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The incommensurately modulated scheelite-like KSm(MoO4)2 structure has been refined in the monoclinic superspace group I2/b(αβ0)00 by the Rietveld method on the basis of synchrotron radiation powder diffraction data. The systematic broadening of satellite reflections has been accounted for by applying anisotropic microstrain line-broadening. The microstructure has been studied by transmission electron microscopy (TEM). The partial disorder of the K and Sm cations in the A position is best approximated by a combination of harmonic and complex crenel functions with (0.952Sm + 0.048K) and (0.952K + 0.048Sm) atomic domains. This combination yields a compositional wave distribution from {KMoO4} to {SmMoO4} observed in the ab structure projection along q. The specific features of KSm(MoO4)2 and degree of the A-cation ordering are discussed in comparison with the previously reported structure of KNd(MoO4)2.

Supporting information

cif

Crystallographic Information File (CIF) https://doi.org/10.1107/S0108768108001870/sn5063sup1.cif
Contains datablocks global, I

rtv

Rietveld powder data file (CIF format) https://doi.org/10.1107/S0108768108001870/sn5063sup2.rtv
Contains datablock I

Computing details top

Program(s) used to refine structure: Jana2006 (Petricek, Dusek & Palatinus, 2006); software used to prepare material for publication: Jana2006 (Petricek, Dusek & Palatinus, 2006).

Figures top
[Figure 1]
[Figure 2]
[Figure 3]
[Figure 4]
[Figure 5]
[Figure 6]
[Figure 7]
[Figure 8]
[Figure 9]
[Figure 10]
[Figure 11]
potassium samarium molybdenum oxide top
Crystal data top
KSm(MoO4)2Z = 2
Mr = 509.3F(000) = 458
Monoclinic, I2/b(αβ0)00†Dx = 4.899 Mg m3
q = 0.56773a* + -0.12677b*Synchrotron radiation, λ = 0.7114 Å
a = 5.5279 (2) ÅT = 273 K
b = 5.2994 (2) Åyellow
c = 11.7841 (1) Åcapillary, ? × 0.5 mm mm
β = 90°Specimen preparation: Prepared at 293 K
V = 345.15 Å3
† Symmetry operations: (1) x1, x2, x3, x4; (2) −x1, −x2+1/2, x3, −x4; (3) −x1, −x2, −x3, −x4; (4) x1, x2+1/2, −x3, x4; (5) x1+1/2, x2+1/2, x3+1/2, x4; (6) −x1+1/2, −x2, x3+1/2, −x4; (7) −x1+1/2, −x2+1/2, −x3+1/2, −x4; (8) x1+1/2, x2, −x3+1/2, x4.

Data collection top
MAR345
diffractometer
Si(111) monochromator
Radiation source: synchrotron, synchrotron2θmin = 1.998°, 2θmax = 40.798°, 2θstep = 0.005°
Refinement top
Rp = 0.02542 parameters
Rwp = 0.0351 restraint
Rexp = 0.000none constraints
R(F) = 0.014Weighting scheme based on measured s.u.'s
χ2 = NOT FOUND(Δ/σ)max = 0.002
7761 data pointsBackground function: 14 Legendre polynoms
Profile function: Pseudo-VoigtPreferred orientation correction: none
Crystal data top
KSm(MoO4)2β = 90°
Mr = 509.3V = 345.15 Å3
Monoclinic, I2/b(αβ0)00†Z = 2
q = 0.56773a* + -0.12677b*Synchrotron radiation, λ = 0.7114 Å
a = 5.5279 (2) ÅT = 273 K
b = 5.2994 (2) Åcapillary, ? × 0.5 mm mm
c = 11.7841 (1) Å
† Symmetry operations: (1) x1, x2, x3, x4; (2) −x1, −x2+1/2, x3, −x4; (3) −x1, −x2, −x3, −x4; (4) x1, x2+1/2, −x3, x4; (5) x1+1/2, x2+1/2, x3+1/2, x4; (6) −x1+1/2, −x2, x3+1/2, −x4; (7) −x1+1/2, −x2+1/2, −x3+1/2, −x4; (8) x1+1/2, x2, −x3+1/2, x4.

