Acta Cryst. (2008). E64, i28 [ doi:10.1107/S1600536808010386 ]
Single crystals of triclinic Pr2(B2O5)(MoO4) were prepared from an incongruently melting flux in the system Pr2O3-MoO3-B2O3 in a platinum crucible in an atmosphere of air. In the crystal structure, distorted edge-sharing [PrO8] square antiprisms form a three-dimensional framework. These are further linked by isolated [MoO4] tetrahedra and isolated pyroborate groups [B2O5], the latter consisting of two independent trigonal [BO3] groups sharing one O atom. The [MoO4] tetrahedra and the [B2O5] groups are arranged in alternating layers parallel to the ab plane.
Single crystals of (I) were obtained by growth from the melt. A homogenized powder mixture of Pr4O11 (99.9%, Alfa Aesar), B2O3 (99.98%, Alfa Aesar) and MoO3 (99.95%, Alfa Aesar) in a molar ratio of 1: 3.33: 7 was heated in a covered platinum crucible in air atmosphere to 1423 K and subsequently cooled at a rate of 3 K h-1 to 1173 K. Transparent, light-green prismatic single crystals of the title compound were separated mechanically from the fine-grained praseodymium borate molybdate matrix.
The final difference Fourier map indicated a positive maximum at a distance of 0.76 Å from Pr1 and a negative maximum at a distance of 0.85 Å from the same atom.
Data collection: MACH3 (Enraf–Nonius, 1993); cell refinement: MACH3 (Enraf–Nonius, 1993); data reduction: MolEN (Fair, 1990); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ATOMS (Dowty, 2002); software used to prepare material for publication: publCIF (Westrip, 2008).
| Pr2(B2O5)(MoO4) | Z = 2 |
| Mr = 543.38 | F000 = 484 |
| Triclinic, P1 | Dx = 5.059 Mg m−3 |
| a = 5.2806 (5) Å | Mo Kα radiation λ = 0.71073 Å |
| b = 7.0278 (5) Å | Cell parameters from 25 reflections |
| c = 10.5824 (9) Å | θ = 20.8–24.2º |
| α = 74.557 (6)º | µ = 15.20 mm−1 |
| β = 76.307 (7)º | T = 291 (2) K |
| γ = 73.065 (6)º | Prism, light green |
| V = 356.69 (5) Å3 | 0.20 × 0.15 × 0.12 mm |
| Enraf–Nonius CAD-4 diffractometer | Rint = 0.019 |
| Radiation source: fine-focus sealed tube | θmax = 30.4º |
| Monochromator: graphite | θmin = 3.1º |
| T = 291(2) K | h = −7→7 |
| ω/2θ scans | k = −10→10 |
| Absorption correction: ψ scan (MolEN; Fair, 1990) | l = −15→15 |
| Tmin = 0.296, Tmax = 0.999 | 3 standard reflections |
| 4733 measured reflections | every 100 reflections |
| 2155 independent reflections | intensity decay: 1.7% |
| 1983 reflections with I > 2σ(I) |
| Refinement on F2 | Secondary atom site location: difference Fourier map |
| Least-squares matrix: full | w = 1/[σ2(Fo2) + (0.0533P)2 + 1.1726P] where P = (Fo2 + 2Fc2)/3 |
