Figure 5.8: Simulation in frequency domain of the Pole Zero Cancellation circuit (PZC) performed with TINA simulatore (the red curves). Above we have the frequency response, below the phase (with respect the input) diagram.

In figure 5.10 it is possible to see the collimation system constituted by a
*tung-sten shield with a 0.5×1mm slit to irradiate the detector. To center the detector*
we use a solid state laser placed instead of the source and we move the detector
by means of micro-metric movements. In figure 5.11 we can see the laser spot
*on the detector. Measurements with very thin beam (in the order of 10µm)*
have been taken at the ESRF institute (Grenoble). Unfortunately because of
the length of the BNC cable (the digitizer was outside the room where is
lo-cated the sample) the data are affected by low frequency noise, as we can see in
figure??.

Figure 5.9: The fist data acquired inred, ingreenthe data acquired at the ESRF institute, and inbluethe last data taken after the noise reduction.

calculate the collection efficiency defined as:

*η(ξ) =* *Q**M**(ξ)*

*Q*0 = *Q*0*F (ξ, t**M**(ξ))*
*Q*0

*= F (ξ, t**M**(ξ)) = F**M**(ξ)* (5.1)
*where ξ = x/L is the relative impact depth and Q**M**(ξ) the maximum of the*
*collected charge, t**M**(ξ) the time when the maximum occurs, and F (ξ, t**M**(ξ)) =*
*F**M**(ξ) the maximum of the collection efficiency function F (ξ, t) calculated in*
the modelization. Finally, after these steps, we colud correct the charge
col-lected.

At the beginning, in the “Hecht case”(no detrapping), as we can see in the
ap-pendix A, we have calculated when the maximum of collected charge occurs in
relation to the transport properties. With these values for the transport
*prop-erties, we have calculated the maximum of the collection efficiency F**M**(ξ).*

When the model is become more complex, with the detrapping mathematical discussion, the idea has always remained the same, with some more complica-tions.

Figure 5.10: Collimator. In figure we can see the laser spot coming out from the slit

### 5.5.2 Fitting

The fundamental problem in the fitting procedure is that we must fit concur-rently all the collected data. The model indeed depends on two kind of param-eters:

*•* The first kind of parameters concerns the singular events. This parameters
*are, for example, the amplitude A and the impact depth x*0. Moreover
other individual parameters are needed to fit the data as the CSP offset
*of and the time when the photon is absorbed t*0. Because these parameters
*(A, x*0*, of , t*0) are related to the singular event, we have four parameters
for each singular collected event, so the number of parameters should be
multiplied by the number of events.

*•* the second kind of parameters is general and concerns the material
*trans-port properties (i.e. the crossing time T related to mobility µ, the lifetime*
*τ**t**, and the detrapping time τ**d* both for electrons and holes) or the
*elec-tronic characteristics (the integration time τ**i*).

Figure 5.11: Centering with laser beam.

The first set of parameters should be determined by means of singular fits, one for each event; the second set should be determined with an overall fitting procedure (global fit). The problems are two:

*•* the two set of parameters are mutual dependent.

*•* A large computational power (and time) is required.

Now we face this two problems.

Fitting procedure To choose the starting values for the fitting parameters we
*initially do a singular fit for each of the N event. In this way we get N ×(A, x*0,
*of , t*0*) singular parameters and N × (T**e**, T**h**, τ**te**, τ**th**, τ**de**, τ**dh*) global parameters

*while τ**i* is fixed because its value is yet known. The first group is the starting
parameter set used for each singular fit. To choose the starting parameters for
the global fitting procedure, instead, we take the mean-value of each group.

Once given the initial parameters, the fitting procedure follows an iterative method in two step:

*•* In the first step we fix all the individual parameters for each event and we
*search the global parameters that minimize the global “chi-square”χ*^{2}* _{G}*.

*•* In the second step we fix the global parameters and we search, event by
event, the new individual parameter set that minimizes each singular
*chi-square χ*^{2}* _{S}*.

This procedure is iterated until the global chi-square convergence. Indeed, be-cause the singular and the global parameter sets depend on each other, the procedure normally converges to the minimum value for the global chi-square.

*To fit the data we exploit the fminuit matlab function*^{1} Using this function we
can fix or release parameters and this is very useful because some parameters,
*as of or t*0, are weakly dependent on the other ones, so, once found them in
the first step, they can be fixed forever to simplify the further minimization.

We can also set the range where a parameter can be varied (when we know this range), to speed up the convergence and avoid false minima

*Computational expedients Because of the number of events fitted N is in*
the order of 1000, the calculation efficiency is very important. Initially, the most
of the computational resources was used to calculate numerically the double
in-tegral expressed in the equation 4.85. This double inin-tegral must be recalculated
every time a parameter have been varied, and, more again, it must be calculated
*for each value of the discrete time t (about 5000). At the beginning the time*
*employed to calculate this 5000 values was about 62.5 seconds. Now the same*
*calculation is completed in 0.025 second.*

*The inner integral in the equation 4.85, that we will call BGI(P, Θ) (i.e. *
*Bessel-Gauss Integral), depends on the two parameters P (ξ) and Θ(ξ, t), so that we*
can avoid to recalculate always the same integral. Indeed, if we calculate this
*integral on a grid of 250×100 (P*_{i}*, Θ*_{j}*) points, every time we need the BGI(x, y)*
value we can interpolate the values in the grid without recalculate every time
the same integral. Only with this ploy we have reduced the calculation time of
a factor 2500.

Other solutions, employed to increase the calculation efficiency, is to use
*vec-torial integration routines (as QU ADV matlab function), to avoid useless or*
*redundant operation, to reduce the required numerical precision (from 1E*^{−6}*to in 1E** ^{−4}*), to reduce the number of calls at the same routine (for example
the integration for the electron and the hole contribution are carried out in the
same call), and to use smooth functions in the integration process. Concerning

1*fminuit is the matlab call for the minuit DLL, a function to search the function minima*
developed at the CERN institute. Fminuit has been created by Dr. Giuseppe Allodi.

this last point, the tabulated values, used to calculate the double integral in eq.

*4.85, are not the BGI(P, Θ) values, but the BGEP (P, Θ) values, where BGEP*
is the quantity in the round brackets in eq. 4.85. This last function is a smooth
function (as we can see in figure 4.3) so it makes easier the integral convergence.

In this way, with all these stratagems, the overall time reduction is estimated greater than 10000 times. Nevertheless for a set of 1000 events the calculation time on my double processor PC is in the order of one month.