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Lets look at a simple cubic lattice containing one
vacancy and connect the atoms by springs symbolizing the bonds. It looks like
this: |
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We have two kinds of springs:
- The red ones connect nearest neighbors and
will heavily influence the vibration frequencies.
- The violet ones, connecting diagonally. They
will have some bearing on the vibration frequency, but since they must be
weaker (the bond is weaker) than the red springs, their influence should be
less pronounced.
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However, without the violet springs you could not make a
stable crystal if you tried to built a model with balls and springs. |
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Lets assign a spring constant D to the red
springs and c · D to the violet springs, with c
< 1, and see what we get for the vibration frequency of an atom
completely surrounded by other atoms and for the atoms around the vacancy. |
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Generally, the resonance (circular)
frequency ω of a particle with mass
m is given by |
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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 |
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. |
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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 |
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The entropy of formation
SF then becomes (note that we only have to sum over
the "afflicted" dimensions): |
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SF |
= |
k · |
6
S
1 |
ln |
ω0
ω1 |
= k · 6 · ln (2)1/2 = 3k
· ln 2 = 3k · 0,693 |
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= |
2,08 k |
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Not bad for such a simple
approximation. But now lets go one step further and add the
violet springs. |
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We have now for the frequency of the lattice
atoms without a vacancy |
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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. |
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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 ω2, and consider the changed
ω0, too. Altogether we have |
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ω02 |
= |
2D + 4cD
m |
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ω12 |
= |
D + 2cD
m |
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ω22 |
= |
2D + 3cD
m |
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The entropy now is |
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SF |
= k · |
( |
6
S
1 |
ln |
ω0
ω1 |
+ |
12
S
1 |
ln |
ω0
ω2 |
) |
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Crunching the numbers gives |
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SF |
= |
3k · ln |
2 + 4c
1 + 2c |
+ 6 · ln |
2 + 4c
2 + 3c |
= 3k ·ln (2) + 6k · ln |
2 + 4c
2 + 3c |
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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 |
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SF |
= |
3k · ln (2) + 6k · ln (1,171) = 2,08 k +
0,947k = 3,027 k |
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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. |
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© H. Föll