Spatial distribution of 6s² lone electron pair in Pb₄Na(PO₄)₃ and stereochemical activity of the 6s² electron cloud in lead-bearing apatites

Single crystals of lacunary apatite structure-type compound Pb₄Na(PO₄)₃ were prepared with a flux/solid-state-reaction technique and consecutive melt-growth to examine spatial distribution and stereochemical activity of the 6s² lone electron pair of Pb²⁺ using X-ray diffraction. The target compound crystallized in space group P6₃/m with unit-cell edge lengths a = 9.7345 (12) Å and c = 7.2130 (18) Å without any indication of configurational ordering of Pb²⁺ and Na⁺ at the crystallographic A1 site. The anion channel running through the structure was confirmed as vacant. Residual density attributable to the deformed 6s² electron cloud was found 0.6 Å apart from the A2-site position on the A2 triangle normal to c. The lack of a channel-site anion induced shrinkage of the unit cell and the O3 trigonal antiprism in the xy plane, and an increase in the twist angle of the A1O₆ trigonal metaprism compensated for the shrinkage of the latter to keep the volume of the A1O₉ coordination polyhedron constant. Systematic comparison of the size of the A2 triangle in Pb₄Na(PO₄)₃ and other apatite-type compounds indicates no contribution from the 6s² electron orbital on the size of the A2 triangle and limited space-filling ability of the orbital. A2 cations are attached on the periphery of a large void formed by the in-plane distortion of the hexagonal close packed arrangement of (BO₄)³⁻ complex anions. In other words, a face-sharing array of A2 octahedra was inserted `as a tube' through the framework. The sizes of A1O₉ and BO₄ polyhedra and their framework define the maximum size of the A2 triangle, while the triangle can easily be shrunk by attraction from an anion, such as F⁻, in the channel.

1. Introduction

After the first reports of the structure of Ca₅(PO₄)₃F by Mehmel (1930) and Náray-Szabó (1930), there are numerous articles to date regarding the structure of apatite-type compounds (hereafter referred to as apatites) for their industrial applications, such as an oxide ion conductor for solid oxide fuel cells operated under moderate temperature (Nakayama et al., 1999; Okudera et al., 2005; Ali et al., 2009; Matsuura & Okudera, 2022). Their crystal chemistry, crystal structures with symmetry lowering (hettotypes), and their descriptions have been summarized in reviews (White & ZhiLi, 2003; Mercier et al., 2005; Pasero et al., 2010), and only a short description of the aristotype structure (space group P6₃/m) is given below. The generalized formula of apatites is written as [A1₂][A2₃](BO₄)₃X (Z = 2), here A1, A2, B and X are designated also as atomic sites. The aristotype of their host structure can be depicted as a hexagonal close packed (hcp) array of (BO₄)³⁻ complex anions with large in-plane distortion. This in-plane distortion forms four smaller and two larger octahedral voids (per unit cell). The smaller ones are arrayed parallel to c and filled by A1 cations at z ≃ 0 and 1/2 (Wyckoff position 4f). This site is coordinated by O1- and O2-site oxide anions to form an A1O₆ trigonal prism with a metaprismatic twist angle φ and three more distant O3-site oxide anions which cap each of the square faces (Fig. 1). White & ZhiLi (2003) and Dong & White (2004) pointed out that the A1O₆ metaprism is a key structural unit along with the BO₄ unit, and the twist angle φ is a useful measure to validate the structure. The larger ones are large enough to accommodate three A2 cations around c at z = 1/4 and 3/4. A regular triangle made of these three A2 cations will be referred to as an A2 triangle. Six A2 cations in adjacent nets form a nearly regular trigonal antiprism with A2 cations on its apices (A2 octahedron). The face-sharing array of this octahedron along c forms an anion channel with the X anion as a guest at (0, 0, z) [z = 0 (Wyckoff position 2b with symmetry 3..), 1/4 (Wyckoff position 2a with symmetry 6..) or somewhere in between (Wyckoff position 4e with symmetry 3.. with a half occupancy)]. Six O3-site oxide anions on the periphery of this large void also form a highly oblate trigonal antiprism by two staggered O3 regular triangles (O3 triangle) normal to c in the middle of each A2 octahedron. Repulsion from O3-site oxide anions affects displacements and even the position of the X anion (Okudera, 2013; Matsuura & Okudera, 2022). Mercier et al. (2005) reported that unit-cell edge length a is governed by constraints inherent to the (A1O₆)–(BO₄) polyhedral arrangement, however, there was no in-detail discussion on repulsive forces inside the channel such as repulsion between O3 and X anions and direct interference of 6s² lone electron pairs at the A2-site Pb²⁺ in lead-bearing apatites.

Figure 1 Structure of Pb4Na(PO4)3. (a) Projection in [001] with atoms in the range −0.6 < x < 0.6, −0.6 < y < 0.6, 0 < z < 1. Symmetry codes on atomic sites are abridged. BO4 polyhedra are drawn in purple. A1O6 trigonal metaprisms are drawn in grey with the definition of angle φ indicated. Edges of the A2 octahedron are drawn with grey lines. (b) Orthogonal projection of the structure with atoms in the range −0.2 < x < 1.2, −0.2 < y < 1.2, 0 < z < 1. A1O9 and BO4 polyhedra are drawn in grey and purple, respectively, to show their edge-sharing manner. All displacement ellipsoids at the 70% probability level. Drawn with VESTA (Momma & Izumi, 2011) software.

Contribution from the stereochemically active 6s² lone electron pair on the stability of lacunary apatite structures has been mentioned by several authors (e.g. Mathew et al., 1980; Krivovichev et al., 2004; Kampf & Housley, 2011). They suggested the role of the space-filling 6s² orbital as a substitute for the X anion to prevent the structure from collapse. Hirano & Okudera (2025) solved the structure of Pb₁₀(PO₄)₆O and found commensurate modulation in sizes of A2 triangles in the manner `large-middle-small-middle(-large)', which caused a 2 × c superstructure. Investigation of the structure showed significant local shrinkage of the channel due to attraction on A2-site Pb²⁺ from X-site O²⁻ in the channel and a simultaneous shift of O²⁻ out from the centre of the A2 triangle. The latter was attributed to the 6s² electron cloud that thrusted O²⁻ out from its ideal position on the triangle plane. However, this structure also left some issues to be considered. First, the smallest A2 triangle [edge length d(A2c–A2c) = 3.871 (3) Å] was highly shrunk, suggesting that repulsive force among 6s² electron clouds was rather weak. Second, the size of the largest A2 triangle [d(A2a–A2a) = 4.359 (3) Å] at the vacant part of the channel was comparable to that in Pb₅(PO₄)₃Cl in which Cl⁻ and presumably space-filling 6s² orbitals coexisted. Indeed, estimated bond valence sums (BVSs) for Cl⁻ in lead chlorapatites commonly exceeded its formal valence, in other words, the channel was tight even for Cl⁻ alone (Okudera, 2013). These observations might indicate rather limited ability of the 6s² orbital to fill the space and also the highly subordinate nature of the size of the A2 octahedron.

