Mathematical, physical, and chemical Fundaments of ...lafactoria.lec.csic.es/mcc/attachments/article/65... · Mathematical, physical, and chemical Fundaments of Crystallography. Thema

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