Chapter 3: 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 Diamond and graphite are two allotropes of carbon: pure forms of the same element that differ in structure. Diamond An extremely hard, transparent crystal with tetrahedral bonding of carbon very high thermal conductivity very low electric conductivity. The large single crystals are typically used as gem stones The small crystals are used to grind/cut other materials diamond thin films hard surface coatings – used for cutting tools, medical devices.

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, C60 Like a soccer ball Carbon nanotubes – sheet of graphite rolled into a tube Ends capped with fullerene hemispheres

Tin, allotropic transformation Tin Disease: Another common metal that experiences an allotropic change is tin. White tin (BCT) transforms into gray tin (diamond cubic) at 13.2 ˚C. The transformation rate is slow, though, the lower the temp, the faster the rate. Due to the volume expansion, the metal disintegrates into the coarse gray powder. In 1850 Russia, a particularly cold winter crumbled the tin buttons on some soldier uniforms. There were also problems with tin church organ pipes.

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.

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

Glass Structure • Basic Unit: Glass is noncrystalline (amorphous) Silicate glass tetrahedron SiO4 Glass is noncrystalline (amorphous) • Fused silica is SiO2 - no impurities have been added. • Other common glasses contain impurity ions such as Na+, Ca2+, Al3+, and B3+ Si 4+ Na + O 2 - • Quartz is crystalline SiO2: (soda glass)

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

Polycrystals Anisotropic • Most engineering materials are polycrystals. 1 mm Isotropic • 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).

Single vs Polycrystals E (diagonal) = 273 GPa E (edge) = 125 GPa • Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: • Polycrystals 200 mm -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)

Crystal Systems Unit cell: smallest repetitive volume that contains the complete lattice pattern of a crystal. 7 crystal systems 14 crystal lattices a, b and c are the lattice constants

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

Unit Cells Types A unit cell is the smallest component of the crystal that reproduces the whole crystal when stacked together. Primitive (P) unit cells contain only a single lattice point. Internal (I) unit cell contains an atom in the body center. Face (F) unit cell contains atoms in the all faces of the planes composing the cell. Centered (C) unit cell contains atoms centered on the sides of the unit cell. Face-Centered Primitive Body-Centered End-Centered Combining  7 Crystal Classes (cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic, trigonal)  with 4 unit cell types (P, I, F, C) symmetry allows for only 14  types of 3-D lattice. 18

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

Points, Directions and Planes in the Unit Cell Miller indices - A shorthand notation to describe certain crystallographic directions and planes in a material. Denoted by [ ], <>, ( ) brackets. A negative number is represented by a bar over the number.

Point Coordinates Coordinates of selected points in the unit cell. The number refers to the distance from the origin in terms of lattice parameters.

Point Coordinates Point Coordinates z x y a b c 000 111 Point coordinates for unit cell center are a/2, b/2, c/2 ½ ½ ½ Point coordinates for unit cell corner are 111 Translation: integer multiple of lattice constants  identical position in another unit cell z 2c y b b

Miller Indices, Directions Determine the Miller indices of directions A, B, and C. (c) 2003 Brooks/Cole Publishing / Thomson Learning™

SOLUTION Direction A 1. Two points are 1, 0, 0, and 0, 0, 0 2. 1, 0, 0, -0, 0, 0 = 1, 0, 0 3. No fractions to clear or integers to reduce 4. [100] Direction B 1. Two points are 1, 1, 1 and 0, 0, 0 2. 1, 1, 1, -0, 0, 0 = 1, 1, 1 4. [111] Direction C 1. Two points are 0, 0, 1 and 1/2, 1, 0 2. 0, 0, 1 -1/2, 1, 0 = -1/2, -1, 1 3. 2(-1/2, -1, 1) = -1, -2, 2

Crystallographic Directions z Algorithm 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvw] y x ex: 1, 0, ½ Lecture 2 ended here => 2, 0, 1 => [ 201 ] -1, 1, 1 where overbar represents a negative index [ 111 ] =>

Families of Directions <uvw> For some crystal structures, several nonparallel directions with different indices are crystallographically equivalent; this means that atom spacing along each direction is the same.

Crystallographic Planes If the plane passes thru origin, either: Construct another plane, or Create a new origin Then, for each axis, decide whether plane intersects or parallels the axis. Algorithm for Miller indices 1.  Read off intercepts of plane with axes in terms of a, b, c 2. Take reciprocals of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses, no commas.

Crystallographic Planes Crystallographic planes are specified by 3 Miller Indices (h k l). All parallel planes have same Miller indices.

Crystallographic Planes z x y a b c example a b c 1. Intercepts 1 1  2. Reciprocals 1/1 1/1 1/ 1 1 0 3. Reduction 1 1 0 4. Miller Indices (110) example a b c z x y a b c 1. Intercepts 1/2   2. Reciprocals 1/½ 1/ 1/ 2 0 0 3. Reduction 2 0 0 4. Miller Indices (200)

Crystallographic Planes example a b c 1. Intercepts 1/2 1 3/4 2. Reciprocals 1/½ 1/1 1/¾ 2 1 4/3 z x y a b c 3. Reduction 6 3 4 4. Miller Indices (634)

Family of Planes Planes that are crystallographically equivalent have the same atomic packing. Also, in cubic systems only, planes having the same indices, regardless of order and sign, are equivalent. Ex: {111} = (111), (111), (111), (111), (111), (111), (111), (111) _ (001) (010), (100), (001), Ex: {100} = (100),

FCC Unit Cell with (110) plane

BCC Unit Cell with (110) plane

SUMMARY Crystallographic points, directions and planes are specified in terms of indexing schemes. Materials can be single crystals or polycrystalline. Material properties generally vary with single crystal orientation (anisotropic), but are generally non-directional (isotropic) in polycrystals with randomly oriented grains. Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy).