Anisotropic broadening correction in X-ray powder diffraction by an ellipsoidal formula is applied on samples with nanosized crystals. Two cases of minerals with largely anisotropic crystallite shapes are presented. The properly applied formalism not only improves the fitting of the theoretical and observed diffraction diagrams but also gives direct information about realistic crystallite shapes and sizes. The approach is demonstrated using the Rietveld refinement program TOPAS and it is easily adaptable to other similar software.
Supporting information
| Crystallographic Information File (CIF) https://doi.org/10.1107/S0021889811055075/ks5301sup1.cif Contains datablocks global, Hematite_phase1, Microcline_phase2, Quartz_phase3, calcite_phase4 |
| Rietveld powder data file (CIF format) https://doi.org/10.1107/S0021889811055075/ks5301UT97_Fig1asup2.rtv Contains datablock UT97_Fig1a |
| Crystallographic Information File (CIF) https://doi.org/10.1107/S0021889811055075/ks5301sup3.cif Contains datablocks global, Hematite_phase1, Microcline_phase2, Quartz_phase3, calcite_phase4 |
| Rietveld powder data file (CIF format) https://doi.org/10.1107/S0021889811055075/ks5301UT97_Fig1bsup4.rtv Contains datablock UT97_Fig1b |
| Crystallographic Information File (CIF) https://doi.org/10.1107/S0021889811055075/ks5301sup5.cif Contains datablocks global, Goethite_phase1, halite_phase2 |
| Rietveld powder data file (CIF format) https://doi.org/10.1107/S0021889811055075/ks5301goet_Fig4asup6.rtv Contains datablock goet_Fig4a |
| Crystallographic Information File (CIF) https://doi.org/10.1107/S0021889811055075/ks5301sup7.cif Contains datablocks global, Goethite_phase1, halite_phase2 |
| Rietveld powder data file (CIF format) https://doi.org/10.1107/S0021889811055075/ks5301goet_Fig4bsup8.rtv Contains datablock goet_Fig4b |
For both compounds, data collection: DiffracPlus Basic Measurement Center, XRD Wizard 2.9, Bruker AXS; cell refinement: Topas 4.1 Bruker AXS; program(s) used to refine structure: Topas 4.1 Bruker AXS.
(Hematite_phase1) Hematite
top
Crystal data top
Fe2O3 | V = 302.09 (2) Å3 |
Mr = 159.7 | Z = 6 |
Trigonal, r_3_c:h | Cu Kα1 radiation, λ = 1.540596 Å |
a = 5.03612 (14) Å | T = 25 K |
c = 13.7535 (6) Å | Specimen preparation: Prepared at 25 K |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 90°, 2θstep = 0.014° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 4.52 | 27 parameters |
Rwp = 6.73 | Weighting scheme based on measured s.u.'s |
Rexp = 3.25 | (Δ/σ)max = 0.001 |
RBragg = 4.95 | Background function: Chebychev 4th order |
5544 data points | |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Fe | 0.00000 | 0.00000 | 0.35530 | 1 | |
O | 0.30590 | 0.00000 | 0.25000 | 1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(Microcline_phase2) microcline
top
Crystal data top
AlKO8Si3 | β = 115.88 (4)° |
Mr = 278.33 | γ = 86.86 (16)° |
Triclinic, C1 | V = 714.9 (18) Å3 |
a = 8.591 (13) Å | Z = 4 |
b = 12.86 (2) Å | Cu Kα1 radiation, λ = 1.540596 Å |
c = 7.206 (3) Å | T = 25 K |
α = 91.17 (10)° | |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 90°, 2θstep = 0.014° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | 27 parameters |
