = 1.0 mm.
• 
  Gaussian fitted FWHM of measured LaB₆ and Si powder reference samples using IPS adaptor with IS = 0.5 mm.
• 
  Gaussian fitted FWHM of measured LaB₆ and Si powder reference samples using IPS adaptor with IS = 1.0 mm.
• 
  Gaussian fitted Integral breadth of measured LaB₆ and Si powder reference samples using Ge220 monochromator with IS = 0.5 mm.
• 
  Gaussian fitted Integral breadth of measured LaB₆ and Si powder reference samples using Ge220 monochromator with IS = 1.0 mm.
• 
  Gaussian fitted Integral breadth of measured LaB₆ and Si powder reference samples using IPS adaptor with IS = 0.5 mm.
• 
  Gaussian fitted Integral breadth of measured LaB₆ and Si powder reference samples using IPS adaptor with IS = 1.0 mm.

case1 - LaB₆ with Ge220 monochromator

LaB6 Ge220 monochromator on incident side

IS = 0.5 mm Orientation 2θ (°) Integral breadth (°) Integral breadth (rad) FWHM (°) FWHM (rad) 100 21.337 0.10739 0.00187 0.10089 0.00176 110 30.3523 0.13819 0.0024 0.12982 0.0023 111 37.4169 0.12657 0.0022 0.1189 0.0021 200 43.4813 0.12415 0.0022 0.11663 0.0020 210 48.9438 0.19641 0.0034 0.12504 0.0022 211 53.9726 0.14188 0.0025 0.13328 0.0023 220 63.2028 0.12865 0.0022 0.12086 0.0021 ? 67.5401 0.12066 0.0021 0.11335 0.0020 ? 71.7371 0.10965 0.0019 0.10301 0.0018 310 75.8311 0.10923 0.0019 0.10262 0.0018 320 87.7842 0.13106 0.0023 0.12312 0.0021 410 99.6432 0.13603 0.0024 0.12779 0.0022 411 103.6716 0.14652 0.0026 0.13765 0.0024 421 116.2506 0.14605 0.0025 0.1372 0.0024 IS = 1.0 mm Orientation 2θ (°) Integral breadth (°) Integral breadth (rad) FWHM (°) FWHM (rad) 100 21.3112 0.24000 0.0042 0.22547 0.0039 110 30.3476 0.23945 0.0042 0.22494 0.0039 111 37.4059 0.22363 0.0039 0.21009 0.0037 200 43.471 0.23381 0.0041 0.21965 0.0038 210 48.9311 0.23146 0.0040 0.21744 0.0038 211 53.9635 0.21833 0.0038 0.2051 0.0036 220 63.2056 0.21683 0.0038 0.20369 0.0036 ? 67.6308 0.22749 0.0040 0.21371 0.0037 ? 71.7340 0.24789 0.0043 0.23288 0.0041 310 75.8186 0.22282 0.0039 0.20933 0.0037 320 87.7794 0.22295 0.0039 0.20945 0.0037 410 99.6445 0.21801 0.0038 0.20481 0.0036 411 103.6522 0.20431 0.0036 0.21748 0.0038 421 116.2526 0.2521 0.0044 0.23683 0.0041

