Day 5


The magnitude of translations in inorganic crystals is on the order of 1 to 10 angstroms, and 1 angstrom is one hundred millionth of a centimeter (10-8 cm); thus a 1 cm thick crystal would have on the order of 100 milion translations! So that means they're small, right? How small? Well (here comes one of those silly examples) ... if you were to draw a lattice for the mineral halite, spacing the motifs that represent atoms 1" apart, your drawing would reach from Auburn to Los Angeles to represent about 1 cm of the mineral. Pretty doggone small.

All directions and planes in a mineral are referenced to a crystallographic coordinate system. This is always a right-handed coordinate system based on the unit cell of the mineral (see Day 2).

Three dimensional lattices can be constructed by adding one additional translation direction (vector) to the plane lattices discussed yesterday. The five types of nets can be stacked ten ways to create 14 possible space lattices or Bravais lattices. The 14 Bravais space lattices represent all possible ways that a motif can be repeated in 3D space.

The bravais lattices also serve as our introduction to the six crystal systems: cubic, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic. Note carefully the angles and axis lengths for each of the bravais lattice types and the relationship to the six crystal systems.

Note - Greek and other symbols may not come out correctly on this page - compare with your text.

Crystal System Constraints to lengths of crystallographic axes Constraints to angles between crystallographic axes
Triclinic no constraints no constraints
Monoclinic no constraints alpha=gamma=90°
Orthorhombic no constraints alpha=beta=gamma=90°
Tetragonal a=b alpha=beta=gamma=90°
Hexagonal a1=a2=a3 alpha=beta=90°; gamma=120°
Isometric a=b=c alpha=beta=gamma=90°

Vector space is referred to the three, familiar, non-coplanar axes x,y, and z; unit cell vectors are referred to as a, b, and c.

Rational directions in a mineral may be located by extending a vector from the lattice point that is the origin of the unit cell to any other lattice point. The direction is labeled with the coordinates of the lattice point placed in square brackets without commas. For example, the direction parallel to the b-axis of a crystal would be [010]. This is the same direction as [020], [030], etc. By convention, [010] is used instead of [020], [030], etc.

Rational planes are perpendicular to corresponding rational directions: Rational planes in a mineral are defined by Miller indices, which may be determined for any plane (or any line in two dimensions) from the intersections of the plane (or line) with the crystallographic axes.

The recipe for Miller indices is:

  1. Determine the intercepts of the plane of interest with the crystallographic axes (i.e., the axes of one unit cell containing the origin) ;
  2. Invert the intercepts (so that x becomes 1/x);
  3. Multiply all terms by the lowest common denominator to clear fractions. The indices determined by this recipe are placed in parentheses without commas.

For example, a plane that intersects the a-axis at 2, the c-axis at 1, and is parallel to the b-axis, would have a Miller index of (102): (2,°,1) - (1/2,1/°,1/1) - (2/2,2/°,2/1) - (102). Note that planes parallel to a crystallographic axis will have a zero in the Miller index for that axis. If the intercepts are negative, a bar is placed over the appropriate index (my html editor won't do this). When saying out loud a Miller index that contains a negative number, one says "bar" before the number.

(° = "infinity" in the text above)

Note in the previous paragraph that Miller indices can be positive and negative; remember that crystal axes are a Cartesian coordinate system, in which positive directions are taken as 'to the front, right, and top.' Therefore, if you have to move back, left, or down to intersect a line or plane, the Miller indices of that feature will contain negative numbers.

Miller indices also conveniently give the equation for the plane of interest. In two dimensions, if A and B are the intercepts of a line on the x and y axes, the equation of the line would be: x/A + y/B = 1 if A = 1/4 and B = 1/3, then the equation of the line becomes 4x + 3y = 1. If A=3 and B=4, the equation becomes x/3 + y/4 = 1 or 4x + 3y = 12 Thus the Miller indices are the rational (whole number) coefficients for the equations for any set of planes with hx +ky + lz = constant

By convention, when one refers to a set of planes for which one or more axis intercept is unspecified, the letters h, k, and l may be used for the unspecified a, b, and c indices, respectively. Thus a general Miller index would be (hkl), which would refer to any or all rational planes. All the planes parallel to a given line in a crystal are said to belong to a single zone. For example, all (h0l) planes are parallel to the b-axis.

Miller indices, in brackets, may also be used as a shorthand representation of forms. For example, if the face (110) is present on a tetragonal crystal, three faces that are symmetrically equivalent to (110) by the 4-fold axis should also be present. These are (1,bar-1,0), (bar-1,10), and (bar-1,bar-1,0). The set of crystal faces in a crystal class that are symmetrically equivalent are called a form for that class and are designated by braces around the Miller indices of one of the faces. For example, the four faces mentioned above may be represented by the one symbol {110}. This form is called a tetragonal prism (see Table 2.5 of K&H).


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