K-site splitting in KTiOPO₄ at room temperature

The room-temperature structure of potassium titanyl phosphate (KTiOPO₄, KTP) with Pna2₁ symmetry has been studied by means of synchrotron radiation. Each of the two crystallographically unique K1 and K2 cations is split over two sites that are shifted along the c direction by 0.287 (13) and 0.255 (13) Å for the K1a/b and K2a/b pairs, respectively. The refined populations of the minor K1b and K2b sites are 0.102 (12) and 0.132 (17), respectively. It is shown that accurate high-resolution synchrotron data (R_merged = 0.015 for 25 010 reflections, 9456 unique, sinθ/λ limit > 1.0) are required for the determination of a reliable structure model.

S1. Comment

The KTiOPO₄ (KTP, potassium titanyl phosphate) structure was determined by Tordjman et al. (1974). The highly asymmetric bonds in the non-centrosymmetric structure indicated its potential as a nonlinear optical (NLO) material, as later confirmed by Zumsteg et al. (1976). Since then, KTP has become one of the best known materials for various NLO applications (Bierlein, 1989; Stucky et al., 1989) and it is the material of choice in applications that utilize the second harmonic generating effect for laser-frequency doubling (Boulanger et al., 1994). The transparent crystalline material has favourable material properties, such as a high thermal stability, chemical stability, high optical nonlinearity and a high optical damage threshold. Additionally, the crystals have a wide optical transmission window that covers both the UV and IR spectra. These properties are also found in some isostructural ATiOBO₄ materials, particularly for those with A = K, Rb, Cs or Tl and B = P or As.

KTP has a well known structure which, upon heating, undergoes a reversible ferroelectric (space group Pna2₁) to paraelectric (space group Pnan) phase transition at ~1207 K (Stefanovich et al., 1996). The exact temperature depends on variations in growth conditions, such as the K/P ratio in the flux (Angert et al., 1995). Belokoneva et al. (1997) have described the temperature-induced phase transition as being both displacive and of an order–disorder nature, unlike the purely displacive phase transition under high pressure at ambient temperature.

Belokoneva et al. (1990) were the first to report on alkali site splitting in KTP isostructures. They refined split sites for the K cations in KFeFPO₄ and mentioned that a similar phenomenon existed for KTP. However, no details were provided of the data collection or the structural parameters for KTP in either their paper or the Inorganic Crystal Structure Database (ICSD, 2001), except that the minor K split positions have a population of 0.1. The description of alkali hole sites and framework pseudo-symmetry for KTiOPO₄ isostructures was later discussed and improved by Thomas et al. (1990) and Thomas & Glazer (1991). More recent investigations relating to the temperature dependence of the K cation site behaviour were carried out by Delarue et al. (1998, 1999) and Norberg and co-workers (Norberg, Sobolev & Streltsov, 2003; Norberg, Gustafsson & Mellander, 2003). The latter studies resulted in a general description of the relationship between the temperature and the structural behaviour of the alkali sites.

The splitting of alkali sites has also been found at room temperature in several KTP isostructures, e.g. RbTiOAsO₄ (Streltsov et al., 2000), CsTiOAsO₄ (Nordborg, 2000), and RbSbOGeO₄ (Belokoneva et al., 1997). It has, on the other hand, only been observed in KTP at temperatures of 673 K and above (Delarue et al., 1999). The thorough data collected by Delarue et al. at 273, 673 and 973 K revealed the splitting of K sites at both 673 K and 973 K but not at 273 K, even though their published residual electron-density maps suggest the presence of splitting. The very detailed structural investigations of KTP at room temperature by Hansen et al. (1991) and Dahaoui et al. (1997) did not reveal any site splitting either, although these studies were mainly concerned with the charge density in the TiO₆ octahedra.

