Anharmonic motions versus dynamic disorder at the Mg ion from the charge densities in pyrope (Mg₃Al₂Si₃O₁₂) crystals at 30 K: six of one, half a dozen of the other

The possible occurrence of static/dynamic disorder at the Mg site in pyrope (Mg₃Al₂Si₃O₁₂), with or without anharmonic contribution to the thermal vibrations even at low temperatures, has been largely debated but conclusions were contrasting. Here a report is given on the experimental charge density distribution, ρ_EXP, of synthetic pyrope at T = 30 K, built through a Stewart multipolar expansion up to l = 5 and based on a very precise and accurate set of in-home measured single-crystal X-ray diffraction amplitudes with a maximum resolution of 0.44 Å. Local and integral topological properties of ρ_EXP are in substantial agreement with those of ρ_THEO, the corresponding DFT-grade quantum charge density of an ideal pyrope crystal, and those derived from synchrotron investigations of chemical bonding in olivines. Relevant thermal atomic displacements, probably anharmonic in nature, clearly affect the whole structure down to 30 K. No significant (> 2.5σ) residual Fourier peaks are detectable from the ρ_EXP distribution around Mg, after least-squares refinement of a multipole model with anharmonic thermal motion at the Mg site. Experimental findings were confirmed by a full analysis of normal vibration modes of the DFT-optimized structure of the perfect pyrope crystal. Mg undergoes wide displacements from its equilibrium position even at very low temperatures, as it is allocated in a ∼ 4.5 Å large dodecahedral cavity and involved in several soft phonon modes. Implications on the interplay among static/dynamic disorder of Mg and lattice vibrational degrees of freedom are discussed.

Computing details

Program(s) used to solve structure: SHELXS97 (Sheldrick, 1990); program(s) used to refine structure: VALRAY2000 (Stewart, Spackman & Flensburg, 2000)'.

(I)

Crystal data

2(AL)3(MG)3(SIO4)	Mo Kα radiation, λ = 0.71073 Å
Mr = 403.16	Cell parameters from 406 reflections
Cubic, Ia3d	θ = 11.0–53.5°
a = 11.4405 (3) Å	µ = 1.22 mm^−1
V = 1497.39 (12) Å^3	T = 30 K
Z = 8	Sphere, colourless
F(000) = 1600	0.45 × 0.45 × 0.45 × 0.45 (radius) mm
Dx = 3.576 Mg m^−3

Data collection

Syntex P1- diffractometer	Rint = 0.013
Graphite monochromator	θmax = 54.5°, θmin = 1.0°
2θ/ω scans	h = 0→26
Absorption correction: for a sphere TBAR, home written program	k = 0→18
Tmin = 0.668, Tmax = 0.687	l = 0→14
7486 measured reflections	3 standard reflections every 97 reflections
788 independent reflections	intensity decay: none
785 reflections with > 0

Refinement

Refinement on F^2	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F^2 > 2σ(F^2)] = 0.008
wR(F^2) = 0.008	(Δ/σ)max = 0.001
S = 0.76	Δρmax = 0.12 e Å^−3
785 reflections	Δρmin = −0.16 e Å^−3
93 parameters	Extinction correction: type 1
0 restraints	Extinction coefficient: 0.248

Special details

Experimental. To determine the cell dimensions 406 reflections were centered in both negative and positive 2theta regions. A least-square fit to the resulting values of (sin(theta))**2 gave the reported cell parameters.

Refinement. Refinement of F² against ALL reflections. To adjust for the deviations from kinematic theory, the formalism of Becker and Coppens was used, assuming type I crystals and a Lorentzian "mosaic" distribution.

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

	x	y	z	Uiso*/Ueq
Al1	0.0000	0.0000	0.0000	0.00217
Mg1	0.0000	0.2500	0.1250	0.00423
Si1	0.0000	0.2500	0.3750	0.00171
O1	0.03282 (3)	0.05077 (3)	0.65326 (3)	0.00319

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Al1	0.00217 (4)	0.00217 (4)	0.00217 (4)	0.00001 (2)	0.00001 (2)	0.00001 (2)
Mg1	0.00485 (7)	0.00485 (7)	0.00298 (8)	0.00096 (5)	0.0000	0.0000
Si1	0.00180 (2)	0.00180 (2)	0.00153 (2)	0.0000	0.0000	0.0000
O1	0.00321 (4)	0.00375 (4)	0.00261 (4)	0.00040 (2)	−0.00057 (2)	0.00009 (2)

