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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

cif

Crystallographic Information File (CIF) https://doi.org/10.1107/S0021889811055075/ks5301sup1.cif
Contains datablocks global, Hematite_phase1, Microcline_phase2, Quartz_phase3, calcite_phase4

rtv

Rietveld powder data file (CIF format) https://doi.org/10.1107/S0021889811055075/ks5301UT97_Fig1asup2.rtv
Contains datablock UT97_Fig1a

cif

Crystallographic Information File (CIF) https://doi.org/10.1107/S0021889811055075/ks5301sup3.cif
Contains datablocks global, Hematite_phase1, Microcline_phase2, Quartz_phase3, calcite_phase4

rtv

Rietveld powder data file (CIF format) https://doi.org/10.1107/S0021889811055075/ks5301UT97_Fig1bsup4.rtv
Contains datablock UT97_Fig1b

cif

Crystallographic Information File (CIF) https://doi.org/10.1107/S0021889811055075/ks5301sup5.cif
Contains datablocks global, Goethite_phase1, halite_phase2

rtv

Rietveld powder data file (CIF format) https://doi.org/10.1107/S0021889811055075/ks5301goet_Fig4asup6.rtv
Contains datablock goet_Fig4a

cif

Crystallographic Information File (CIF) https://doi.org/10.1107/S0021889811055075/ks5301sup7.cif
Contains datablocks global, Goethite_phase1, halite_phase2

rtv

Rietveld powder data file (CIF format) https://doi.org/10.1107/S0021889811055075/ks5301goet_Fig4bsup8.rtv
Contains datablock goet_Fig4b

Computing details top

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
Fe2O3V = 302.09 (2) Å3
Mr = 159.7Z = 6
Trigonal, r_3_c:hCu Kα1 radiation, λ = 1.540596 Å
a = 5.03612 (14) ÅT = 25 K
c = 13.7535 (6) ÅSpecimen preparation: Prepared at 25 K
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 90°, 2θstep = 0.014°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 4.5227 parameters
Rwp = 6.73Weighting scheme based on measured s.u.'s
Rexp = 3.25(Δ/σ)max = 0.001
RBragg = 4.95Background function: Chebychev 4th order
5544 data points
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Fe0.000000.000000.355301
O0.305900.000000.250001
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
(Microcline_phase2) microcline top
Crystal data top
AlKO8Si3β = 115.88 (4)°
Mr = 278.33γ = 86.86 (16)°
Triclinic, C1V = 714.9 (18) Å3
a = 8.591 (13) ÅZ = 4
b = 12.86 (2) ÅCu Kα1 radiation, λ = 1.540596 Å
c = 7.206 (3) ÅT = 25 K
α = 91.17 (10)°
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 90°, 2θstep = 0.014°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycle27 parameters
Rp = 4.52Weighting scheme based on measured s.u.'s
Rwp = 6.72(Δ/σ)max = 0.001
Rexp = 3.25Background function: Chebychev 4th order
RBragg = 1.73Preferred 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
xyzBiso*/BeqOcc. (<1)
K0.282600.992700.136501.6
Al10.009800.188000.216800.60.962
Si10.009800.188000.216800.60.038
Al20.009900.819500.232600.60.023
Si20.009900.819500.232600.60.977
Al30.711600.120400.340200.60.008
Si30.711600.120400.340200.60.992
Al40.706100.885600.350900.60.008
Si40.706100.885600.350900.60.992
O10.000500.144400.983501
O20.636900.005700.285200.9
O30.821100.148100.220101.2
O40.831500.857600.240101.3
O50.035200.320100.251600.9
O60.039000.695700.270800.9
O70.190500.122800.405100.9
O0.175400.874300.412101.1
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
(Quartz_phase3) microcline top
Crystal data top
O2SiV = 113.01 Å3
Mr = 60.09Z = 3
Trigonal, P3221Cu Kα1 radiation, λ = 1.540596 Å
a = 4.9137 ÅT = 25 K
c = 5.4047 Å
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 90°, 2θstep = 0.014°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 4.5227 parameters
Rwp = 6.72Weighting scheme based on measured s.u.'s
Rexp = 3.25(Δ/σ)max = 0.001
RBragg = 8.47Background function: Chebychev 4th order
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Si10.48694100.6666670.4
O10.4085630.2642230.7902670.9
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
(calcite_phase4) microcline top
Crystal data top
CCaO3V = 366.86 (14) Å3
Mr = 100.09Z = 6
Trigonal, r_3_c:hCu Kα1 radiation, λ = 1.540596 Å
a = 4.9848 (8) ÅT = 25 K
c = 17.048 (3) Å
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 90°, 2θstep = 0.014°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 4.5227 parameters
Rwp = 6.72Weighting scheme based on measured s.u.'s
Rexp = 3.25(Δ/σ)max = 0.001
RBragg = 4.12Background function: Chebychev 4th order
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Ca10.000000.000000.000001
C10.000000.000000.250001
O10.256700.000000.250001
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
 
