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How can we measure directly the type and concentration of point defects
and, if we do it as function of temperature, extract the formation energies and formation entropies? |
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Simple question - but there is essentially only
one direct method: Measure the change of
the lattice constant a, i.e. Δa, and the change in the specimen dimension,
Δl, (one dimension is sufficient)
simultaneously as a function of temperature. |
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What you have then is the differential thermal
expansion method also called the Δl/l – Δa/a method. |
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This method was invented by Simmons and
Balluffi around 1960. |
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The basic idea is that Δl/l – Δa/a (with l = length of
the specimen = l(T, defects)) contains the regular thermal
expansion and the dimensional change from
point defects, especially vacancies. |
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This is so because for every vacancy in the
crystal an atom must be added at the surface; the total volume of the vacancies
must be compensated by an approximately equal additional volume and therefore
an additional Δl. |
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If we subtract the regular thermal expansion,
which is simply given by the change in lattice parameter, whatever is left can
only be caused by point defects. The
difference then gives directly the vacancy concentration. |
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For a cubic crystal with negligible relaxation
of the atoms into the vacancy (so the total
volume of the vacancy provides added volume of the crystal), we have |
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( |
Δl
l |
– |
Δa
a |
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= |
cV – ci |
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With cV = vacancy
concentration, ci = interstitial concentration. |
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We have to take the difference of the concentration because interstitial
atoms (coming from a vacancy) do not add
volume. |
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This is quite ingenious and
straightforward, but not so easy to measure in practice. |
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The measurements of both parameters have to be
very precise (in the 10– 5 range); you also may have to
consider the double vacancies. |
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But successful measurements have been made for
most simple crystals including all important metals, and it is this method that
supplied the formation energies and entropies for most important materials.
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The link shows a
successful measurement of
Δl/l – Δa/a for Ag + 4% Sb. |
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Some values mostly obtained with that
method are shown in the following table (after Seeger): |
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| Element |
cV at Tm |
HF [eV] |
SF [k] |
| Cu |
2 x 10–4 |
1,04 |
0,3 |
| Ag |
1,7 x 10–4 |
0,99 |
0,5 |
| Au |
7,2 x 10–4 |
0,92 |
0,9 |
| Al |
9 x 10–4 |
0,65 |
0,8 |
| Pb |
1,7 x 10-4 |
0,5 |
0,7 |
| Na |
7 x 10–4 |
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| Li |
4 x 10–4 |
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| Cd |
6,2 x 10–4 |
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| Kr |
3 x 10–3 |
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| Si |
no values, Δl/l – Δa/a = 0 even at ultra-high
precision |
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A somewhat exotic, but still rather
direct method is measuring the time constant for positron
annihilation as a function of temperature to obtain
information about vacancies in thermal equilibrium. |
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What you do is to shoot
positrons into your sample
and measure how long it takes for them to disappear by annihilation with an
electron in a burst of γ - rays. The time
from entering the sample to the end of the positron is its (mean) life time
τ. |
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It is rather short (about 10–10
seconds), but long enough to be measured, and it
varies with the concentration of vacancies in the sample. Since
electrons are needed for annihilation and a certain overlap of the wave
functions has to occur, the life time τ is
directly related to the average electron concentration available for
annihilation. |
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A nice feature of these technique is that the positron is
usually generated by some radioactive decay event, and then announces its birth
by some specific radiation emitted simultaneously. Its death is also marked by
specific γ rays, so all you have to do is to
measure the time between two special bursts of radiation. |
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Vacancies are areas with low electron densities.
Moreover, they are kind of attractive to a positron because they form a
potential well for a positron - once it falls in there, it will be trapped for
some time. |
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Since an average life time of 10–10 s
is large enough for the positron, even after it has been thermalized, to cover
rather large distances on an atomic scale, some positrons will be trapped
inside vacancies and their percentage will depend on the vacancy
concentration. |
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Inside a vacancy the electron density is smaller than in the
lattice, the trapped positrons will enjoy a somewhat longer life span. The
average life time of all positrons will thus go up with an increasing number of
vacancies, i.e. with increasing temperature. |
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This can be easily quantified in a good
approximation as follows. |
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Lets assume that on the average we have
n0 (thermalized) positrons in the lattice, split into
n1 "free" positrons, and
n2 positrons trapped in vacancies; i.e. |
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The free positrons will either decay with a fixed rate
λ given by λ1 = 1/τ1, (with τ1 = (average) lifetime), or are trapped
with a probability ν by vacancies being
present in a concentration cV. |
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The trapped positrons then decays with a rate
λ2 which will be somewhat smaller
then λ1 because it lives a little
longer; its average lifetime is now τ2. |
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The change in the partial concentration then becomes |
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dn1
dt |
= |
– (λ1 +
ν · cV) ·
n1 |
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dn2
dt |
= |
– λ2 ·
n2 + ν · cV
· n1 |
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This system of coupled differential equation is
easily solved (we will do that as an
exercise), the starting
conditions are |
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| n1(t = 0) |
= |
n0 |
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| n2(t = 0) |
= |
0 |
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The average lifetime τ,
which is the weighted average of the decay paths and what the experiment
provides, will be |
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| τ |
= τ1 ·
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( |
1 + τ2 ·
ν · cV
1 + τ1 ·
ν · cV |
) |
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The probability ν for a positron to get trapped by a vacancy can be
estimated with relative ease, the following principal "S" - curve is
expected. By now, it comes as no surprise that no effect was found for
Si. |
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The advantage of positron annihilation experiments
is its relatively high sensitivity for low vacancy concentrations
(10–6 - 10–7 is a good value), the
obvious disadvantage that a quantitative evaluation of the data needs the
trapping probability, or cross section for positron capture. |
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Some examples of real measurements and further
information are given in the links:
Life time of positrons in Ag
Life time of positrons in Si and
Ge.
Paper (in German):
Untersuchung von
Kristalldefekten mit Hilfe der Positronenannihilation |
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A large table containing values for
HF as determined by positron annihilition (and compared
to values obtained otherwise) can be found in the link |
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There isn't much. Some occasionally
used methods are |
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Measurements of the resistivity. Very suitable to ionic crystals if the
mechanism of conduction is ionic transport via point defects. But you never
know for sure if you are measuring intrinsic equilibrium because
"doping"
by impurities may have occurred. |
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Specific heat as a
function of T. While there should be some dependence on the
concentration of point defects, it is experimentally very difficult to handle
with the required accuracy. |
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Measuring electronic
noise. This is a relatively new method which relies on very
sophisticated noise measurements. It is more suited for measuring diffusion
properties, but might be used for equilibrium conditions, too. The illustration
in the link shows a noise
measurement obtained upon annealing frozen-in point defects. |
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However, the view presented above (and in the
chapters before) is not totally unchallenged. There are serious scientists out
there who claim that things are quite different, especially with respect to
equilibrium concentrations of vacancies in refractory metals, because the
formation entropy is much higher than assumed. |
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The method of choice to look at this is calorimetry at high temperature, i.e. the
measurement of the specific heat. A champion of this viewpoint is Y.
Kraftmakher, who just published a book to this
point. |
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© H. Föll
