
Generally, the resonance (circular)
frequency w of a particle with mass
m is given by 




In the most simple approximation,
only accounting for the red springs, a regular atom would feel the force of
two springs per
direction and thus vibrate in any of the three dimensions with 


. 




The six atoms (for three dimensions)
surrounding the vacancy and missing one red
spring each (in one dimension), in contrast, would vibrate in one of the three
dimensions with 




The entropy of formation
S_{F} then becomes (note that we only have to sum over
the "afflicted" dimensions): 


S_{F} 
= 
k · 
6
S
1 
ln 
w_{0}
w_{1} 
= k · 6 · ln (2)^{1/2} = 3k
· ln 2 = 3k · 0,693 








= 
2,08 k 



Not bad for such a simple
approximation. But now lets go one step further and add the
violet springs. 


We have now for the frequency of the lattice
atoms without a vacancy 





and we simply include the factor
(1/2)^{–1/2}, that would give us the component of the
violet springs in the direction considered, into c; we thus have
c < 0,707. 

We now have to consider the 6
atoms with a missing red spring and 2 missing violet springs separately
from the 12 atoms just missing one violet spring which are vibrating
with w_{2}, and consider the changed
w_{0}, too. Altogether we have 


w_{0}^{2} 
= 
2D + 4cD
m 



w_{1}^{2} 
= 
D + 2cD
m 



w_{2}^{2} 
= 
2D + 3cD
m 



The entropy now is 


S_{F} 
= k · 
æ
ç
è 
6
S
1 
ln 
w_{0}
w_{1} 
+ 
12
S
1 
ln 
w_{0}
w_{2} 
ö
÷
ø 




Crunching the numbers gives 


S_{F} 
= 
3k · ln 
2 + 4c
1 + 2c 
+ 6 · ln 
2 + 4c
2 + 3c 
= 3k ·ln (2) + 6k · ln 
2 + 4c
2 + 3c 




For c = 0 we must obtain our old
result which indeed we do (check it), and for c = 0,707, the most
extreme case possible, we find 


S_{F} 
= 
3k · ln (2) + 6k · ln (1,171) = 2,08 k +
0,947k = 3,027 k 



In other words: For realistic
c values, the correction is negligible and we can confidently
claim that the formation entropy of a monovacancy in a cubic primitive lattice
is around 2 k in our ball and spring model approximation. 