Data collection top
MAR345
diffractometer
2θmin = 1.998°, 2θmax = 40.798°, 2θstep = 0.005°
Refinement top
Rp = 0.025χ2 = NOT FOUND
Rwp = 0.0357761 data points
Rexp = 0.00042 parameters
R(F) = 0.0141 restraint
Special details top

Refinement. Two complementary waves, K and Sm, have been used for the occupation modulation of the (K,Sm) position

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/UeqOcc. (<1)
K0.50.250.8661360.0251*0.5
Sm0.50.250.8780 (5)0.0001 (3)*0.5
Mo0.50.250.37801 (8)0.0126 (2)*
O10.3543 (3)0.0109 (4)0.29180 (17)0.0093 (8)*
O20.7632 (5)0.3985 (4)0.04196 (18)0.0225 (9)*
Geometric parameters (Å, º) top
AverageMinimumMaximum
K—O1i2.575 (6)2.185 (6)2.918 (6)
K—O1ii2.573 (6)2.188 (7)2.918 (6)
K—O1iii2.523 (6)2.094 (5)2.851 (5)
K—O1iv2.522 (6)2.095 (5)2.852 (5)
K—O2v2.645 (6)2.487 (5)2.806 (6)
K—O2vi2.644 (6)2.487 (5)2.807 (5)
K—O2vii2.635 (7)2.474 (8)2.788 (7)
K—O2viii2.637 (7)2.474 (8)2.788 (7)
Sm—O1i2.566 (7)2.395 (6)2.896 (6)
Sm—O1ii2.569 (7)2.395 (6)2.893 (6)
Sm—O1iii2.545 (6)2.386 (6)2.899 (6)
Sm—O1iv2.543 (6)2.386 (6)2.899 (6)
Sm—O2v2.546 (6)2.442 (7)2.779 (7)
Sm—O2vi2.550 (6)2.443 (7)2.779 (7)
Sm—O2vii2.563 (7)2.378 (6)2.824 (8)
Sm—O2viii2.565 (7)2.378 (6)2.827 (9)
Mo—O11.807 (5)1.765 (5)1.847 (5)
Mo—O1ix1.807 (5)1.765 (5)1.847 (5)
Mo—O1x3.095 (5)2.931 (5)3.262 (5)
Mo—O1xi3.092 (5)2.931 (5)3.262 (5)
Mo—O2xii2.970 (5)2.687 (5)3.254 (5)
Mo—O2xiii2.970 (5)2.687 (5)3.254 (5)
Mo—O2xiv1.812 (5)1.783 (5)1.843 (5)
Mo—O2xv1.812 (5)1.783 (5)1.843 (5)
O1i—K—O1ii137.92 (17)130.09 (15)151.06 (18)
O1i—K—O1iii77.64 (17)64.00 (17)94.8 (2)
O1i—K—O1iv71.27 (18)65.26 (18)78.6 (2)
O1i—K—O2v71.48 (18)70.9 (2)72.30 (17)
O1i—K—O2vi147.73 (19)137.17 (17)158.17 (16)
O1i—K—O2vii100.3 (2)92.9 (3)107.2 (3)
O1i—K—O2viii95.39 (19)88.6 (2)103.5 (3)
O1ii—K—O1iii71.30 (18)65.26 (18)78.6 (2)
O1ii—K—O1iv77.71 (17)64.00 (17)94.8 (2)
O1ii—K—O2v147.70 (19)137.17 (17)158.17 (16)
O1ii—K—O2vi71.49 (18)71.0 (2)72.30 (17)
O1ii—K—O2vii95.45 (19)88.6 (2)103.4 (3)
O1ii—K—O2viii100.2 (2)92.9 (3)107.2 (3)
O1iii—K—O1iv80.86 (17)75.25 (15)90.8 (2)
O1iii—K—O2v124.5 (2)112.5 (2)139.2 (3)
O1iii—K—O2vi126.5 (2)122.8 (3)130.1 (2)
O1iii—K—O2vii153.17 (19)143.52 (16)166.83 (18)
O1iii—K—O2viii73.52 (19)68.26 (19)80.0 (3)
O1iv—K—O2v126.5 (2)122.8 (3)130.1 (2)
O1iv—K—O2vi124.6 (2)112.5 (2)139.2 (3)
O1iv—K—O2vii73.55 (19)68.3 (2)80.1 (3)
O1iv—K—O2viii153.14 (19)143.51 (16)166.83 (18)
O2v—K—O2vi78.10 (15)65.80 (15)89.04 (16)
O2v—K—O2vii77.83 (18)66.63 (15)87.31 (16)
O2v—K—O2viii64.97 (18)61.64 (18)68.8 (2)
O2vi—K—O2vii65.00 (18)61.64 (18)68.8 (2)
O2vi—K—O2viii77.80 (18)66.64 (15)87.31 (16)
O2vii—K—O2viii132.13 (16)119.25 (16)142.56 (15)
O1i—Sm—O1ii133.4 (2)129.3 (3)135.91 (19)
O1i—Sm—O1iii75.0 (2)68.8 (2)83.0 (2)
