Crystal structures and thermal expansion of α-MgSO4 and β-MgSO4 from 4.2 to 300 K by neutron powder diffraction

Fortes, A.D.; Wood, I.G.; Vočadlo, L.; Brand, H.E.A.; Knight, K.S.

doi:10.1107/S0021889807029937

Crystal structures and thermal expansion of α-MgSO₄ and β-MgSO₄ from 4.2 to 300 K by neutron powder diffraction

A. D. Fortes, I. G. Wood, L. Vočadlo, H. E. A. Brand and K. S. Knight

Detailed neutron powder diffraction measurements have been carried out on two polymorphs of anhydrous magnesium sulfate, α-MgSO₄ and β-MgSO₄. α-MgSO₄ is orthorhombic, space group Cmcm (Z = 4); at 4.2 K the unit-cell dimensions are a = 5.16863 (3), b = 7.86781 (5), c = 6.46674 (5) Å, V = 262.975 (2) Å³ [ρ_calc = 3040.16 (2) kg m⁻³], and at 300 K, a = 5.17471 (3), b = 7.87563 (5), c = 6.49517 (5) Å, V = 264.705 (2) Å³ [ρ_calc = 3020.29 (2) kg m⁻³]. The axial and volumetric thermal expansion coefficients are positive at all temperatures and exhibit no unusual behaviour. Structures were refined at 4.2 and 300 K to R_P < 3%; less precise structural parameters were determined during warming from 4.2 to 300 K. β-MgSO₄ has a more complex structure, crystallizing in space group Pbnm (Z = 4); the unit-cell dimensions at 4.2 K are a = 4.73431 (8), b = 8.58170 (12), c = 6.67266 (11) Å, V = 271.100 (5) Å³ [ρ_calc = 2949.04 (5) kg m⁻³], and at 300 K, a = 4.74598 (7), b = 8.58310 (10), c = 6.70933 (10) Å, V = 273.306 (4) Å³ [ρ_calc = 2925.42 (4) kg m⁻³]. The thermal expansivities of the a and c axes, and the volumetric thermal expansion coefficient, are positive at all temperatures and normally behaved. However, the thermal expansion of the b axis is both very small and negative below ∼125 K. Structural and thermal motion parameters for β-MgSO₄ as a function of temperature are also reported.

Keywords: magnesium sulfate; polymorphism; thermal expansion; neutron powder diffraction.

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Supporting information

	Crystallographic Information File (CIF) https://doi.org/10.1107/S0021889807029937/db5013sup1.cif Contains datablocks global, ALPHAMGSO4_300KNEW_publ, ALPHAMGSO4_300KNEW_overall, ALPHAMGSO4_300KNEW_phase_1, ALPHAMGSO4_300KNEW_p_01, ALPHAMGSO4_300KNEW_p_02, ALPHAMGSO4BEST4KFIT_publ, ALPHAMGSO4BEST4KFIT_overall, ALPHAMGSO4BEST4KFIT_phase_1, ALPHAMGSO4BEST4KFIT_p_01, ALPHAMGSO4BEST4KFIT_p_02, BETAMGSO4_300KPROFIL_publ, BETAMGSO4_300KPROFIL_overall, BETAMGSO4_300KPROFIL_phase_1, BETAMGSO4_300KPROFIL_p_01, BETAMGSO4_300KPROFIL_p_02, BETAMGSO4_4KPROFILE_publ, BETAMGSO4_4KPROFILE_overall, BETAMGSO4_4KPROFILE_phase_1, BETAMGSO4_4KPROFILE_p_01, BETAMGSO4_4KPROFILE_p_02
	Zip compressed file https://doi.org/10.1107/S0021889807029937/db5013sup2.zip DAT files and GSAS refinement files for α-MgSO<inf>4</inf>
	Zip compressed file https://doi.org/10.1107/S0021889807029937/db5013sup3.zip DAT files and GSAS refinement files for β-MgSO<inf>4</inf>

