AMME1362 Chap.10 Crystal Directions, Planes and Anisotropy
Crystal Directions, Planes and Anisotropy
Crystal directions describe vectors, while crystal planes are indexed from reciprocal intercepts. Both require consistent lattice axes, reduction to the smallest integers and clear notation for negative values. Orientation matters because slip, elastic response and fracture can depend on direction.
In polycrystals, preferred orientation or texture carries single-crystal anisotropy into rolling, transverse and thickness properties.
What this chapter covers
- 01
Direction indexing
- 02
Plane intercepts and reciprocals
- 03
Negative indices
- 04
Families of directions and planes
- 05
Slip systems
- 06
Texture and bulk anisotropy
Worked example · free
Index a crystal plane
- 1Write intercepts as 1, infinity and one half in lattice units.
- 1Take reciprocals to obtain 1, 0 and 2.
- 1The indices are therefore (102), already in the smallest integers.
Key terms
- Crystal direction
- A vector expressed through components along lattice axes.
- Miller indices
- The smallest integer reciprocal-intercept notation for a crystal plane.
- Slip system
- A permitted combination of slip plane and slip direction.
- Texture
- A nonrandom distribution of crystal orientations in a polycrystal.
- Plane family
- A symmetry-related set of crystallographic planes grouped by brace notation. It differs from one specific plane and helps organise equivalent slip, packing or diffraction possibilities in a crystal of stated symmetry.
- Resolved shear
- The component of applied loading acting as shear along a candidate slip direction on a candidate plane. Orientation controls this component, while slip begins only when it also reaches the required critical resistance.
- Negative index
- An index marking a direction component or plane intercept on the negative side of a lattice axis. Losing the sign changes the orientation, so it must survive coordinate subtraction or reciprocal-intercept construction.
Crystal Directions, Planes and Anisotropy FAQ
Why does a parallel plane get index zero?
Its intercept is at infinity, and the reciprocal of infinity is zero. This geometric step distinguishes a parallel axis from an axis crossed at a finite lattice distance.
Can a direction be translated in a unit cell?
Yes. A vector can be moved parallel to itself to a convenient origin without changing direction. Rotating it or reversing its arrow changes the indexed direction.
How does texture affect engineering properties?
If many grains share preferred orientations, direction-dependent single-crystal responses no longer average randomly. Sheet can then show different yielding, forming or fracture behaviour along rolling and transverse directions.
Exam move
Keep two indexing routines on different sides of the page. For directions, choose a tail and head, subtract coordinates in lattice units, preserve negative signs, clear fractions and reduce to the smallest integer set. Translation parallel to the vector is allowed; rotation is not. Reverse several arrows deliberately and confirm that every sign reverses.
For planes, draw intercepts on the lattice axes, record infinity for a parallel axis, take reciprocals, clear fractions and reduce. Practise planes with one negative intercept and write the negative value with the correct notation. Check the answer by converting indices back to relative intercepts. This reverse step catches confusion between a zero intercept and a plane parallel to an axis.
Build a family exercise by sketching symmetry-related directions or planes in a cubic cell. Distinguish one specific orientation from a family without treating brackets and parentheses as decoration. When a plane passes through the origin, translate to a parallel member before reading intercepts; do not rotate it to make the arithmetic convenient. Connect indexing to slip.
For a proposed plane-direction pair, verify that the direction lies in the plane and that both belong to an allowed slip system supplied by the course. Then sketch tensile loading and resolve why orientation changes shear on that system. Keep geometric eligibility separate from the critical stress needed to activate motion. Add a single-crystal versus polycrystal comparison.
In one crystal, modulus, yielding or fracture can change sharply with orientation. In a randomly oriented polycrystal, many directions average, while strong processing texture preserves bulk differences between rolling, transverse and thickness directions. Link a drawing or forming observation to that preferred orientation rather than declaring every polycrystal isotropic. Use physical models when the indices remain abstract.
Mark axes on a transparent cube, tape a plane across the edges and view it from several directions. Count only lattice-unit changes, not apparent perspective lengths. Recreate the same plane in a two-dimensional projection and explain which information was lost. Finish with mixed timed drills where a prompt does not announce whether it gives a plane or direction. Identify the notation and geometric object before calculating.
After each answer, add one property connection: slip, planar density, diffraction, anisotropic stiffness or texture. The final recall task is to teach both algorithms aloud without using the words basically or just. Precise verbs such as subtract, reciprocate, translate and reduce should make each step unambiguous.
