X-ray diffraction study of the atomic interactions, anharmonic displacements and inner-crystal field in orthorhombic KNbO₃

An X-ray diffraction study aimed at establishing the subtle details of the electron density and anharmonicity of the atomic vibrations in a stoichiometric monodomain single crystal of potassium niobate, KNbO₃, has been conducted at room tem­per­ature (orthorhombic ferroelectric phase Amm2). The cation and anion displacements obtained from the experiment are weakly anharmonic without any manifestation of structural disorder. The chemical bond and interatomic interactions inside and between crystal substructures at the balance of intracrystalline forces are characterized in detail. The role of each of the ions in the formation of the ferroelectric phase was studied and the features of the electron-density deformation in the niobium and oxygen substructures, and the role of each of them in the occurrence of spontaneous polarization are established. The position-space distribution of electrostatic and quantum forces in KNbO₃ is restored. It is emphasized that for the com­pleteness of the analysis of the nature of the ferroelectric properties it is necessary to consider both static and kinetic electronic factors, which are of a quantum origin. The experimental results and theoretical estimations by the Kohn–Sham calculation with periodic boundary conditions are in reasonable agreement, thus indicating the physical significance of the findings of this study.

Computing details

Data collection: CAD-4-PC V1.2 (Enraf-Nonius ,1993); cell refinement: CAD-4-PC V1.2 (Enraf-Nonius ,1993); data reduction: CAD-4-PC V1.2 (Enraf-Nonius ,1993); program(s) used to solve structure: SHELXT-2014/4 (Sheldrick, 2014); program(s) used to refine structure: SHELXL2018/3 (Sheldrick, 2018).

(I)

Crystal data

KNbO3	Dx = 4.620 Mg m^−3
Mr = 180.01	Mo Kα radiation, λ = 0.71073 Å
Orthorhombic, Amm2	Cell parameters from 24 reflections
a = 3.9719 (7) Å	θ = 45.5–47.3°
b = 5.6943 (9) Å	µ = 5.99 mm^−1
c = 5.7209 (9) Å	T = 293 K
V = 129.39 (4) Å^3	Plate, colorless
Z = 2	0.20 × 0.12 × 0.04 mm
F(000) = 168

Data collection

Enraf-Nonius CAD-4 diffractometer	1097 reflections with I > 2σ(I)
Radiation source: fine-focus tube	Rint = 0.032
β-filter monochromator	θmax = 59.9°, θmin = 5.1°
ω/2θ scans	h = −9→9
Absorption correction: analytical (see. N.W. Alcock (1970). Cryst. Computing, p271)	k = −13→13
Tmin = 0.498, Tmax = 0.795	l = −13→13
11554 measured reflections	3 standard reflections every 60 min
1098 independent reflections	intensity decay: 0.0%

Refinement

Refinement on F^2	w = 1/[σ^2(Fo^2) + (0.011P)^2 + 0.0047P] where P = (Fo^2 + 2Fc^2)/3
Least-squares matrix: full	(Δ/σ)max = 0.001
R[F^2 > 2σ(F^2)] = 0.007	Δρmax = 0.47 e Å^−3
wR(F^2) = 0.017	Δρmin = −0.91 e Å^−3
S = 1.08	Extinction correction: SHELXL-2018/3 (Sheldrick 2018), Fc^*=kFc[1+0.001xFc^2λ^3/sin(2θ)]^-1/4
1098 reflections	Extinction coefficient: 0.091 (3)
19 parameters	Absolute structure: Flack x determined using 520 quotients [(I+)-(I-)]/[(I+)+(I-)] (Parsons, Flack and Wagner, Acta Cryst. B69 (2013) 249-259).
0 restraints	Absolute structure parameter: 0.099 (13)

Special details

Experimental. Initial single crystal of KNbO₃

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds involving l.s. planes.

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

	x	y	z	Uiso*/Ueq
Nb1	0.500000	0.000000	0.500000	0.00419 (2)
K1	0.000000	0.000000	0.01529 (10)	0.00796 (5)
O1	0.000000	0.000000	0.53578 (13)	0.00726 (7)
O2	0.500000	0.24702 (9)	0.28431 (9)	0.00733 (5)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Nb1	0.00533 (2)	0.00388 (2)	0.00335 (3)	0.000	0.000	0.000
K1	0.00794 (4)	0.00830 (4)	0.00763 (16)	0.000	0.000	0.000
O1	0.00442 (12)	0.00921 (16)	0.00814 (18)	0.000	0.000	0.000
O2	0.00866 (12)	0.00695 (11)	0.00639 (13)	0.000	0.000	0.00207 (11)

Geometric parameters (Å, º)

