mathematical, physical, and chemical fundaments of...
TRANSCRIPT
Mathematical, physical, and chemical
Fundaments of Crystallography.
Thema 1:
Periodic distributions.
Crystal.
A crystal is a periodic repetition in the three-dimensional space of a group of atomos, ions,
molecules.
The crystalline state is the highest ordered of the matter states,
correlations among different points are higher than other states of
the matter, having larger range.
.
The order is reflected on the properties, which are anisotropic
and discontinues.
Chabazite, Ca6[Al12Si24O72]· 40H2O
Cinabrium, HgS
Daily periodic structuresBrick walls
Alhambra’s tiles
Gymnastics
(Crystallinity degree)
rugs
Daily periodic structuresBrick walls
Alhambra’s tiles
Gymnastics
(Crystallinity degree)
rugs
Daily periodic structuresBrick walls
Alhambra’s tiles
Gymnastics
(Crystallinity degree)
rugs
Daily periodic structuresBrick walls
Alhambra’s tiles
Gymnastics
(Crystallinity degree)
rugs
Daily periodic structuresBrick walls
Alhambra’s tiles
Gymnastics
(Crystallinity degree)
rugs
Daily periodic structuresBrick walls
Alhambra’s tiles
Gymnastics
(Crystallinity degree)
rugs
Daily periodic structuresBrick walls
Alhambra’s tiles
Gymnastics
(Crystallinity degree)
rugs
Daily periodic structuresBrick walls
Alhambra’s tiles
Gymnastics
(Crystallinity degree)
rugs
Daily periodic structuresBrick walls
Alhambra’s tiles
Gymnastics
(Crystallinity degree)
rugs
Periodic structure
Cyistal
Disordered distribution
Amorphous/Glass
To describe periodic structures crystalline net and lattice concepts are used. We make used
of them as reference system but they do not exist.
Starting from any chosen point and taking two different directions to explore, all equivalent
points (identical surroundings) are lattice’s points. There are infinite lattices.
We have to choose the more adequated
Lattice points are independent of the real atoms or motives.
Atomic coordinates
x/a y/b1 0.0[1.0;1.0;0.0] 0.0[0.0;1.0;1.0]
2 2/3[2/3] 0.0[1.0]
3 1/6 0.5
4 ½-1/6 0.5
Cell (2D). Polyhedron limited by lattice points or nodes.
Parameters: a (x), b (y), (angle between x and y). By
translation along the two (x and y) chosen directions, the
periodic structure is reproduced.
Unit Cell. It is that which require less parameters to
describe the structure. Minimal surface + maximal
symmetry.
a = b ; = 60º; m1 (x1,y1), m2 (x2, y2); S
a ≠ b ; = 90º; m1 (x1,y1), m2 (x2, y2 =y1), m3 (x3, y3=0), m4 (x4,y4=0); S = 2 · S
C: (x, y) => (x+1/2, y+1/2) [translational symmetry]
P: (x, y)
m m => 2
6 => (3) ; m m => 2
a (= b) ; = 60º; P; m1 (1/3,1/3); S
a ≠b ; = 90º; C; m1 (x1,1/2); S = 2 · S
( 2 0 0 ) ; ( 0 1 0 )
( 3 1 0 ) ; ( 1 1 0 )0
b
a
(a || x b || y c || z)
Plane’s Families
Red Reciproca
λ = 2d sen θ ; λ = 2d sen θ => d (hkl) <=> cte / d*(hkl)
El diagrama de difracción
es una proyección de la
Red Recíproca
Familias de planos (espacio real) Puntos (espacio recíproico)
0
b
a
Chem. Eur. J., 14 (2008), 8555
Chem. Eur. J., 14 (2008), 8555
Solutions