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Bragg Intensity, Structure Factor, Unit Cell. Intensity, I hkl , is proportional to square of the structure factor, F hkl , modulus:. Press any button for next slide. Press any button.
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Bragg Intensity, Structure Factor, Unit Cell Intensity, Ihkl, is proportional to square of the structure factor, Fhkl, modulus: Press any button for next slide Press any button fj, xj, yj, zj – form factorand crystallographic coordinates of each j-th atom in the crystal Since and if m, n and o – integers, |Fhkl|2 can be calculated using the coordinates of atoms located only within a unity-sized (0≤x,y,z≤1) portion of the crystal. This portion – the unit cell – is chosen in such a way that when the atoms within it are translated along the x, y and z axes (by adding m, n and o to their coordinates) they map all atoms within the crystal. The intensity value, Ihkl, is then obtained by multiplying (|Fhkl|2)unit cell by the number of unit cells within the crystal (included in “Scale factor” variable in Rietveld codes). Conventional description of crystal structures does not include coordinates of all atoms in the unit cell. Instead only coordinates of atoms within the asymmetric unit of the unit cell are given together with the space group. The latter defines non-translational symmetry operators using which the rest of the unit cell is filled.
Filling Unit Cell (all atoms in general positions) a=6.197Å b=12.697Å c=18.542Å β=95.458 P21/n 4e general positions: (G3)LiAsF6 x, y, z -x+1/2, y+1/2,- z+1/2 -x, -y, -z Li G3 x+1/2, -y+1/2, z+1/2 AsF6 Y 1 asymmetric unit β c a x-1 x+1 equivalent to b when translation along x applied by -1 (G3)1Li1(AsF6)1 and by +1 Z Press any button X
Press any button Filling Unit Cell (all atoms in general positions) a=6.197Å b=12.697Å c=18.542Å β=95.458 P21/n 4e general positions: (G3)LiAsF6 x, y, z -x+1/2, y+1/2,- z+1/2 -x, -y, -z Li G3 x+1/2, -y+1/2, z+1/2 AsF6 Y 1 asymmetric unit β c 4 asymmetric units 3 asymmetric units 2 asymmetric units a Completed b Z Press any button for next slide (G3)4Li4(AsF6)4 (G3)3Li3(AsF6)3 (G3)2Li2(AsF6)2 X
Filling Unit Cell (atoms in both general and special positions) a=12.3788Å b=22.512Å c=12.3431Å 8d general positions: Pbcn (G4)LiAsF6 x, y, z -x, -y, -z x+1/2, y+1/2,- z+1/2 -x+1/2, -y+1/2, z+1/2 G4 -x, y, -z+1/2 x, -y, z+1/2 Li -x+1/2, y+1/2, z x+1/2, -y+1/2, -z AsF6 1 asymmetric unit 2 asymmetric units If x=0 (1/2)and z=1/4 (3/4) the general positions degenerate into 4c special positions: 3 asymmetric units a c Shared with neighbouring unit cell 0, y, 1/4 1/2, -y+1/2, 3/4 Shared by 2 a.u. Shared with neighbouring unit cell 0, -y, 3/4 b 1/2, y+1/2, 1/4 Shared by 2 a.u. Press any button (G4)1Li2As2F7 (G4)2Li4As4F14 (G4)3Li4As4F19
Filling Unit Cell (atoms in both general and special positions) a=12.3788Å b=22.512Å c=12.3431Å 8d general positions: Pbcn (G4)LiAsF6 x, y, z -x, -y, -z x+1/2, y+1/2,- z+1/2 -x+1/2, -y+1/2, z+1/2 G4 -x, y, -z+1/2 x, -y, z+1/2 Li -x+1/2, y+1/2, z x+1/2, -y+1/2, -z AsF6 4 asymmetric units 5 asymmetric units 3 asymmetric units 6 asymmetric units Press any button (G4)4Li4(AsF6)4 (G4)5Li6As6F31 (G4)3Li4As4F19 (G4)6Li8As8F38
Filling Unit Cell (atoms in both general and special positions) a=12.3788Å b=22.512Å c=12.3431Å 8d general positions: Pbcn (G4)LiAsF6 x, y, z -x, -y, -z x+1/2, y+1/2,- z+1/2 -x+1/2, -y+1/2, z+1/2 G4 -x, y, -z+1/2 x, -y, z+1/2 Li -x+1/2, y+1/2, z x+1/2, -y+1/2, -z AsF6 7 asymmetric units 8 asymmetric units 6 asymmetric units Completed Press any button (G4)8Li8(AsF6)8 (G4)6Li8As8F38 (G4)7Li8As8F43
