Volumen II - SAM

Congreso SAM/CONAMET 2009 Buenos Aires, 19 al 23 de Octubre de 2009
In consequence the (unknown) parameters of background function must be such as to minimize the area
between the curves. Therefore, this area can be used to define the penalty function for the minimization
problem,
Tmax
∫
F ( M 2B
) = ( M 2 ( T ) − M 2B
( T )) dT
(4)
Tmin
where Tmin and Tmax are the measurement temperatures range limits. The function F depends, of course, on
measured data M2, but also on the parameters which define the background, see eq. (3). Therefore, the
method for estimate the background implies to find the M2B parameters which minimize the function F,
subjected to the additional condition that, at every point of the integration domain, the integrand must be
positive. This additional requirement is obviously necessary in order to prevent a mathematical solution
(without physical meaning) where the background is higher than the peak.
This method has two important advantages: in first place, there are no assumptions about the form of the
function that describes the peak, except that it decreases to zero at both sides of the maximum. Furthermore,
no assumptions are made about where the peak has decreased enough to be negligible. On the other hand, if
there are many different measurements which have the same background, this function can be estimated for
all the curves simultaneity. In fact, it is possible to define a penalty function F as the sum of the different
areas between the curves and the background, and perform a single minimization procedure.
3. RESULTS
The procedure proposed in this work was applied to the study of molybdenum loss modulus peaks. The
samples were single crystals prepared from zone refined single-crystal rods of molybdenum in A.E.R.E.,
Harwell, UK. The residual resistivity of the samples was about 8000, tungsten being the main residual
impurity. Samples with the crystallographic tensile axis were selected, being more favourable for
deformation by single slip. Samples were irradiated with 2 MeV electrons at room temperature with a
fluence of 10 16 electrons/cm 2 , at the J. J. Thomson Phys. Lab. Reading University, UK. An estimation by the
Kinchin-Pease model indicates about 30 ppm of defects promoted by the irradiation [9].
Mechanical spectroscopy studies were performed in an inverted torsion pendulum, under a vacuum of about
10 -5 Pa [10]. The equipment can also apply a bias stress or “in situ” deformation. The maximum strain on the
surface of the sample was 5 x 10 -5 . The measurement frequency was around 1 Hz, except for the
determination of the frequency dependence of the peak temperature. The heating and cooling rates in the test
were of 1K/minute; there was no hold time once the maximum temperature had been achieved. A heating
ramp and its corresponding cooling run will be called hereafter a thermal cycle.
Figure 3 shows one of the experimental curves -dot points-, measured after several thermal cycles. It can be
seen that the experimental curve may be considered as the superposition of a temperature peak and a high
temperature background. In order to analyze the peak behavior, it is necessary to subtract the background.
The estimation of the background function was performed by the method developed in the theory section. A
Cole-Cole function was proposed and the parameters were obtained by minimizing the area between the
curves. The resulting curve is the line shown in Figure 3, and the adjusted activation energy is 4.3 eV. This
value is consistent with values reported in literature [11].
Once the background is subtracted, the isolated peak can be analyzed. The dots in Figure 4 represent the
experimental values after the subtraction of the adjusted background function. To describe this peak, the
imaginary part of a Havriliak-Negami (HN) function is proposed [12, 13]:
δM
M ( T,
ω)
= Im[
(5)
[ ( ) ] ]
α
1+
iω
τ exp( H / kT)
2 β
0
The parameters α and β are both positive. They are related to the width of the distribution function of
relaxation times; α describes a symmetrical broadening (in comparison to a Debye peak) and β is related to
the asymmetry of this distribution. It is easy to show that the Cole-Cole function of eq. (3) is a particular case
of the HN function with β equal to 1.
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