We reinvestigated the KTP structure using high-resolution synchrotron data, with the specific purpose of confirming the earlier results reported by Belokoneva et al. (1990). In this paper, we report that each of KTP's crystallographically independent K cations is undoubtedly split over two nearby sites at room temperature. Reasons for why this has been missed in previous studies of the KTP structure are discussed by analyzing the correlation between the quality of the diffraction data and the details of the refined structure.

The data were refined utilizing the Xtal3.7 software package (Hall et al., 2000), first as KTP without any K site splitting and with atomic parameters from Norberg et al. (2000) (model I). This model converged to R = 0.021, wR = 0.027, S = 1.078 and residual electron densities Δρ_max/Δρ_min of 1.56/-0.91 e Å⁻³ [σ(Δρ) = 0.09 e Å⁻³]. The resulting residual electron-density maps were extremely flat and clean of peaks except close to each K site. Peaks of 1.56 e Å⁻³ and 1.31 e Å⁻³ were found 0.496 (1) Å from K1 and 0.474 (1) Å from K2, respectively, both corresponding to splitting along the c direction (Figs. 1a and 1b). The residual electron-density peaks in model I prompted a second refinement with split K sites (model II). This model converged to R = 0.019, wR = 0.025, S = 0.996 and residual electron densities Δρ_max/Δρ_min of 1.02/-0.64 e Å⁻³ [σ(Δρ) = 0.09 e Å⁻³]. The minor K1b and K2b sites are located 0.287 (13) and 0.255 (13) Å from the original K1a and K2a sites, with occupancies of 0.102 (12) and 0.132 (17), respectively. The surrounding residual electron densities are as shown in Figs. 1c and 1d.

The K1a and K1b cations occupy slightly smaller cation cages and have rather irregular coordination polyhedra with all shorter K—O bonds roughly parallel to the (001) plane. The K1a coordination environment includes four shorter bonds [between 2.7187 (14) and 2.7593 (14) Å] and four longer bonds [between 2.858 (2) and 3.0425 (18) Å]. Cation K1b is shifted from K1a along the positive c direction and is, thus, barely coordinated to six O atoms with four shorter bonds [between 2.657 (7) and 2.764 (7) Å] and two longer bonds of 3.034 (10) and 3.122 (14) Å. Displacement ellipsoid plots of the K1O_x polyhedra are shown in Fig. 2. The K2O_x polyhedra are similar.

Our results confirm the presence of split K sites in KTP at room temperature. Energy-dispersive X-ray analysis did not indicate any contamination by other alkali cations and the coordination environments of the split sites do not correspond to other alkali metals such as Na, for instance, which would require much shorter Na—O bond lengths. The splitting of K sites has also been confirmed by another data collection and structure refinement carried out on a KTP crystal from a different crystal growth experiment. This second refinement [R_int of 0.042 due to a shorter scan time, R = 0.031, wR = 0.041, S = 1.003, Δρ_max/Δρ_min of 0.81/-0.66 e Å⁻³, σ(Δρ) = 0.11 e Å⁻³] yielded almost exactly the same structure. The K1b site is shifted by 0.266 (16) Å from the K1a site with an occupancy of 0.121 (19), and K2b is likewise located 0.241 (17) Å away from K2a with an occupancy of 0.15 (2). All features of the residual electron-density map are also remarkably similar to those in the KTP structure reported in this paper.

It should be noted that an alternative to a split-cation model is an anharmonic model using a Gram–Charlier expansion series for the modelling of thermal displacement parameters. However, Delarue et al. (1999) concluded that such a single-site model is not adequate for KTP, and that a more sophisticated model with split K sites, with the distance between the split sites decreasing to zero at 0 K, is needed.