Computing details top

Figures top

(ALPHAMGSO4_300KNEW_phase_1) alpha-magnesium sulfate top

Crystal data top

SO₄²⁻·MgO₆²⁺	V = 264.71 (1) Å³
M_r = 120.37	Z = 4
Orthorhombic, Cmcm	Melting point: 1200 K
a = 5.17471 (3) Å	? radiation, λ = ? Å
b = 7.87563 (5) Å	T = 300 K
c = 6.49517 (5) Å	?; ?, ?; ? × ?; ? mm

Data collection top

HRPD, ISIS Facility
diffractometer

Radiation source: Pulsed neutron spallation

Refinement top

Least-squares matrix: full	4404 data points
R_p = 0.027	Profile function: TOF Profile function number 3 with 21 terms Profile coefficients for exponential pseudovoigt convolution Von Dreele, 1990 (unpublished) #1 (alp ) = 0.3423 #2 (bet-0) = 0.026201 #3 (bet-1) = 0.009785 #4 (sig-0) = 0.0 #5 (sig-1) = 955.3 #6 (sig-2) = 0.0 #7 (gam-0) = 0.00 #8 (gam-1) = 34.24 #9 (gam-2) = 0.00 #10(gsf ) = 0.00 #11(g1ec ) = 0.00 #12(g2ec ) = 0.00 #13(rstr ) = 0.000 #14(rsta ) = 0.000 #15(rsca ) = 0.000 #16(L11) = 0.473 #17(L22) = 0.365 #18(L33) = 1.445 #19(L12) = 0.108 #20(L13) = -0.168 #21(L23) = -0.280 Peak tails are ignored where the intensity is below 0.0010 times the peak Aniso. broadening axis 0.0 0.0 1.0
R_wp = 0.032	50 parameters
R_exp = 0.027	0 restraints
R(F²) = 0.09217	(Δ/σ)_max = 0.04
χ² = 3.133	Background function: GSAS Background function number 1 with 5 terms. Shifted Chebyshev function of 1st kind 1: 1.42228 2: 0.520597 3: -1.141080E-02 4: -2.737090E-02 5: -1.666800E-02

Crystal data top

SO₄²⁻·MgO₆²⁺	V = 264.71 (1) Å³
M_r = 120.37	Z = 4
Orthorhombic, Cmcm	? radiation, λ = ? Å
a = 5.17471 (3) Å	T = 300 K
b = 7.87563 (5) Å	?; ?, ?; ? × ?; ? mm
c = 6.49517 (5) Å

Data collection top

HRPD, ISIS Facility
diffractometer

Refinement top

R_p = 0.027	χ² = 3.133
R_wp = 0.032	4404 data points
R_exp = 0.027	50 parameters
R(F²) = 0.09217	0 restraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Mg1	0.0	0.0	0.0	0.01254
S	0.0	0.3508 (2)	0.25	0.0105 (4)*
O1	0.0	0.24939 (8)	0.06357 (8)	0.01275
O2	0.23321 (13)	0.46184 (7)	0.25	0.01188

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Mg1	0.0144 (6)	0.0135 (7)	0.0097 (7)	0.0	0.0	−0.0002 (4)
O1	0.0125 (4)	0.0129 (4)	0.01288 (33	0.0	0.0	−0.00367 (29
O2	0.0093 (4)	0.0148 (4)	0.0116 (4)	−0.00340 (31	0.0	0.0

Geometric parameters (Å, º) top

Mg1—Mg1ⁱ	3.2476 (1)	S—Mg1ⁱⁱ	3.2045 (14)
Mg1—Mg1ⁱⁱ	3.2476 (1)	S—O1	1.4505 (10)
Mg1—S	3.2045 (14)	S—O1^viii	1.4505 (10)
Mg1—Sⁱ	3.2045 (14)	S—O2	1.4904 (10)
Mg1—O1	2.0071 (6)	S—O2^ix	1.4904 (10)
Mg1—O1ⁱⁱⁱ	2.0071 (6)	O1—Mg1	2.0071 (6)
Mg1—O2^iv	2.1524 (4)	O1—S	1.4505 (10)
Mg1—O2^v	2.1524 (4)	O2—Mg1^x	2.1524 (4)
Mg1—O2^vi	2.1524 (4)	O2—Mg1^xi	2.1524 (4)
Mg1—O2^vii	2.1524 (4)	O2—S	1.4904 (10)
S—Mg1	3.2045 (14)