Location, or deformation, of the 6s² electron cloud and systematic variation in the sizes of the A2 triangle with- and without such a lone electron pair seem to be the keys to solve these questions. In spite of the long research history of apatite-type compounds, no accurate X-ray investigation of the 6s² electron density had been reported to date. It is not surprising since Pb is a heavy X-ray absorber, and the presence of the X anion in the channel further hinders the precise investigation of electron density distribution inside the channel. Here, we report electron density distribution inside the vacant channel in one of the lacunary apatites, Pb₄Na(PO₄)₃, after structure refinement with single-crystal X-ray diffraction. Stereochemical activity of the 6s² lone electron pair at the A2 site will be briefly discussed based on relationships found among unit-cell edge length a, species of the X anion, and sizes of A2 and O3 triangles in the subjected compound and selected apatite-type compounds hitherto reported.

2. Experimental

2.1. Sample preparation

Pb₄Na(PO₄)₃ (hereafter referred to as PNP) single crystals were prepared in two steps. A pre-grown powder of PNP was prepared by a flux/solid-state-reaction method. A mixture of PbO (99.5%; FUJIFILM Wako Pure Chemical Co., Japan), Bi₂O₃ (98.0%; Kanto Chemical Co., Inc., Japan), and NH₄H₂PO₄ (99.0%; FUJIFILM Wako) reagents were mixed to attain Pb, Bi and P in a 9:1:6 molar ratio. Bi source was mixed in the starting material to check if we could incorporate Bi³⁺ in the apatite structure. Na₂B₄O₇·10H₂O (99.0%; FUJIFILM Wako) was added at 10 wt% of the mixture as a flux and also a Na source; the molar ratio of Pb and Na in the starting material was 4:0.74. The mixture was ground, charged into a porcelain crucible and heated in air. The temperature was raised from room temperature to 850°C over 1 h, held at this temperature for 24 h and slowly lowered to 600°C over 24 h. After holding the temperature for 2 h, the furnace was cooled down by turning the power off. This pre-grown powdery specimen was washed with water to remove the soluble component. All powder X-ray diffraction peaks from the specimen could be successfully indexed with a hexagonal apatite cell and a trace amount of cubic Pb₃Bi(PO₄)₃ (Sahoo & Guru Row, 2010). A single-crystal specimen was grown from the melt of this pre-grown powdery specimen: the yielded powder was lightly ground, charged into a Pt capsule and heated in air. The temperature was raised from room temperature to 1100°C over 1 h and held at this temperature for 10 h. Then, the temperature was slowly lowered to 500°C over 120 h before turning the power off. Transparent crystals in euhedral shape (hexagonal prism with a hexagonal cap at one end) associated with a tiny amount of translucent white fragments were obtained. Weissenberg photographs of the transparent crystals indicated an aristotype apatite lattice of the crystal. The white fragments were confirmed as cubic Pb₃Bi(PO₄)₃ with powder X-ray diffractometry.

In contrast to a recent report of Pb_7.4Bi_0.3Na_2.3(PO₄)₆ (Hamdi et al., 2007), preliminary qualitative- and quantitative-chemical analyses (a JEOL JSM-6010LV scanning electron microscope with an integrated energy dispersive spectrometer) showed no sign of bis­muth nor boron in transparent crystals from the batch. The averaged atomic ratio of Pb:Na:P at four points on two crystals was 3.90 (15):1.08 (3):3.03 (6) with reference to 12 oxygen atoms. Non-occurrence of Na-deficiency in the crystal could be explained as the formation of Pb₃Bi(PO₄)₃ removed equal amounts of Pb and (PO₄) from the system and caused overabundance of Na and (PO₄) with respect to Pb in the system. Since the P:O ratio matched with that in the target composition and no Na-excess structure was possible for charge neutrality, we concluded at this stage that the single crystals from the batch had the target composition with no X-site anion nor point defect at cation sites.

2.2. Data collection and structure analysis

A single-crystal specimen was ground into a sphere of d = 195 µm. The intensities of Bragg reflections and values of θ were measured at room temperature on a Rigaku AFC-5S automated four-circle diffractometer with graphite-monochromatized Mo Kα radiation. The ω–2θ scan method with scan width 1.5° + 0.35tanθ and scan speed of 4° min⁻¹ was employed at the data collection. 19672 (19658 for space group P6₃/m) reflections in a full reciprocal sphere were measured up to 2θ = 90°. Space groups P31c and P3c1 were ruled out for violation of systematic absence of hh2hl for odd l and hh0l for odd l. Relationships found among Bragg positions and observed intensities suggested the Laue symmetry 6/m. There was no apparent scattering power at 000l with l ≠ 2n (n: integers) Bragg positions, which restricts the possible space group of the specimen to be P6₃/m or P6₃. Agreements among symmetry-equivalent reflections under 6/m (centrosymmetric) and 6 (noncentrosymmetric) point groups were virtually the same in spite of relatively large anomalous dispersion coefficients of Pb, and space group P6₃/m was employed in the following examinations. Thus, ordering of Na at the A1 site was ruled-out at this stage. The least-squares fitting of the peak positions of 11 intense reflections in the range 43.9° < 2θ < 47.9° resulted in the unit-cell edge lengths of a = 9.7345 (12) Å and c = 7.2130 (18) Å after calibration with Si (Okada & Tokumaru, 1984). See Table 1 for crystallographic details and further experimental conditions.

Table 1
Experimental details

Crystal data
Chemical formula	Pb4Na(PO4)3
Mr	1136.70
Crystal system, space group	Hexagonal, P63/m
Temperature (K)	294
a, c (Å)	9.7345 (12), 7.2130 (18)
V (Å^3)	591.93 (16)
Z	2
Dx (g cm^−3)	6.38
Radiation type	Mo Kα
μ (mm^−1)	57.316
Crystal shape	Sphere
Crystal radius (mm)	0.095
Data collection
Diffractometer	Rigaku AFC5S
Data collection method	ω–2θ scan
Scan speed (° min^−1)	4
Repeat scan	Up to 3 times until |Fobs| > 10σ(Fobs)
Absorption correction	Spherical
Tmin, Tmax	0.006, 0.009
No. of measured, independent and |Fobs| ≥ 3σ(Fobs) reflections	19658, 1729, 707
Rint for all reflections collected	0.101
(sin θ/λ)max (Å^−1)	0.995
Refinement
R(F), wR(F), S	0.017, 0.019, 1.56
No. of reflections	627
Rint for used reflections	0.033
No. of parameters	39
No. of restraints	3
Δρmax, Δρmin (e Å^−3)	2.64, −1.80
Computer programs: Rigaku control software, LSGCEX (Kihara, 1990), VESTA (Momma & Izumi, 2011).