O1i—Sm—O1iv69.50 (18)66.74 (18)72.3 (2)
O1i—Sm—O2v73.5 (2)61.60 (19)83.2 (2)
O1i—Sm—O2vi151.7 (2)137.5 (2)164.6 (2)
O1i—Sm—O2vii101.3 (2)85.6 (2)118.4 (3)
O1i—Sm—O2viii95.4 (2)91.1 (2)99.4 (3)
O1ii—Sm—O1iii69.49 (18)66.74 (18)72.3 (2)
O1ii—Sm—O1iv75.2 (2)68.81 (19)83.0 (2)
O1ii—Sm—O2v151.4 (2)137.5 (2)164.8 (2)
O1ii—Sm—O2vi73.2 (2)61.60 (18)83.2 (2)
O1ii—Sm—O2vii95.3 (2)91.1 (2)99.4 (3)
O1ii—Sm—O2viii100.9 (2)85.6 (2)118.4 (3)
O1iii—Sm—O1iv78.04 (19)74.7 (2)79.83 (20)
O1iii—Sm—O2v125.5 (3)119.5 (2)130.33 (18)
O1iii—Sm—O2vi127.4 (2)117.6 (3)140.6 (3)
O1iii—Sm—O2vii149.8 (2)143.43 (19)152.8 (3)
O1iii—Sm—O2viii73.3 (2)67.3 (2)78.2 (2)
O1iv—Sm—O2v127.6 (2)117.6 (3)140.6 (3)
O1iv—Sm—O2vi125.6 (3)119.4 (2)130.33 (17)
O1iv—Sm—O2vii73.4 (2)67.4 (2)78.2 (2)
O1iv—Sm—O2viii149.8 (2)143.48 (19)152.8 (3)
O2v—Sm—O2vi79.33 (19)75.59 (18)89.7 (2)
O2v—Sm—O2vii78.8 (2)72.42 (18)91.19 (19)
O2v—Sm—O2viii67.1 (2)64.0 (2)70.9 (2)
O2vi—Sm—O2vii67.1 (2)64.0 (2)70.9 (2)
O2vi—Sm—O2viii78.64 (19)72.41 (18)91.29 (19)
O2vii—Sm—O2viii134.8 (2)131.5 (2)143.4 (3)
O1—Mo—O1ix111.2 (2)106.7 (2)116.0 (2)
O1—Mo—O1x69.22 (19)59.89 (19)78.89 (19)
O1—Mo—O1xi67.97 (18)63.34 (18)72.24 (18)
O1—Mo—O2xii75.00 (19)71.71 (19)78.31 (19)
O1—Mo—O2xiii172.98 (19)167.34 (19)177.32 (19)
O1—Mo—O2xiv107.8 (2)102.8 (2)112.7 (2)
O1—Mo—O2xv106.2 (2)105.0 (2)106.9 (2)
O1ix—Mo—O1x68.03 (18)63.34 (18)72.24 (18)
O1ix—Mo—O1xi69.37 (19)59.89 (19)78.88 (19)
O1ix—Mo—O2xii172.90 (19)167.34 (19)177.32 (19)
O1ix—Mo—O2xiii74.93 (19)71.71 (19)78.31 (19)
O1ix—Mo—O2xiv106.2 (2)105.0 (2)106.9 (2)
O1ix—Mo—O2xv107.7 (2)102.8 (2)112.7 (2)
O1x—Mo—O1xi99.18 (14)97.84 (15)100.75 (14)
O1x—Mo—O2xii113.82 (13)110.06 (13)117.71 (13)
O1x—Mo—O2xiii115.95 (15)112.35 (15)119.43 (15)
O1x—Mo—O2xiv169.33 (19)166.4 (2)171.62 (19)
O1x—Mo—O2xv72.03 (19)67.19 (19)76.75 (19)
O1xi—Mo—O2xii115.88 (15)112.34 (15)119.43 (15)
O1xi—Mo—O2xiii113.87 (13)110.06 (13)117.71 (13)
O1xi—Mo—O2xiv72.00 (19)67.19 (19)76.75 (19)
O1xi—Mo—O2xv169.35 (19)166.4 (2)171.62 (19)
O2xii—Mo—O2xiii98.93 (15)96.79 (14)101.30 (17)
O2xii—Mo—O2xiv72.62 (19)70.46 (19)74.80 (19)
O2xii—Mo—O2xv67.7 (2)60.9 (2)74.7 (2)
O2xiii—Mo—O2xiv67.7 (2)60.9 (2)74.7 (2)
O2xiii—Mo—O2xv72.59 (19)70.46 (19)74.80 (19)
O2xiv—Mo—O2xv117.2 (2)110.2 (2)124.6 (3)
Symmetry codes: (i) x1+1/2, x2+1/2, x3+1/2, x4; (ii) x1+1/2, x2, x3+1/2, x4; (iii) x1+1, x2, x3+1, x4; (iv) x1, x2+1/2, x3+1, x4; (v) x1, x2, x3+1, x4; (vi) x1+1, x2+1/2, x3+1, x4; (vii) x1+1, x2+1, x3+1, x4; (viii) x1, x21/2, x3+1, x4; (ix) x1+1, x2+1/2, x3, x4; (x) x1+1/2, x2+1/2, x3+1/2, x4; (xi) x1+1/2, x2, x3+1/2, x4; (xii) x11/2, x21/2, x3+1/2, x4; (xiii) x1+3/2, x2+1, x3+1/2, x4; (xiv) x1+3/2, x2+1/2, x3+1/2, x4; (xv) x11/2, x2, x3+1/2, x4.