Geometric parameters (Å, º)

Al1—O1^i	1.8848 (3)	Mg1—O1^v	2.3326 (4)
Al1—O1^ii	1.8848 (3)	Mg1—O1^xvi	2.3326 (4)
Al1—O1^iii	1.8848 (3)	Mg1—O1^xvii	2.3326 (4)
Al1—O1^iv	1.8848 (3)	Mg1—Si1	2.8601 (1)
Al1—O1^v	1.8848 (3)	Mg1—Si1^v	2.8601 (1)
Al1—O1^vi	1.8848 (3)	Mg1—Al1^xviii	3.1977 (1)
Al1—Mg1^vii	3.1977 (1)	Mg1—Al1^xix	3.1977 (1)
Al1—Mg1^viii	3.1977 (1)	Si1—O1^xx	1.6354 (4)
Al1—Mg1^ix	3.1977 (1)	Si1—O1^xiii	1.6354 (4)
Al1—Mg1^x	3.1977 (1)	Si1—O1^xxi	1.6354 (4)
Al1—Mg1^xi	3.1977 (1)	Si1—O1^xii	1.6354 (4)
Al1—Mg1	3.1977 (1)	Si1—Mg1^xvi	2.8601 (1)
Mg1—O1^xii	2.1960 (4)	O1—Si1^xxii	1.6354 (4)
Mg1—O1^xiii	2.1960 (4)	O1—Al1^vi	1.8848 (3)
Mg1—O1^iii	2.1960 (4)	O1—Mg1^xxii	2.1960 (4)
Mg1—O1^xiv	2.1960 (4)	O1—Mg1^xvi	2.3326 (4)
Mg1—O1^xv	2.3326 (4)
O1^i—Al1—O1^ii	180.0 (2)	O1^xiii—Mg1—O1^v	93.640 (10)
O1^i—Al1—O1^iii	87.70 (2)	O1^iii—Mg1—O1^v	70.370 (10)
O1^ii—Al1—O1^iii	92.30 (2)	O1^xiv—Mg1—O1^v	124.330 (10)
O1^i—Al1—O1^iv	92.30 (2)	O1^xv—Mg1—O1^v	164.080 (10)
O1^ii—Al1—O1^iv	87.70 (2)	O1^xii—Mg1—O1^xvi	124.330 (10)
O1^iii—Al1—O1^iv	180.0 (2)	O1^xiii—Mg1—O1^xvi	70.370 (10)
O1^i—Al1—O1^v	87.70 (2)	O1^iii—Mg1—O1^xvi	93.640 (10)
O1^ii—Al1—O1^v	92.30 (2)	O1^xiv—Mg1—O1^xvi	73.070 (10)
O1^iii—Al1—O1^v	87.70 (2)	O1^xv—Mg1—O1^xvi	109.490 (10)
O1^iv—Al1—O1^v	92.30 (2)	O1^v—Mg1—O1^xvi	72.830 (10)
O1^i—Al1—O1^vi	92.30 (2)	O1^xii—Mg1—O1^xvii	70.370 (10)
O1^ii—Al1—O1^vi	87.70 (2)	O1^xiii—Mg1—O1^xvii	124.330 (10)
O1^iii—Al1—O1^vi	92.30 (2)	O1^iii—Mg1—O1^xvii	73.070 (10)
O1^iv—Al1—O1^vi	87.70 (2)	O1^xiv—Mg1—O1^xvii	93.640 (10)
O1^v—Al1—O1^vi	180.0 (2)	O1^xv—Mg1—O1^xvii	72.830 (10)
O1^i—Al1—Mg1^vii	87.690 (10)	O1^v—Mg1—O1^xvii	109.490 (10)
O1^ii—Al1—Mg1^vii	92.310 (10)	O1^xvi—Mg1—O1^xvii	164.080 (10)
O1^iii—Al1—Mg1^vii	137.990 (10)	O1^xii—Mg1—Si1	34.700 (10)
O1^iv—Al1—Mg1^vii	42.010 (10)	O1^xiii—Mg1—Si1	34.700 (10)
O1^v—Al1—Mg1^vii	133.770 (10)	O1^iii—Mg1—Si1	145.300 (10)
O1^vi—Al1—Mg1^vii	46.230 (10)	O1^xiv—Mg1—Si1	145.300 (10)