Nb1—O2	1.8711 (5)	K1—O1^viii	2.7433 (11)
Nb1—O2^i	1.8712 (5)	K1—O2^ix	2.7866 (6)
Nb1—O1^ii	1.9965 (4)	K1—O2^x	2.7866 (6)
Nb1—O1	1.9965 (4)	K1—O2^xi	2.7866 (6)
Nb1—O2^iii	2.1727 (5)	K1—O2^xii	2.7866 (6)
Nb1—O2^iv	2.1727 (5)	K1—O1^xiii	2.8496 (5)
Nb1—K1^ii	3.4108 (6)	K1—O1^xii	2.8496 (5)
Nb1—K1	3.4108 (6)	K1—O2	2.8794 (6)
Nb1—K1^v	3.4725 (4)	K1—O2^i	2.8795 (6)
Nb1—K1^iv	3.4725 (4)	K1—O2^xiv	2.8795 (6)
Nb1—K1^vi	3.4725 (4)	K1—O2^xv	2.8795 (6)
Nb1—K1^vii	3.4725 (4)	K1—O1	2.9777 (11)
O2—Nb1—O2^i	97.48 (4)	O1^viii—K1—O1^xii	92.36 (2)
O2—Nb1—O1^ii	93.876 (14)	O2^ix—K1—O1^xii	60.198 (14)
O2^i—Nb1—O1^ii	93.876 (14)	O2^x—K1—O1^xii	122.412 (17)
O2—Nb1—O1	93.877 (14)	O2^xi—K1—O1^xii	122.412 (17)
O2^i—Nb1—O1	93.876 (14)	O2^xii—K1—O1^xii	60.198 (14)
O1^ii—Nb1—O1	168.23 (4)	O1^xiii—K1—O1^xii	175.29 (4)
O2—Nb1—O2^iii	89.729 (13)	O1^viii—K1—O2	122.309 (15)
O2^i—Nb1—O2^iii	172.79 (3)	O2^ix—K1—O2	175.96 (3)
O1^ii—Nb1—O2^iii	85.599 (16)	O2^x—K1—O2	60.620 (10)
O1—Nb1—O2^iii	85.598 (16)	O2^xi—K1—O2	119.502 (10)
O2—Nb1—O2^iv	172.79 (3)	O2^xii—K1—O2	90.828 (12)
O2^i—Nb1—O2^iv	89.729 (13)	O1^xiii—K1—O2	59.331 (14)
O1^ii—Nb1—O2^iv	85.599 (16)	O1^xii—K1—O2	117.782 (19)
O1—Nb1—O2^iv	85.598 (16)	O1^viii—K1—O2^i	122.309 (15)
O2^iii—Nb1—O2^iv	83.06 (3)	O2^ix—K1—O2^i	119.502 (10)
O2—Nb1—K1^ii	57.578 (14)	O2^x—K1—O2^i	90.828 (12)
O2^i—Nb1—K1^ii	57.579 (13)	O2^xi—K1—O2^i	175.96 (3)
O1^ii—Nb1—K1^ii	60.27 (2)	O2^xii—K1—O2^i	60.620 (10)
O1—Nb1—K1^ii	131.49 (2)	O1^xiii—K1—O2^i	117.782 (19)
O2^iii—Nb1—K1^ii	127.490 (12)	O1^xii—K1—O2^i	59.331 (13)
O2^iv—Nb1—K1^ii	127.490 (12)	O2—K1—O2^i	58.48 (2)
O2—Nb1—K1	57.578 (14)	O1^viii—K1—O2^xiv	122.309 (15)
O2^i—Nb1—K1	57.579 (14)	O2^ix—K1—O2^xiv	60.620 (10)
O1^ii—Nb1—K1	131.49 (2)	O2^x—K1—O2^xiv	175.96 (3)
O1—Nb1—K1	60.27 (2)	O2^xi—K1—O2^xiv	90.828 (12)
O2^iii—Nb1—K1	127.490 (12)	O2^xii—K1—O2^xiv	119.502 (10)
O2^iv—Nb1—K1	127.490 (12)	O1^xiii—K1—O2^xiv	117.782 (19)
K1^ii—Nb1—K1	71.219 (17)	O1^xii—K1—O2^xiv	59.331 (14)
O2—Nb1—K1^v	53.147 (16)	O2—K1—O2^xiv	115.38 (3)
O2^i—Nb1—K1^v	129.271 (14)	O2^i—K1—O2^xiv	87.21 (2)
O1^ii—Nb1—K1^v	55.146 (7)	O1^viii—K1—O2^xv	122.309 (15)
O1—Nb1—K1^v	124.494 (7)	O2^ix—K1—O2^xv	90.828 (11)
O2^iii—Nb1—K1^v	55.773 (15)	O2^x—K1—O2^xv	119.502 (10)
O2^iv—Nb1—K1^v	121.652 (14)	O2^xi—K1—O2^xv	60.620 (10)
K1^ii—Nb1—K1^v	71.789 (8)	O2^xii—K1—O2^xv	175.96 (3)
K1—Nb1—K1^v	110.700 (12)	O1^xiii—K1—O2^xv	59.331 (14)