Experimental details

Crystal data
Chemical formulaKSm(MoO4)2
Mr509.3
Crystal system, space groupMonoclinic, I2/b(αβ0)00†
Temperature (K)273
Wave vectorsq = 0.56773a* + -0.12677b*
a, b, c (Å)5.5279 (2), 5.2994 (2), 11.7841 (1)
γ (°) 91.1388 (5)
V3)345.15
Z2
Radiation typeSynchrotron, λ = 0.7114 Å
Specimen shape, size (mm)Capillary, ? × 0.5 mm
Data collection
DiffractometerMAR345
diffractometer
Specimen mounting?
Data collection mode?
Scan method?
2θ values (°)2θmin = 1.998 2θmax = 40.798 2θstep = 0.005
Refinement
R factors and goodness of fitRp = 0.025, Rwp = 0.035, Rexp = 0.000, R(F) = 0.014, χ2 = NOT FOUND
No. of data points7761
No. of parameters42
No. of restraints1

† Symmetry operations: (1) x1, x2, x3, x4; (2) −x1, −x2+1/2, x3, −x4; (3) −x1, −x2, −x3, −x4; (4) x1, x2+1/2, −x3, x4; (5) x1+1/2, x2+1/2, x3+1/2, x4; (6) −x1+1/2, −x2, x3+1/2, −x4; (7) −x1+1/2, −x2+1/2, −x3+1/2, −x4; (8) x1+1/2, x2, −x3+1/2, x4.

Computer programs: Jana2006 (Petricek, Dusek & Palatinus, 2006).

