Lecture 3: More on Structures, Miller Indeces, Stereographic Projection

PHYS 430/603 material

Laszlo Takacs

UMBC Department of Physics

Crystal structure data fromhttp://cst-www.nrl.navy.mil/lattice/

The three most essential crystal structures

1. FCC, Cu , A1, cF4

2. HCP, Mg, A3, hP2

3. BCC, W, A2, cI2

What other information is there?• Space group/ set of symmetries• Number of space group (arbitrary)• Other substances with the same

structure• Primitive translational vector of the

lattice (of mathematical points)• Basis vectors describing the position

of atoms in the basis• Different ways to view the structure

Some compound structures that derive from fcc with insterstitial sites filled:

TiN (NaCl structure)N at octahedral sites,as many as atoms

TiH2 (CaF2 structure)H at tetrahedral sites,twice as many as atoms

TiH (ZnS, zinc blende)

H only in every other tetrahedral site;related to diamond

Ti alone is hcp.

The size of atoms varies amazingly little, given the large increase in the number of electrons. Especially true for the meals used in typical alloys.

Notice the small sizes of H, B, C, N.

Defining the size of atoms is not trivial, they are not rigid balls. The atom-atom distance depends on the character of the bond.

Ionic radii are much different, small for positive,

Large for negative.

The metal lattice is rearranged to create a larger interstitial site. The Al2Cu (C16) structure of Fe2B

The NRL site on Al2Cu

• Al2Cu prototype.

• tI12: Body centered tetragonal with 12 atoms in the unit cell.

• C16: The 16th A2B structure.

• I4/mcm: Space group symbol.• 140: Arbitrary number assigned to the

space group.• X, Y, Z: Unit vectors in the directions of

the unit cell axes; a, c lattice parameters.• Atom positions within the unit cell are

given for only one of the equivalents. Notice that x has to be determined from measurement.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

The elementary vectors of translation, i.e. the edges of the elementary cell, define the unit vectors of our coordinate system. Directions and planes are defined in this coordinate system. It is identical to the ordinary rectangular coordinate system with identical scales only for cubic structures.

a

b

c

Defining directions and planes

Directions defined by a vector: r = ua + vb + wcUsually we are interested in directions where u, v, w are small integers.

Standard notation for a direction: [u v w]Notation for all equivalent directions: < u v w >There is no comma between numbers, “-” sign put over the numbers.

Set of parallel planes described by Miller indeces• Note the intersections with the axes in units of a, b, c; typically small

integers. Put down infinity, if the plane is parallel to the axis.• Take the reciprocal of the three numbers, 0 for infinity.• Multiply with the smallest number that gives a set of integers.

Notation for a single plane: (m n q)Notation for a set of equivalent planes: {m n q}There is no comma between numbers, “-” sign put over the numbers.

In the cubic system (and only there) [m n q] is the normal of (m n q).

Some planes of the [0 0 1] zone(A zone is the set of all planes that are parallel to the given direction.

Weiss zone law: hu + kv + lw =0)

Complications with the hcp structure

In the ideal case, lattice parameters a and c are related:a = edge of hexagon = diameter of atoms c = twice the distance between layersThe red tetrahedron is regular, thus

c/a = 1.62 for Mg, Co; 1.86 for Zn; 1.58 for Ti

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c = 2 a2 − a / 3( )2= a

8

3≈1.63 a

In order to reflect symmetry in the basal plane, rather than (h k l) Miller indeces defined with two primitive vectors in the basal plane, the four-index Miller-Bravais notation is used:

(h k -(h+k) l)

Similarly, a four-index notation is used for directions:

[m n p q] with m+n+p = 0 and

r = ma1 + na2 + pa3 +qc

In order to represent a direction (or the normal of a plane) in a stereographic projection, intersect the reference sphere with the direction (P) then project P from the “South pole” of the sphere onto the equatorial plane.

[0 0 1] direction points up. The intersection of directions with the sphere are projected to the equatorial plane from the [0 0 -1] point; the upper hemisphere is imaged.

The more detailed standard [0 0 1] projection of a cubic lattice.Every type of direction appears in the shaded triangle, the rest relates by symmetry operations.

Projection from the [0 1 1] direction

[0 1 1]

x

yz

Pole figure of rolled aluminum. The sample is looked at from the normal direction, the rolling direction (RD) and the transverse direction (TD) are the vertical and horizontal axis. Shade shows he probability of the indicated direction pointing in the given direction.

http://www.labosoft.com.pl/texture_standards.htm

Texture of iron processed by ECAE, Equal Channel Angular Extrusion

M.A. Gibbs, K.T. Hartwig, R.E. Goforth, Department of Mechanical Engineering, Texas A&M UniversityE.A. Payzant, HTML, Oak Ridge National Laboratory

TF