O1—Mg1—O1ⁱⁱⁱ	179.972	O2^v—Mg1—O2^vii	100.21 (3)
O1—Mg1—O2^iv	88.936 (19)	O2^vi—Mg1—O2^vii	79.79 (3)
O1—Mg1—O2^v	88.936 (19)	O1—S—O1^xii	113.20 (12)
O1—Mg1—O2^vi	91.064 (19)	O1—S—O2	108.849 (16)
O1—Mg1—O2^vii	91.064 (19)	O1—S—O2^ix	108.849 (16)
O1ⁱⁱⁱ—Mg1—O2^iv	91.064 (19)	O1^xii—S—O2	108.849 (16)
O1ⁱⁱⁱ—Mg1—O2^v	91.064 (19)	O1^xii—S—O2^ix	108.849 (16)
O1ⁱⁱⁱ—Mg1—O2^vi	88.936 (19)	O2—S—O2^ix	108.13 (12)
O1ⁱⁱⁱ—Mg1—O2^vii	88.936 (19)	Mg1—O1—S	135.27 (6)
O2^iv—Mg1—O2^v	79.79 (3)	Mg1^x—O2—Mg1^xi	97.95 (3)
O2^iv—Mg1—O2^vi	100.21 (3)	Mg1^x—O2—S	126.96 (2)
O2^iv—Mg1—O2^vii	179.9657	Mg1^xi—O2—S	126.96 (2)
O2^v—Mg1—O2^vi	179.972

Symmetry codes: (i) x, −y, z−1/2; (ii) x, −y, z+1/2; (iii) −x, −y, −z; (iv) x−1/2, y−1/2, z; (v) −x+1/2, y−1/2, z; (vi) x−1/2, −y+1/2, z−1/2; (vii) −x+1/2, −y+1/2, z−1/2; (viii) −x, y, −z+1/2; (ix) −x, y, z; (x) x+1/2, y+1/2, z; (xi) x+1/2, −y+1/2, z+1/2; (xii) −x−1, y−1, −z+3/2.

(ALPHAMGSO4BEST4KFIT_phase_1) alpha-magnesium sulfate top

Crystal data top

SO₄²⁻·MgO₆²⁺	V = 262.98 (1) Å³
M_r = 120.37	Z = 4
Orthorhombic, Cmcm	Melting point: 1200 K
a = 5.16863 (3) Å	? radiation, λ = ? Å
b = 7.86781 (5) Å	T = 4 K
c = 6.46674 (5) Å	?; ?, ?; ? × ?; ? mm

Data collection top

HRPD, ISIS Facility diffractometer	Specimen mounting: slab can
Radiation source: Pulsed neutron spallation

Refinement top

Least-squares matrix: full	4404 data points
R_p = 0.029	Profile function: TOF Profile function number 3 with 21 terms Profile coefficients for exponential pseudovoigt convolution Von Dreele, 1990 (unpublished) #1 (alp ) = 0.3423 #2 (bet-0) = 0.026201 #3 (bet-1) = 0.009785 #4 (sig-0) = 0.0 #5 (sig-1) = 1212.2 #6 (sig-2) = 0.0 #7 (gam-0) = 0.00 #8 (gam-1) = 35.70 #9 (gam-2) = 0.00 #10(gsf ) = 0.00 #11(g1ec ) = 0.00 #12(g2ec ) = 0.00 #13(rstr ) = 0.000 #14(rsta ) = 0.000 #15(rsca ) = 0.000 #16(L11) = 0.509 #17(L22) = 0.543 #18(L33) = 1.926 #19(L12) = 0.163 #20(L13) = -0.337 #21(L23) = -0.386 Peak tails are ignored where the intensity is below 0.0010 times the peak Aniso. broadening axis 0.0 0.0 1.0
R_wp = 0.036	41 parameters
R_exp = 0.027	0 restraints
R(F²) = 0.08920	(Δ/σ)_max = 0.17
χ² = 3.881	Background function: GSAS Background function number 1 with 5 terms. Shifted Chebyshev function of 1st kind 1: 1.24219 2: 0.442743 3: -4.282540E-02 4: -1.056000E-02 5: -1.243180E-02