The earlier failures correctly to characterize the K site splitting in KTP are either due to a low resolution or to a lack of intensity in the diffraction data, or a combination of both. KTP isostructures are characterized by a remarkably large number of weak reflections (Streltsov et al., 2000), so studies based on X-ray sealed-tube data will certainly suffer from the low statistical significance of the weak reflections. The effect of limited resolution was investigated in the present work by refining the diffraction data using sinθ/λ limits of 0.6, 0.7, 0.8, 0.9, and 1.0, respectively. The results show conclusively that using data with a limited resolution greatly decreases the likelihood of successfully identifying the split K sites (Fig. 3). For instance, a sinθ/λ limit of 0.6 (2θ_max = 53.5°) allows the refinement of a structure model with unsplit K sites to a final R value of 0.011, without any disturbing residual electron-density peaks (Fig. 3a).

The effect of different sinθ/λ limits on the refinement of the structure model with unsplit K sites (model I) is shown in Table 2. The significant difference between |Δρ_max| and |Δρ_min| appearing around sinθ/λ = 0.7 suggests the failure of this model. However, these data still do not allow a reliable refinement of the split sites. As described in Table 3, the splitting of K sites (model II) is only correctly refined with a resolution of sinθ/λ = 1.0 and a higher resolution seems to yield little improvement to the structure model. However, it should be noted that our data are not only of high resolution (2θ_max = 110.02°) but also of good statistical quality, as indicated by the low R_int value of 0.015. It is likely, therefore, that data with less accurate statistics will need an even higher resolution for the correct identification of the K site splitting in KTP.

The recent investigation of split alkali sites in RbTiOAsO₄ by Streltsov et al. (2000) was also carried out by means of synchrotron radiation at room temperature. The Rb1a—Rb1b and Rb2a—Rb2b distances of 0.312 (3) and 0.233 (4) Å, respectively, are similar to those in the present report. Further investigations of KTP isostructures using synchrotron data are likely to result in more structures with accurately determined alkali site splitting at room temperature. Previous determinations of alkali site splitting might also benefit from reinvestigation, since we have demonstrated the importance of high-resolution synchrotron data for such studies. Finally, it is noteworthy that no charge-density analysis of the KTP structure has taken the splitting of K sites into account, and such structural disorder might be of value to improve current charge-density models.

S2. Experimental

Single crystals of KTP were grown by spontaneous nucleation in a self-flux containing TiO₂, KH₂PO₄ and K₂HPO₄ in the molar ratio of 1:3:2. The chemicals were thoroughly mixed in a 35 ml platinum crucible and heated to 1373 K over 1 d and held at that temperature for 48 h. The temperature was then lowered to 973 K at 3.33 K h⁻¹ and then quickly to room temperature. Water was used to dissolve the flux and transparent colourless crystals (from µm to mm in size) were recovered. Multiple crystals were analysed using an energy-dispersive X-ray spectrometer (Jeol JED-2001 and JSM-6100) and no sample contamination was observed.

S3. Refinement

The Flack (1983) parameter refined to 0.426 (1) for both models, indicating that the crystal contained a mixture of domains with opposite polarization.

Computing details

Data collection: DIFF14A (Vaalsta & Hester, 1997); cell refinement: LATCON in Xtal3.7 (Hall et al., 2000); data reduction: DIFDAT, ADDREF, SORTRF and ABSORB in Xtal3.7; program(s) used to solve structure: atomic parameters from Norberg et al. (2000); program(s) used to refine structure: CRYLSQ in Xtal3.7; molecular graphics: DIAMOND (Brandenburg, 2001), and FOURR, SLANT and CONTRS in Xtal3.7; software used to prepare material for publication: BONDLA, ATABLE and CIFIO in Xtal3.7.

