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) = QM(ξ)
Q0 = Q0F (ξ, tM(ξ)) Q0
= F (ξ, tM(ξ)) = FM(ξ) (5.1) where ξ = x/L is the relative impact depth and QM(ξ) the maximum of the collected charge, tM(ξ) the time when the maximum occurs, and F (ξ, tM(ξ)) = FM(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 FM(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 x0. Moreover other individual parameters are needed to fit the data as the CSP offset of and the time when the photon is absorbed t0. Because these parameters (A, x0, of , t0) 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 τt, and the detrapping time τd both for electrons and holes) or the elec-tronic characteristics (the integration time τi).
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, x0, of , t0) singular parameters and N × (Te, Th, τte, τth, τde, τdh) global parameters
while τi 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”χ2G.
• 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 χ2S.
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 function1 Using this function we can fix or release parameters and this is very useful because some parameters, as of or t0, 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 (Pi, Θj) 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
1fminuit 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.