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)*.

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:

+x2\ /-x3 \ / \ / \ / \ / -x1 \ / +x1 ----------------+---------------- / \ / \ / \ / \ / \ +x3/ \-x2

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.

- 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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