The regularity of the external form of crystals suggests that crystals are formed by the regular repetition of identical building blocks (Figure E.4). When a crystal grows in a constant environment, the shape remains unchanged during growth, as if identical building blocks were being added continuously. The building blocks are atoms or groups of atoms: A crystal is a three-dimensional periodic array of atoms. This was learned in the eighteenth century when mineralogists discovered that the index numbers of the directions of all faces of a crystal are exact integers.

There is an extensive formalism to the description of crystal systems. For our purpose we can regard them as lattices of atoms. Figure E.5 gives a few of the conventional cells, and Figure E.6 shows the notation used in Table E.2 to describe the atoms.

When crystals are well formed, it should be possible to identify them as belonging to one of the seven crystal systems. The chemical substance usually has already been unequivocally assigned to one system on the basis of x-ray diffraction analysis. Six of these systems, diagramed in Figure E.5, are defined as follows:

Cubic: Three mutually perpendicular axes with equal atomic spacing.

Tetragonal: Two mutually perpendicular axes having equal spacing, with different spacing along the axis perpendicular to the other two.

Fig. E.4. A scanning electron microscope (SEM) micrograph of a crystal of magnesium oxide (MgO). The crystal faces and the growth steps are evident on the cubic crystal. The legend at the bottom shows that the microscope was operated at 3.0 kilovolts (kV) with a magnification of 60,000 (60.0K), and the white dot markers indicate a length of 500 nanometers (0.5 microns), which indicates that the crystal is somewhat over 1 micron in size with a top growth face of 0.5 by 0.5 microns. (Micrograph by Bob Roberts, Arizona State University.)

Hexagonal: Two axes 120° apart with equal atomic spacing and different spacing along the axis perpendicular to the plane of the other two.

Orthorhombic: Three mutually perpendicular axes with unequal atomic spacing.

Monoclinic: Two perpendicular axes with unequal spacing, the third nonperpendicular axis with different spacing.

Triclinic: Three oblique axes with unequal spacing.

Inorganic pigments are characterized by the presence of crystalline particles. These particles have a particular crystal structure as given in Figure E.5 and Table E.4. Four structures, rhombohedral, hexagonal,

Fig. E.5. Six crystal systems that are found in pigment crystallites.

tetragonal, and monoclinic, are characteristic of the pigments. Even though two red pigments may have the same structure—vermilion and red ochre are both rhombohedral—the spacings between atoms and planes are different. X-ray diffraction allows one to measure the planar spacings and hence to identify the pigments.

TABLE E.4 | ||

System |
Axial Lengths and Anles |
Bravais Lattice |

Cubic |
Three equal axes at right angles |
Simple |

a = b = c, a = fi = y = 90° |
Body-centered | |

Face-centered | ||

Tetragonal |
Three axes at right angles, two equal |
Simple |

a = b ± c, a = fi = y = 90° |
Body-centered | |

Orthorhombic |
Three unequal axes at right angles |
Simple |

a ± b ± c, a = fi = y = 90° |
Body-centered | |

Base-centered | ||

Face-centered | ||

Rhombohedral |
Three equal axes, equally inclined |
Simple |

a = b = c, a = fi = y ± 90° | ||

Hexagonal |
Two equal coplanar axes at 120°, |
Simple |

third axis at right angles | ||

a = b ± c, a = fi = 90°, y = 120° | ||

Monoclinic |
Three unequal axes, |
Simple |

one pair not at right angles |
Base-centered | |

a ± b ± c, a = y = 90° ± fi | ||

Triclinic |
Three unequal axes, unequally inclined |
Simple |

and none at right angles | ||

a ± b ± c, a ± fi ± y ± 90° | ||

Seven crystal systems indicating axial lengths, angles, and Bravais lattice. |

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