The intensity data were converted to |F_obs| and their standard uncertainties (s.u.s: σ), after applying Lorentz, polarization and spherical absorption corrections (μr = 5.56). Averages over equivalent reflections for the 6/m point group were taken, and the averages which obeyed conditions |F_obs| ≥ 3σ(F_obs) and |F_obs|_max < 1.5|F_obs|_min among equivalents were used in the structure refinements with weights of σ⁻². The least-squares program LSGCEX (Kihara, 1990) was used for structure refinements with variables including one scale and one isotropic extinction factor [type I with the Lorentzian mosaic spread of Becker & Coppens (1974)]. Neutral form factors and a low-angle threshold for diffraction data (0.25 ≤ sinθ/λ) were employed after examinations noted in Okudera (2013). Neutral form factors for respective atoms and their anomalous dispersion terms were taken from International Tables for Crystallography, Vol. C.

The refinements started from the atomic coordinates and anisotropic displacement parameters (ADPs) given for Pb₅(PO₄)₃Cl [OP-4 in Okudera (2013)] but no X site. In the following calculations the ratio of Pb:Na was fixed as 4:1 and all atomic sites were assumed fully occupied based on the results of quantitative chemical analyses. As a first attempt, the A1 site was equi-partitioned by Pb and Na and the A2 site was assumed solely occupied by Pb as reported by Koumiri et al. (2000), and coordinates of atomic sites with ADPs were refined. The calculation converged at R(F), R(F²) and wR(F) = 0.024, 0.040, 0.028 for 627 independent reflections with 38 parameters. Next, occupation of the A2 site by Na was allowed under above-mentioned constraints and restraints. The calculation converged at R(F), R(F²) and wR(F) = 0.017, 0.030, 0.019 with 39 parameters, indicating that a certain amount of Na was located at the A2 site as was also reported by Toumi & Mhiri (2008). Minimum and maximum Δρ were −1.80 e Å⁻³ at (0.97, 0.69, 0.80) and 2.64 e Å⁻³ at (0.33, 0.66, 0.58), respectively. No particular positive residual density attributable to an X-site anion was found on the c axis. The ADPs at the A2 site were refined also with third-rank tensors after Gram–Charlier expansion formalism (Johnson & Levy, 1974) for a highly skewed coordination environment at the site. Absolute values of some of third-rank tensors exceeded three times of their s.u.s. The largest among their absolute values was −3.4 (4) × 10⁻⁵ for b²²² as expected from its coordination environment, namely, the absence of the X anion (Kuhs, 2003). Their absolute values, however, were small, and indeed no clear difference was found on Δρ maps after the refinements with and without those third-rank tensors. Details of structure refinement and refined structural parameters after the refinement with harmonic ADPs are given in Tables 1 and 2, respectively. Selected interatomic distances, bond angles, polyhedral volumes and bond-valence sums are given in Table 3. The residual density map after the refinement is shown in Fig. 2.

Table 2
Extinction factor, atomic coordinates, and anisotropic displacement parameters (Ų)

U₂₂ = U₁₁ and U₁₂ = 1/2U₁₁ at the A1 site, U₁₃ = U₂₃ = 0 at A1, A2, B, O1 and O2 sites.

Extinction factor	0.012 (6)
Site and site symmetry	A1 and 4f 3..
Occupancy Pb	0.557 (2)
Occupancy Na	0.443
x	1/3
y	2/3
z	0.00877 (9)
U11	0.01344 (16)
U33	0.0166 (3)
Site and site symmetry	A2 and 6h m..
Occupancy Pb	0.962
Occupancy Na	0.038
x	0.00130 (3)
y	0.25558 (3)
z	1/4
U11	0.01270 (11)
U22	0.01554 (12)
U33	0.01904 (12)
U12	0.00706 (9)
Site and site symmetry	B and 6h m..
Occupancy P	1
x	0.3996 (2)
y	0.3785 (2)
z	1/4
U11	0.0099 (6)
U22	0.0088 (6)
U33	0.0100 (6)
U12	0.0049 (5)
Site and site symmetry	O1 and 6h m..
Occupancy O	1
x	0.3232 (7)
y	0.4827 (7)
z	1/4
U11	0.025 (3)
U22	0.013 (2)
U33	0.017 (2)
U12	0.013 (2)
Site and site symmetry	O2 and 6h m..
Occupancy O	1
x	0.5204 (8)
y	0.1019 (8)
z	1/4
U11	0.019 (3)
U22	0.019 (3)
U33	0.071 (6)
U12	0.015 (2)
Site and site symmetry	O3 and 12i 1
Occupancy O	1
x	0.3479 (7)
y	0.2685 (6)
z	0.0808 (6)
U11	0.044 (3)
U22	0.0221 (18)
U33	0.0151 (17)
U12	0.023 (2)
U13	−0.0100 (17)
U23	−0.0072 (14)

Table 3
Selected interatomic distances (Å), bond angles (°), polyhedral volume (ų) and bond-valence sums

Bond-valence parameters were taken from Brown & Altermatt (1985) (B&A), Krivovichev & Brown (2001) (K&B) and Gagné & Hawthorne (2015) (G&H).

A1 site
A1–O1 (×3)	2.463 (6)
A1–O2 (×3)	2.717 (7)
A1–O3 (×3)	2.913 (7)
Mean value	2.696 (6)
BVS for Pb^2+	2.09 (B&A), 2.16 (K&B), 2.16 (G&H)
BVS for Na^+	0.91 (B&A), 0.91 (G&H)
A2 site
A2–O1	2.790 (6)
A2–O2	2.262 (13)
A2–O3 (×2)	2.612 (5)
A2–O3 (×2)	2.529 (5)
Mean value	2.556 (6)
BVS for Pb^2+	1.99 (B&A), 1.89 (K&B), 1.95 (G&H)
BVS for Na^+	0.86 (B&A), 0.83 (G&H)
B site
B–O1	1.528 (9)
B–O2	1.536 (10)
B–O3 (×2)	1.533 (5)
Mean value	1.533 (7)
BVS	5.03 (B&A)
O1–B–O2	111.2 (5)
O1–B–O3 (×2)	111.4 (3)
O2–B–O3 (×2)	108.5 (3)
O3–B–O3	105.4 (3)
Volume (Å^3)	1.84 (1)

Figure 2 Residual density, Δρ, map with atoms in the range −0.5 < x < 0.5, −0.5 < y < 0.5, −0.1 < z < 1.1. B and O1–O3 atoms are abridged for clarity. O3 triangles and A2 octahedra are drawn in red and grey, respectively. All displacement ellipsoids are drawn at the 70% probability level. Purple and blue tetrahedra are (PO4)^3− and (SiO4)^4− complex anions, respectively. Isosurface: ± 1.5 e Å^−3. Yellow: positive; pale blue: negative. (a) Pb4Na(PO4)3. (b) La9.33(SiO4)6O2 (Okudera et al., 2005). X-site oxide anions are shown in red. Drawn with VESTA (Momma & Izumi, 2011).