Crystal data top

SO₄²⁻·MgO₆²⁺	V = 262.98 (1) Å³
M_r = 120.37	Z = 4
Orthorhombic, Cmcm	? radiation, λ = ? Å
a = 5.16863 (3) Å	T = 4 K
b = 7.86781 (5) Å	?; ?, ?; ? × ?; ? mm
c = 6.46674 (5) Å

Data collection top

HRPD, ISIS Facility
diffractometer

Specimen mounting: slab can

Refinement top

R_p = 0.029	4404 data points
R_wp = 0.036	41 parameters
R_exp = 0.027	0 restraints
R(F²) = 0.08920	(Δ/σ)_max = 0.17
χ² = 3.881

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Mg1	0.0	0.0	0.0	0.0084 (3)*
S	0.0	0.3510 (2)	0.25	0.0079 (4)*
O1	0.0	0.24966 (8)	0.06264 (9)	0.00822 (18)*
O2	0.23393 (14)	0.46246 (7)	0.25	0.00833 (19)*

Geometric parameters (Å, º) top

Mg1—Mg1ⁱ	3.2334 (1)	S—Mg1ⁱⁱ	3.2003 (14)
Mg1—Mg1ⁱⁱ	3.2334 (1)	S—Mg1^x	3.2659 (6)
Mg1—S	3.2003 (14)	S—Mg1^xi	3.2659 (6)
Mg1—Sⁱ	3.2003 (14)	S—Mg1^xii	3.2659 (6)
Mg1—Sⁱⁱⁱ	3.2659 (6)	S—Mg1^xiii	3.2659 (6)
Mg1—S^iv	3.2659 (6)	S—O1	1.4506 (10)
Mg1—S^v	3.2659 (6)	S—O1^xiv	1.4506 (10)
Mg1—S^vi	3.2659 (6)	S—O2	1.4935 (11)
Mg1—O1	2.0056 (6)	S—O2^xv	1.4935 (11)
Mg1—O1^vii	2.0056 (6)	O1—Mg1	2.0056 (6)
Mg1—O2ⁱⁱⁱ	2.1429 (4)	O1—S	1.4506 (10)
Mg1—O2^viii	2.1429 (4)	O2—Mg1^xi	2.1429 (4)
Mg1—O2^v	2.1429 (4)	O2—Mg1^xiii	2.1429 (4)
Mg1—O2^ix	2.1429 (4)	O2—S	1.4935 (11)
S—Mg1	3.2003 (14)

O1—Mg1—O1^vii	180.0	O2^viii—Mg1—O2^ix	100.16 (3)
O1—Mg1—O2ⁱⁱⁱ	89.005 (19)	O2^v—Mg1—O2^ix	79.84 (3)
O1—Mg1—O2^viii	89.005 (19)	O1—S—O1^xvi	113.28 (11)
O1—Mg1—O2^v	90.995 (19)	O1—S—O2	108.830 (16)
O1—Mg1—O2^ix	90.995 (19)	O1—S—O2^xv	108.830 (16)
O1^vii—Mg1—O2ⁱⁱⁱ	90.995 (19)	O1^xvi—S—O2	108.830 (16)
O1^vii—Mg1—O2^viii	90.995 (19)	O1^xvi—S—O2^xv	108.830 (16)
O1^vii—Mg1—O2^v	89.005 (19)	O2—S—O2^xv	108.12 (12)
O1^vii—Mg1—O2^ix	89.005 (19)	Mg1—O1—S	135.01 (6)
O2ⁱⁱⁱ—Mg1—O2^viii	79.84 (3)	Mg1^xi—O2—Mg1^xiii	97.95 (3)
O2ⁱⁱⁱ—Mg1—O2^v	100.16 (3)	Mg1^xi—O2—S	126.90 (2)
O2ⁱⁱⁱ—Mg1—O2^ix	180.0	Mg1^xiii—O2—S	126.90 (2)
O2^viii—Mg1—O2^v	180.0