Figures

	Fig. 1. Δρ maps in the (010) plane through the K1 and K2 sites for Model I with unsplit sites (a, c) and Model II with split sites (b, d). Contour intervals are 0.2 e Å^−3 [σ(Δρ) = 0.08 e Å^−3]. The positive and negative contours are shown as solid and short dashed lines, respectively. Map borders are 4 × 4 Å.
	Fig. 2. (a) The K1a and (b) the K1b coordination polyhedra, with displacement ellipsoids drawn at the 80% probability level. Symmetry codes are as given in Table 1.
	Fig. 3. Δρ maps in the (010) plane through the K2 site refined as Model I with sinθ/λ limits of 0.6 (a), 0.7 (b), 0.8 (c), 0.9 (d), ?.? (e) and 1.0 (f). Contour intervals are 0.2 e Å^−3 [σ(Δρ) = 0.08 e Å^−3]. The positive and negative contours are shown as solid and short dashed lines, respectively. Map borders are 4 × 4 Å. Please provide information for part (e).

potassium titanyl phosphate

Crystal data

K2O10P2Ti2	F(000) = 768
Mr = 395.88	Dx = 3.026 Mg m^−3
Orthorhombic, Pna21	Synchrotron radiation, λ = 0.75052 (1) Å
Hall symbol: p 2c -2n	Cell parameters from 17 reflections
a = 12.816 (3) Å	θ = 27.9–49.0°
b = 6.4045 (5) Å	µ = 3.62 mm^−1
c = 10.5889 (8) Å	T = 295 K
V = 869.2 (2) Å^3	Rectangular, colourless
Z = 4	0.04 × 0.04 × 0.03 mm

Data collection

Bl14a 4-circle diffractometer	9211 reflections with F > 2σ(F)
Radiation source: Photon Factory bl14a	Rint = 0.015
Si(111) monochromator	θmax = 55.0°, θmin = 3.4°
ω–2θ scans	h = −27→27
Absorption correction: analytical (Alcock, 1974)	k = −13→9
Tmin = 0.943, Tmax = 0.978	l = −23→23
25010 measured reflections	4 standard reflections every 200 reflections
9456 independent reflections	intensity decay: none

Refinement

Refinement on F	w = 1/[σ^2(F) + 0.02(F)^2]
Least-squares matrix: full	(Δ/σ)max = 0.001
R[F^2 > 2σ(F^2)] = 0.019	Δρmax = 1.02 (8) e Å^−3
wR(F^2) = 0.025	Δρmin = −0.64 (8) e Å^−3
S = 1.02	Extinction correction: isotropic Gaussian, Zachariasen, 1967; Larson (1970), Eq. 22 p. 292
9211 reflections	Extinction coefficient: 7400 (400)
154 parameters	Absolute structure: Flack (1983), with how many Friedel pairs
0 restraints	Absolute structure parameter: 0.426 (11)
14 constraints

Special details

Refinement. The following constraints were used during the refinement:

H-atom parameters constrained u11(K1b)=0.0 + 1.0*u11(K1a) H-atom parameters constrained u22(K1b)=0.0 + 1.0*u22(K1a) H-atom parameters constrained u33(K1b)=0.0 + 1.0*u33(K1a) H-atom parameters constrained u12(K1b)=0.0 + 1.0*u12(K1a) H-atom parameters constrained u13(K1b)=0.0 + 1.0*u13(K1a) H-atom parameters constrained u23(K1b)=0.0 + 1.0*u23(K1a) H-atom parameters constrained pop(K1b)=1.0–1.0*pop(K1a) H-atom parameters constrained u11(K2b)=0.0 + 1.0*u11(K2a) H-atom parameters constrained u22(K2b)=0.0 + 1.0*u22(K2a) H-atom parameters constrained u33(K2b)=0.0 + 1.0*u33(K2a) H-atom parameters constrained u12(K2b)=0.0 + 1.0*u12(K2a) H-atom parameters constrained u13(K2b)=0.0 + 1.0*u13(K2a) H-atom parameters constrained u23(K2b)=0.0 + 1.0*u23(K2a) H-atom parameters constrained pop(K2b)=1.0–1.0*pop(K2a)

No restraints used.