Rp = 4.52 | Weighting scheme based on measured s.u.'s |
Rwp = 6.72 | (Δ/σ)max = 0.001 |
Rexp = 3.25 | Background function: Chebychev 4th order |
RBragg = 1.73 | Preferred orientation correction: March-Dollase function
Direction : 0 0 1
0.42(2) |
Profile function: fundamental parameters | |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | Occ. (<1) |
K | 0.28260 | 0.99270 | 0.13650 | 1.6 | |
Al1 | 0.00980 | 0.18800 | 0.21680 | 0.6 | 0.962 |
Si1 | 0.00980 | 0.18800 | 0.21680 | 0.6 | 0.038 |
Al2 | 0.00990 | 0.81950 | 0.23260 | 0.6 | 0.023 |
Si2 | 0.00990 | 0.81950 | 0.23260 | 0.6 | 0.977 |
Al3 | 0.71160 | 0.12040 | 0.34020 | 0.6 | 0.008 |
Si3 | 0.71160 | 0.12040 | 0.34020 | 0.6 | 0.992 |
Al4 | 0.70610 | 0.88560 | 0.35090 | 0.6 | 0.008 |
Si4 | 0.70610 | 0.88560 | 0.35090 | 0.6 | 0.992 |
O1 | 0.00050 | 0.14440 | 0.98350 | 1 | |
O2 | 0.63690 | 0.00570 | 0.28520 | 0.9 | |
O3 | 0.82110 | 0.14810 | 0.22010 | 1.2 | |
O4 | 0.83150 | 0.85760 | 0.24010 | 1.3 | |
O5 | 0.03520 | 0.32010 | 0.25160 | 0.9 | |
O6 | 0.03900 | 0.69570 | 0.27080 | 0.9 | |
O7 | 0.19050 | 0.12280 | 0.40510 | 0.9 | |
O | 0.17540 | 0.87430 | 0.41210 | 1.1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(Quartz_phase3) microcline
top
Crystal data top
O2Si | V = 113.01 Å3 |
Mr = 60.09 | Z = 3 |
Trigonal, P3221 | Cu Kα1 radiation, λ = 1.540596 Å |
a = 4.9137 Å | T = 25 K |
c = 5.4047 Å | |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 90°, 2θstep = 0.014° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 4.52 | 27 parameters |
Rwp = 6.72 | Weighting scheme based on measured s.u.'s |
Rexp = 3.25 | (Δ/σ)max = 0.001 |
RBragg = 8.47 | Background function: Chebychev 4th order |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Si1 | 0.486941 | 0 | 0.666667 | 0.4 | |
O1 | 0.408563 | 0.264223 | 0.790267 | 0.9 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(calcite_phase4) microcline
top
Crystal data top
CCaO3 | V = 366.86 (14) Å3 |
Mr = 100.09 | Z = 6 |
Trigonal, r_3_c:h | Cu Kα1 radiation, λ = 1.540596 Å |
a = 4.9848 (8) Å | T = 25 K |
c = 17.048 (3) Å | |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 90°, 2θstep = 0.014° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 4.52 | 27 parameters |
Rwp = 6.72 | Weighting scheme based on measured s.u.'s |
Rexp = 3.25 | (Δ/σ)max = 0.001 |
RBragg = 4.12 | Background function: Chebychev 4th order |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Ca1 | 0.00000 | 0.00000 | 0.00000 | 1 | |
C1 | 0.00000 | 0.00000 | 0.25000 | 1 | |
O1 | 0.25670 | 0.00000 | 0.25000 | 1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(Hematite_phase1) Hematite
top
Crystal data top
Fe2O3 | V = 302.08 (2) Å3 |
Mr = 159.7 | Z = 6 |
Trigonal, r_3_c:h | Cu Kα1 radiation, λ = 1.540596 Å |
a = 5.03599 (8) Å | T = 25 K |
c = 13.7536 (6) Å | Specimen preparation: Prepared at 25 K |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 90°, 2θstep = 0.014° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 3.8 | 28 parameters |
Rwp = 5.31 | Weighting scheme based on measured s.u.'s |
Rexp = 3.25 | (Δ/σ)max = 0.001 |
RBragg = 2.56 | Background function: Chebychev 4th order |