| R[F2 > 2σ(F2)] = 0.031 | (Δ/σ)max < 0.001 |
| wR(F2) = 0.082 | Δρmax = 2.55 e Å−3 |
| S = 1.13 | Δρmin = −1.73 e Å−3 |
| 2155 reflections | Extinction correction: SHELXL97 (Sheldrick, 2008), Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4 |
| 127 parameters | Extinction coefficient: 0.0319 (13) |
| Primary atom site location: structure-invariant direct methods |
| Pr2(B2O5)(MoO4) | γ = 73.065 (6)º |
| Mr = 543.38 | V = 356.69 (5) Å3 |
| Triclinic, P1 | Z = 2 |
| a = 5.2806 (5) Å | Mo Kα |
| b = 7.0278 (5) Å | µ = 15.20 mm−1 |
| c = 10.5824 (9) Å | T = 291 (2) K |
| α = 74.557 (6)º | 0.20 × 0.15 × 0.12 mm |
| β = 76.307 (7)º |
| Enraf–Nonius CAD-4 diffractometer | 1983 reflections with I > 2σ(I) |
| Absorption correction: ψ scan (MolEN; Fair, 1990) | Rint = 0.019 |
| Tmin = 0.296, Tmax = 0.999 | 3 standard reflections |
| 4733 measured reflections | every 100 reflections |
| 2155 independent reflections | intensity decay: 1.7% |
| R[F2 > 2σ(F2)] = 0.031 | 127 parameters |
| wR(F2) = 0.082 | Δρmax = 2.55 e Å−3 |
| S = 1.13 | Δρmin = −1.73 e Å−3 |
| 2155 reflections |
Experimental. A suitable single-crystal was carefully selected under a polarizing microscope and mounted in a glass capillary. |
Geometry. All e.s.d.'s (except the e.s.d. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell e.s.d.'s are taken into account individually in the estimation of e.s.d.'s in distances, angles and torsion angles; correlations between e.s.d.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell e.s.d.'s is used for estimating e.s.d.'s involving l.s. planes. |
| x | y | z | Uiso*/Ueq | ||
| Pr1 | 0.37016 (5) | 0.07689 (4) | −0.31461 (2) | 0.00706 (11) | |
| Pr2 | 0.15410 (5) | −0.79427 (4) | 0.04666 (2) | 0.00693 (11) | |
| Mo | 0.03514 (8) | 0.70782 (6) | −0.41958 (4) | 0.00758 (12) | |
| O7 | 0.1449 (8) | 0.4427 (5) | −0.3778 (4) | 0.0144 (7) | |
| O5 | 0.2335 (7) | 0.7903 (6) | −0.5706 (3) | 0.0152 (7) | |
| O6 | 0.0805 (7) | 0.8248 (6) | −0.2991 (4) | 0.0141 (7) | |
| O2 | 0.3079 (7) | 0.1829 (5) | −0.5554 (3) | 0.0111 (6) | |
| B1 | 0.4573 (10) | −0.4118 (7) | −0.1373 (5) | 0.0077 (8) | |
| O4 | 0.4775 (7) | −0.2184 (5) | −0.1465 (3) | 0.0114 (6) | |
| O8 | 0.2350 (7) | −0.4758 (5) | −0.0732 (3) | 0.0114 (6) | |
| O9 | 0.6870 (7) | −0.5435 (5) | −0.1886 (4) | 0.0121 (7) | |
| B2 | −0.2634 (10) | −0.7520 (8) | −0.1570 (5) | 0.0091 (8) | |
| O1 | 0.5437 (7) | −0.8575 (5) | −0.1392 (3) | 0.0094 (6) | |
| O3 | −0.0094 (7) | −0.8664 (5) | −0.1355 (3) | 0.0086 (6) |
| U11 | U22 | U33 | U12 | U13 | U23 | |
