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

The geometry factor (always for a single vacancy) was defined as
 
g  =  ½ · Σi ( Δxi
a 		) 2
 
With Δxi = 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 Δxi = a/2
We thus have
 
g fcc  =  ½ · 8 · (  1 
2		 ) 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 Δxi = a/2 in x-direction, again we have
 
g bcc  =  ½ · 8 · (  1 
2		 ) 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 Δxi = a/4 in x-direction, and we obtain
 
g diamond  =  ½ · 4 · (  1 
4		 ) 2  =  1/8
 
 

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