5544 data points | |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Fe | 0.00000 | 0.00000 | 0.35530 | 1 | |
O | 0.30590 | 0.00000 | 0.25000 | 1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(Microcline_phase2) microcline
top
Crystal data top
AlKO8Si3 | β = 115.90 (3)° |
Mr = 278.33 | γ = 87.63 (11)° |
Triclinic, C1 | V = 719.1 (12) Å3 |
a = 8.575 (9) Å | Z = 4 |
b = 12.947 (16) Å | Cu Kα1 radiation, λ = 1.540596 Å |
c = 7.2069 (18) Å | T = 25 K |
α = 90.45 (7)° | |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 90°, 2θstep = 0.014° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | 28 parameters |
Rp = 3.8 | Weighting scheme based on measured s.u.'s |
Rwp = 5.31 | (Δ/σ)max = 0.001 |
Rexp = 3.25 | Background function: Chebychev 4th order |
RBragg = 1.62 | Preferred orientation correction: March-Dollase function
direction : 0 0 1
0.44(1) |
Profile function: fundamental parameters | |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | Occ. (<1) |
K | 0.28260 | 0.99270 | 0.13650 | 1.6 | |
Al1 | 0.00980 | 0.18800 | 0.21680 | 0.6 | 0.962 |
Si1 | 0.00980 | 0.18800 | 0.21680 | 0.6 | 0.038 |
Al2 | 0.00990 | 0.81950 | 0.23260 | 0.6 | 0.023 |
Si2 | 0.00990 | 0.81950 | 0.23260 | 0.6 | 0.977 |
Al3 | 0.71160 | 0.12040 | 0.34020 | 0.6 | 0.008 |
Si3 | 0.71160 | 0.12040 | 0.34020 | 0.6 | 0.992 |
Al4 | 0.70610 | 0.88560 | 0.35090 | 0.6 | 0.008 |
Si4 | 0.70610 | 0.88560 | 0.35090 | 0.6 | 0.992 |
O1 | 0.00050 | 0.14440 | 0.98350 | 1 | |
O2 | 0.63690 | 0.00570 | 0.28520 | 0.9 | |
O3 | 0.82110 | 0.14810 | 0.22010 | 1.2 | |
O4 | 0.83150 | 0.85760 | 0.24010 | 1.3 | |
O5 | 0.03520 | 0.32010 | 0.25160 | 0.9 | |
O6 | 0.03900 | 0.69570 | 0.27080 | 0.9 | |
O7 | 0.19050 | 0.12280 | 0.40510 | 0.9 | |
O | 0.17540 | 0.87430 | 0.41210 | 1.1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(Quartz_phase3) microcline
top
Crystal data top
O2Si | V = 113.01 Å3 |
Mr = 60.09 | Z = 3 |
Trigonal, P3221 | Cu Kα1 radiation, λ = 1.540596 Å |
a = 4.9137 Å | T = 25 K |
c = 5.4047 Å | |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 90°, 2θstep = 0.014° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 3.8 | 28 parameters |
Rwp = 5.31 | Weighting scheme based on measured s.u.'s |
Rexp = 3.25 | (Δ/σ)max = 0.001 |
RBragg = 8.31 | Background function: Chebychev 4th order |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Si1 | 0.486941 | 0 | 0.666667 | 0.4 | |
O1 | 0.408563 | 0.264223 | 0.790267 | 0.9 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(calcite_phase4) microcline
top
Crystal data top
CCaO3 | V = 366.80 (12) Å3 |
Mr = 100.09 | Z = 6 |
Trigonal, r_3_c:h | Cu Kα1 radiation, λ = 1.540596 Å |
a = 4.9843 (7) Å | T = 25 K |
c = 17.049 (3) Å | |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 90°, 2θstep = 0.014° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 3.8 | 28 parameters |
Rwp = 5.31 | Weighting scheme based on measured s.u.'s |
Rexp = 3.25 | (Δ/σ)max = 0.001 |
RBragg = 3.06 | Background function: Chebychev 4th order |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Ca1 | 0.00000 | 0.00000 | 0.00000 | 1 | |
C1 | 0.00000 | 0.00000 | 0.25000 | 1 | |
