Untitled Quiz

Crystallographic Directions

• A crystallographic direction is a line between two points, also known as a vector.
• Determining crystallographic directions involves these steps:
  • Position a vector of a convenient length through the origin of the coordinate system. Maintaining parallelism throughout the crystal lattice is crucial.
  • Measure the vector's projections onto each of the three axes (a, b, and c) in terms of the unit cell dimensions.
  • Reduce the three values to their smallest integer values by multiplication or division by a common factor.
  • Enclose the reduced indices within square brackets, e.g., [uvw], where u, v, and w correspond to the reduced projections along the x, y, and z axes, respectively.

Negative Indices

• Negative coordinates are possible and are represented by a bar over the appropriate index.
• Reversing the signs of all indices defines an antiparallel direction.
• For example, [111] is the direct opposite of [111].

Equivalent Directions

• Some crystal structures have multiple non-parallel directions with different indices but have equivalent atomic spacing.
• These equivalent directions are grouped together in angle brackets, e.g., (100).
• In cubic crystals, directions with the same indices (irrespective of order or sign), are equivalent. For example, [123] and [213] are equivalent.

Crystallographic Planes

• The orientations of crystal planes are also represented similarly.
• In most crystal systems (except hexagonal), crystallographic planes are specified by three Miller indices (hkl).
• Determining indices follows these steps:
  • If the plane passes through the chosen origin, a parallel plane must be constructed within the unit cell. Otherwise, a new origin must be created.
  • Determine the intercepts of the plane with each of the three axes (x, y, and z) in terms of the lattice parameters (a, b, and c).
  • Calculate the reciprocals of these intercepts.
  • Simplify the reciprocals to their smallest integral values via multiplication or division by a common factor.
  • Enclose the integer indices in parentheses to represent the plane, e.g., (hkl).
• Intercepts on the negative side of the origin are indicated by a bar (e.g., (123)). Reversing all the indices gives the opposite plane.
• Planes with the same indices (e.g., (100)) in a cubic crystal are perpendicular to each other

Atomic Arrangement

• Atomic arrangements on crystallographic planes depend on the crystal structure (e.g., FCC or BCC).
• Circles in diagrams represent atoms located at the centers of full-sized hard spheres in the crystallographic planes.
• A "family" of planes includes all equivalent planes having the same atomic packing.
• In cubic crystals, equivalent planes share the same indices, irrespective of order and any sign. For example both (123) and (312) belong to the {123} family.

Crystalline and Non-crystalline Materials

• A single crystal has a perfect, repeating atomic arrangement throughout the entire specimen.
• Single crystals can occur naturally or be created artificially.
• Growth of single crystals is often difficult and requires careful control of the environment.
• Polycrystalline materials are composed of many small crystals/grains that grow during solidification.
• These crystals have random orientations, indicated by square grids.
• Adjacent grains in a polycrystalline material impinge on one another and have some atomic mismatch forming grain boundaries.

Anisotropy

• Anisotropy describes the phenomenon where physical properties of single crystals depend on the crystallographic direction in which the measurements are taken.
• Properties like elastic modulus, electrical conductivity, and refractive index can vary based on direction.
• Isotropic materials have properties that are independent of direction.
• The extent and magnitude of anisotropic effects increase with decreasing structural symmetry in crystalline materials. Triclinic structures are highly anisotropic.