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Intercepts

Intercepts. Intercepts measure where a crystal face hits a crystal axis. The location on the axes corresponding to unit lengths is arbitrary and chosen for simplicity and convenience. Axes usually radiate from the center in a right hand rule arrangement

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Intercepts

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  1. Intercepts • Intercepts measure where a crystal face hits a crystal axis. The location on the axes corresponding to unit lengths is arbitrary and chosen for simplicity and convenience Axes usually radiate from the center in a right hand rule arrangement Axes pass through centers or edges

  2. Intercepts are relative sizes • “The intercepts of a face have no relation to its size, for a face may be moved parallel to itself for any distance without changing the relative values of its intersections with the crystallographic axes.” • K&D p. 133

  3. Miller Indices from Intercepts • “The Miller Indices of a face consist of a series of whole numbers that have been derived from the intercepts by • inverting, and if necessary • by the clearing of fractions.” • “The Miller Indices [also] express a ratio ….” K&D p133

  4. Problem: find the Miller Index for a face with intercepts 2a, 2b, 2/3c • Invert the indices: • ½ ½ 3/2 • 2. This is a ratio. If we multiply all terms by a constant, the ratio remains the same. • Let’s multiply by 2 to clear the fractions: • (1 1 3)

  5. Miller Indices for horizontal and vertical faces • A face perpendicular to one axis may be considered to intersect the others at infinity. • For example, for a face perpendicular to the c-axis (aka a3-axis) the [positive side] index would be (001).

  6. The colored face is parallel to a1 and a2, meeting them only at ∞ Problem: find the Miller Index for this face with intercepts ∞a1, ∞a2, 1a3 • Invert the indices: • Since 1/∞ = 0 and 1/1 = 1 • we have 0/1 0/1 1/1 • 2. Clear fractions by multiplying • through by 1 • (0 0 1)

  7. Miller Indices for faces parallel to two axes • A face parallel to two axes may be considered to intersect the other at unity. • For example, for a face parallel to the a-axis and c-axis (or a1 and a3) the [positive side] index would be (010).

  8. Miller Indices for faces parallel to one axis • If a face is parallel to one of the crystallographic axes, a zero “0” is used (because 1/infinity = 0) • For example, for a face parallel to the a-axis, the [positive side] Miller Index could be (011)

  9. Faces that intersect axes on their negative side. • “For faces that intersect negative ends of crystallographic axes, a bar is placed over the appropriate number…. The bar represents the minus sign in a negative number

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