aula08-Fundamentos do Estado Cristalino

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h, k, l are Miller indices
 a, b, c are unit cell distances
 , ,  are angles between the lattice directions
Complexity of calculations is dependent on the symmetry of the crystal system.
Reciprocal lattice
d* is perpendicular to the corresponding crystallographic planes
An infinite number of crystallographic planes in the direct lattice, is replaced by
a single vector or by a point at the end of the vector in the reciprocal lattice.
The 230 Space Groups
Operation of the translation operations on the 32 point groups produces the (somewhat intimidating) space groups (listed on following slides). 
Screw Axes combine rotation about an axis with translation parallel to it. Rotations can be 180º, 120º, 90º or 60º defining 2-, 3-, 4- and 6-fold axes respectively. 
Glide Planes combine reflection across a plane combined with translation parallel to it. Glides are expressed as a direction (a,b,c) with a subscript indicating how many glides occur in one unit distance.
Shubnikov symmetry groups – 1651 space groups. 
rules that govern space group symbolic
The international crystallographic space group symbols begin with a
capital letter designating Bravais lattice, i.e. P, A, B, C, I, F or R
The point group symbol, in which rotation axes and mirror planes can be
substituted with allowed screw axes or glide planes, respectively, is added as the second part of the international space group symbol.
In some cases, the second and the third symmetry elements of the point
group can be switched, e.g. point group symmetry
produces two different space groups without introducing glide planes and/or screw axes in the space group symbol, i.e.
		
Space group diagrams
Example: P222
a
b
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Space group diagrams
Example: P222
a
b
x,y,z
x,y,z
x,y,z
x,y,z
General equipoint: 4u (x,y,z) (x,y,z) (x,y,z) (x,y,z)
Special equipoints: 1a (0,0,0); 1b (1/2,0,0); … 2i (x,0,0), (x,0,0); 2j (x,0,1/2), (x,0,1/2); … 2m (0,y,z) (0,y,0); …
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Space group tables
 International Tables for Crystallography, Vol. A
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Space Group Information
Usual Name
Crystal System
Full Name
General positions
(diagram including handedness)
Location and type of symmetry elements
General positions
(coordinates)
Special positions
(coordinates)
Limiting conditions for observed reflections (this tells us what types of reflections we can see from this space group ).
Plane groups of special projections
Wyckoff letters are assigned to site positions starting with "a" for the site with the lowest multiplicity and the highest symmetry.