O1 | 0.25670 | 0.00000 | 0.25000 | 1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(Goethite_phase1) Hematite
top
Crystal data top
FeHO2 | V = 138.91 (4) Å3 |
Mr = 88.85 | Z = 4 |
Orthorhombic, Pbnm | Cu Kα1 radiation, λ = 1.540596 Å |
a = 4.6141 (9) Å | T = 25 K |
b = 9.9566 (13) Å | Specimen preparation: Prepared at 25 K |
c = 3.0237 (4) Å | |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 70°, 2θstep = 0.02° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 6.58 | 19 parameters |
Rwp = 8.36 | H-atom parameters not defined? |
Rexp = 7.18 | Weighting scheme based on measured s.u.'s |
RBragg = 1.69 | (Δ/σ)max = 0.001 |
3048 data points | Background function: Chebychev 4th order |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Fe1 | 0.04893 | 0.853657 | 0.25 | 1 | |
O1 | 0.7057 | 0.19914 | 0.25 | 1 | |
O2 | 0.1987 | 0.05298 | 0.25 | 1 | |
H1 | 0.379 | 0.082 | 0.25 | 1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(halite_phase2) microcline
top
Crystal data top
NaCl | V = 179.48 (17) Å3 |
Mr = 58.44 | Z = 4 |
Cubic, Fm3m | Cu Kα1 radiation, λ = 1.540596 Å |
a = 5.6408 (18) Å | T = 25 K |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 70°, 2θstep = 0.02° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 6.58 | 19 parameters |
Rwp = 8.36 | Weighting scheme based on measured s.u.'s |
Rexp = 7.18 | (Δ/σ)max = 0.001 |
RBragg = 1.82 | Background function: Chebychev 4th order |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Na1 | 0 | 0 | 0 | 1 | |
Cl1 | 0.5 | 0.5 | 0.5 | 1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(Goethite_phase1) Hematite
top
Crystal data top
FeHO2 | V = 138.89 (3) Å3 |
Mr = 88.85 | Z = 4 |
Orthorhombic, Pbnm | Cu Kα1 radiation, λ = 1.540596 Å |
a = 4.6141 (8) Å | T = 25 K |
b = 9.9556 (10) Å | Specimen preparation: Prepared at 25 K |
c = 3.0237 (3) Å | |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 70°, 2θstep = 0.02° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 6.10 | 21 parameters |
Rwp = 7.72 | H-atom parameters not defined? |
Rexp = 7.18 | Weighting scheme based on measured s.u.'s |
RBragg = 2.98 | (Δ/σ)max = 0.001 |
3048 data points | Background function: Chebychev 4th order |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Fe1 | 0.04893 | 0.853657 | 0.25 | 1 | |
O1 | 0.7057 | 0.19914 | 0.25 | 1 | |
O2 | 0.1987 | 0.05298 | 0.25 | 1 | |
H1 | 0.379 | 0.082 | 0.25 | 1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |
(halite_phase2) microcline
top
Crystal data top
NaCl | V = 179.38 (15) Å3 |
Mr = 58.44 | Z = 4 |
Cubic, Fm3m | Cu Kα1 radiation, λ = 1.540596 Å |
a = 5.6398 (16) Å | T = 25 K |
Data collection top
Ge monochromator | 2θmin = 10°, 2θmax = 70°, 2θstep = 0.02° |
Scan method: step | |
Refinement top
Least-squares matrix: full with fixed elements per cycle | Profile function: fundamental parameters |
Rp = 6.10 | 21 parameters |
Rwp = 7.72 | Weighting scheme based on measured s.u.'s |
Rexp = 7.18 | (Δ/σ)max = 0.001 |
RBragg = 1.60 | Background function: Chebychev 4th order |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Biso*/Beq | |
Na1 | 0 | 0 | 0 | 1 | |
Cl1 | 0.5 | 0.5 | 0.5 | 1 | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
? | ? | ? | ? | ? | ? | ? |