3. Results and discussion

3.1. Structure

The PNP specimen crystallized in an aristotype P6₃/m apatite host structure with a vacant channel. In contrast to the previously reported structure, 11.4% of total Na was located at the A2 site in the present specimen instead of only 2.1% in Toumi & Mhiri (2008). Calculated BVSs for Pb²⁺ and Na⁺ at the A2 site were 1.99 and 0.86 after Brown & Altermatt (1985), 1.95 and 0.83 after Gagné & Hawthorne (2015), and 1.89 for Pb²⁺ after Krivovichev & Brown (2001), indicating that the refined coordinates represented the position of Pb, and an actual position of Na would be a little closer to the O2-site position to use up its charge. The mean-square displacements (MSDs), 〈u²〉 (Ų), at the B site were nearly isotropic and the smallest among all atomic sites as in most of the structures hitherto reported. Refined ADP values at O sites were larger and more anisotropic than those in natural pyromorphite [Pb₅(PO₄)₃Cl], and those were closer to the values in natural vanadinite [Pb₅(VO₄)₃Cl] (Okudera, 2013) and artificial La_9.33(SiO₄)₆O₂ (Okudera et al., 2005). The libration mode of the (PO₄)³⁻ complex anion in PNP was an intermediate of roto-oscillation around the B—O1 bond in the above-mentioned natural lead chlorapatites and cradle-like motion with B–O2 as the unique axis in La_9.33(SiO₄)₆O₂. Since no particular mode was dominant, the refined shape of the BO₄ tetrahedron in PNP was fairly regular with a quadratic elongation index of 1.0014 and bond-angle variance of 5.73° despite its large libration amplitude. Displacement ellipsoids at A sites were slightly elongated (23% at A1 and 35% at A2) in [001], while their anisotropy was small. Virtually isotropic atomic displacements at the A1 site ascertained full occupation of the site by cations, otherwise the MSDs reflect local displacement of the A1-site cation (at 1/3, 2/3, z) toward one of its neighbouring A1-site positions (at 1/3, 2/3, z ± 1/2) when such a position is vacant as in the case of Nd_9.33(SiO₄)₆O₂ (Okudera et al., 2004) and La_9.33(SiO₄)₆O₂.

3.2. Residual density

Residual density, Δρ, is shown in Fig. 2 with the isosurface of 1.5 e Å⁻³. The structure of La_9.33(SiO₄)₆O₂ (Okudera et al., 2005) was re-refined for comparison purpose with a low-angle threshold 0.25 ≤ sinθ/λ and the same extinction formalism employed in the present study. The |F_obs|_max < 1.1|F_obs|_min threshold on structure refinement was kept unchanged. No notable difference was found on convergence of the least-squares cycles, refined structure and residual density from previously reported ones. Residual density after the calculation is also shown in Fig. 2.

There were some characteristic positive and negative residues in the vicinity of the A2 site position. Two accumulations of positive residue, separated on the xz plane for a mirror at z = 1/4, and two accumulations of negative residue, also separated from z = 1/4 but slightly out from the xz plane, were commonly found on PNP and La_9.33(SiO₄)₆O₂ despite absence of 6s² electrons at A2-site La³⁺ in the latter. The appearance of positive residues on the triad axis which run through the A1 site could not be ascribed to 6s² electrons at the site for the same reason. On the other hand, accumulation of positive residue at z = 1/4 in the vicinity of the A2 site was unique in PNP. Its maximum (2.61 e Å⁻³) was located at (−0.01, 0.19, 1/4), which was separated 0.6 Å from the A2 site position on the line to the centre of the A2 triangle, in contrast to speculations on an appearance of 6s² electrons at the position opposite to the A2—O2 bond (e.g. Mathew et al., 1980; Krivovichev et al., 2004). Indeed, the appearance of this accumulation had a striking resemblance to electron localization function in the vicinity of A2-site Pb²⁺ in hexagonal Pb₅(AsO₄)₃Cl after density functional theory calculations (Cametti et al., 2022). The peak of observed accumulation was found on the opposite side with respect to the line, but spatial difference was small. So, the residue had attributes expected for the 6s² electron cloud. More consideration, however, seemed necessary on its separation from the A2 site position. Similarly, prominent accumulation of positive residue was found at approximately 1.0 Å from the Bi core in Bi₂WO₆ (Okudera et al., 2018) despite expectedly close attractions on electrons from respective atom cores. Stereochemical activity of 6s² electrons will be nonetheless discussed in the following sections.

3.3. Unit-cell edge lengths and some characteristic values in PNP and chemically similar apatites

Some geometric parameter values in the PNP structure were compared first with those in closely related compounds, namely, normal and lacunary lead phosphate apatites (Table 4). Pb₅(PO₄)₃Cl (Okudera, 2013) and Pb₅(PO₄)₃Br (Liu et al., 2011) in Table 4 have the X anion at z ≃ 0 with BVS values of 1.25 and 1.50, respectively. Unit-cell edge lengths a and c of the present PNP specimen were the smallest among these compounds. Variations in c showed that the presence or absence, and even the size, of the X anion in the channel had little effect on unit-cell edge length c. Differences in volume of the A1O₉ polyhedron could be ascribed to differences in the sizes of Na⁺, Pb²⁺ and K⁺ at the A1 site in an increasing order (Shannon, 1976). A high population (25%) of point defects at the A1 site had little effect on the volume of the polyhedron in Pb₉(PO₄)₃, and the unit-cell edge length a followed the increasing order of the sizes of A1-site cations among these lacunary apatites.

Table 4
Unit-cell edge lengths (Å), polyhedron A1O₉ volume (ų), A2 and O3 triangle edge lengths (Å), and twist angle φ (°) of the A1O₆ trigonal metaprism in selected lead phosphate apatites

	a	c	v(A1O9)	d(A2–A2)	d(O3–O3)	φ	Reference
Pb4Na(PO4)3	9.7345 (12)	7.1230 (18)	37.45 (10)	4.2982 (7)	5.324 (9)	24.2 (4)	This study
Pb9(PO4)6	9.826 (4)	7.357 (3)	39.3 (3)	4.311 (3)	5.35 (3)	27.3 (12)	Hata et al. (1980)
Pb4K(PO4)3	9.827 (1)	7.304 (1)	40.19 (18)	4.372 (2)	5.402 (14)	28.8 (5)	Mathew et al. (1980)
Pb5(PO4)3Cl†	9.983 (1)	7.341 (1)	38.41 (13)	4.3525 (11)	5.643 (10)	18.0 (4)	Okudera (2013)
Pb5(PO4)3Br	10.0622 (5)	7.3575 (3)	39.70 (15)	4.5252 (16)	5.668 (14)	17.0 (4)	Liu et al. (2011)

†Averages of two natural specimens.