| Pr1 | 0.00931 (15) | 0.00715 (15) | 0.00456 (15) | −0.00225 (10) | −0.00160 (10) | −0.00032 (10) |
| Pr2 | 0.00839 (16) | 0.00662 (15) | 0.00584 (15) | −0.00215 (10) | −0.00193 (10) | −0.00046 (9) |
| Mo | 0.0094 (2) | 0.0072 (2) | 0.00587 (19) | −0.00159 (14) | −0.00209 (14) | −0.00082 (13) |
| O7 | 0.0190 (18) | 0.0082 (15) | 0.0112 (15) | −0.0002 (12) | −0.0023 (14) | 0.0025 (12) |
| O5 | 0.0167 (17) | 0.0188 (17) | 0.0100 (15) | −0.0069 (14) | 0.0023 (13) | −0.0039 (13) |
| O6 | 0.0165 (17) | 0.0179 (17) | 0.0109 (15) | −0.0060 (13) | −0.0036 (13) | −0.0052 (13) |
| O2 | 0.0110 (15) | 0.0136 (16) | 0.0096 (15) | −0.0016 (12) | −0.0034 (12) | −0.0038 (12) |
| B1 | 0.011 (2) | 0.008 (2) | 0.0034 (19) | −0.0022 (16) | −0.0003 (16) | −0.0011 (15) |
| O4 | 0.0159 (16) | 0.0096 (15) | 0.0101 (15) | −0.0046 (12) | −0.0044 (13) | −0.0008 (11) |
| O8 | 0.0093 (15) | 0.0086 (14) | 0.0130 (16) | −0.0015 (11) | 0.0004 (13) | 0.0005 (12) |
| O9 | 0.0120 (16) | 0.0075 (15) | 0.0142 (17) | −0.0020 (12) | 0.0011 (13) | −0.0012 (12) |
| B2 | 0.009 (2) | 0.009 (2) | 0.008 | −0.0028 (16) | −0.0004 (17) | −0.0010 (16) |
| O1 | 0.0099 (15) | 0.0107 (15) | 0.0088 (14) | −0.0036 (12) | −0.0030 (12) | −0.0017 (11) |
| O3 | 0.0076 (14) | 0.0106 (15) | 0.0071 (14) | −0.0025 (11) | 0.0007 (11) | −0.0024 (11) |
| Pr1—O4 | 2.370 (3) | Pr2—O1ix | 2.585 (3) |
| Pr1—O3i | 2.430 (3) | Pr2—O6x | 2.645 (4) |
| Pr1—O1i | 2.450 (3) | Pr2—B2viii | 3.021 (5) |
| Pr1—O5ii | 2.461 (3) | Pr2—Pr2vii | 3.9237 (6) |
| Pr1—O7 | 2.480 (3) | Pr2—Pr1vi | 3.9702 (5) |
| Pr1—O2 | 2.529 (3) | Pr2—Pr1iv | 3.9893 (5) |
| Pr1—O2iii | 2.557 (3) | Mo—O5 | 1.748 (3) |
| Pr1—O6iv | 2.610 (3) | Mo—O7 | 1.748 (3) |
| Pr1—Mov | 3.6714 (6) | Mo—O6 | 1.782 (4) |
| Pr1—Pr2vi | 3.9702 (5) | Mo—O2v | 1.803 (3) |
| Pr1—Pr2i | 3.9893 (5) | Mo—Pr1v | 3.6714 (6) |
| Pr1—Pr2vii | 4.0171 (5) | B1—O8 | 1.345 (6) |
| Pr2—O8 | 2.364 (3) | B1—O4 | 1.370 (6) |
| Pr2—O8vii | 2.375 (3) | B1—O9 | 1.387 (6) |
| Pr2—O4vi | 2.456 (3) | B2—O9xi | 1.373 (6) |
| Pr2—O3 | 2.506 (3) | B2—O1xi | 1.378 (6) |
| Pr2—O1 | 2.513 (3) | B2—O3 | 1.384 (6) |
| Pr2—O3viii | 2.585 (3) | B2—Pr2viii | 3.021 (5) |
| O4—Pr1—O3i | 77.59 (12) | O4vi—Pr2—O1ix | 64.22 (10) |
| O4—Pr1—O1i | 67.58 (11) | O3—Pr2—O1ix | 102.94 (11) |
| O3i—Pr1—O1i | 73.79 (11) | O1—Pr2—O1ix | 75.10 (12) |
| O4—Pr1—O5ii | 112.06 (12) | O3viii—Pr2—O1ix | 54.11 (11) |
| O3i—Pr1—O5ii | 139.81 (11) | O8—Pr2—O6x | 117.87 (11) |
| O1i—Pr1—O5ii | 74.69 (11) | O8vii—Pr2—O6x | 68.69 (12) |
| O4—Pr1—O7 | 149.37 (12) | O4vi—Pr2—O6x | 79.69 (11) |
| O3i—Pr1—O7 | 75.46 (11) | O3—Pr2—O6x | 126.73 (10) |
