Material Engineering - Lecture - MillerIndices, Lecture notes of Material Engineering

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Miller Indices

Allotropes

• Two or more distinct crystal structures for the same material

(allotropy/polymorphism).

• Allotropy (Gr. allos , other, and tropos , manner) or allotropism

is a behavior exhibited by certain chemical elements that can

exist in two or more different forms, known as allotropes of that

element.

• In each allotrope, the element's atoms are bonded together in a

different manner.

• Note that allotropy refers only to different forms of an element

within the same phase or state of matter (i.e. different solid,

liquid or gas forms) - the changes of state between solid, liquid

and gas in themselves are not considered allotropy.

Polymorphic Forms of Carbon

Graphite

• (^) a soft, black, flaky solid, with a layered structure – parallel hexagonal arrays of carbon atoms
• (^) weak van der Waal’s forces between layers
• (^) planes slide easily over one another

Polymorphic Forms of Carbon Fullerenes

and Nanotubes

• (^) Fullerenes – spherical cluster of 60 carbon atoms, C 60
  • (^) Like a soccer ball
• (^) Carbon nanotubes – sheet of graphite rolled into a tube
  • (^) Ends capped with fullerene hemispheres

Single Crystals

• (^) For a crystalline solid, when the periodic and repeated arrangement of atoms extends throughout without interruption, the result is a single crystal.
• (^) The crystal lattice of the entire sample is continuous and unbroken with no grain boundaries.
• (^) For a variety of reasons, including the distorting effects of impurities, crystallographic defects and dislocations, single crystals of meaningful size are exceedingly rare in nature, and difficult to produce in the laboratory under controlled conditions. Huge KDP (monopotassium phosphate) crystal grown from a seed crystal in a supersaturated aqueous solution at LLNL. Below, silicon boule.

• Some engineering applications require single crystals:
• Properties of crystalline materials often related to crystal structure. -- Ex: Quartz fractures more easily along some crystal planes than others. -- diamond Single crystals for abrasives -- turbine blades

Crystals as Building Blocks

Anisotropy

• The physical properties of single crystals of

some substances depend on the

crystallographic direction in which

measurements are taken.

• For example, the elastic modulus, electrical

conductivity, and the index of refraction may

have different values in the [100] and [111]

directions.

• The directionality of the properties is termed

anisotropy and is associated with the atomic

spacing.

Isotropic

• If measured properties are independent of the

direction of measurement then they are

isotropic.

• For many polycrystalline materials, the

crystallographic orientations of the individual

grains are totally random.

• So, though, a specific grain may be anisotropic,

when the specimen is composed of many

grains, the aggregate behavior may be isotropic.

Polycrystals

• Most crystalline solids are composed of many

small crystals (also called grains).

• Initially, small crystals (nuclei) form at various

positions.

• These have random orientations.

• The small grains grow and begin to impinge on

one another forming grain boundaries.

Micrograph of a polycrystalline stainless steel showing grains and grain boundaries

• Most engineering materials are polycrystals.
  • Nb-Hf-W plate with an electron beam weld.
  • Each "grain" is a single crystal.
  • If grains are randomly oriented, overall component properties are not directional.
  • Grain sizes typical range from 1 nm to 2 cm (from a few to millions of atomic layers). 1 mm

Polycrystals

Isotropic Anisotropic

Crystal Systems

7 crystal systems 14 crystal lattices

Unit cell: smallest repetitive volume that

contains the complete lattice pattern of a crystal.

a, b and c are the lattice constants

The fourteen (14) types of Bravais lattices grouped in seven (7) systems. (c) 2003 Brooks/Cole Publishing / Thomson Learning™

(c) 2003 Brooks/Cole Publishing / Thomson Learning™ Lattice parameters in cubic, orthorhombic and hexagonal crystal systems.