3.14/3.40J/22.71J Recap Basic Crystallography and Miller Indices Ting-Yun Huang 2012 Sep. 18th

Outline Basic Structures Cubic: FCC/BCC HCP Miller Indices From 3D to 4D: Miller indices for hexagonal crystals From 3D to 2D: Stereographic Projection Zone axis The basic structures here will have brief introduction of the cubic and hexagonal structure, including the common metal with that kind of structure. For the Miller indices, here present basic knowledge of miller indices and stereographic projection, zone axis.

BCC/FCC /HCP BASIC STRUCTURES

Basic Structures: BCC Coordination number = 8 Number of atoms in unit cell = 2 α-Fe(room temperature) , δ-Fe(high T), β-Ti(high T), Zr (high T) High temperature tends to form more open structures, (From FCC/HCP to BCC) The packing density for this structure is 68%, which is lower than FCC and HCP structure. (Which is 74%) Materials at high temperature tends to form more open structure. In most of the cases, when materials transform from low temperature to high temperature, FCC phase will transform to BCC phase. (what about the alpha ferrite to gamma phase?)

Basic Structures: FCC Coordination number = 12 Number of atoms in unit cell = 4 Closest packing ABCABC packing γ-Fe, Cu, Ag, Au, Al, Ni, Pb, Pt The transition of iron from BCC to FCC,( which contradicts with the statement that high temperature will form more open structure) is due to the change of magnetic properties, alpha ferrite is of paramagnetic properties while gamma iron is not. Usually FCC structures metals will have better ductility, see exmples like Au, Ag, Cu…

Basic Structures: HCP Coordination number = 12 Number of atoms in unit cell = 2(*) Closest packing ABAB packing Mg, Zn, α-Ti, Be, Cd, Zr (low T) The number of atoms in a unit cell actually depends on how you chose the unit cell, in here, I choose the primitive unit cell of HCP structure, which is the parallelpiped. Thought the packing density is same as FCC and the way it packs is pretty similar to the FCC structure, HCP structure has less close-packed planes. In FCC, the closest packing planes are the octahedral planes, but in HCP, only the basal plane. Thus the plastic deformation properties are much more directional for the HCP structures.

Notations/Hexagonal Systems/Stereographic Projection MILLER INDICES

Miller Indices-Notations Directions Specific Direction: [u1 u2 u3] Family of Directions: <u1 u2 u3> Planes Reciprocal of the intercepts Specific Plane: (u1 u2 u3) Family of Planes: {u1 u2 u3} The most commonly used Miller indices is just the same as the use of cartesian coordinate system. The notation/meaning for directions are exactly the same as the vectors in the cartesian coordinates. Just need to pay attention to the different brackets. For Planes, the indices for planes are the reciprocal of the intercepts of axis. () parenthesis [] square bracket Angle brackets {} curly brackets

From 3D to 4D- Miller indices for HCP structures Constraints: [u1u2u3v]→u1a1+u2a2+u3a3+vc u1+u2+u3 = 0 Redundant coordinate system which is not orthogonal But we can still perform inner product. Due to the floating degree of freedom, set the constraints of a1+a2+a3=0 This constraint actually allows us to use the convenience of dot product calculation in the cartesian coordinate system. Which I don’t want to repeat the math here.

From 3D to 2D-Stereographic Projection   Here, n denotes the 3D direction and M is the projection direction. P(n) lies in the unit circle for any n due to n is normalized. But the mark of n to p actually has degenarcy of two, thus use the close and open symbol to distinguish the front and rear hemisphere. The closed symbol is for the front hemisphere, while the open symbol is for the rear hemisphere.

Zone axis In this TEM diffraction, the structure is FCC and the zone axis is [112]. The (11-1) and (2-20) plane have the same zone axis [112]. _ (111) _ (220) Here presents an example of the use of zone axis and it’s meaning.

From 3D to 2D-Stereographic Projection Due to the symmetry, usually consider only the front hemisphere. two ways to label a plane: 1. A dot of plane normal. (e.g. the 1-10 pole here) 2. A line of characteristic ds on the plane e.g. in here, we can take 111 direction as 111 pole to denote the plan, or we can also use the (1-10) as line to represent the 111 plane.

Thank you!