In contrast to the volume of the A1O₉ polyhedron, unit-cell edge length a, edge lengths of A2 and O3 triangles [d(A2–A2) and d(O3–O3)] in PNP and Pb₉(PO₄)₆ (Hata et al., 1980) are virtually the same. These two edge lengths are a little larger in Pb₉K(PO₄)₆, while differences were still small. d(A2–A2) in lacunary lead phosphate apatites are close also to that in Pb₅(PO₄)₃Cl. O3 triangles are apparently larger in Pb₅(PO₄)₃Cl and Pb₅(PO₄)₃Br, validating the repulsion of O3-site O²⁻ from the X anion at z ≃ 0. As twist angle φ of the A1O₆ trigonal metaprism indicated, these expansions are compensated by shift/rotation of the BO₄ unit and a cooperative untwist of the metaprism. Size relationships in a and d(O3–O3) are concordant with that in a and the radius of the X anions (e.g. Sudarsanan & Young, 1974). These two compounds, however, showed a contrast in expansion of the A2 triangle: increase in d(A2–A2) was 1% in Pb₅(PO₄)₃Cl and 5% in Pb₅(PO₄)₃Br with reference Pb₉(PO₄)₆. Intuitive understanding is that Br⁻ is too large to settle in the A2 octahedron of the Pb₅(PO₄)₃ host structure without expanding the channel whereas Cl⁻ fits in the A2 octahedron. However, the A2 octahedron in Pb₅(PO₄)₃Cl is still small for Cl⁻ as its BVS value indicated. The A2 octahedron can be larger also in Pb₅(PO₄)₃Cl to realize the ideal coordination environment for Cl⁻, and a larger A2 octahedron would also be beneficial for 6s² electrons. However, Cl⁻ is encapsulated in a small cavity in the Pb₅(PO₄)₃Cl structure. Therefore, 6s² electrons of Pb²⁺ at the A2 site should not be so active as to strongly affect the geometry of the host structure. Space-filling ability of those electrons would also be rather limited.

3.4. Stereochemical activity of 6s² electrons at A2-site Pb²⁺

The above consideration on d(A2–A2) supports the idea that the size of the A2 triangle is primarily a subordinate of the sizes of A1O₉ and BO₄ polyhedra: the apatitic host structure is made of an hcp array of BO₄ units with in-plane distortion primarily for the accommodation of the A1 cation in the smaller void, and face-sharing array of the A2 octahedron is inserted as a tube in resultant larger voids. To validate the idea, the sizes of A2 triangles were compared among chlor- and fluorapatites. Variation in size of the A1O₉ polyhedron in mixed-cation cases will be reflected in unit-cell edge length a. The relationship between a and d(A2–A2) in lead apatites will differ from those in other apatites when interference from 6s² electrons affects the structure.

Chlor- and fluorapatites (A2 = Ca, Pb, Sr and Ba, B = P, Mn, As and V) reported from 1971 onward were employed together with lacunary apatites and the 2 × c superstructure of Pb₁₀(PO₄)₃O (Hirano & Okudera, 2025) for the following comparison. The 2 × c superstructure of Pb₁₀(PO₄)₃O reported by Krivovichev & Engel (2023) was not used here for the reasons described in our previous study. When structures of solid-solution series were reported, only structures of end members were used for comparison. Up to 13% of heteroatom was allowed at the A2 site. Structures with multiple X sites are not used in this discussion. Unit-cell edge lengths, d(A2–A2) and the position of the X site in selected structures are listed in Table 5, together with the position and the BVS for the X anion after Brown & Altermatt (1985) for Ba²⁺—F⁻ and Ca²⁺—F⁻, and Brese & O'Keeffe (1991) for the other combinations. Variations in d(A2–A2) with unit-cell edge length a are shown in Fig. 3 with guide for the eye for each combination of A2 and X ions. As can be seen in Fig. 3, reported dependences of unit-cell edge length a on sizes of the A1 cation (e.g. Badraoui et al., 2006), the B cation (e.g. Reinen et al., 1986; Flis et al., 2010) and the X anion (e.g. Sudarsanan & Young, 1974; Piotrowski et al., 2002) held among compounds listed in Tables 4 and 5. BVS values for Cl⁻ exceeded its formal valence in all compounds and varied in the range 1.05∼1.40. Large d(A2–A2) values on calcium chlorapatites could be ascribed to the smaller size of Ca²⁺ under the inserted tube consideration. Despite large variations in BVS values, d(A2–A2) in all chlorapatites except A2 = Ca were found on single wide line. No characteristic feature was found on lead chlorapatites and the effect from the in-plane collision of 6s² orbitals, if any, should be small. Sizes of A2 triangles in lacunary apatites including PNP were found on the extension of the line on the smaller a side as if the size of the triangle is a simple function of a. It is interesting to note that the largest one, d(A2a–A2a), in Pb₁₀(PO₄)₆O also fell on the extension of this wide line: the size of the largest A2 triangle was not expanded as a counteraction for shrinkage of the other two triangles. Change in the size of the A2 triangle with a different X anion will be examined below by comparing its size in chlor- and fluorapatites.

Table 5
Unit-cell edge lengths (Å), A2 triangle edge length (Å), z-coordinate of the X site and BVS for the X anion in hitherto reported chlorapatites, fluorapatites and selected related apatites

z-coordinate of X site was taken at 0 ≤ z < 1/2. z < 1/4 when possible.