O1^i—Al1—Mg1^viii	133.770 (10)	O1^xv—Mg1—Si1	82.040 (9)
O1^ii—Al1—Mg1^viii	46.230 (10)	O1^v—Mg1—Si1	82.040 (9)
O1^iii—Al1—Mg1^viii	87.690 (10)	O1^xvi—Mg1—Si1	97.960 (8)
O1^iv—Al1—Mg1^viii	92.310 (10)	O1^xvii—Mg1—Si1	97.960 (8)
O1^v—Al1—Mg1^viii	137.990 (10)	O1^xii—Mg1—Si1^v	145.300 (10)
O1^vi—Al1—Mg1^viii	42.010 (10)	O1^xiii—Mg1—Si1^v	145.300 (10)
Mg1^vii—Al1—Mg1^viii	66.4220 (10)	O1^iii—Mg1—Si1^v	34.700 (10)
O1^i—Al1—Mg1^ix	46.230 (10)	O1^xiv—Mg1—Si1^v	34.700 (10)
O1^ii—Al1—Mg1^ix	133.770 (10)	O1^xv—Mg1—Si1^v	97.960 (9)
O1^iii—Al1—Mg1^ix	92.310 (10)	O1^v—Mg1—Si1^v	97.960 (9)
O1^iv—Al1—Mg1^ix	87.690 (10)	O1^xvi—Mg1—Si1^v	82.040 (9)
O1^v—Al1—Mg1^ix	42.010 (10)	O1^xvii—Mg1—Si1^v	82.040 (9)
O1^vi—Al1—Mg1^ix	137.990 (10)	Si1—Mg1—Si1^v	180.0
Mg1^vii—Al1—Mg1^ix	113.5780 (10)	O1^xii—Mg1—Al1^xviii	35.060 (10)
Mg1^viii—Al1—Mg1^ix	180.000	O1^xiii—Mg1—Al1^xviii	94.780 (10)
O1^i—Al1—Mg1^x	137.990 (10)	O1^iii—Mg1—Al1^xviii	97.530 (10)
O1^ii—Al1—Mg1^x	42.010 (10)	O1^xiv—Mg1—Al1^xviii	127.180 (10)
O1^iii—Al1—Mg1^x	133.770 (10)	O1^xv—Mg1—Al1^xviii	78.112 (9)
O1^iv—Al1—Mg1^x	46.230 (10)	O1^v—Mg1—Al1^xviii	94.702 (9)
O1^v—Al1—Mg1^x	87.690 (10)	O1^xvi—Mg1—Al1^xviii	159.386 (9)
O1^vi—Al1—Mg1^x	92.310 (10)	O1^xvii—Mg1—Al1^xviii	35.707 (9)
Mg1^vii—Al1—Mg1^x	66.4220 (10)	Si1—Mg1—Al1^xviii	63.4350 (10)
Mg1^viii—Al1—Mg1^x	66.4220 (10)	Si1^v—Mg1—Al1^xviii	116.5650 (10)
Mg1^ix—Al1—Mg1^x	113.5780 (10)	O1^xii—Mg1—Al1^xix	94.780 (10)
O1^i—Al1—Mg1^xi	42.010 (10)	O1^xiii—Mg1—Al1^xix	35.060 (10)
O1^ii—Al1—Mg1^xi	137.990 (10)	O1^iii—Mg1—Al1^xix	127.180 (10)
O1^iii—Al1—Mg1^xi	46.230 (10)	O1^xiv—Mg1—Al1^xix	97.530 (10)
O1^iv—Al1—Mg1^xi	133.770 (10)	O1^xv—Mg1—Al1^xix	94.702 (9)
O1^v—Al1—Mg1^xi	92.310 (10)	O1^v—Mg1—Al1^xix	78.112 (9)
O1^vi—Al1—Mg1^xi	87.690 (10)	O1^xvi—Mg1—Al1^xix	35.707 (9)
Mg1^vii—Al1—Mg1^xi	113.5780 (10)	O1^xvii—Mg1—Al1^xix	159.386 (9)
Mg1^viii—Al1—Mg1^xi	113.5780 (10)	Si1—Mg1—Al1^xix	63.4350 (10)