case2 - LaB₆ with IPS adaptor

LaB6 IPS on incident side

IS = 0.5 mm Orientation 2θ (°) Integral breadth (°) Integral breadth (rad) FWHM (°) FWHM (rad) 100 21.3385 0.16688 0.0029 0.15677 0.0027 110 30.3784 0.7473 0.0030 0.16415 0.0029 111 37.4424 0.18775 0.0033 0.17638 0.0031 200 43.5157 0.21383 0.0037 0.20088 0.0035 210 48.9575 0.16613 0.0029 0.15607 0.0027 211 53.9741 0.13757 0.0024 0.12924 0.0023 220 63.2098 0.16157 0.0028 0.15179 0.0026 ? 67.5436 0.13682 0.0024 0.12853 0.0022 ? 71.74000 0.13819 0.0024 0.12982 0.0023 310 75.84990 0.15249 0.0027 0.14326 0.0025 320 87.79290 0.13792 0.0024 0.12956 0.0023 410 99.6461 0.14402 0.0025 0.13529 0.0024 411 103.6695 0.14404 0.0025 0.13532 0.0024 421 116.2607 0.16243 0.0028 0.15259 0.0027 IS = 1.0 mm Orientation 2θ (°) Integral breadth (°) Integral breadth (rad) FWHM (°) FWHM (rad) 100 21.3354 0.25476 0.0044 0.23933 0.0042 110 30.3763 0.24934 0.0044 0.23424 0.0041 111 37.4454 0.25983 0.0045 0.24409 0.0043 200 43.5167 0.26947 0.0047 0.25315 0.0044 210 48.9694 0.26146 0.0046 0.24562 0.0043 211 53.9866 0.24426 0.0043 0.22947 0.0040 220 63.2297 0.27671 0.0048 0.25995 0.0045 ? 67.5594 0.27847 0.0049 0.26161 0.0046 ? 71.7568 0.26005 0.0045 0.2443 0.0043 310 75.8530 0.27326 0.0048 0.25671 0.0045 320 87.7921 0.25544 0.0045 0.23997 0.0042 410 99.6463 0.23764 0.0041 0.22324 0.0039 411 103.664 0.23902 0.0042 0.22454 0.0039 421 116.2617 0.25501 0.0045 0.23957 0.0042

case3 - Si powder with Ge220 monochromator

Si powder Ge220 monochromator on incident side

IS = 0.5 mm Orientation 2θ (°) Integral breadth (°) Integral breadth (rad) FWHM (°) FWHM (rad) 111 28.4124 0.15459 0.0027 0.14522 0.0025 220 47.2829 0.13319 0.0023 0.12512 0.0022 311 56.1009 0.1342 0.0023 0.12608 0.0022 331 76.3771 0.17568 0.0031 0.16504 0.0029 511 88.0381 0.14264 0.0025 0.1340 0.0023 440 - - - - - 531 - - - - - 422 - - - - - IS = 1.0 mm Orientation 2θ (°) Integral breadth (°) Integral breadth (rad) FWHM (°) FWHM (rad) 111 28.3944 0.2392 0.0042 0.22471 0.0039 220 47.2802 0.20108 0.0035 0.1889 0.0033 311 56.1069 0.22110 0.0039 0.20771 0.0036 331 76.3585 0.20981 0.0037 0.1971 0.0034 422 88.0277 0.2230 0.0039 0.20949 0.0037 511 94.9552 0.23689 0.0041 0.22255 0.0039 440 106.6949 0.19242 0.0034 0.18076 0.0032 531 114.0880 0.22245 0.0039 0.20898 0.0036

case4 - Si powder with IPS adaptor

Si powder IPS on incident side

IS = 0.5 mm Orientation 2θ (°) Integral breadth (°) Integral breadth (rad) FWHM (°) FWHM (rad) 111 28.4386 0.23378 0.0041 0.14883 0.0026 220 47.3197 0.22441 0.0039 0.21082 0.0037 311 56.1199 0.14714 0.0026 0.13823 0.0024 331 76.3703 0.13962 0.0024 0.13116 0.0023 511 88.0354 0.15322 0.0027 0.14394 0.0025 440 94.9602 0.15685 0.0027 0.14735 0.0026 531 106.7178 0.16969 0.0030 0.15941 0.0028 422 114.1167 0.18994 0.0033 0.17844 0.0031 IS = 1.0 mm Orientation 2θ (°) Integral breadth (°) Integral breadth (rad) FWHM (°) FWHM (rad) 111 28.4311 0.24618 0.0043 0.23127 0.0040 220 47.3122 0.27804 0.0049 0.2612 0.0046 311 56.1430 0.27204 0.0047 0.25557 0.0045 331 76.3978 0.27933 0.0049 0.26241 0.0046 422 88.0267 0.24578 0.0043 0.23089 0.0040 511 94.957 0.26134 0.0046 0.24551 0.0043 440 106.7393 0.29928 0.0052 0.28116 0.0049 531 114.1079 0.2957 0.0052 0.27779 0.0048