# Solution to Exercise 3.1-1: "Calculate Geometry Factors"

| The geometry factor (always for a single vacancy) was defined as
g | = ½ · Si | æ ç è | D xi a | ö ÷ ø | 2
|  | With D xi = component of the jump in x-direction.
| Looking at the fcc lattice we realize that there are 12 possibilities for a jump because there are 12 next neighbors.
|  | 8 of the possible jumps have a component in x (or – x ) -direction, and D xi = a/2
|  | We thus have
g fcc | = ½ · | 8 · | æ ç è | 1 2 | ö ÷ ø | 2 | = 1
| Looking at the bcc lattice we realize that there are 8 possibilities for a jump because there are 8 next neighbors.
|  | All 8 possible jumps have the component D xi = a/2 in x-direction, again we have
g bcc | = ½ · | 8 · | æ ç è | 1 2 | ö ÷ ø | 2 | = 1
| Looking at the diamond lattice we realize, after a bit more thinking (or drawing, or looking at a ball and stick model), that there are 4 possible jumps.
|  | All 4 jumps have the component D xi = a/4 in x-direction, and we obtain
g diamond | = ½ · | 4 · | æ ç è | 1 4 | ö ÷ ø | 2 | = 1/8

With frame

Lattice and Crystal

Ionic Crystals

Exercise 3.1-1: Calculate the Geometry Factor

Exercise 3.3-1 Quick Questions

© H. Föll (Defects - Script)