Selected geometric parameters (Å, º) top
AverageMinimumMaximum
K—O1i2.575 (6)2.185 (6)2.918 (6)
K—O1ii2.573 (6)2.188 (7)2.918 (6)
K—O1iii2.523 (6)2.094 (5)2.851 (5)
K—O1iv2.522 (6)2.095 (5)2.852 (5)
K—O2v2.645 (6)2.487 (5)2.806 (6)
K—O2vi2.644 (6)2.487 (5)2.807 (5)
K—O2vii2.635 (7)2.474 (8)2.788 (7)
K—O2viii2.637 (7)2.474 (8)2.788 (7)
Sm—O1i2.566 (7)2.395 (6)2.896 (6)
Sm—O1ii2.569 (7)2.395 (6)2.893 (6)
Sm—O1iii2.545 (6)2.386 (6)2.899 (6)
Sm—O1iv2.543 (6)2.386 (6)2.899 (6)
Sm—O2v2.546 (6)2.442 (7)2.779 (7)
Sm—O2vi2.550 (6)2.443 (7)2.779 (7)
Sm—O2vii2.563 (7)2.378 (6)2.824 (8)
Sm—O2viii2.565 (7)2.378 (6)2.827 (9)
Mo—O11.807 (5)1.765 (5)1.847 (5)
Mo—O1ix1.807 (5)1.765 (5)1.847 (5)
Mo—O1x3.095 (5)2.931 (5)3.262 (5)
Mo—O1xi3.092 (5)2.931 (5)3.262 (5)
Mo—O2xii2.970 (5)2.687 (5)3.254 (5)
Mo—O2xiii2.970 (5)2.687 (5)3.254 (5)
Mo—O2xiv1.812 (5)1.783 (5)1.843 (5)
Mo—O2xv1.812 (5)1.783 (5)1.843 (5)
O1i—K—O1ii137.92 (17)130.09 (15)151.06 (18)
O1i—K—O1iii77.64 (17)64.00 (17)94.8 (2)
O1i—K—O1iv71.27 (18)65.26 (18)78.6 (2)
O1i—K—O2v71.48 (18)70.9 (2)72.30 (17)
O1i—K—O2vi147.73 (19)137.17 (17)158.17 (16)
O1i—K—O2vii100.3 (2)92.9 (3)107.2 (3)
O1i—K—O2viii95.39 (19)88.6 (2)103.5 (3)
O1ii—K—O1iii71.30 (18)65.26 (18)78.6 (2)
O1ii—K—O1iv77.71 (17)64.00 (17)94.8 (2)
O1ii—K—O2v147.70 (19)137.17 (17)158.17 (16)
O1ii—K—O2vi71.49 (18)71.0 (2)72.30 (17)
O1ii—K—O2vii95.45 (19)88.6 (2)103.4 (3)
O1ii—K—O2viii100.2 (2)92.9 (3)107.2 (3)
O1iii—K—O1iv80.86 (17)75.25 (15)90.8 (2)
O1iii—K—O2v124.5 (2)112.5 (2)139.2 (3)
O1iii—K—O2vi126.5 (2)122.8 (3)130.1 (2)
O1iii—K—O2vii153.17 (19)143.52 (16)166.83 (18)
O1iii—K—O2viii73.52 (19)68.26 (19)80.0 (3)
O1iv—K—O2v126.5 (2)122.8 (3)130.1 (2)
O1iv—K—O2vi124.6 (2)112.5 (2)139.2 (3)
O1iv—K—O2vii73.55 (19)68.3 (2)80.1 (3)
O1iv—K—O2viii153.14 (19)143.51 (16)166.83 (18)
O2v—K—O2vi78.10 (15)65.80 (15)89.04 (16)
O2v—K—O2vii77.83 (18)66.63 (15)87.31 (16)
O2v—K—O2viii64.97 (18)61.64 (18)68.8 (2)
O2vi—K—O2vii65.00 (18)61.64 (18)68.8 (2)
O2vi—K—O2viii77.80 (18)66.64 (15)87.31 (16)
O2vii—K—O2viii132.13 (16)119.25 (16)142.56 (15)
O1i—Sm—O1ii133.4 (2)129.3 (3)135.91 (19)
O1i—Sm—O1iii75.0 (2)68.8 (2)83.0 (2)
O1i—Sm—O1iv69.50 (18)66.74 (18)72.3 (2)
O1i—Sm—O2v73.5 (2)61.60 (19)83.2 (2)
O1i—Sm—O2vi151.7 (2)137.5 (2)164.6 (2)
O1i—Sm—O2vii101.3 (2)85.6 (2)118.4 (3)
O1i—Sm—O2viii95.4 (2)91.1 (2)99.4 (3)
O1ii—Sm—O1iii69.49 (18)66.74 (18)72.3 (2)
O1ii—Sm—O1iv75.2 (2)68.81 (19)83.0 (2)
O1ii—Sm—O2v151.4 (2)137.5 (2)164.8 (2)
