1 Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University 02/24/ /24/2015 DEE3517 Solid State Physics (1) Lecture 1c: Bravais Lattice: Three Dimensions (3-D)

2 How to count the number of atoms in a cell? A complete circle (spanned by 360 o ) in 2-D = one atom A complete sphere in 3-D = one atom Summing up from corner to corner and within

3 Keep in Mind: Packing density of a conventional unit cell = number of atoms in the conventional unit cell divided by its volume Packing density of a primitive unit cell #1 = 1/the primitive unit cell #1 volume (Note: only one atom in the primitive unit cell #1) Packing density of a primitive unit cell #2 with a M-atom basis = M/the primitive unit cell #2 volume 1.A conventional unit cell (thus a primitive unit cell #1 with a single atom) may have another conventional unit cell (hence another primitive unit cell #2 in a basis having M atoms). 2.Above packing densities must have the same value.

4 from Wikipedia

5 You should be able to produce this table, except those of second neighbors.

6 BCC

7 FCC

8 fcc or bcc = sc (simple cubic) + a M-atom basis

9 Diamond = a simple fcc + two atoms (0,0,0) and (1/4, 1/4, 1/4) as basis M = 2

10 HCP = a simple Hexagonal + 2 atoms as bases

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