A small transparent crystal, with KTP morphology as described by Bolt & Bennema (1990), was used for the data collection at Photon Factory, Tsukuba, Japan, using the beamline 14 A four-circle diffractometer (Satow & Iitaka, 1989). Vertically polarized X-ray radiation from a vertical wiggler was monochromated by a double Si(111) perfect crystal monochromator, and focused using a curved fused-quartz mirror coated with Pt. The beam optics are automatically adjusted every 20 min for maximum flux. A high-speed avalanche photodiode detector with counting linearity up to 10⁸ c.p.s. was used (Kishimoto et al., 1998). The experimental set-up was similar to that described by Streltsov et al. (1998). Diffraction intensities were measured at room temperature with λ = 0.75052 (1) Å using ω/2θ continuous time scans with an ω-scan width of 0.30°. The collected X-ray intensities were corrected for experimental intensity variations as indicated by measured standard reflections, and for absorption using an analytical model (Alcock, 1974). Anomalous scattering factors were taken from Sasaki (1989) and the linear absorption coefficient µ was calculated using mass attenuation coefficients for neutral atoms at 0.75 Å (Sasaki, 1990). An isotropic extinction parameter (Zachariasen, 1967) using Larson's implementation (Larson, 1970) was refined for both models and about 1.3% of the reflections were affected by extinction. The maximum correction of y = 0.94 was used for the 800 reflection (the observed structure factor is F_obs = yF_kin, where F_kin is the kinematic value).

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
K1a	0.37733 (7)	0.21992 (9)	0.3120 (2)	0.0165 (2)	0.898 (12)
K1b	0.3847 (6)	0.2141 (9)	0.3374 (13)	0.0165 (2)	0.102 (12)
K2a	0.10551 (5)	0.30033 (12)	0.0658 (3)	0.0175 (2)	0.868 (17)
K2b	0.1040 (4)	0.3034 (10)	0.0897 (12)	0.0175 (2)	0.132 (17)
Ti1	0.372929 (11)	0.50000 (2)	0.00206	0.00513 (4)
Ti2	0.246673 (11)	0.73058 (2)	0.25338 (2)	0.00508 (4)
P1	0.49807 (2)	0.66366 (4)	0.26196 (3)	0.00518 (7)
P2	0.18085 (2)	0.49797 (4)	0.51447 (3)	0.00545 (7)
O1	0.48579 (7)	0.51335 (14)	0.15194 (8)	0.0092 (2)
O2	0.51002 (6)	0.53487 (15)	0.38504 (8)	0.0091 (2)
O3	0.40024 (6)	0.80068 (13)	0.28127 (8)	0.0079 (2)
O4	0.59349 (6)	0.80668 (13)	0.24263 (8)	0.0086 (2)
O11	0.27538 (6)	0.53332 (13)	0.14527 (8)	0.0081 (2)
O12	0.22371 (7)	0.95876 (14)	0.39198 (8)	0.0088 (2)
O5	0.11247 (7)	0.68910 (12)	0.54297 (8)	0.0084 (2)
O6	0.11138 (7)	0.30838 (13)	0.48910 (9)	0.0094 (2)
O7	0.25274 (7)	0.46062 (14)	0.62981 (8)	0.0092 (2)
O8	0.25297 (7)	0.53914 (14)	0.40123 (8)	0.0091 (2)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
K1a	0.02199 (15)	0.01008 (9)	0.0174 (4)	−0.00383 (9)	0.00435 (19)	−0.00060 (13)
K1b	0.02199 (15)	0.01008 (9)	0.0174 (4)	−0.00383 (9)	0.00435 (19)	−0.00060 (13)
K2a	0.01293 (9)	0.02021 (12)	0.0194 (5)	−0.00499 (8)	0.00049 (17)	0.0007 (2)
K2b	0.01293 (9)	0.02021 (12)	0.0194 (5)	−0.00499 (8)	0.00049 (17)	0.0007 (2)
Ti1	0.00545 (4)	0.00508 (4)	0.00485 (4)	0.00012 (3)	−0.00021 (4)	−0.00062 (4)
Ti2	0.00503 (4)	0.00561 (4)	0.00460 (4)	0.00031 (3)	0.00046 (4)	0.00012 (4)
P1	0.00423 (6)	0.00594 (7)	0.00538 (8)	0.00002 (6)	−0.00061 (6)	−0.00032 (6)
P2	0.00683 (7)	0.00446 (7)	0.00508 (8)	0.00024 (6)	0.00032 (6)	0.00069 (5)
O1	0.0087 (2)	0.0101 (3)	0.0087 (2)	0.0015 (2)	−0.00232 (19)	−0.0033 (2)
O2	0.0087 (2)	0.0106 (3)	0.0081 (2)	−0.0015 (2)	−0.00277 (19)	0.0032 (2)
O3	0.0052 (2)	0.0084 (2)	0.0101 (2)	0.00105 (17)	0.00017 (17)	−0.00138 (19)
O4	0.0051 (2)	0.0094 (2)	0.0112 (3)	−0.00155 (16)	0.00054 (18)	0.0008 (2)
O11	0.0080 (2)	0.0084 (2)	0.0079 (2)	−0.00053 (19)	0.00218 (18)	−0.00218 (19)
O12	0.0090 (2)	0.0089 (2)	0.0085 (2)	−0.0006 (2)	0.00264 (19)	−0.0018 (2)
O5	0.0100 (2)	0.0052 (2)	0.0100 (3)	0.00102 (18)	0.0022 (2)	0.00002 (18)
O6	0.0111 (2)	0.0057 (2)	0.0115 (3)	−0.00123 (18)	−0.0022 (2)	−0.0001 (2)
O7	0.0105 (3)	0.0097 (3)	0.0073 (2)	−0.0019 (2)	−0.00251 (19)	0.0034 (2)
O8	0.0103 (3)	0.0101 (3)	0.0069 (2)	0.0028 (2)	0.00306 (18)	0.0030 (2)