O2—Nb1—K1^iv	129.271 (14)	O1^xii—K1—O2^xv	117.782 (19)
O2^i—Nb1—K1^iv	53.147 (16)	O2—K1—O2^xv	87.21 (2)
O1^ii—Nb1—K1^iv	124.494 (7)	O2^i—K1—O2^xv	115.38 (3)
O1—Nb1—K1^iv	55.146 (6)	O2^xiv—K1—O2^xv	58.48 (2)
O2^iii—Nb1—K1^iv	121.652 (15)	O1^viii—K1—O1	180.0
O2^iv—Nb1—K1^iv	55.773 (15)	O2^ix—K1—O1	118.307 (16)
K1^ii—Nb1—K1^iv	110.700 (12)	O2^x—K1—O1	118.307 (16)
K1—Nb1—K1^iv	71.789 (8)	O2^xi—K1—O1	118.307 (16)
K1^v—Nb1—K1^iv	177.113 (18)	O2^xii—K1—O1	118.307 (16)
O2—Nb1—K1^vi	53.147 (16)	O1^xiii—K1—O1	87.64 (2)
O2^i—Nb1—K1^vi	129.271 (14)	O1^xii—K1—O1	87.64 (2)
O1^ii—Nb1—K1^vi	124.494 (7)	O2—K1—O1	57.691 (15)
O1—Nb1—K1^vi	55.146 (6)	O2^i—K1—O1	57.691 (15)
O2^iii—Nb1—K1^vi	55.773 (15)	O2^xiv—K1—O1	57.691 (15)
O2^iv—Nb1—K1^vi	121.652 (14)	O2^xv—K1—O1	57.691 (15)
K1^ii—Nb1—K1^vi	110.700 (12)	Nb1^xv—O1—Nb1	168.23 (4)
K1—Nb1—K1^vi	71.789 (8)	Nb1^xv—O1—K1^xvi	95.88 (2)
K1^v—Nb1—K1^vi	69.768 (13)	Nb1—O1—K1^xvi	95.88 (2)
K1^iv—Nb1—K1^vi	110.155 (13)	Nb1^xv—O1—K1^iv	89.758 (3)
O2—Nb1—K1^vii	129.271 (14)	Nb1—O1—K1^iv	89.758 (3)
O2^i—Nb1—K1^vii	53.147 (16)	K1^xvi—O1—K1^iv	92.36 (2)
O1^ii—Nb1—K1^vii	55.146 (6)	Nb1^xv—O1—K1^vi	89.758 (3)
O1—Nb1—K1^vii	124.494 (7)	Nb1—O1—K1^vi	89.758 (3)
O2^iii—Nb1—K1^vii	121.652 (14)	K1^xvi—O1—K1^vi	92.36 (2)
O2^iv—Nb1—K1^vii	55.773 (15)	K1^iv—O1—K1^vi	175.29 (4)
K1^ii—Nb1—K1^vii	71.789 (8)	Nb1^xv—O1—K1	84.12 (2)
K1—Nb1—K1^vii	110.700 (12)	Nb1—O1—K1	84.12 (2)
K1^v—Nb1—K1^vii	110.155 (13)	K1^xvi—O1—K1	180.0
K1^iv—Nb1—K1^vii	69.768 (13)	K1^iv—O1—K1	87.64 (2)
K1^vi—Nb1—K1^vii	177.113 (18)	K1^vi—O1—K1	87.64 (2)
O1^viii—K1—O2^ix	61.693 (16)	Nb1—O2—Nb1^xiii	172.79 (3)
O1^viii—K1—O2^x	61.693 (16)	Nb1—O2—K1^v	94.35 (2)
O2^ix—K1—O2^x	123.39 (3)	Nb1^xiii—O2—K1^v	90.702 (18)
O1^viii—K1—O2^xi	61.693 (16)	Nb1—O2—K1^vi	94.35 (2)
O2^ix—K1—O2^xi	62.25 (2)	Nb1^xiii—O2—K1^vi	90.702 (18)
O2^x—K1—O2^xi	90.91 (2)	K1^v—O2—K1^vi	90.91 (2)
O1^viii—K1—O2^xii	61.693 (16)	Nb1—O2—K1	89.155 (19)
O2^ix—K1—O2^xii	90.91 (2)	Nb1^xiii—O2—K1	85.628 (19)
O2^x—K1—O2^xii	62.25 (2)	K1^v—O2—K1	175.96 (3)
O2^xi—K1—O2^xii	123.39 (3)	K1^vi—O2—K1	90.828 (12)
O1^viii—K1—O1^xiii	92.36 (2)	Nb1—O2—K1^ii	89.155 (19)
O2^ix—K1—O1^xiii	122.412 (17)	Nb1^xiii—O2—K1^ii	85.628 (19)
O2^x—K1—O1^xiii	60.198 (14)	K1^v—O2—K1^ii	90.828 (11)
O2^xi—K1—O1^xiii	60.198 (14)	K1^vi—O2—K1^ii	175.96 (3)
O2^xii—K1—O1^xiii	122.412 (17)	K1—O2—K1^ii	87.21 (2)

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