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

(BETAMGSO4_300KPROFIL_phase_1) alpha-magnesium sulfate top

Crystal data top

SO₄²⁻·MgO₆²⁺	V = 273.31 (1) Å³
M_r = 120.37	Z = 4
Orthorhombic, Pbnm	Melting point: 1200 K
a = 4.74599 (7) Å	? radiation, λ = ? Å
b = 8.5831 (1) Å	T = 300 K
c = 6.70934 (10) Å	?; ?, ?; ? × ?; ? mm

Data collection top

HRPD, ISIS Facility
diffractometer

Radiation source: Pulsed neutron spallation

Refinement top

Least-squares matrix: full	4404 data points
R_p = 0.038	Profile function: TOF Profile function number 3 with 21 terms Profile coefficients for exponential pseudovoigt convolution Von Dreele, 1990 (unpublished) #1 (alp ) = 0.3423 #2 (bet-0) = 0.026201 #3 (bet-1) = 0.009785 #4 (sig-0) = 0.0 #5 (sig-1) = 1213.0 #6 (sig-2) = 823.6 #7 (gam-0) = 0.00 #8 (gam-1) = 54.16 #9 (gam-2) = 0.00 #10(gsf ) = 0.00 #11(g1ec ) = 0.00 #12(g2ec ) = 0.00 #13(rstr ) = 0.000 #14(rsta ) = 0.000 #15(rsca ) = 0.000 #16(L11) = 1.294 #17(L22) = 0.264 #18(L33) = 1.480 #19(L12) = 0.242 #20(L13) = 0.499 #21(L23) = 0.340 Peak tails are ignored where the intensity is below 0.0010 times the peak Aniso. broadening axis 0.0 1.0 0.0
R_wp = 0.045	61 parameters
R_exp = 0.033	3 restraints
R(F²) = 0.07342	(Δ/σ)_max = 0.27
χ² = 4.410	Background function: GSAS Background function number 1 with 5 terms. Shifted Chebyshev function of 1st kind 1: 1.19515 2: 0.318323 3: 3.560090E-02 4: -1.899200E-02 5: -2.705870E-03

Crystal data top

SO₄²⁻·MgO₆²⁺	V = 273.31 (1) Å³
M_r = 120.37	Z = 4
Orthorhombic, Pbnm	? radiation, λ = ? Å
a = 4.74599 (7) Å	T = 300 K
b = 8.5831 (1) Å	?; ?, ?; ? × ?; ? mm
c = 6.70934 (10) Å

Data collection top

HRPD, ISIS Facility
diffractometer

Refinement top

R_p = 0.038	4404 data points
R_wp = 0.045	61 parameters
R_exp = 0.033	3 restraints
R(F²) = 0.07342	(Δ/σ)_max = 0.27
χ² = 4.410

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O3	0.3375 (2)	0.12661 (15)	0.06959 (15)	0.01483
O2	0.4641 (4)	0.35188 (15)	0.25	0.01412
O1	0.7709 (3)	0.1265 (3)	0.25	0.01333
S	0.4780 (5)	0.1812 (2)	0.25	0.0129 (7)*
Mg	0.0	0.0	0.0	0.01312

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
O3	0.0143 (7)	0.0268 (7)	0.0034 (5)	−0.0027 (7)	−0.0022 (5)	0.0007 (8)
O2	0.0300 (14)	0.0034 (5)	0.0089 (10)	0.0055 (10)	0.0	0.0
O1	0.0034 (5)	0.0073 (10)	0.0292 (10)	−0.0029 (8)	0.0	0.0
Mg	0.0186 (13)	0.0110 (8)	0.0098 (11)	−0.0035 (10)	0.0028 (8)	0.0073 (11)