| O1i—Pr1—O7 | 90.93 (12) | O1—Pr2—O6x | 154.52 (11) |
| O5ii—Pr1—O7 | 80.77 (12) | O3viii—Pr2—O6x | 69.74 (10) |
| O4—Pr1—O2 | 141.05 (11) | O1ix—Pr2—O6x | 82.67 (11) |
| O3i—Pr1—O2 | 121.14 (11) | O8—Pr2—Pr2vii | 34.21 (8) |
| O1i—Pr1—O2 | 146.59 (11) | O8vii—Pr2—Pr2vii | 34.03 (8) |
| O5ii—Pr1—O2 | 76.77 (11) | O4vi—Pr2—Pr2vii | 97.53 (8) |
| O7—Pr1—O2 | 67.53 (11) | O3—Pr2—Pr2vii | 93.69 (8) |
| O4—Pr1—O2iii | 76.63 (11) | O1—Pr2—Pr2vii | 103.65 (8) |
| O3i—Pr1—O2iii | 145.67 (11) | O3viii—Pr2—Pr2vii | 140.80 (7) |
| O1i—Pr1—O2iii | 115.72 (11) | O1ix—Pr2—Pr2vii | 161.72 (7) |
| O5ii—Pr1—O2iii | 71.93 (11) | O6x—Pr2—Pr2vii | 93.54 (8) |
| O7—Pr1—O2iii | 133.79 (11) | B2viii—Pr2—Pr2vii | 166.65 (10) |
| O2—Pr1—O2iii | 70.32 (12) | O8—Pr2—Pr1vi | 113.11 (9) |
| O4—Pr1—O6iv | 69.35 (11) | O8vii—Pr2—Pr1vi | 125.58 (9) |
| O3i—Pr1—O6iv | 72.69 (11) | O4vi—Pr2—Pr1vi | 33.96 (8) |
| O1i—Pr1—O6iv | 129.69 (11) | O3—Pr2—Pr1vi | 139.70 (8) |
| O5ii—Pr1—O6iv | 147.48 (12) | O1—Pr2—Pr1vi | 90.73 (8) |
| O7—Pr1—O6iv | 115.17 (12) | O3viii—Pr2—Pr1vi | 79.17 (7) |
| O2—Pr1—O6iv | 83.51 (11) | O1ix—Pr2—Pr1vi | 36.78 (7) |
| O2iii—Pr1—O6iv | 77.21 (11) | O6x—Pr2—Pr1vi | 63.84 (8) |
| O4—Pr1—Pr2vi | 35.36 (8) | B2viii—Pr2—Pr1vi | 57.28 (10) |
| O3i—Pr1—Pr2vi | 89.53 (8) | Pr2vii—Pr2—Pr1vi | 126.063 (11) |
| O1i—Pr1—Pr2vi | 39.18 (8) | O8—Pr2—Pr1iv | 82.46 (9) |
| O5ii—Pr1—Pr2vi | 81.40 (9) | O8vii—Pr2—Pr1iv | 115.15 (9) |
| O7—Pr1—Pr2vi | 129.95 (9) | O4vi—Pr2—Pr1iv | 115.41 (8) |
| O2—Pr1—Pr2vi | 149.14 (8) | O3—Pr2—Pr1iv | 35.45 (7) |
| O2iii—Pr1—Pr2vi | 82.37 (8) | O1—Pr2—Pr1iv | 35.98 (7) |
| O6iv—Pr1—Pr2vi | 104.66 (8) | O3viii—Pr2—Pr1iv | 88.62 (7) |
| Mov—Pr1—Pr2vi | 173.822 (12) | O1ix—Pr2—Pr1iv | 89.26 (7) |
| O4—Pr1—Pr2i | 68.51 (8) | O6x—Pr2—Pr1iv | 157.63 (7) |
| O3i—Pr1—Pr2i | 36.74 (8) | B2viii—Pr2—Pr1iv | 87.02 (10) |
| O1i—Pr1—Pr2i | 37.06 (8) | Pr2vii—Pr2—Pr1iv | 100.262 (11) |
| O5ii—Pr1—Pr2i | 108.19 (8) | Pr1vi—Pr2—Pr1iv | 118.956 (9) |
| O7—Pr1—Pr2i | 81.16 (8) | O5—Mo—O7 | 107.10 (17) |
| O2—Pr1—Pr2i | 147.30 (8) | O5—Mo—O6 | 107.70 (16) |
| O2iii—Pr1—Pr2i | 142.36 (8) | O7—Mo—O6 | 111.77 (17) |
| O6iv—Pr1—Pr2i | 102.41 (8) | O5—Mo—O2v | 105.62 (17) |
| Mov—Pr1—Pr2i | 124.637 (12) | O7—Mo—O2v | 118.20 (16) |
| Pr2vi—Pr1—Pr2i | 61.044 (9) | O6—Mo—O2v | 105.89 (16) |
| O4—Pr1—Pr2vii | 53.61 (8) | Pr1i—O6—Pr2x | 99.70 (12) |
| O3i—Pr1—Pr2vii | 38.13 (7) | Mov—O2—Pr1 | 114.85 (15) |
| O1i—Pr1—Pr2vii | 91.90 (8) | Mov—O2—Pr1iii | 122.00 (16) |
| O5ii—Pr1—Pr2vii | 164.08 (9) | Pr1—O2—Pr1iii | 109.68 (12) |
| O7—Pr1—Pr2vii | 108.49 (8) | O8—B1—O4 | 122.1 (4) |