O1ii—Sm—O2vi73.2 (2)61.60 (18)83.2 (2)
O1ii—Sm—O2vii95.3 (2)91.1 (2)99.4 (3)
O1ii—Sm—O2viii100.9 (2)85.6 (2)118.4 (3)
O1iii—Sm—O1iv78.04 (19)74.7 (2)79.83 (20)
O1iii—Sm—O2v125.5 (3)119.5 (2)130.33 (18)
O1iii—Sm—O2vi127.4 (2)117.6 (3)140.6 (3)
O1iii—Sm—O2vii149.8 (2)143.43 (19)152.8 (3)
O1iii—Sm—O2viii73.3 (2)67.3 (2)78.2 (2)
O1iv—Sm—O2v127.6 (2)117.6 (3)140.6 (3)
O1iv—Sm—O2vi125.6 (3)119.4 (2)130.33 (17)
O1iv—Sm—O2vii73.4 (2)67.4 (2)78.2 (2)
O1iv—Sm—O2viii149.8 (2)143.48 (19)152.8 (3)
O2v—Sm—O2vi79.33 (19)75.59 (18)89.7 (2)
O2v—Sm—O2vii78.8 (2)72.42 (18)91.19 (19)
O2v—Sm—O2viii67.1 (2)64.0 (2)70.9 (2)
O2vi—Sm—O2vii67.1 (2)64.0 (2)70.9 (2)
O2vi—Sm—O2viii78.64 (19)72.41 (18)91.29 (19)
O2vii—Sm—O2viii134.8 (2)131.5 (2)143.4 (3)
O1—Mo—O1ix111.2 (2)106.7 (2)116.0 (2)
O1—Mo—O1x69.22 (19)59.89 (19)78.89 (19)
O1—Mo—O1xi67.97 (18)63.34 (18)72.24 (18)
O1—Mo—O2xii75.00 (19)71.71 (19)78.31 (19)
O1—Mo—O2xiii172.98 (19)167.34 (19)177.32 (19)
O1—Mo—O2xiv107.8 (2)102.8 (2)112.7 (2)
O1—Mo—O2xv106.2 (2)105.0 (2)106.9 (2)
O1ix—Mo—O1x68.03 (18)63.34 (18)72.24 (18)
O1ix—Mo—O1xi69.37 (19)59.89 (19)78.88 (19)
O1ix—Mo—O2xii172.90 (19)167.34 (19)177.32 (19)
O1ix—Mo—O2xiii74.93 (19)71.71 (19)78.31 (19)
O1ix—Mo—O2xiv106.2 (2)105.0 (2)106.9 (2)
O1ix—Mo—O2xv107.7 (2)102.8 (2)112.7 (2)
O1x—Mo—O1xi99.18 (14)97.84 (15)100.75 (14)
O1x—Mo—O2xii113.82 (13)110.06 (13)117.71 (13)
O1x—Mo—O2xiii115.95 (15)112.35 (15)119.43 (15)
O1x—Mo—O2xiv169.33 (19)166.4 (2)171.62 (19)
O1x—Mo—O2xv72.03 (19)67.19 (19)76.75 (19)
O1xi—Mo—O2xii115.88 (15)112.34 (15)119.43 (15)
O1xi—Mo—O2xiii113.87 (13)110.06 (13)117.71 (13)
O1xi—Mo—O2xiv72.00 (19)67.19 (19)76.75 (19)
O1xi—Mo—O2xv169.35 (19)166.4 (2)171.62 (19)
O2xii—Mo—O2xiii98.93 (15)96.79 (14)101.30 (17)
O2xii—Mo—O2xiv72.62 (19)70.46 (19)74.80 (19)
O2xii—Mo—O2xv67.7 (2)60.9 (2)74.7 (2)
O2xiii—Mo—O2xiv67.7 (2)60.9 (2)74.7 (2)
O2xiii—Mo—O2xv72.59 (19)70.46 (19)74.80 (19)
O2xiv—Mo—O2xv117.2 (2)110.2 (2)124.6 (3)
Symmetry codes: (i) x1+1/2, x2+1/2, x3+1/2, x4; (ii) x1+1/2, x2, x3+1/2, x4; (iii) x1+1, x2, x3+1, x4; (iv) x1, x2+1/2, x3+1, x4; (v) x1, x2, x3+1, x4; (vi) x1+1, x2+1/2, x3+1, x4; (vii) x1+1, x2+1, x3+1, x4; (viii) x1, x21/2, x3+1, x4; (ix) x1+1, x2+1/2, x3, x4; (x) x1+1/2, x2+1/2, x3+1/2, x4; (xi) x1+1/2, x2, x3+1/2, x4; (xii) x11/2, x21/2, x3+1/2, x4; (xiii) x1+3/2, x2+1, x3+1/2, x4; (xiv) x1+3/2, x2+1/2, x3+1/2, x4; (xv) x11/2, x2, x3+1/2, x4.
 

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