Geometric parameters (Å, º) top

O3—S	1.4592 (12)	S—Mgⁱⁱ	3.2113 (14)
O3—Mg	1.9910 (11)	S—Mg^vi	3.2214 (18)
O2—S	1.4666 (16)	Mg—O3	1.9910 (11)
O2—Mgⁱ	2.1116 (8)	Mg—O3^vii	1.9910 (11)
O2—Mgⁱⁱ	2.1116 (8)	Mg—O2^viii	2.1116 (8)
O1—S	1.4673 (18)	Mg—O2^ix	2.1116 (8)
O1—Mgⁱⁱⁱ	2.2749 (13)	Mg—O1^x	2.2749 (13)
O1—Mg^iv	2.2749 (13)	Mg—O1^xi	2.2749 (13)
S—O3	1.4592 (12)	Mg—S	3.2214 (18)
S—O3^v	1.4592 (12)	Mg—S^viii	3.2113 (14)
S—O2	1.4666 (16)	Mg—S^ix	3.2113 (14)
S—O1	1.4673 (18)	Mg—S^xii	3.2214 (18)
S—Mg	3.2214 (18)	Mg—Mg^xii	3.3547 (1)
S—Mgⁱ	3.2113 (14)	Mg—Mg^vi	3.3547 (1)

S—O3—Mg	137.50 (11)	O3—Mg—O2^viii	94.44 (6)
S—O2—Mgⁱ	126.72 (3)	O3—Mg—O2^ix	85.56 (6)
S—O2—Mgⁱⁱ	126.72 (3)	O3—Mg—O1^x	87.19 (6)
Mgⁱ—O2—Mgⁱⁱ	105.19 (6)	O3—Mg—O1^xi	92.81 (6)
S—O1—Mgⁱⁱⁱ	127.27 (5)	O3^vii—Mg—O2^viii	85.56 (6)
S—O1—Mg^iv	127.27 (5)	O3^vii—Mg—O2^ix	94.44 (6)
Mgⁱⁱⁱ—O1—Mg^iv	95.01 (7)	O3^vii—Mg—O1^x	92.81 (6)
O3—S—O3^xiii	112.11 (18)	O3^vii—Mg—O1^xi	87.19 (6)
O3—S—O2	107.47 (10)	O2^viii—Mg—O2^ix	179.9802
O3—S—O1	109.29 (10)	O2^viii—Mg—O1^x	74.95 (5)
O3^xiv—S—O2	107.47 (10)	O2^viii—Mg—O1^xi	105.05 (5)
O3^v—S—O1	109.29 (10)	O2^ix—Mg—O1^x	105.05 (5)
O2—S—O1	111.2 (2)	O2^ix—Mg—O1^xi	74.95 (5)
O3—Mg—O3^vii	180.0	O1^x—Mg—O1^xi	180.0

Symmetry codes: (i) −x+1/2, y+1/2, z; (ii) x+1/2, −y+1/2, z+1/2; (iii) x+1, y, z; (iv) −x+1, −y, z+1/2; (v) x, y, −z+1/2; (vi) −x, −y, z+1/2; (vii) −x, −y, −z; (viii) −x+1/2, y−1/2, z; (ix) x−1/2, −y+1/2, z−1/2; (x) x−1, y, z; (xi) −x+1, −y, z−1/2; (xii) −x, −y, z−1/2; (xiii) x−1, y−1, −z+3/2; (xiv) x−1, y−1, −z+1/2.

(BETAMGSO4_4KPROFILE_phase_1) alpha-magnesium sulfate top

Crystal data top

SO₄²⁻·MgO₆²⁺	V = 271.10 (1) Å³
M_r = 120.37	Z = 4
Orthorhombic, Pbnm	Melting point: 1200 K
a = 4.73431 (8) Å	? radiation, λ = ? Å
b = 8.58171 (12) Å	T = 4 K
c = 6.67266 (11) Å	?; ?, ?; ? × ?; ? mm