| O2—Pr1—Pr2vii | 118.48 (8) | O8—B1—O9 | 121.5 (4) |
| O2iii—Pr1—Pr2vii | 107.57 (8) | O4—B1—O9 | 116.2 (4) |
| O6iv—Pr1—Pr2vii | 40.47 (8) | B1—O4—Pr1 | 129.0 (3) |
| Mov—Pr1—Pr2vii | 97.798 (12) | B1—O4—Pr2vi | 113.9 (3) |
| Pr2vi—Pr1—Pr2vii | 82.769 (10) | Pr1—O4—Pr2vi | 110.68 (12) |
| Pr2i—Pr1—Pr2vii | 62.010 (9) | B1—O8—Pr2 | 134.1 (3) |
| O8—Pr2—O8vii | 68.24 (13) | B1—O8—Pr2vii | 113.4 (3) |
| O8—Pr2—O4vi | 79.16 (11) | Pr2—O8—Pr2vii | 111.76 (13) |
| O8vii—Pr2—O4vi | 113.81 (11) | B2xii—O9—B1 | 125.1 (4) |
| O8—Pr2—O3 | 95.79 (11) | O9xi—B2—O1xi | 123.8 (4) |
| O8vii—Pr2—O3 | 90.33 (12) | O9xi—B2—O3 | 119.3 (4) |
| O4vi—Pr2—O3 | 150.59 (11) | O1xi—B2—O3 | 116.8 (4) |
| O8—Pr2—O1 | 72.10 (11) | Pr1iv—O1—Pr2 | 106.97 (12) |
| O8vii—Pr2—O1 | 134.09 (11) | B2xii—O1—Pr2ix | 94.4 (3) |
| O4vi—Pr2—O1 | 79.52 (11) | Pr1iv—O1—Pr2ix | 104.05 (12) |
| O3—Pr2—O1 | 71.43 (11) | Pr2—O1—Pr2ix | 104.90 (12) |
| O8—Pr2—O3viii | 167.27 (12) | B2—O3—Pr1iv | 123.2 (3) |
| O8vii—Pr2—O3viii | 108.01 (11) | B2—O3—Pr2 | 114.9 (3) |
| O4vi—Pr2—O3viii | 113.02 (11) | Pr1iv—O3—Pr2 | 107.81 (12) |
| O3—Pr2—O3viii | 71.81 (12) | B2—O3—Pr2viii | 94.2 (3) |
| O1—Pr2—O3viii | 105.54 (10) | Pr1iv—O3—Pr2viii | 106.39 (12) |
| O8—Pr2—O1ix | 134.34 (11) | Pr2—O3—Pr2viii | 108.19 (12) |
| O8vii—Pr2—O1ix | 150.80 (11) |
| Symmetry codes: (i) x, y+1, z; (ii) −x+1, −y+1, −z−1; (iii) −x+1, −y, −z−1; (iv) x, y−1, z; (v) −x, −y+1, −z−1; (vi) −x+1, −y−1, −z; (vii) −x, −y−1, −z; (viii) −x, −y−2, −z; (ix) −x+1, −y−2, −z; (x) −x, −y, −z; (xi) x−1, y, z; (xii) x+1, y, z. |
| Pr1—O4 | 2.370 (3) | Pr2—O3vii | 2.585 (3) |
| Pr1—O3i | 2.430 (3) | Pr2—O1viii | 2.585 (3) |
| Pr1—O1i | 2.450 (3) | Pr2—O6ix | 2.645 (4) |
| Pr1—O5ii | 2.461 (3) | Mo—O5 | 1.748 (3) |
| Pr1—O7 | 2.480 (3) | Mo—O7 | 1.748 (3) |
| Pr1—O2 | 2.529 (3) | Mo—O6 | 1.782 (4) |
| Pr1—O2iii | 2.557 (3) | Mo—O2x | 1.803 (3) |
| Pr1—O6iv | 2.610 (3) | B1—O8 | 1.345 (6) |
| Pr2—O8 | 2.364 (3) | B1—O4 | 1.370 (6) |
| Pr2—O8v | 2.375 (3) | B1—O9 | 1.387 (6) |
| Pr2—O4vi | 2.456 (3) | B2—O9xi | 1.373 (6) |
| Pr2—O3 | 2.506 (3) | B2—O1xi | 1.378 (6) |
| Pr2—O1 | 2.513 (3) | B2—O3 | 1.384 (6) |
| Symmetry codes: (i) x, y+1, z; (ii) −x+1, −y+1, −z−1; (iii) −x+1, −y, −z−1; (iv) x, y−1, z; (v) −x, −y−1, −z; (vi) −x+1, −y−1, −z; (vii) −x, −y−2, −z; (viii) −x+1, −y−2, −z; (ix) −x, −y, −z; (x) −x, −y+1, −z−1; (xi) x−1, y, z. |
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under project BE 2147/6–1.
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Lysanova, G. V., Dzhurinskii, B. F., Komova, M. G. & Tananaev, I. V. (1983). Russ. J. Inorg. Chem. 28, 1344–1349.
