Changes

== Detailed Characterization ==

== Detailed Characterization ==

Since the SiPM performance is sensitive to the bias voltage applied and the ambient temperature, a measurement of SiPM properties as functions of bias voltage (V_b) and temperature (T) was performed on the SSPM-06. By this point, the SSPM-06 was judged to be a better sensor for the tagger microscope, owing to [[:Image:Image:BCF20,LED,SiPMs comp.png|higher sensitivity in the blue-green range]] and better active area match to the fiber cross-section.  Aside from gains in efficiency and dynamic range of the resulting readout, higher photon detection implies better time resolution because of the scintillation decay time of 2.7ns in the fiber.

Since the SiPM performance is sensitive to the bias voltage applied and the ambient temperature, a measurement of SiPM properties as functions of bias voltage (V_b) and temperature (T) was performed on the SSPM-06. By this point, the SSPM-06 was judged to be a better sensor for the tagger microscope, owing to [[:Image:Image:BCF20,LED,SiPMs comp.png|higher sensitivity in the blue-green range]] and better active area match to the fiber cross-section.  Higher photon detection efficiency is not required for the sake of pulse-height resolution, but it is important in that it improves the time resolution because of the intrinsic scintillation decay time of 2.7 ns in the fiber.

The range of interest for these operating variables were:

The range of interest for these operating variables were:

* V_b: from 0.5V below to 0.5V above the operating range, yielding a ranger of interest: 19V-21V

* V_b: from 0.5 V below to 0.5 V above the operating range, yielding a range of interest: 19-21 V

* T: 0-above room temp., in practice 3°C (to avoid growing snow) to 35°C

* T: 3°C (to avoid accumulating frost) up to 35°C

 

=== Histogram Fitting Method ===

=== Histogram Fitting Method ===

The solution to this was to abandon the manual location of pedestals, peak spacing etc. Instead, a model was created by Prof. Richard Jones based on which fitting of the histograms was performed. It has the form:

It was found that the individual photon peaks were very indistinct at bias voltages below 20 V and also at temperatures above 20°C.  This merging of the peaks is explained by the reduction in pixel gain that occurs for lower bias voltages and higher temperatures, while the electronic noise from the preamplifier remains relatively constant. The solution to this was to abandon the manual location of pedestals, peak spacing etc. Instead, a parametrized model was created by Prof. Richard Jones based on which fitting of the histograms was performed. It has the form:

<math>

<math>

f(q) = \sum_{p,s} \left(\frac{e^{-\lambda_{(p)}} \lambda_{(p)}^p}{p!}\right)\left(\frac{e^{-p \lambda_{(s)}}(p \lambda_{(s)})^s}{s!}\right) \left(\frac{\exp \left(-\frac{1}{2}\; \frac{\left[q-(p+s)\right]^2}{\sigma_0^2+(p+s)\sigma_1^2}\right) } {\sqrt{2\pi}\left[\sigma_0^2+(p+s)\sigma_1^2\right]^{\frac{1}{2}} }\right)

f(q) = \sum_{p,s} \left(\frac{e^{-\lambda} \lambda^p}{p!}\right)\left(\frac{e^{-p \mu}(p \mu)^s}{s!}\right) \left(\frac{\exp \left(-\frac{1}{2}\; \frac{\left[q-(p+s)\right]^2}{\sigma_0^2+(p+s)\sigma_1^2}\right) } {\sqrt{2\pi}\left[\sigma_0^2+(p+s)\sigma_1^2\right]^{\frac{1}{2}} }\right)

</math>

</math>

|<math>x\quad</math> || || is the real integral value (in Vs) and <math>x_0</math> is the pedestal offset (location of first peak).

|<math>x\quad</math> || || is the real integral value (in Vs) and <math>x_0</math> is the pedestal offset (location of first peak).

|-

|-

|<math>p, \lambda_{(p)}\quad</math> || || is the pixel count and Poisson average pixel count

|<math>p, \lambda\quad</math> || || is the pixel count and Poisson average pixel count

|-

|-

|<math>s, \lambda_{(s)}\quad</math> || ||are multi-Poisson factors that take into account Poisson distributions of secondary pixel counts per each real hit from the set of p hits.

|<math>s, \mu\quad</math> || ||are multi-Poisson factors that take into account Poisson distributions of secondary pixel counts per each real hit from the set of p hits.

|-

|-

|<math>\sigma_0, \sigma_1\quad</math> || || are random noise parameters.

|<math>\sigma_0, \sigma_1\quad</math> || || are random noise parameters.

<math>\lambda_{(p)}, \lambda_{(s)}, \sigma_0, \sigma_1, g, x_0 \quad</math> are the fit parameters. Note the absence of a vertical scale parameter. The vertical scale depends on the number of samples collected, whereas the equation in this model is normalized.  Rescaling works as follows:

<math>\lambda, \mu, \sigma_0, \sigma_1, g, x_0 \quad</math> are the fit parameters. Note the absence of a vertical scale parameter. The vertical scale depends on the number of samples collected, whereas the equation in this model is normalized.  Rescaling works as follows:

If <math>f(x)=T\,f(q)</math>, where <math>T</math> is a vertical scaling parameter and since <math>dq = dx/g</math>,  

If <math>f(x)=T\,f(q)</math>, where <math>T</math> is a vertical scaling parameter and since <math>dq = dx/g</math>,  

<math>\int_{-\infty}^{\infty} f(x)\, dx = Tg \int_{-\infty}^{\infty} f(q)\, dq = Tg </math> implies that Tg is the number of events collected times the bin width (in Vs).  

<math>\int_{-\infty}^{\infty} f(x)\, dx = Tg \int_{-\infty}^{\infty} f(q)\, dq = Tg </math> implies that Tg is the number of events collected times the bin width (in Vs).  

Now, with this powerful instrument at hand used with a fitter in Paw, the histograms collected as function of T and V_b were analyzed. It turned out that even histograms with nearly indistinguishable peaks  yielded a best fit to this model and suggested the appropriate gain and other parameters.

Now, with this powerful instrument at hand used with a fitter in Paw, the histograms collected as function of T and V_b were analyzed. It turned out that even histograms with nearly indistinguishable peaks  yielded a reasonable fit to this model.

=== Results ===

=== Results ===