Using powder XRD and pair distribution function to determine anisotropic atomic displacement parameters of orthorhombic tridymite and tetragonal cristobalite

Determination of the crystal structures of low-temperature tridymite and cristobalite using single-crystal XRD has been challenging because they generally occur as metastable fine-grained crystals in the geological environment. Therefore, using powder diffraction and scattering techniques is critical to study the low-temperature crystals. Synchrotron powder X-ray diffraction (XRD), pair distribution function (PDF) and transmission electron microscopy were used to investigate the structure of orthorhombic tridymite with C222₁ symmetry and tetragonal cristobalite with P4₁2₁2 symmetry, including their anisotropic atomic displacement parameters (ADPs). Rietveld refinement was used to determine the unit-cell parameters, fractional coordinates and isotropic atomic displacement parameters (U_iso) of the tridymite and cristobalite. The PDF method was used to determine ADPs for each atom. The results suggest that the crystal structure with high quality ADP values can be obtained using the combined methods of XRD and PDF analyses. The method can be used for characterizing crystals that are not suitable for single-crystal XRD.

Computing details

(tridymite)

Crystal data

SiO2	b = 5.0335 (3) Å
Orthorhombic, C2221	c = 8.2118 (4) Å
a = 8.7549 (5) Å	T = 550 K

Data collection

Radiation source: Synchrotron	k = →
θmax = 14.1°, θmin = 0.8°	l = →
h = →

Refinement

24 parameters

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
O1	0.3315 (5)	0.000000	0.500000	0.039000*
O2	0.000000	0.5587 (7)	0.250000	0.027667*
O3	0.2522 (3)	0.3139 (4)	0.2729 (3)	0.018667*
Si1	0.1672 (2)	0.5459 (2)	0.1877 (1)	0.029333*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
O1	0.049 (6)	0.043 (5)	0.025 (2)	0.00000	0.00000	0.003 (1)
O2	0.022 (3)	0.026 (4)	0.035 (4)	0.00000	0.006 (2)	0.00000
O3	0.019 (2)	0.015 (2)	0.022 (3)	0.002 (1)	−0.006 (2)	0.001 (1)
Si1	0.033 (3)	0.031 (3)	0.024 (2)	0.003 (1)	0.004 (1)	0.007 (2)

(cristobalite)

Crystal data

SiO2	c = 6.9257 (5) Å
Tetragonal, P41212	T = 298 K
a = 4.9727 (6) Å

Data collection

Radiation source: Synchrotron	k = →
θmax = 14.1°, θmin = 0.8°	l = →
h = →

Refinement

15 parameters

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Biso*/Beq
O	0.2390 (5)	0.1041 (2)	0.1787 (4)	1.476493*
Si	0.3007 (3)	0.3007 (3)	0.000000	2.729275*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
O	0.0289 (6)	0.0094 (3)	0.0178 (4)	−0.002 (1)	0.006 (2)	0.001 (1)
Si	0.0089 (2)	0.0086 (2)	0.0088 (2)	−0.001 (1)	0.002 (1)	−0.002 (1)