
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. 


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 g  rays. The time
from entering the sample to the end of the positron is its (mean) life time
t. 


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 t is
directly related to the average electron concentration available for
annihilation. 


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 g rays, so all you have to do is to
measure the time between two special bursts of radiation. 

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. 


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. 


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. 

This can be easily quantified in a good
approximation as follows. 


Lets assume that on the average we have
n_{0} (thermalized) positrons in the lattice, split into
n_{1} "free" positrons, and
n_{2} positrons trapped in vacancies; i.e. 





The free positrons will either decay with a fixed rate
l given by l_{1} = 1/t_{1}, (with t_{1} = (average) lifetime), or are trapped
with a probability n by vacancies being
present in a concentration c_{V}. 


The trapped positrons then decays with a rate
l_{2} which will be somewhat smaller
then l_{1} because it lives a little
longer; its average lifetime is now t_{2}. 


The change in the partial concentration then becomes 


dn_{1}
dt_{ } 
= 
– (l_{1} + n
· c_{V}) · n_{1} 



dn_{2}
dt_{ } 
= 
– l_{2} · n_{2} +
n · c_{V} ·
n_{1} 



This system of coupled differential equation is
easily solved (we will do that as an
exercise), the starting
conditions are 


n_{1}(t = 0) 
= 
n_{0} 



n_{2}(t = 0) 
= 
0 




The average lifetime t,
which is the weighted average of the decay paths and what the experiment
provides, will be 


t_{ } 
= t_{1}
· 
æ
ç
è 
1 + t_{2}
· n · c_{V}
1 + t_{1} ·
n · c_{V} 
ö
÷
ø 



The probability n 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. 




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. 

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 


A large table containing values for
H_{F} as determined by positron annihilition (and compared
to values obtained otherwise) can be found in the link 