Composition	Method	A1	a	c	d(A2–A2)	z of X	BVS for X	Reference
Chlorapatites
A2 = Ca
Ca4.78Na0.22(PO4)3Cl0.78	Single	Ca	9.5773 (13)	6.8033 (6)	4.3149 (10)	0.0527 (6)	1.13	Matsuura & Okudera (2022)
Ca5(PO4)3Cl	Single	Ca	9.598 (2)	6.776 (4)	4.2736 (15)	0.0677 (4)	1.31	Hughes et al. (1989)
Ca5(PO4)3Cl_U†	Single	Ca	9.6233 (2)	6.7784 (3)	4.3229 (15)	0.060 (1)‡‡	1.18	Luo et al. (2009)
Ca5(PO4)3Cl	Single	Ca	9.628 (5)	6.764 (5)	4.2864 (19)	0.0562 (6)	1.23	Mackie et al. (1972) (averaged structure)
Ca5(PO4)3Cl_Th‡	Single	Ca	9.6330 (2)	6.7834 (2)	4.3229 (15)	0.059 (1)	1.20	Luo et al. (2009)
Ca5(AsO4)3Cl	Single	Ca	10.076 (1)	6.807 (1)	4.5446 (14)	0.1263 (5)	1.15	Wardojo & Hwu (1996)
Ca5(VO4)3Cl	Single	Ca	10.1490 (13)	6.7957 (18)	4.5865 (17)	0.1691 (6)	1.26	Matsuura & Okudera (2022)
A2 = Sr
Sr5(PO4)3Cl_Th§	Single	Sr	9.8562 (3)	7.2095 (4)	4.3531 (8)	0	1.27	Luo et al. (2009)
Sr5(PO4)3Cl	Single	Sr	9.859 (1)	7.206 (2)	4.3401 (5)	0	1.26	Sudarsanan & Young (1974)
Sr5(AsO4)3Cl	Rietveld	Sr	10.1969 (1)	7.28108 (9)	4.466 (11)	0	1.05	Bell et al. (2009)
Sr5(VO4)3Cl	Single	Sr	10.2047 (11)	7.3040 (5)	4.4530 (10)	0	1.05	Matsuura & Okudera (unpublished)
Sr5(VO4)3Cl	Rietveld	Sr	10.2073 (1)	7.3067 (1)	4.417 (6)	0	1.07	Beck et al. (2006)
A2 = Pb
Ca2Pb3(PO4)3Cl	Single	Ca	9.857 (1)	7.130 (2)	4.325 (2)	0	1.40	Kampf et al. (2006)
Pb5(PO4)3Cl	Single	Pb	9.977 (1)	7.351 (2)	4.3487 (18)	0	1.10	Dai & Hughes (1989)
Pb5(PO4)3Cl (OP-4)	Single	Pb	9.9791 (14)	7.3439 (11)	4.3493 (11)	0	1.25	Okudera (2013)
Pb5(PO4)3Cl (OP-1)	Single	Pb	9.9856 (10)	7.3318 (11)	4.3557 (10)	0	1.24	Okudera (2013)
Pb5(PO4)3Cl	Rietveld	Pb	9.9938 (1)	7.3397 (1)	4.3595 (9)	0	1.24	Flis et al. (2010)
Pb5(PO4)3Cl	Single	Pb	10.0017 (19)	7.3413 (16)	4.3620 (15)	0	1.23	Mills et al. (2012)
Ca2Pb3(AsO4)3Cl	Single	Ca	10.140 (3)	7.185 (4)	4.414 (5)	0	1.23	Rouse et al. (1984)
Pb5(AsO4)3Cl (OM-3)	Single	Pb	10.2382 (14)	7.4502 (12)	4.4154 (15)	0	1.10	Okudera (2013)
Pb5(AsO4)3Cl (OM-6)	Single	Pb	10.2396 (13)	7.4405 (24)	4.4358 (13)	0	1.11	Okudera (2013)
Pb5(VO4)3Cl	Single	Pb	10.250 (2)	7.454 (1)	4.426 (9)	0	1.08	Calos et al. (1990)
Pb5(AsO4)3Cl	Rietveld	Pb	10.2518 (2)	7.4482 (2)	4.4145 (11)	0	1.10	Flis et al. (2010)
Pb5(VO4)3Cl	Single	Pb	10.2990 (2)	7.3080 (1)	4.4347 (6)	0	1.14	Laufek et al. (2006)
Pb5(VO4)3Cl	Single	Pb	10.3090 (10)	7.3735 (17)	4.4675 (15)	0	1.06	Matsuura & Okudera (unpublished)
Pb5(VO4)3Cl	Single	Pb	10.315 (6)	7.337 (3)	4.3487 (18)	0	1.10	Dai & Hughes (1989)
Pb5(VO4)3Cl (OV-5)	Single	Pb	10.3217 (13)	7.3407 (13)	4.4530 (13)	0	1.10	Okudera (2013)
Pb5(VO4)3Cl (OV-1)	Single	Pb	10.3231 (9)	7.3399 (7)	4.4530 (13)	0	1.10	Okudera (2013)
A2 = Ba
Ba5(PO4)3Cl	Single	Ba	10.284 (2)	7.651 (3)	4.509 (3)	0	1.39	Hata et al. (1979)
Sr2Ba3(AsO4)3Cl	Single	Sr	10.390 (1)	7.575 (1)	4.5318 (12)	0	1.39	Đordevic et al. (2008)
Ba5(MnO4)3Cl	Single	Ba	10.469 (1)	7.760 (1)	4.571 (3)	0	1.23	Reinen et al. (1986)
Ba5(VO4)3Cl	Rietveld	Ba	10.5468 (1)	7.7437 (1)	4.591 (5)	0	1.22	Beck et al. (2006)
Ba5(VO4)3Cl	Single	Ba	10.5492 (13)	7.7534 (8)	4.5942 (7)	0	1.20	Matsuura & Okudera (unpublished)
Ba5(VO4)3Cl	Single	Ba	10.5565 (1)	7.7584 (1)	4.5982 (9)	0	1.19	Roh & Hong (2005)
Ba5(AsO4)3Cl	Rietveld	Ba	10.5570 (1)	7.73912 (8)	4.590 (10)	0	1.21	Bell et al. (2008)
Fluorapatites
A2 = Ca
(Ca0.879Mn0.121)5(PO4)3F0.74	Single	Ca/Mn	9.343 (2)	6.8227 (10)	3.9677 (13)	0.25	0.89	Hughes et al. (1991)
(Ca0.9582Mn0.0422)5(PO4)3F0.93	Single	Ca/Mn	9.3596 (10)	6.8603 (10)	3.9745 (11)	0.25	0.88	Hughes et al. (1991)
Ca5(PO4)3F	Single	Ca	9.367 (1)	6.884 (1)	3.8634 (10)	0.25	1.05	Sudarsanan et al. (1972)
Ca5(PO4)3F_U¶	Single	Ca	9.3709 (2)	6.8849 (2)	3.9625 (12)	0.25	0.90	Luo et al. (2009)
Ca5(PO4)3F_Th∥	Single	Ca	9.375 (2)	6.883 (3)	3.9632 (10)	0.25	0.90	Luo et al. (2009)
(Ca0.9704Sr0.0296)5(PO4)3F0.89	Single	Ca/Sr	9.379 (2)	6.8922 (7)	3.9827 (13)	0.25	0.87	Hughes et al. (1991)
(Ca0.9372Sr0.0628)5(PO4)3F	Single	Ca/Sr	9.390 (2)	6.9011 (8)	3.9822 (11)	0.25	0.87	Hughes et al. (1991)
Ca5(PO4)3F	Single	Ca	9.397 (3)	6.878 (4)	4.0025 (19)	0.25	0.84	Hughes et al. (1989)
Ca5(AsO4)3F	Rietveld	Ca	9.6873 (5)	6.9815 (3)	4.167 (3)§§	0.245 (2)	0.83	Baikie et al. (2007)
Ca5(VO4)3F	Rietveld	Ca	9.6960 (3)	7.0170 (2)	4.03 (3)‡	0.252 (2)	0.80	Baikie et al. (2007)
Ca5(VO4)3F	Rietveld	Ca	9.7371 (1)	7.0063 (1)	4.096 (6)	0.25	0.73	Dong & White (2004)
A2 = Sr
(Na,Ce)2Sr3(PO4)3F#	Single	Na/Ce	9.659 (1)	7.182 (1)	4.1343 (12)	0.211 (1)‡	1.04	Rakovan & Hughes (2000)
Sr5(PO4)3F	Single	Sr	9.678 (3)	7.275 (5)	4.146 (3)	0.25	1.09	Swafford & Holt (2002)
Sr5(PO4)3F_Th††	Single	Sr	9.7038 (4)	7.2723 (7)	4.1593 (8)	0.25	1.07	Luo et al. (2009)
(Sr0.992Nd0.005)5(PO4)3F	Single	Sr	9.7156 (4)	7.2810 (3)	4.1505 (15)	0.25	1.08	Corker et al. (1995)
Sr5(PO4)3F	Rietveld	Sr	9.7211 (2)	7.2869 (1)	4.160 (2)	0.2486 (3)	1.07	Aissa et al. (2004)
PbSr9(PO4)3F2	Rietveld	Sr	9.7268 (3)	7.2871 (1)	4.167 (3)	0.226 (3)	1.04	Badraoui et al. (2006)
Sr5(PO4)3F	Single	Sr	9.845 (7)	7.383 (4)	4.211 (5)	0.25	0.98	Pekov et al. (2010)
Sr5(AsO4)3F	Single	Sr	9.990 (1)	7.395 (1)	4.2139 (11)	0.25	0.98	Đordevic et al. (2008)
(Sr0.982Nd0.012)5(VO4)3F	Single	Sr	10.0077 (6)	7.434 (4)	4.2349 (15)	0.25	0.95	Corker et al. (1995)
Sr5(VO4)3F	Rietveld	Sr	10.01267 (7)	7.43169 (4)	4.246 (3)	0.220 (3)	0.91	Oka et al. (2022)
A2 = Pb
Ca2Pb3(PO4)3F	Single	Ca	9.6402 (12)	7.0121 (8)	3.9298 (8)	0	0.62	Kampf & Housley (2011)
Pb5(PO4)3F	Single	Pb	9.7638 (6)	7.2866 (4)	3.9526 (19)	0.039 (4)	0.59	Fleet et al. (2010)
Pb9Sr(PO4)6F2	Rietveld	Pb	9.7662 (4)	7.2929 (2)	3.987 (6)	0.051 (3)	0.60	Badraoui et al. (2006)
Pb3(VO4)3F	Rietveld	Pb	10.10647 (15)	7.35582 (8)	4.1000 (3)	0.052 (3)	0.90	Oka et al. (2022)
A2 = Ba
Ba5(PO4)3F	Single	Ba	10.153 (2)	7.733 (3)	4.383 (3)	0.219 (5)	1.16	Mathew et al. (1979)
Ba5(PO4)3F	Rietveld	Ba	10.1611 (2)	7.7322 (1)	4.389 (4)	0.2019 (7)	1.10	Aissa et al. (2004)
Ba5(VO4)3F	Rietveld	Ba	10.43080 (5)	7.86002 (4)	4.468 (3)	0.223 (3)	1.02	Oka et al. (2022)
Lacunary and superstructure apatites
Pb7.36Bi0.32Na2.08Li0.24(PO4)6	Rietveld	Pb/Na	9.6916 (8)	7.1751 (7)	4.2639 (11)			Hamdi et al. (2007)
Pb7.4Bi0.3Na2.3(PO4)6	Rietveld	Pb/Na	9.7065 (7)	7.1705 (6)	4.3026 (9)			Hamdi et al. (2007)
Pb10(PO4)6O	Single	Pb	9.8151 (15)	14.8458 (11)	4.359 (3)			Hirano & Okudera (2025); d(A2a–A2a)
					4.128 (6)			Hirano & Okudera (2025); d(A2b–A2b)
					3.781 (3)			Hirano & Okudera (2025); d(A2c–A2c)