Geometric parameters (Å, º)

K1a—O1	2.8870 (17)	K2b—O2^ii	3.129 (10)
K1a—O2	2.7495 (13)	K2b—O4^iv	2.980 (9)
K1a—O3^i	2.7205 (10)	K2b—O11	2.709 (6)
K1a—O11	2.9751 (17)	K2b—O5^v	2.819 (6)
K1a—O12^i	2.7187 (14)	K2b—O7^ii	2.893 (6)
K1a—O5^ii	2.858 (2)	K2b—O8^ii	3.195 (9)
K1a—O7^ii	3.0425 (18)	Ti1—O1	2.1490 (9)
K1a—O8	2.7593 (14)	Ti1—O2^vi	1.9585 (9)
K1a—K1b	0.287 (13)	Ti1—O11	1.9770 (8)
K1b—O1	3.034 (10)	Ti1—O12^ii	1.7212 (9)
K1b—O2	2.656 (7)	Ti1—O5^ii	2.0463 (8)
K1b—O3^i	2.721 (6)	Ti1—O6^vii	1.9900 (8)
K1b—O12^i	2.695 (8)	Ti2—O3	2.0402 (9)
K1b—O5^ii	3.122 (14)	Ti2—O4^iv	1.9810 (9)
K1b—O8	2.764 (7)	Ti2—O11	1.7440 (9)
K2a—O1^iii	2.6876 (16)	Ti2—O12	2.0919 (9)
K2a—O2^ii	2.957 (2)	Ti2—O7^vii	1.9704 (9)
K2a—O3^ii	3.013 (3)	Ti2—O8	1.9902 (9)
K2a—O4^iv	3.141 (2)	P1—O1	1.5195 (9)
K2a—O11	2.7703 (15)	P1—O2	1.5499 (9)
K2a—O12^ii	3.034 (2)	P1—O3	1.5441 (8)
K2a—O5^v	2.8049 (13)	P1—O4	1.5416 (8)
K2a—O7^ii	2.9144 (14)	P2—O5	1.5355 (9)
K2a—O8^ii	3.020 (2)	P2—O6	1.5295 (9)
K2a—K2b	0.255 (13)	P2—O7	1.5485 (9)
K2b—O1^iii	2.616 (7)	P2—O8	1.5368 (9)
O1—K1a—O2	52.40 (3)	O12^ii—K2a—O1^iii	150.20 (7)
O1—K1a—O11	54.84 (4)	O12^ii—K2a—O4^iv	97.41 (3)
O1—K1a—O8	89.82 (3)	O7^ii—K2a—O8^ii	49.19 (3)
O1—K1a—O3^i	121.36 (7)	O7^ii—K2a—O1^iii	73.67 (4)
O1—K1a—O12^i	158.76 (8)	O7^ii—K2a—O4^iv	119.34 (9)
O1—K1a—O5^ii	55.81 (5)	O8^ii—K2a—O1^iii	97.16 (4)
O1—K1a—O7^ii	104.30 (7)	O8^ii—K2a—O4^iv	144.80 (3)
O2—K1a—O11	86.77 (3)	O1^iii—K2a—O4^iv	111.63 (9)
O2—K1a—O8	73.58 (4)	O11—K2b—O5^v	145.8 (2)
O2—K1a—O3^i	133.70 (5)	O11—K2b—O2^ii	145.9 (4)
