1000 Chemistry Lecture 10: Crystal structures (continued)
Marc R. Roussel
Packing fraction
Recall:
A corner atom is shared between 8 unit cells 1 of an atom is inside any given cell. 8 A facial atom is shared between 2 unit cells 1 of an atom is inside any given cell. 2
For the bcc lattice, we have 8
1 8
+ 1 = 2 atoms.
The diagonal of the bcc unit cell = 4r .
D
4r a
A a x B a C
In the triangle ABC, x 2 = a2 + a2 = 2a2 . In the triangle ACD, x 2 = (4r )2 a2 . Combining the two equations, we get 2a2 = 16r 2 a2 , or r = 43 a.
4 The atoms occupy 2 atoms 3 r 3 =
Since the total volume of the unit the unit cell occupied by atoms is
3 3 8 a . cell is a3 , the 3 8 = 0.68.
fraction of
Coordination number
Number of atoms in direct contact with a given atom in a lattice. In bcc, this is 8. (Think about the central atom in the unit cell.) In fcc and hcp, this is 12.
Holes in close-packed lattices
Tetrahedron: Octahedron:
tetrahedral hole
octahedral hole
For a close-packed with lattice N atoms, there are 2N tetrahedral holes N octahedral holes
Challenges
1. Show that the packing fraction in a simple cubic crystal is 6 = 0.52. 2. Convince yourself that the coordination number in a simple cubic structure is 6.
X-ray diraction
Xray beam
sample scattered Xrays detector
Bragg diraction condition
X-rays scatter after interacting with an atom. X-rays which interact with atoms in dierent planes of the crystal travel dierent distances. We will observe X-rays emerging from the crystal when X-rays which interact at dierent depths are in phase.
P x
Q x
d
...