Bravais Lattices in 2D In 2D there are five ways to order atoms in a lattice Primitive unit cell: contains only one atom (but 4 points?) Are the dotted lattices primitive? Non-primitive unit cells sometimes useful if orthogonal coordinate system can be used Special case where angles go to 90  a=b Special case where point halfway a=b

2 α a1a1 a2a2 O y x b) Crystal lattice obtained by identifying all the atoms in (a) a) Situation of atoms at the corners of regular hexagons Defining Crystal Structure (In your groups) determine the lattice vectors and basis for the smallest possible unit cell (more than one possible answer) If you finish, draw the lattice.

Structure of Solids Objectives By the end of this section you should be able to: Compare bcc, fcc and hcp crystal structures Calculate atomic packing factors (HW) Determine/understand coordination numbers Identify primitive unit cell lattice parameters Be able to build the Wigner-Seitz cell for a lattice

Solid Models: Close-Packed Spheres Many atoms or ions forming solids have spherical symmetry (e.g. noble gases and simple metals) Considering the atoms or ions as solid spheres we can imagine crystals as closely packed spheres How can we pack them?

Simple cubic The simple cubic structure is a Bravais lattice. How many atoms are in each unit cell? a1a1 a2a2 a3a3 R = n 1 a 1 + n 2 a 2 + n 3 a 3

Simple cubic We can also simply count the atoms we see in one unit cell. But we have to keep track of how many unit cells share these atoms.

APF for the simple cubic structure = 0.52 ATOMIC PACKING FACTOR (APF) contains 8 x 1/8 = 1atom/unit cell Lattice constant close-packed directions a R=0.5a

Simple cubic A simple cubic structure is not efficient at packing spheres (atoms occupy only 52% of the total volume). Marbles will not resemble. Only two single elements crystallize in the simple cubic structure (F and O).

Another Reason Simple Cubic Structure is Rare Groups: Using the spheres (like atoms) and magnetic sticks (like bonds between atoms), create a simple cubic lattice. How does this compare to a triangular pyramid structure?

Vertex(corner) atom shared by 8 cells  1 / 8 atom per cell Three Cubic Unit Cell Types in 3D Often drawn as different colors (for easy viewing) but these are NOT different atoms!

# of Atoms/Unit Cell For atoms in a cubic unit cell: Atoms in corners are ⅛ within the cell

# of Atoms/Unit Cell For atoms in a cubic unit cell: Atoms on faces are ½ within the cell Atoms on Faces Face Centered Cubic (FCC)

 Vertex(corner) atom shared by 8 cells  1 / 8 atom per cell  Edge atom shared by 4 cells  1 / 4 atom per cell  Face atom shared by 2 cells  1 / 2 atom per cell  Body unique to 1 cell  1 atom per cell Three Cubic Unit Cell Types in 3D Are these primitive unit cells?

Crystal Structure14 UNIT CELLS PrimitiveConventional & Non-primitive § Single lattice point per cell § Smallest area in 2D, or §Smallest volume in 3D § More than one lattice point per cell § Integral multiples of the area of primitive cell Body centered cubic(bcc) Conventional ≠ Primitive cell Simple cubic(sc) Conventional = Primitive cell

What is the close packed direction? --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. BODY CENTERED CUBIC STRUCTURE (BCC) How would we calculate the atomic packing factor?

Better packing than SC In the body-centred cubic (bcc) structure 68% of the total volume is occupied. Next-nearest neighbors relatively close by – make structure stable in some instances. Examples: Alkali metals, Ba, V, Nb, Ta, W, Mo, Cr, Fe Is this cube a primitive lattice? No. The bcc structure is a Bravais lattice but the edges of the cube are not the primitive lattice vectors. Not smallest Vol.

--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. FACE CENTERED CUBIC STRUCTURE (FCC) What is the close packed direction? What are the lattice directions of the primitive unit cell? APF = 0.74

Simple Crystal FCC Another view

Homework 4.1 A. simple cubic with additional points in the horizontal faces C. simple cubic with additional points at the midpoints of lines joining nearest neighbors Note: green and orange are same atoms (only different colors for clarity) What does it mean to ask if it’s a Bravais lattice? That orientation looks the same from any point. Is it the smallest possible unit cell? If you think not a Bravais lattice, define the crystal structure.

Groups: Fill in this Table for Cubic Structures SCBCCFCC Volume of conventional cella3a3 a3a3 a3a3 # of atoms per cubic cell124 Volume, primitive cella3a3 ½ a 3 ¼ a 3 # of nearest neighbors6812 Nearest-neighbor distancea ½ a  3a/  2 # of second neighbors1266 Second neighbor distance a2a2 aa

Wigner-Seitz Method for Defining a Primitive Unit Cell (points are closest to each other) 1. Pick a center atom (origin) within the lattice 2. Draw perp. bisector to all neighbors of reciprocal lattice 3. Draw smallest polyhedron enclosed by bisectors Always a hexagon in 2D, unless the lattice is rectangular.

Wigner-Seitz for BCC & FCC Looks a little different in FCC. Why? Is this BCC or FCC?

Close packed crystals A plane B plane C plane A plane …ABCABCABC… packing [Face Centered Cubic (FCC)] …ABABAB… packing [Hexagonal Close Packing (HCP)]

Close-packed structures: fcc and hcp hcp ABABAB... fcc ABCABCABC... If time allows: In groups, build these two differing crystal structures.

ABAB... Stacking Sequence APF = 0.74 (same as fcc) What is the packing direction? 3D Projection 2D Projection HEXAGONAL CLOSE-PACKED STRUCTURE (HCP) For ideal packing, c/a ratio of However, in most metals, ratio deviates from this value

Lattice Planes and Miller Indices Hexagonal structure: a-b plane (2D hexagon) can be defined by 3 vectors in plane ( hkl ) 3D structure can be defined by 4 miller indices ( h k l m ) Third miller index not independent: h + k = -l Have more on HCP planes in the Additional Materials tab of website e h k l m

Other non close-packed structures In covalently bonded materials, bond direction is more important than packing diamond (only 34 % packing) graphite Packing isn’t the only consideration when building a lattice.

Simple Crystal Structures Diamond Crystal class T d (tetrahedral) - Each atom has 4 nearest- neighbors (nn). Can be interpreted as two combined fcc structures – One atom at origin – Other atom displaced along diagonal (¼, ¼, ¼) Includes C, Si, Ge,  -Sn

Diamond & Zincblende Crystal Structure Basis set: 2 atoms. Lattice  face centered cubic (fcc). The fcc primitive lattice is generated by r = n 1 a 1 +n 2 a 2 +n 3 a 3 with lattice vectors: a 1 = a(0,1,0)/2, a 2 = a(1,0,1)/2, a 3 = a(1,1,0)/2 NOTE: The a i ’s are NOT mutually orthogonal! Diamond: 2 identical atoms in basis (e.g. 2 C) fcc lattice Zincblende: 2 different atoms in basis and fcc lattice For FCC 2 atom ABCABC stacking, it is called zinc blende

How would we calculate the atomic packing factor of diamond or zincblende?