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Zobetz, E. (1982). Z. Kristallogr. 160, 81–92.
The existence of several compounds in the systems RE2O3— B2O3—MoO3 (RE = rare earth element) has been studied by Lysanova et al. (1983) by means of X-ray powder diffraction data and was partly corroborated by Dzhurinskii & Lysanova (1998). Among these pseudo-ternary compounds, rare earth pyroborate molybdates of the type RE2(B2O5)(MoO4) were reported for RE = Pr - Tb (excluding Pm) (Lysanova et al., 1983), all with incongruent melting behaviour. However, no structural information about these compounds has been given so far. In the course of our investigations of the systems RE2O3—B2O3—MO3 (M = Mo, W) we grew single crystals of the Pr-compound Pr2(B2O5)(MoO4), (I), as a representative of the rare earth pyroborate molybdate series.
In the crystal structure of (I) the two symmetrically non-equivalent Pr atoms show a distinct eightfold coordination by oxygen atoms, both with a slightly distorted square antiprismatic coordination polyhedron, and with Pr—O bond lengths ranging from 2.370 (3) Å to 2.610 (3) Å for Pr1, and from 2.364 (3) Å to 2.645 (4) Å for Pr2 (Fig. 2). The [PrO8] polyhedra are connected via common edges, where Pr2 is connected to six neighbouring [PrO8] polyhedra with Pr—Pr distances ranging from 3.9237 (5) Å to 4.1242 (5) Å, while Pr1 is connected to only four [PrO8] polyhedra with Pr—Pr distances between 3.9702 (6) Å and 4.1583 (6) Å. From the different connection schemes of the two Pr atoms a three-dimensional framework of [PrO8] polyhedra with interstitial voids results (Fig. 1). In these voids nearly undistorted and isolated [MoO4] tetrahedra, that are arranged in layers parallel to the ab plane are positioned (Fig. 1).
The two crystallographically different B atoms are threefold coordinated by O atoms. The two [BO3] groups are linked by a common oxygen ligand O9 (see Fig. 2), thus forming isolated pyroborate groups [B2O5]. The pyroborate groups are bent with an angle (B1— O9— B2) of 125.1 (4)°, while the individual [BO3] groups show no unusual distortions (Zobetz, 1982). The oxygen ligands of the pyroborate group (apart from the bridging oxygen O9, see Fig. 2) each belong either to two different [PrO8] polyhedra (being simultaneously ligands of B1) or to three different [PrO8] polyhedra (being simultaneously ligands of B2). All [B2O5] groups are arranged in double layers that extend parallel to the ab-plane and alternate with layers of [MoO4] tetrahedra (see Fig. 1).