†(Ca_0.994U_0.006)₂(Ca_0.993U_0.007)₃(PO₄)₃Cl.
‡(Ca_0.977Th_0.023)₂(Ca_0.979Th_0.021)₃(PO₄)₃Cl.
§(Sr_0.985Th_0.015)₂(Sr_0.990Th_0.010)₃(PO₄)₃Cl.
¶Ca₂(Ca_0.986U_0.014)₃(PO₄)₃F.
∥(Ca_0.999)₂(Ca_0.987Th_0.013)₃(PO₄)₃F.
#(Na_2.12Ce_1.18La_0.64Nd_0.34)(Sr_5.74Ba_0.22)(P_6.06Si_0.22)O₂₄(F_1.96OH_0.02Cl_0.02).
††(Sr_0.993Th_0.007)₂(Sr_0.993Th_0.007)₃(PO₄)₃F.
‡‡From CIF.
§§Averaged d(A2–A2) in P1 structure.

Figure 3 Variations in edge length of the A2 triangle with unit-cell edge length a in fluor-, chlor- and lacunary apatites together with those in Pb10(PO4)6O. Solid circles with solid lines: chlorapatites; solid triangles with dashed lines: fluorapatites; blue open squares: lacunarly apatites; black solid squares: three A2 triangles with different sizes in Pb10(PO4)6O. Purple: A2 = Ba; blue: A2 = Pb; green: A2 = Sr; red: A2 = Ca. Solid and dashed lines are drawn for guides for the eye after a least-squares fit for each combination of A2 and X ions.

