Bragg Intensity, Structure Factor, Unit Cell
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[ 2 ( h + + ) ] =cos [ 2 ( h + + ) ] + ∙ sin [ 2 ( h + + ) ] Bragg Intensity, Structure Factor, Unit Cell h ∝ | h | 2 Intensity, I hkl , is proportional to square of the structure factor, F hkl , modulus: h = ∑ =1 ∙ [ 2 ( h + + ) ] f j , x j , y j , z j – form factor and crystallographic coordinates of each j-th atom in the crystal if m, n and o – integers cos [ 2 ( h + + ) ] =cos [ 2 ( h { ± } + { ± } + { ± } ) ] sin [ 2 ( h + + ) ] =sin [ 2 ( h { ± } + { ± } + { ± } ) ] Since and |F hkl | 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, I hkl , is then obtained by multiplying (|F hkl | 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. Press any button Press any button for next slide
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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. - PowerPoint PPT Presentation
Transcript of Bragg Intensity, Structure Factor, Unit Cell
𝑒𝑥𝑝 [2𝜋 𝑖 ( h𝑥 𝑗+𝑘𝑦 𝑗+𝑙 𝑧 𝑗 ) ]=cos [2𝜋 (h 𝑥 𝑗+𝑘 𝑦 𝑗+𝑙 𝑧 𝑗 ) ]+ 𝑖∙ sin [2𝜋 (h 𝑥 𝑗+𝑘 𝑦 𝑗+𝑙 𝑧 𝑗 ) ]
Bragg Intensity, Structure Factor, Unit Cell𝐼 h𝑘𝑙∝|𝐹 h𝑘𝑙|
2Intensity, Ihkl, is proportional to square of the structure factor, Fhkl, modulus:
𝐹 h𝑘𝑙=∑𝑗=1
𝑁
𝑓 𝑗 ∙𝑒𝑥𝑝 [2𝜋 𝑖 (h 𝑥 𝑗+𝑘 𝑦 𝑗+𝑙 𝑧 𝑗 ) ]fj, xj, yj, zj – form factor and crystallographic coordinates of each j-th atom in the crystal
if m, n and o – integers,
cos [2𝜋 ( h𝑥 𝑗+𝑘𝑦 𝑗+𝑙 𝑧 𝑗 ) ]=cos [2𝜋 (h {𝑥 𝑗±𝑚}+𝑘 {𝑦 𝑗±𝑛}+ 𝑙 {𝑧 𝑗±𝑜 }) ]sin [2𝜋 (h 𝑥 𝑗+𝑘 𝑦 𝑗+𝑙 𝑧 𝑗 ) ]=sin [2𝜋 (h {𝑥 𝑗±𝑚}+𝑘 {𝑦 𝑗±𝑛 }+𝑙 {𝑧 𝑗±𝑜 }) ]
Since and
|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.
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X
Z
Ya c
b
β
(G3)1Li1(AsF6)1
(G3)LiAsF6
G3AsF6
Li
a=6.197Å b=12.697Å c=18.542Å β=95.458 P21/n 4e general positions:x, y, z
-x+1/2, y+1/2,- z+1/2
-x, -y, -zx+1/2, -y+1/2, z+1/2
x-1
x+1equivalent to
when translation along x appliedby -1
and by +1
Filling Unit Cell (all atoms in general positions)
1 asymmetric unit
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X
Z
Ya c
b
β
Filling Unit Cell (all atoms in general positions)(G3)LiAsF6
G3AsF6
Li
a=6.197Å b=12.697Å c=18.542Å β=95.458 P21/n 4e general positions:x, y, z
-x+1/2, y+1/2,- z+1/2
-x, -y, -zx+1/2, -y+1/2, z+1/2
(G3)2Li2(AsF6)2 (G3)3Li3(AsF6)3 (G3)4Li4(AsF6)4
Completed
1 asymmetric unit2 asymmetric units 3 asymmetric units 4 asymmetric units
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a
b
c
G4
(G4)LiAsF6
AsF6
Li
a=12.3788Å b=22.512Å c=12.3431Å 8d general positions: Pbcn
x, y, zx+1/2, y+1/2,- z+1/2
-x, -y, -z
x+1/2, -y+1/2, -z
Filling Unit Cell (atoms in both general and special positions)
-x+1/2, -y+1/2, z+1/2
-x, y, -z+1/2 x, -y, z+1/2-x+1/2, y+1/2, z
If x=0 (1/2)and z=1/4 (3/4)the general positions degenerate
into 4c special positions:
0, y, 1/4
1/2, -y+1/2, 3/4
0, -y, 3/4
1/2, y+1/2, 1/4
1 asymmetric unit
Shared by 2 a.u.
Shared by 2 a.u.
Shared with neighbouringunit cell
Shared with neighbouringunit cell
2 asymmetric units 3 asymmetric units
(G4)1Li2As2F7 (G4)2Li4As4F14 (G4)3Li4As4F19
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G4
(G4)LiAsF6
AsF6
Li
a=12.3788Å b=22.512Å c=12.3431Å 8d general positions: Pbcn
x, y, zx+1/2, y+1/2,- z+1/2
-x, -y, -z
x+1/2, -y+1/2, -z
Filling Unit Cell (atoms in both general and special positions)
-x+1/2, -y+1/2, z+1/2
-x, y, -z+1/2 x, -y, z+1/2-x+1/2, y+1/2, z
3 asymmetric units4 asymmetric units 5 asymmetric units
(G4)4Li4(AsF6)4 (G4)3Li4As4F19(G4)5Li6As6F31
6 asymmetric units
(G4)6Li8As8F38
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6 asymmetric units
G4
(G4)LiAsF6
AsF6
Li
a=12.3788Å b=22.512Å c=12.3431Å 8d general positions: Pbcn
x, y, zx+1/2, y+1/2,- z+1/2
-x, -y, -z
x+1/2, -y+1/2, -z
Filling Unit Cell (atoms in both general and special positions)
-x+1/2, -y+1/2, z+1/2
-x, y, -z+1/2 x, -y, z+1/2-x+1/2, y+1/2, z
(G4)6Li8As8F38
7 asymmetric units
(G4)7Li8As8F43 (G4)8Li8(AsF6)8
Completed
8 asymmetric units
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