Crystals And Structure

1. Crystals And Structure

2. Solids Solids grow as crystalline, regular structures. The repetitive pattern where atoms reside is known as the lattice. There is a limited number of such symmetries which can produce solids. The simplest form are 2d lattices. The idea of a lattice is to leave no uncovered area behind. Unit cells describe the smallest unit whose repetition creates the lattice.

3. The Reciprocal Lattice The reciprocal lattice is useful when x-rays are employed to determine crystal lattice structures. Wigner-Seitz cells, and their equivalents in reciprocal space, Brillouin zones, are the smallest unit to be investigated by x-ray structure analysis.

4. 17 plane 2d groups Perhaps surprisingly there are only 17 ways to create 2d solid lattices. The symbols in the drawings represent rotational symmetry operations, and the lines mirror planes. The acronyms below the figures are the names of the structures.

5. 3d lattices Analogous ideas lead to the 3d lattices. There are 14 basic types, known as the Bravais lattices.

6. A systematic summary of the 3d lattice types:

7. Symmetry The simple cubic structure shows a high degree of symmetry exactly because of its simplicity. Square symbols represent points of 4-fold symmetry in the structure, triangles 3-fold, Ovals 2-fold. In some structures there is also 6-fold symmetry represented by hexagons. The cube possesses also many mirror planes. The name explains itself:

8. Densest atom packing In solids, atoms are packed very densely. Different lattice structures arise from varying stacking orders of densely packed planes. An example how stacking order changes hexagonal into rhombohedral order:

9. Stacking and coordination polyhedra The details of the stacking order lead to the nearest neighbor configuration of an atom in the structure. These are polyhedra and are useful in understanding oxidation states, conductivity pathways, electron orbit overlap, etc.

10. Miller Indices To describe directions and planes within a 3d lattice a mathematical language is introduced. The (hkl) triple describes distances in a Cartesian coordinate system.

11. Examples of Miller indices describing certain planes in a lattice.

19. 3d Wigner-Seitz cells

20. 3d reciprocal lattice

21. Example: monoclinic cell

23. 3d Brillouin zones

24. Space groups

30. X-Ray Diffraction Structure determination

31. Diffraction of Light

42. Indexing Lines

43. Atomic scattering factor

44. The structure factor