O2—K1a—O12^i	144.24 (9)	O11—K2b—O7^ii	82.30 (17)
O2—K1a—O5^ii	107.63 (6)	O11—K2b—O8^ii	87.62 (18)
O2—K1a—O7^ii	156.60 (8)	O11—K2b—O1^iii	146.8 (5)
O11—K1a—O8	56.61 (3)	O11—K2b—O4^iv	57.52 (16)
O11—K1a—O3^i	129.97 (8)	O5^v—K2b—O2^ii	55.11 (15)
O11—K1a—O12^i	106.38 (4)	O5^v—K2b—O7^ii	131.6 (2)
O11—K1a—O5^ii	58.37 (5)	O5^v—K2b—O8^ii	117.7 (4)
O11—K1a—O7^ii	75.61 (5)	O5^v—K2b—O1^iii	59.17 (12)
O8—K1a—O3^i	146.57 (5)	O5^v—K2b—O4^iv	92.09 (19)
O8—K1a—O12^i	86.03 (5)	O2^ii—K2b—O7^ii	88.9 (2)
O8—K1a—O5^ii	114.78 (6)	O2^ii—K2b—O8^ii	62.9 (2)
O8—K1a—O7^ii	107.77 (4)	O2^ii—K2b—O1^iii	58.52 (15)
O3^i—K1a—O12^i	60.54 (3)	O2^ii—K2b—O4^iv	144.7 (2)
O3^i—K1a—O5^ii	78.92 (5)	O7^ii—K2b—O8^ii	47.60 (11)
O3^i—K1a—O7^ii	56.27 (3)	O7^ii—K2b—O1^iii	75.06 (18)
O12^i—K1a—O5^ii	107.53 (5)	O7^ii—K2b—O4^iv	125.8 (4)
O12^i—K1a—O7^ii	57.66 (3)	O8^ii—K2b—O1^iii	94.5 (2)
O5^ii—K1a—O7^ii	49.88 (4)	O8^ii—K2b—O4^iv	144.0 (2)
O1—K1b—O2	51.41 (14)	O1^iii—K2b—O4^iv	119.1 (4)
O1—K1b—O8	86.77 (19)	O1—Ti1—O11	81.66 (4)
O1—K1b—O3^i	116.2 (4)	O1—Ti1—O2^vi	87.49 (4)
O1—K1b—O12^i	148.1 (5)	O1—Ti1—O12^ii	172.13 (4)
O1—K1b—O5^ii	51.8 (2)	O1—Ti1—O5^ii	79.68 (3)
O2—K1b—O8	74.96 (17)	O1—Ti1—O6^vii	86.76 (4)
O2—K1b—O3^i	138.6 (3)	O11—Ti1—O2^vi	169.12 (4)
O2—K1b—O12^i	153.0 (5)	O11—Ti1—O12^ii	94.64 (4)
O2—K1b—O5^ii	102.9 (3)	O11—Ti1—O5^ii	90.03 (3)
O8—K1b—O3^i	146.2 (3)	O11—Ti1—O6^vii	90.55 (4)
O8—K1b—O12^i	86.4 (2)	O2^vi—Ti1—O12^ii	96.04 (4)
O8—K1b—O5^ii	106.9 (3)	O2^vi—Ti1—O5^ii	87.30 (4)
O3^i—K1b—O12^i	60.82 (13)	O2^vi—Ti1—O6^vii	89.55 (4)