Data collection top

HRPD, ISIS Facility
diffractometer

Radiation source: Pulsed neutron spallation

Refinement top

Least-squares matrix: full	4404 data points
R_p = 0.042	Profile function: TOF Profile function number 3 with 21 terms Profile coefficients for exponential pseudovoigt convolution Von Dreele, 1990 (unpublished) #1 (alp ) = 0.3423 #2 (bet-0) = 0.026201 #3 (bet-1) = 0.009785 #4 (sig-0) = 0.0 #5 (sig-1) = 3231.3 #6 (sig-2) = 974.1 #7 (gam-0) = 0.00 #8 (gam-1) = 52.99 #9 (gam-2) = 0.00 #10(gsf ) = 0.00 #11(g1ec ) = 0.00 #12(g2ec ) = 0.00 #13(rstr ) = 0.000 #14(rsta ) = 0.000 #15(rsca ) = 0.000 #16(L11) = 2.214 #17(L22) = 0.514 #18(L33) = 2.557 #19(L12) = 0.555 #20(L13) = -0.218 #21(L23) = -0.087 Peak tails are ignored where the intensity is below 0.0010 times the peak Aniso. broadening axis 0.0 1.0 0.0
R_wp = 0.050	48 parameters
R_exp = 0.030	0 restraints
R(F²) = 0.08309	(Δ/σ)_max = 0.25
χ² = 6.200	Background function: GSAS Background function number 1 with 5 terms. Shifted Chebyshev function of 1st kind 1: 1.14930 2: 0.339049 3: -7.066090E-03 4: -1.534120E-02 5: -8.436520E-04

Crystal data top

SO₄²⁻·MgO₆²⁺	V = 271.10 (1) Å³
M_r = 120.37	Z = 4
Orthorhombic, Pbnm	? radiation, λ = ? Å
a = 4.73431 (8) Å	T = 4 K
b = 8.58171 (12) Å	?; ?, ?; ? × ?; ? mm
c = 6.67266 (11) Å

Data collection top

HRPD, ISIS Facility
diffractometer

Refinement top

R_p = 0.042	4404 data points
R_wp = 0.050	48 parameters
R_exp = 0.030	0 restraints
R(F²) = 0.08309	(Δ/σ)_max = 0.25
χ² = 6.200

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O3	0.3385 (2)	0.12604 (12)	0.06902 (13)	0.0082 (3)*
O2	0.4628 (3)	0.35178 (14)	0.25	0.0068 (4)*
O1	0.7759 (3)	0.1266 (3)	0.25	0.0079 (4)*
S	0.4817 (7)	0.1804 (3)	0.25	0.0089 (7)*
Mg	0.0	0.0	0.0	0.0059 (4)*

Geometric parameters (Å, º) top

O3—S	1.4612 (18)	S—Mgⁱⁱ	3.2115 (19)
O3—Mg	1.9875 (10)	S—Mg^vi	3.222 (3)
O2—S	1.474 (3)	Mg—O3	1.9875 (10)
O2—Mgⁱ	2.1051 (8)	Mg—O3^vii	1.9875 (10)
O2—Mgⁱⁱ	2.1051 (8)	Mg—O2^viii	2.1051 (8)
O1—S	1.468 (4)	Mg—O2^ix	2.1051 (8)
O1—Mgⁱⁱⁱ	2.2557 (12)	Mg—O1^x	2.2557 (12)
O1—Mg^iv	2.2557 (12)	Mg—O1^xi	2.2557 (12)
S—O3	1.4612 (18)	Mg—S	3.222 (3)
S—O3^v	1.4612 (18)	Mg—S^viii	3.2115 (19)
S—O2	1.474 (3)	Mg—S^ix	3.2115 (19)
S—O1	1.468 (4)	Mg—S^xii	3.222 (3)
S—Mg	3.222 (3)	Mg—Mg^xii	3.3363 (1)
S—Mgⁱ	3.2115 (19)	Mg—Mg^vi	3.3363 (1)