(Hematite_phase1) Hematite top
Crystal data top
Fe2O3V = 302.08 (2) Å3
Mr = 159.7Z = 6
Trigonal, r_3_c:hCu Kα1 radiation, λ = 1.540596 Å
a = 5.03599 (8) ÅT = 25 K
c = 13.7536 (6) ÅSpecimen preparation: Prepared at 25 K
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 90°, 2θstep = 0.014°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 3.828 parameters
Rwp = 5.31Weighting scheme based on measured s.u.'s
Rexp = 3.25(Δ/σ)max = 0.001
RBragg = 2.56Background function: Chebychev 4th order
5544 data points
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Fe0.000000.000000.355301
O0.305900.000000.250001
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
(Microcline_phase2) microcline top
Crystal data top
AlKO8Si3β = 115.90 (3)°
Mr = 278.33γ = 87.63 (11)°
Triclinic, C1V = 719.1 (12) Å3
a = 8.575 (9) ÅZ = 4
b = 12.947 (16) ÅCu Kα1 radiation, λ = 1.540596 Å
c = 7.2069 (18) ÅT = 25 K
α = 90.45 (7)°
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 90°, 2θstep = 0.014°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycle28 parameters
Rp = 3.8Weighting scheme based on measured s.u.'s
Rwp = 5.31(Δ/σ)max = 0.001
Rexp = 3.25Background function: Chebychev 4th order
RBragg = 1.62Preferred 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
xyzBiso*/BeqOcc. (<1)
K0.282600.992700.136501.6
Al10.009800.188000.216800.60.962
Si10.009800.188000.216800.60.038
Al20.009900.819500.232600.60.023
Si20.009900.819500.232600.60.977
Al30.711600.120400.340200.60.008
Si30.711600.120400.340200.60.992
Al40.706100.885600.350900.60.008
Si40.706100.885600.350900.60.992
O10.000500.144400.983501
O20.636900.005700.285200.9
O30.821100.148100.220101.2
O40.831500.857600.240101.3
O50.035200.320100.251600.9
O60.039000.695700.270800.9
O70.190500.122800.405100.9
O0.175400.874300.412101.1
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
(Quartz_phase3) microcline top
Crystal data top
O2SiV = 113.01 Å3
Mr = 60.09Z = 3
Trigonal, P3221Cu Kα1 radiation, λ = 1.540596 Å
a = 4.9137 ÅT = 25 K
c = 5.4047 Å
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 90°, 2θstep = 0.014°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 3.828 parameters
Rwp = 5.31Weighting scheme based on measured s.u.'s
Rexp = 3.25(Δ/σ)max = 0.001
RBragg = 8.31Background function: Chebychev 4th order
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Si10.48694100.6666670.4
O10.4085630.2642230.7902670.9
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
(calcite_phase4) microcline top
Crystal data top
CCaO3V = 366.80 (12) Å3
Mr = 100.09Z = 6
Trigonal, r_3_c:hCu Kα1 radiation, λ = 1.540596 Å
a = 4.9843 (7) ÅT = 25 K
c = 17.049 (3) Å
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 90°, 2θstep = 0.014°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 3.828 parameters
Rwp = 5.31Weighting scheme based on measured s.u.'s
Rexp = 3.25(Δ/σ)max = 0.001
RBragg = 3.06Background function: Chebychev 4th order
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Ca10.000000.000000.000001
C10.000000.000000.250001
O10.256700.000000.250001
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
 
(Goethite_phase1) Hematite top
Crystal data top
FeHO2V = 138.91 (4) Å3
Mr = 88.85Z = 4
Orthorhombic, PbnmCu Kα1 radiation, λ = 1.540596 Å
a = 4.6141 (9) ÅT = 25 K
b = 9.9566 (13) ÅSpecimen preparation: Prepared at 25 K
c = 3.0237 (4) Å
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 70°, 2θstep = 0.02°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 6.5819 parameters
Rwp = 8.36H-atom parameters not defined?
Rexp = 7.18Weighting scheme based on measured s.u.'s
RBragg = 1.69(Δ/σ)max = 0.001
3048 data pointsBackground function: Chebychev 4th order
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Fe10.048930.8536570.251
O10.70570.199140.251
O20.19870.052980.251
H10.3790.0820.251
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
(halite_phase2) microcline top
Crystal data top
NaClV = 179.48 (17) Å3
Mr = 58.44Z = 4
Cubic, Fm3mCu Kα1 radiation, λ = 1.540596 Å
a = 5.6408 (18) ÅT = 25 K
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 70°, 2θstep = 0.02°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 6.5819 parameters
Rwp = 8.36Weighting scheme based on measured s.u.'s
Rexp = 7.18(Δ/σ)max = 0.001
RBragg = 1.82Background function: Chebychev 4th order
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Na10001
Cl10.50.50.51
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
 
(Goethite_phase1) Hematite top
Crystal data top
FeHO2V = 138.89 (3) Å3
Mr = 88.85Z = 4
Orthorhombic, PbnmCu Kα1 radiation, λ = 1.540596 Å
a = 4.6141 (8) ÅT = 25 K
b = 9.9556 (10) ÅSpecimen preparation: Prepared at 25 K
c = 3.0237 (3) Å
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 70°, 2θstep = 0.02°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 6.1021 parameters
Rwp = 7.72H-atom parameters not defined?
Rexp = 7.18Weighting scheme based on measured s.u.'s
RBragg = 2.98(Δ/σ)max = 0.001
3048 data pointsBackground function: Chebychev 4th order
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Fe10.048930.8536570.251
O10.70570.199140.251
O20.19870.052980.251
H10.3790.0820.251
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
(halite_phase2) microcline top
Crystal data top
NaClV = 179.38 (15) Å3
Mr = 58.44Z = 4
Cubic, Fm3mCu Kα1 radiation, λ = 1.540596 Å
a = 5.6398 (16) ÅT = 25 K
Data collection top
Ge monochromator2θmin = 10°, 2θmax = 70°, 2θstep = 0.02°
Scan method: step
Refinement top
Least-squares matrix: full with fixed elements per cycleProfile function: fundamental parameters
Rp = 6.1021 parameters
Rwp = 7.72Weighting scheme based on measured s.u.'s
Rexp = 7.18(Δ/σ)max = 0.001
RBragg = 1.60Background function: Chebychev 4th order
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzBiso*/Beq
Na10001
Cl10.50.50.51
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
???????
 

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