O3^i—K1b—O5^ii	74.4 (3)	O12^ii—Ti1—O5^ii	93.43 (4)
O12^i—K1b—O5^ii	101.1 (3)	O12^ii—Ti1—O6^vii	100.27 (4)
O11—K2a—O5^v	142.73 (6)	O5^ii—Ti1—O6^vii	166.19 (4)
O11—K2a—O2^ii	154.13 (10)	O3—Ti2—O11	92.92 (4)
O11—K2a—O3^ii	108.82 (7)	O3—Ti2—O12	83.13 (3)
O11—K2a—O12^ii	55.76 (3)	O3—Ti2—O8	89.00 (4)
O11—K2a—O7^ii	80.88 (4)	O3—Ti2—O7^vii	85.87 (4)
O11—K2a—O8^ii	90.10 (4)	O3—Ti2—O4^iv	172.26 (4)
O11—K2a—O1^iii	138.43 (12)	O11—Ti2—O12	174.99 (4)
O11—K2a—O4^iv	54.95 (4)	O11—Ti2—O8	93.53 (4)
O5^v—K2a—O2^ii	57.28 (4)	O11—Ti2—O7^vii	96.03 (4)
O5^v—K2a—O3^ii	83.66 (7)	O11—Ti2—O4^iv	94.82 (4)
O5^v—K2a—O12^ii	131.16 (10)	O12—Ti2—O8	83.35 (4)
O5^v—K2a—O7^ii	131.21 (4)	O12—Ti2—O7^vii	86.79 (4)
O5^v—K2a—O8^ii	124.18 (8)	O12—Ti2—O4^iv	89.14 (3)
O5^v—K2a—O1^iii	58.57 (3)	O8—Ti2—O7^vii	169.37 (4)
O5^v—K2a—O4^iv	89.03 (4)	O8—Ti2—O4^iv	90.64 (4)
O2^ii—K2a—O3^ii	48.80 (5)	O7^vii—Ti2—O4^iv	93.19 (4)
O2^ii—K2a—O12^ii	99.26 (9)	O1—P1—O2	108.53 (5)
O2^ii—K2a—O7^ii	91.92 (4)	O1—P1—O3	112.18 (5)
O2^ii—K2a—O8^ii	66.98 (6)	O1—P1—O4	110.90 (5)
O2^ii—K2a—O1^iii	60.27 (3)	O2—P1—O3	105.75 (5)
O2^ii—K2a—O4^iv	145.28 (3)	O2—P1—O4	110.46 (5)
O3^ii—K2a—O12^ii	53.92 (5)	O3—P1—O4	108.91 (5)
O3^ii—K2a—O7^ii	104.38 (7)	O5—P2—O6	109.58 (5)
O3^ii—K2a—O8^ii	55.84 (6)	O5—P2—O7	107.93 (5)
O3^ii—K2a—O1^iii	109.03 (7)	O5—P2—O8	111.09 (5)
O3^ii—K2a—O4^iv	126.46 (6)	O6—P2—O7	111.24 (5)
O12^ii—K2a—O7^ii	86.63 (4)	O6—P2—O8	110.44 (5)
O12^ii—K2a—O8^ii	53.29 (4)	O7—P2—O8	106.50 (5)

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