The d(A2–A2) values in barium fluorapatites are also found on the wide line mentioned above. The unit-cell edge length a was reduced with change of X from Cl⁻ to F⁻ as it was expected from the reduction of the repulsion on O3-site O²⁻ from the X anion. Reduction in the d(A2–A2) values with the exchange was small but common among barium apatites and was approximately 0.12 Å. As positions of F⁻ with BVS values close to 1.0 indicated, F⁻ anions were located at their ideal positions in barium fluorapatites. This small but noticeable difference on d(A2–A2) indicated forced expansion of the Ba²⁺ octahedron due to repulsion from Cl⁻ and in fact how small it was. This difference was doubled in strontium apatites with BVS values for F⁻ in the range 0.9∼1.1 at z = 1/4. A gap between guides for the eye (Fig. 3) for barium and strontium fluorapatites indicated that the fluoride anion at the centre of the A2 triangle shrunk the triangle in the latter to make its coordination environment ideal as happened in the structures of Ca₅(AsO₄)₃F and Ca₅(VO₄)₃F (Baikie et al., 2007) and that this effect was larger on the larger d(A2–A2) side. This simple interpretation, however, did not apply to lead fluorapatites. As the residual density map for the PNP structure suggested, the 6s² electron cloud could be directional and located inside the A2 triangle (Fig. 2). The far smaller d(A2–A2) on lead fluorapatites than their strontium analogues with the out-of-plane position of the X anion indicated that attraction on Pb²⁺ from F⁻ shrunk the A2 triangle and that 6s² electron clouds at A2-site Pb²⁺ repelled F⁻ from the centre of the triangle. The size of the smallest A2 triangle in the Pb₁₀(PO₄)₆O structure indicated how much the Pb²⁺ triangle could shrink even with 6s² electrons inside. However, the observed relationship between a and d(A2–A2) in lacunary apatites indicated that the triangle was not shrunk for lack of a large X anion. Simple and the same trends on lacunary apatites and chlorapatites indicated the subordinate nature on the size of the A2 octahedron to the (A1O₉)—(BO₄) framework which defines the maximum size of the A2 triangle. A2 cations stick on the periphery of the larger void as `decorations on the wall' with reference to Krivovichev et al. (2004) in the distorted hexagonal net of (BO₄)³⁻ complex anions, while they are easily pulled by the X anion for attraction.

In summary, the change in size of the A2 triangle was found primarily as a function of unit-cell edge length a. As BVS values for Cl⁻ indicated, the A2 octahedron was hard to expand by repulsion from the X anion, In other words, the anion channel was a face-sharing array of A2 octahedra inserted in the (A1O₉)—(BO₄) framework with geometric restraints, as suggested by Mercier et al. (2005). On the other hand, the triangle could be shrunk rather easily by attraction from the X anion. Repulsion among directional 6s² electron orbitals might exist inside of the anion channel in lead-bearing apatites, but this repulsion had no effect on unit-cell edge length a nor the size of the A2 triangle. However, stereochemical activity of the A2-site 6s² electrons was still potent enough to repel the X anion from the centre of the A2 triangle at z = 1/4. In this sense, directional X-anion conduction through the channel is hard to realize on lead apatites, irrespective of A1, B and X ions.

Supporting information

Computing details

(I)

Crystal data

NaO12P3Pb4	Dx = 6.378 Mg m^−3
Mr = 1136.70	Mo Kα radiation, λ = 0.71069 Å
Hexagonal, P63/m	Cell parameters from 11 reflections
Hall symbol: -P 6 c	θ = 43.9–47.9°
a = 9.7345 (12) Å	µ = 57.32 mm^−1
c = 7.2130 (18) Å	T = 294 K
V = 591.93 (16) Å^3	Sphere, colourless
Z = 2	0.20 × 0.20 × 0.20 × 0.10 (radius) mm
F(000) = 960

Data collection

Rigaku AFC5S diffractometer	707 reflections with F > 3σigma(F) & (|Fobs.max|−|Fobs.min|)/|Fobs.min| < 0.5 among equivalents
Radiation source: X-ray tube, 50kV 25mA	Rint = 0.101
Graphite (002) monochromator	θmax = 45.0°, θmin = 3.7°
integrated with \v/2θ scans	h = −19→19
Absorption correction: for a sphere spherical absorption correction	k = −19→19
Tmin = 0.006, Tmax = 0.009	l = −14→14
19658 measured reflections	3 standard reflections every 200 reflections
1729 independent reflections	intensity decay: none

Refinement

Refinement on F	9 constraints
Least-squares matrix: full	Weighting scheme based on measured s.u.'s weight proportional to sigma^-2
R[F^2 > 2σ(F^2)] = 0.017	(Δ/σ)max < 0.001
wR(F^2) = 0.019	Δρmax = 2.64 e Å^−3
S = 1.56	Δρmin = −1.80 e Å^−3
627 reflections	Extinction correction: B-C type 1 Lorenzian isotropic
39 parameters	Extinction coefficient: 0.012 (6)
3 restraints

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
Pb1	0.33333	0.66667	0.0088 (9)	0.0145	0.56 (2)
Na1	0.33333	0.66667	0.0088	0.0145	0.44
Pb2	0.00130 (3)	0.25558 (4)	0.25000	0.0158	0.96
Na2	0.00130	0.25558	0.25000	0.0158	0.04
P	0.3996 (2)	0.3785 (2)	0.25000	0.0095
O1	0.3232 (7)	0.4827 (7)	0.25000	0.017
O2	0.5204 (8)	0.1018 (9)	0.25000	0.034
O3	0.3479 (7)	0.2685 (6)	0.0808 (6)	0.024

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb1	0.01344 (16)	0.01344	0.0166 (3)	0.00675	0.00000	0.00000
Na1	0.01344	0.01344	0.0166	0.00675	0.00000	0.00000
Pb2	0.01270 (11)	0.01554 (12)	0.01904 (12)	0.00706 (9)	0.00000	0.00000
Na2	0.01270	0.01554	0.01913	0.00706	0.00000	0.00000
P	0.0099 (6)	0.0088 (6)	0.0100 (6)	0.0049 (5)	0.00000	0.00000
O1	0.025 (2)	0.014 (2)	0.017 (2)	0.013 (2)	0.00000	0.00000
O2	0.019 (3)	0.019 (3)	0.071 (6)	0.015 (2)	0.00000	0.00000
O3	0.044 (3)	0.0221 (18)	0.0151 (17)	0.023 (2)	−0.0100 (17)	−0.0072 (14)

Geometric parameters (Å, º)

Pb1—O1	2.463 (6)	Pb2—O3^ii	2.612 (5)
Pb1—O2^i	2.717 (7)	Pb2—O3^i	2.529 (5)
Pb1—O3^i	2.913 (7)	P—O1	1.528 (9)
Pb2—O1	2.790 (6)	P—O2^iii	1.536 (10)
Pb2—O2^ii	2.262 (13)	P—O3	1.533 (5)
O1—P—O2^iii	111.2 (5)	O2^iii—P—O3	108.5 (3)
O1—P—O3	111.4 (3)	O3—P—O3	105.5 (3)

Symmetry codes: (i) x−y, x, −z; (ii) −y, x−y, z; (iii) −x+y+1, −x+1, z.

 

Footnotes

‡Present address: R&D centre, Noritake Co., Limited, Japan.