Mg1^ix—Al1—Mg1^xi	66.4220 (10)	Si1^v—Mg1—Al1^xix	116.5650 (10)
Mg1^x—Al1—Mg1^xi	180.000	Al1^xviii—Mg1—Al1^xix	126.870 (2)
O1^i—Al1—Mg1	92.310 (10)	O1^xx—Si1—O1^xiii	114.57 (2)
O1^ii—Al1—Mg1	87.690 (10)	O1^xx—Si1—O1^xxi	99.69 (2)
O1^iii—Al1—Mg1	42.010 (10)	O1^xiii—Si1—O1^xxi	114.57 (2)
O1^iv—Al1—Mg1	137.990 (10)	O1^xx—Si1—O1^xii	114.57 (2)
O1^v—Al1—Mg1	46.230 (10)	O1^xiii—Si1—O1^xii	99.69 (2)
O1^vi—Al1—Mg1	133.770 (10)	O1^xxi—Si1—O1^xii	114.57 (2)
Mg1^vii—Al1—Mg1	180.000	O1^xx—Si1—Mg1	130.150 (10)
Mg1^viii—Al1—Mg1	113.5780 (10)	O1^xiii—Si1—Mg1	49.850 (10)
Mg1^ix—Al1—Mg1	66.4220 (10)	O1^xxi—Si1—Mg1	130.150 (10)
Mg1^x—Al1—Mg1	113.5780 (10)	O1^xii—Si1—Mg1	49.850 (10)
Mg1^xi—Al1—Mg1	66.4220 (10)	O1^xx—Si1—Mg1^xvi	49.850 (10)
O1^xii—Mg1—O1^xiii	69.390 (10)	O1^xiii—Si1—Mg1^xvi	130.150 (10)
O1^xii—Mg1—O1^iii	114.1670 (10)	O1^xxi—Si1—Mg1^xvi	49.850 (10)
O1^xiii—Mg1—O1^iii	160.490 (10)	O1^xii—Si1—Mg1^xvi	130.150 (10)
O1^xii—Mg1—O1^xiv	160.490 (10)	Mg1—Si1—Mg1^xvi	180.000
O1^xiii—Mg1—O1^xiv	114.170 (10)	Si1^xxii—O1—Al1^vi	130.43 (2)
O1^iii—Mg1—O1^xiv	69.390 (10)	Si1^xxii—O1—Mg1^xxii	95.45 (2)
O1^xii—Mg1—O1^xv	93.640 (10)	Al1^vi—O1—Mg1^xxii	102.93 (2)
O1^xiii—Mg1—O1^xv	73.070 (10)	Si1^xxii—O1—Mg1^xvi	122.99 (2)
O1^iii—Mg1—O1^xv	124.330 (10)	Al1^vi—O1—Mg1^xvi	98.07 (2)
O1^xiv—Mg1—O1^xv	70.370 (10)	Mg1^xxii—O1—Mg1^xvi	101.31 (2)
O1^xii—Mg1—O1^v	73.070 (10)

Symmetry codes: (i) z−1/2, −x, y; (ii) −z+1/2, x, −y; (iii) y, z−1/2, −x; (iv) −y, −z+1/2, x; (v) −x, y, z−1/2; (vi) x, −y, −z+1/2; (vii) −x, −y, −z; (viii) z−1/4, −y+1/4, −x−1/4; (ix) −z+1/4, y−1/4, x+1/4; (x) −x−1/4, z−1/4, −y+1/4; (xi) x+1/4, −z+1/4, y−1/4; (xii) −z+3/4, −y+1/4, x+1/4; (xiii) z−3/4, y+1/4, x+1/4; (xiv) −y, −z+1, −x; (xv) x, −y+1/2, z−1/2; (xvi) y−1/4, −x+1/4, −z+3/4; (xvii) −y+1/4, x+1/4, −z+3/4; (xviii) −y+1/4, −x+1/4, −z+1/4; (xix) x−1/4, z+1/4, −y+1/4; (xx) y, −z+1, −x+1/2; (xxi) −y, z−1/2, −x+1/2; (xxii) z−1/4, −y+1/4, −x+3/4.