S—O3—Mg	137.66 (12)	O3—Mg—O2^viii	94.46 (5)
S—O2—Mgⁱ	126.73 (4)	O3—Mg—O2^ix	85.54 (5)
S—O2—Mgⁱⁱ	126.73 (4)	O3—Mg—O1^x	86.89 (5)
Mgⁱ—O2—Mgⁱⁱ	104.82 (5)	O3—Mg—O1^xi	93.11 (5)
S—O1—Mgⁱⁱⁱ	126.72 (6)	O3^vii—Mg—O2^viii	85.54 (5)
S—O1—Mg^iv	126.72 (6)	O3^vii—Mg—O2^ix	94.46 (5)
Mgⁱⁱⁱ—O1—Mg^iv	95.38 (7)	O3^vii—Mg—O1^x	93.11 (5)
O3—S—O3^xiii	111.5 (2)	O3^vii—Mg—O1^xi	86.89 (5)
O3—S—O2	106.88 (13)	O2^viii—Mg—O2^ix	179.9802
O3—S—O1	109.87 (13)	O2^viii—Mg—O1^x	75.18 (5)
O3^xiv—S—O2	106.88 (13)	O2^viii—Mg—O1^xi	104.82 (5)
O3^v—S—O1	109.87 (13)	O2^ix—Mg—O1^x	104.82 (5)
O2—S—O1	111.8 (2)	O2^ix—Mg—O1^xi	75.18 (5)
O3—Mg—O3^vii	180.0	O1^x—Mg—O1^xi	180.0

Experimental details

	(ALPHAMGSO4_300KNEW_phase_1)	(ALPHAMGSO4BEST4KFIT_phase_1)	(BETAMGSO4_300KPROFIL_phase_1)	(BETAMGSO4_4KPROFILE_phase_1)
Crystal data
Chemical formula	SO₄²⁻·MgO₆²⁺	SO₄²⁻·MgO₆²⁺	SO₄²⁻·MgO₆²⁺	SO₄²⁻·MgO₆²⁺
M_r	120.37	120.37	120.37	120.37
Crystal system, space group	Orthorhombic, Cmcm	Orthorhombic, Cmcm	Orthorhombic, Pbnm	Orthorhombic, Pbnm
Temperature (K)	300	4	300	4
a, b, c (Å)	5.17471 (3), 7.87563 (5), 6.49517 (5)	5.16863 (3), 7.86781 (5), 6.46674 (5)	4.74599 (7), 8.5831 (1), 6.70934 (10)	4.73431 (8), 8.58171 (12), 6.67266 (11)
V (Å³)	264.71 (1)	262.98 (1)	273.31 (1)	271.10 (1)
Z	4	4	4	4
Radiation type	?, λ = ? Å	?, λ = ? Å	?, λ = ? Å	?, λ = ? Å
Specimen shape, size (mm)	?; ?, ?; ? × ?; ?	?; ?, ?; ? × ?; ?	?; ?, ?; ? × ?; ?	?; ?, ?; ? × ?; ?

Data collection
Data collection method	HRPD, ISIS Facility	HRPD, ISIS Facility	HRPD, ISIS Facility	HRPD, ISIS Facility
Specimen mounting	?; ?	Slab can	?; ?	?; ?
Data collection mode	?; ?	?; ?	?; ?	?; ?
Scan method	?; ?	?; ?	?; ?	?; ?
2θ values (°)	2θ_min = ? 2θ_max = ? 2θ_step = ?	2θ_min = ? 2θ_max = ? 2θ_step = ?	2θ_min = ? 2θ_max = ? 2θ_step = ?	2θ_min = ? 2θ_max = ? 2θ_step = ?

Refinement
R factors and goodness of fit	R_p = 0.027, R_wp = 0.032, R_exp = 0.027, R(F²) = 0.09217, χ² = 3.133	R_p = 0.029, R_wp = 0.036, R_exp = 0.027, R(F²) = 0.08920, χ² = 3.881	R_p = 0.038, R_wp = 0.045, R_exp = 0.033, R(F²) = 0.07342, χ² = 4.410	R_p = 0.042, R_wp = 0.050, R_exp = 0.030, R(F²) = 0.08309, χ² = 6.200
No. of data points	4404	4404	4404	4404
No. of parameters	50	41	61	48
No. of restraints	0	0	3	0
(Δ/σ)_max	0.04	0.17	0.27	0.25

Computer programs: GSAS.

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