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The jump rate of a vacancy is
identical to that of an atom next to the vacancy. It
was given by |
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ν |
= ν0 · exp
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Gm
kT |
»
ν0 · exp |
Hm
kT |
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The time ta needed so
that all the atoms with a vacancy next to them will make one jump thus is |
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ta |
= |
1
ν |
= |
1
ν0 |
· exp |
Hm
kT |
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After that time ta, the fraction
of all atoms that had a vacancy a a neighbor, has made one jump. |
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If you now wait another
ta, a second set
of atoms can now make a jump. This second set may include atoms from the first
set which simply jump back to their old position, but we ignore this effect for
a rough estimate. |
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If all atoms of the crystal are supposed to make
one jump, you have to wait for a time tc that is a
defined multiple of ta. It is simply |
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Because the multiplier m is of
course the inverse of the vacancy concentration cV = exp
(HF)/kT) |
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tc is the
quantity we we are looking for, it is |
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tc |
= |
1
ν0 |
· exp |
Hm
kT |
· exp |
HF
kT |
= |
1
ν0 |
· exp |
Hm + HF
kT |
= |
1
ν0 |
· exp |
HSD
kT |
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With HSD = enthalpy of
self diffusion. |
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We
may replace 1/ν0 by 1/ν0 = g · a2/
DSD and use the diffusion coefficient for self-diffusion
to obtain values for specific materials, but lets just look at what we get in a
very simple approximation with ν0 =
1013 Hz |
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Shown is tc
on a (rather far-reaching) log scale versus Hm +
HF = HSD, i.e. the self-diffusion
enthalpy HSD, with the temperature as a
parameter. |
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For Hm +
HF = 0, tc is
1013 s - as it should be. |
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For sensible values. e.g. HSD
= 2 eV, you must be very patient at room temperature, but at 800
oC, your crystal has a different identity after 1 second!
Take Si, with HSD »
5 eV and a melting point of roughly 1700 K, and again no atom will
be where it was after a rather short time. |
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Using better values for ν0 from the self-diffusion coefficient as
stated above, just shifts the whole set of curves a "little bit" on
the t - axis and thus tc by the same
(logarithmic) amount |
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© H. Föll