Steven Dutch, Natural and Applied Sciences, University
of Wisconsin - Green Bay

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What are the equations of lines AB and OP?. The generic equation of a line is y = mx +
b, where b is the y-intercept and m is the slope. For line AB, the slope is -tan q, or
-b/a. Thus we have: Line AB: y = -bx/a + b, or, x/a + y/b = 1. If OP is perpendicular to AB, its y- intercept is obviously zero, and we have y = x tan p, or y = xa/b, or xa = yb. Thus, a general line has a simple equation in terms of its intercepts, and a line through the origin perpendicular to the line also has a simple equation in terms of the intercepts. As a general rule, if two lines are perpendicular, the product of their slopes is -1. |

The equation for a plane in three dimensions is exactly analogous for the line in two
dimensions: x/a + y/b + z/c = 1. If line OP is perpendicular to the plane, its equation is also analogous to the two-dimensional case: xa = yb = zc. If we know a plane, we know the line through the origin perpendicular to it, called the
Line OP makes angles A, B, and C with the axes. The angles are related as follows: cos |

Cos A, cos B and cos C are called the *direction cosines* of the line. They are
obviously also direction numbers. If you know any arbitrary direction numbers a, b and c,
then cos A = a/s, cos B = b/s and cos c = c/s, where s^{2} = a^{2} + b^{2}
+ c^{2}

Now that we know the equation of a plane in space, the rules for Miller Indices are a little more intelligible. They are:

- Determine the intercepts of the face along the crystallographic axes,
*in terms of unit cell dimensions.* - Take the reciprocals
- Clear fractions
- Reduce to lowest terms

For example, if the x-, y-, and z- intercepts are 2,1, and 3, the Miller indices are calculated as:

- Take reciprocals: 1/2, 1/1, 1/3
- Clear fractions {multiply by 6}: 3, 6, 2
- Reduce to lowest terms (already there)

Thus, the Miller indices are 3,6,2. If a plane is parallel to an axis, its intercept is
at infinity and its Miller index is zero. A generic Miller index is denoted by *
{hkl}*. The convention is to use curly brackets for face indices.

If a plane has negative intercept, the negative number is denoted by a bar above the
number. *Never alter negative numbers.* For example, do not divide -1, -1, -1 by -1
to get 1,1,1. This implies symmetry that the crystal may not have!

For hexagonal and trigonal minerals, there are three possible axis directions, spaced 120 degrees apart:

Obviously, any two intercepts specify the face. Also, there will be two intercepts of
one sign and one of the other. The Miller indices for a hexagonal mineral are often
written *{hikl}*. Indices h, i and k are related by h + i + k = 0. Some modern texts
dispense with the i term and treat hexagonal minerals like all others.

The proof is simple:
Imagine a hexagonal axis system as shown and a face cutting each axis with intercepts a, b, and c. Now construct CE parallel to x1 and CD parallel to x2. Obviously triangles AOB, ADC and CEB are similar. Therefore a/b = {a-c}/c = a/c -1. Dividing by a and rearranging, we have 1/a + 1/b = 1/c. Define h = 1/a, k = 1/b and i = -1/c. Then h + i + k =0. |

- If a Miller index is zero, the plane is parallel to that axis.
- The smaller a Miller index, the more nearly parallel the plane is to the axis.
- The larger a Miller index, the more nearly perpendicular a plane is to that axis.
- Multiplying or dividing a Miller index by a constant has no effect on the orientation of the plane
- Miller indices are almost always small.

- Using reciprocals spares us the complication of infinite intercepts.
- Formulas involving Miller indices are very similar to related formulas from analytical geometry.
- Specifying dimensions in unit cell terms means that the same label can be applied to any face with a similar stacking pattern, regardless of the crystal class of the crystal. Face 111 always steps the same way whether the crystal is isometric or triclinic.

The Miller Index of a line is about as simple as it can be: if the line passes through {h, k, l}, its Miller Index is [hkl], written in brackets to distinguish it from a face.

A family of faces all parallel to some particular line is called a *zone*, and the
line is called the *zone axis*. Two faces {hkl} and {pqr} belong to zone [kr-lq,
lp-hr, hq-kp]. Note the similarity of this formula to the cross-product formula from
vector mechanics. Any other face {def} belongs to the same zone if its indices
are some linear combination of {hkl} and {pqr}, for example, d = 2h-3p, e = 2k-3q, etc.

For example, faces {110} and {010} belong to [1*0-0*1, 0*0-1*0, 1*1-1*0], or [001]. The
final zero in the face indices is a tipoff that they are both parallel to the z-axis, and
the zone index [001] *is* the z-axis. Any other face whose index is some linear
combination of {110} and {010} is also a member of that zone. Obviously the final index
must be zero.

What about faces {211} and {124}? Their zone axis is [1*4-1*2, 1*1-2*4, 2*2-1*1] or [2,-7,1]. Faces {335}, {546}, {1,-1,-3}, etc. also belong to this zone.

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*Created 9 Oct 1997, Last Update 14 Oct 1997*

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